Research ArticleA Location-Inventory-Routing Problem in Forward andReverse Logistics Network Design

Qunli Yuchi12 Zhengwen He12 Zhen Yang12 and Nengmin Wang12

1School of Management Xirsquoan Jiaotong University No 28 Xianning Road Xirsquoan Shaanxi 710049 China2The Key Lab of the Ministry of Education for Process Control amp Efficiency Engineering No 28 Xianning Road XirsquoanShaanxi 710049 China

Correspondence should be addressed to Nengmin Wang nengminwang163com

Received 1 February 2016 Revised 25 May 2016 Accepted 6 June 2016

Academic Editor Gabriella Bretti

Copyright copy 2016 Qunli Yuchi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study a new problem of location-inventory-routing in forward and reverse logistic (LIRP-FRL) network design whichsimultaneously integrates the location decisions of distribution centers (DCs) the inventory policies of openedDCs and the vehiclerouting decision in serving customers in which new goods are produced and damaged goods are repaired by a manufacturer andthen returned to themarket to satisfy customersrsquo demands as newonesOur objective is tominimize the total costs ofmanufacturingand remanufacturing goods building DCs shipping goods (new or recovered) between the manufacturer and opened DCs anddistributing new or recovered goods to customers and ordering and storage costs of goods A nonlinear integer programmingmodelis proposed to formulate the LIRP-FRL A new tabu search (NTS) algorithm is developed to achieve near optimal solution of theproblem Numerical experiments on the benchmark instances of a simplified version of the LIRP-FRL the capacitated locationrouting problem and the randomly generated LIRP-FRL instances demonstrate the effectiveness and efficiency of the proposedNTS algorithm in problem resolution

1 Introduction

The need for designing distribution network to achieve avariety of the supply chain objects with the overall produc-tivity has received considerable attentions and becomes moreand more stronger in recent years As Javid and Azad [1]said location allocation problem inventory control problemand the vehicle routing problem are the most consideredsubproblems in designing a distribution network Liu and Lee[2] showed that the routing and inventory decisions affectthe location decision Also Viswanathan and Mathur [3]found that transportation costs will increase when the orderquantity during each production runs decreases Similarconclusions have also been found by Zhang et al [4] theypointed out that the above three subproblems are stronglycorrelated and mutually affect each other Considering thethree subproblems at the same time may provide a compre-hensive estimation in constructing an efficient production-distribution network This gives rise to the researches on thelocation-inventory-routing problem (LIRP)

Differing in inventory policies used the types of facilitieslocated and the vehicle routing strategies adopted lots ofresearches on LIRP in forward logistics network design havebeen reported Liu and Lee [2] studied a LIRP using order upto level inventory policy A route-first and location-allocationsecond heuristicmethodwas proposed by them to solve LIRPAmbrosino and Scutella [5] studied a LIRP with the con-sideration of customers service level Using (119876 119877) inventorypolicy Shen andQi [6] proposed a Lagrangian relaxation anda rank and search algorithm to solve the LIRP Following theresearches of [6] Javid and Azad [1] solved the same problemby using a tabu search and simulated annealing Guerrero etal [7] proposed a column generation Lagrangian relaxationand local search combined heuristic for a multiperiod LIRP

As Ravi et al [8] said the forces of economic factors leg-islation corporate citizenship and environmental protectionproblems are driving the increasing interests and investmentsof enterprises in reverse logistics These motivate researchersto develop optimization models for reverse supply networkdesign in the past decades (Vahdani and Naderi-Beni [9])

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 3475369 18 pageshttpdxdoiorg10115520163475369

2 Discrete Dynamics in Nature and Society

Chen et al [10] concluded that the consideration ofreverse logistics in distribution network design can greatlyreduce the logistics costs without interrupting the forwardflows Taking both the forward logistics and reverse logisticsinto account many researches on facility location vehiclerouting and inventory management had been presented

Vahdani et al [11] focused on a fuzzy facility location-allocation problem in closed loop logistics Differing from[11] Vahdani et al [12 13] proposed a fuzzy possibility pro-gramming method to solve multiobjective facility problemsin forward and reverse logistic

Under the background of forward and reverse logisticdifferent inventory models had also been developed in theliterature Richter [14] proposed the first inventory modelin a closed logistic using EOQ model It assumed thatcustomersrsquo demands can be satisfied by newly made productsor by repaired products For a given fixed waste disposalrate the optimal minimum cost expression was derived Shiet al [15] considered an inventory policy under uncertaincustomer demand A Lagrangian relaxation based approachwas proposed

Vehicle routing decision is another important researchsubject in forward and reverse logisticmanagement Zachari-adis and Kiranoudis [16] studied a VRP problem under thebackground of forward and reverse logistic A tabu searchmethod was proposed For a similar VRP problem Goksal etal [17] developed a particle swarm optimization (PSO) basedapproach for its resolution in which a variable neighborhooddescent (VND) algorithm was used as a local search toexplore solution space

Different combination of facility location decision inven-torymanagement decision and vehicle routing decision givesrise to different research problems in forward and reverselogistics For example Liu et al [18] proposed a hybridparticle swarm optimization to solve a location routingproblem using grey systems theory in reverse logistics Yuand Lin [19] presented a location routing problem withsimultaneous pickup and delivery which was solved via amultistart simulated annealing algorithm with multistart hillclimbing Modeling the problem as a bilevel programmingproblemWang et al [20] studied realistic location-inventoryproblem in reverse logistic of a China B2C company Li etal [21] solved a location and inventory control problem bya two-stage heuristic algorithm that combined Lagrangianrelaxation method with an ant colony algorithm

Due to the intrinsic complexity of the problem inmathematics to our best knowledge only a few researchesexcept that of Li et al [22] simultaneously considered thedecisions of the facility location the inventory managementand vehicle routing in forward and reverse logistics in theliterature In their model they consider retrieving thoseproducts that customers dislike to theDCs As these productsare newor unused they can be sold to customers directlywithno need to be returned to manufacturer for repair

However as the work of Adler et al [23] indicated used(old or damaged) products (eg the engines of CaterpillarCompany) can also be retrieved recovered and returnedto the market to satisfy customersrsquo demands as new ones

This realistic new feature motivates us to focus on goodsproduction distribution and recycling

A location-inventory-routing problem in forward andreverse logistic (LIRP-FRL) network design is studied in thispaper The LIRP-FRL simultaneously integrates the locationdecisions of distribution centers (DCs) the inventory policiesof opened DCs and the vehicle routing decision in servingcustomers in which new goods are produced and damagedgoods are repaired by a manufacturer and then transportedto opened DCs Vehicles that start from and end in thesame DC distribute new or recovered goods to satisfy thedemands of customers and retrieve damaged goods Ourobjective is to minimize the total costs of manufacturingand remanufacturing goods building DCs shipping goods(new or recovered) between the manufacturer and openedDCs distributing new or recovered goods to customers andretrieving damaged goods from customers to DCs and theinventory costs of DCs including ordering new or recoveredgoods and holding these goods

Without taking the inventory strategy of DCs intoaccount our problem is simplified to the classical capacitatedlocation routing problem (CLRP) [24] which is NP-hardin strong sense Consequently our problem is also NP-hard Further the LIRP-FRL is a nonlinear problem as theEOQ inventory policy is adopted Due to its complexitywe focus on finding near optimal solutions for the LIRP-FRL A new tabu search (NTS) algorithm is proposed tofind approximately optimal solutions This algorithm prob-abilistically accepts a second best solution to change searchdirection to achieve this goal Numerical experiments of theCLRP benchmarks demonstrate the effectiveness and theefficiency of the proposed NTS method in obtaining highquality solutions in a very reasonable time Computationaland sensitivity analysis results on randomly generated LIRP-FRL instances are also reported

The rest of this paper is organized as follows In Section 2a mathematical model is developed to describe the studiedLIRP-FRL A tabu search that seeks to search near optimalsolution for the LIRP-FRL is proposed in Section 3 Algo-rithm performance evaluations are conducted in Section 4Finally we conclude the paper in Section 5

2 Mathematical Model

Given a set of potential sites of distribution centers (DCs) thestudied location-inventory-routing problem in forward andreverse logistic (LIRP-FRL) network design is to optimallydetermine the locations of DCs and their order quantitiesfrom amanufacturer the product flows (forward and reverse)between the manufacturer and the opened DCs and thepaths of vehicles in serving customers We assume that eachroute starts from and ends in the same DC by which newor recovered products are delivered to satisfy the demandsof customers for each cycle and damaged products areretrieved The objective of the LIRP-FRL is to minimize thesum of the opening costs and the inventory costs of DCs thesetup costs of vehicles and the transportation costs of (newrepaired and damaged) products An example of the studiedLIRP-FRL is illustrated in Figure 1

Discrete Dynamics in Nature and Society 3

Nonselected DCSelected DCManufacturer

CustomerForward logisticsReverse logistics

Figure 1 Location-inventory-routing problem in forward andreverse logistic

21 Notations The notations are described as follows

119868 the index set of customers119869 the index set of potential distribution centers119870 the index set of vehicles119878 the merged set of customers and potential distribu-tion centers (119868 cup 119869)119881119895 the maximum capacity of the distribution center119895 119895 isin 119869FC119895 the fixed opening cost of establishing distribu-tion center 119895 119895 isin 119869119876 the maximum capacity of homogeneous vehicle119888119894119895 the transportation cost of a vehicle on arc(119894 119895) 119894119895 isin 119878 119894 = 119895119891119895 the setup cost of using a vehicle at distributioncenter 119895 119895 isin 119869

119888119895 the product unit transportation cost from themanufacturer to distribution center 119895 119895 isin 119869119860119898 the order cost of new products of DCs per cycle

119860119903 the order cost of recovered or repaired products

of DCs per cycle119888119898 the cost of manufacturing a new product119888119903 the cost of repairing a damaged product to a newproduct119863119894 the demand of customers per cycleℎ119895 the unit holding cost of distribution center 119895 119895 isin 119869119903 the rate of return in percentage

119888119889 the unit cost of disposal of useless retrievedproducts119906 the percentage of reuse of retrieved products

22 Decision Variables The decision variables are describedas follows

119876119898119895 is the order quantity of new products of DC 119895 isin 119869

for each period119876119903119895 is the order quantity of recovered products of DC

119895 isin 119869 for each period119872119895 is the order times of new products from DC 119895 isin 119869

during each period119877119895 is the order times of recovered products from DC119895 isin 119869 during each period119909119894119895119896 = 1 if arc(119894 119895) is travelled by vehicle 119896 0otherwise (119894 119895 isin 119878 119896 isin 119870)119910119894119895 = 1 if customer 119894 isin 119868 is served by distributioncenter 119895 isin 119869 0 otherwise119885119895 = 1 if distribution center 119895 isin 119869 is established 0otherwise

23 Cost Analysis To simplify the presentation we firstabbreviate the costs related to the LIRP-FRL as follows

(1) The total fixed cost of opening distribution centers is

FIXCOST = sum

119895isin119869

FC119895 sdot 119911119895 (1)

(2) The total cost of ordering new and recovered productsis

OC = sum

119895isin119869

119860119898sdotsum119894isin119868119863119894 (1 minus 119906) 119910119894119895

119876119898119895+ 119860119903sdotsum119894isin119868119863119894119906119910119894119895

119876119903119895 (2)

(3) The total cost of producing new products and recov-ering retrieved products is

MR = sum

119895isin119869

sum

119894isin119868

119863119894 sdot ((1 minus 119906) sdot 119888119898+ 119906 sdot 119888

119903) sdot 119910119894119895 (3)

(4) The total cost of disposal of useless retrieved productsis

DISP = sum

119895isin119869

sum

119894isin119868

119863119894 sdot (119903 minus 119906) sdot 119888119889sdot 119910119894119895 (4)

(5) The transportation cost between the manufacturerand distribution centers is

TRAN = sum

119895isin119869

119888119895sum

119894isin119868

119863119894 sdot (1 + 119906) sdot 119910119894119895 (5)

(6) The transportation cost between distribution centersand customers is

TRC = sum

119896isin119870

sum

119895isin119878

sum

119894isin119878

119891119895 sdot 119909119894119895119896 + sum

119896isin119870

sum

119895isin119878

sum

119894isin119868

119888119894119895 sdot 119909119894119895119896 (6)

4 Discrete Dynamics in Nature and Society

Given the locations of DCs and the routes of vehicles theLIRP-FRL is reduced to the inventory model of Teunter [25]Some of their valuable conclusions can be directly applied toour LIRP-FRL

Proposition 1 (Teunter [25]) ldquoIt is reasonable to restrict ourattention to those policies with either119872 = 1 or 119877 = 1rdquo

Teunter [25] concluded that the policies with either119872 =

1 or 119877 = 1 can obtain near optimal solution with lesscomputational time by comparing with that of achievingoptimal solution Since our problem is more complicatedthan that of Teunter [25] to simplify the resolution of LIRP-FRL we consider only the two cases in our implementationThe two cases are as follows

Case 1 (1M) That means 119877 = 1 in this case the order timesof recovered products for each DC are equal to one

Case 2 (R 1) That means119872 = 1 in this case the order timesof new products for each DC are equal to one

For Case 1 (1M) the holding cost is

HM = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895 + 119876119903119895 sdot

119906

119903) sdot 119911119895 (7)

Consequently the total cost is denoted as

TCM = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HM(8)

For Case 2 (R 1) the holding cost is

HR = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895

+ (119903 (119876119898119895 + 119876

119903119895) minus (119903 minus 119906) sdot (119876

119898119895 +

119906

1 minus 119906119876119898119895 ))

119906

119903)

sdot 119911119895

(9)

The corresponding total cost is

TCR = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HR(10)

24 Mathematical Model The LIRP-FRL can be formulatedas the following nonlinearmixed integer programming prob-lemmin TCMTCR (11)

st sum

119896isin119870

sum

119894isin119878

119909119894119895119896 = 1 119895 isin 119868 (12)

sum

119895isin119878

119909119894119895119896 minussum

119895isin119878

119909119895119894119896 = 0 119894 isin 119878 119896 isin 119870 (13)

sum

119895isin119869

sum

119894isin119868

119909119894119895119896 le 1 119896 isin 119870 (14)

sum

119895isin119869

sum

119894isin119868

119863119894119909119894119895119896 le 119876 119896 isin 119870 (15)

sum

119894isin119868

119863119894119910119894119895 minus 119881119895 sdot 119911119895 le 0 119895 isin 119869 (16)

119910119894119895 + 1 ge sum

119892isin119868

119909119894119892119896 + sum

119892isin119878119895

119909119892119895119896

119894 isin 119869 119895 isin 119868 119896 isin 119870

(17)

sum

119895isin119882

sum

119894isin119882

119909119894119895119896 le |119882| minus 1 119882 sube 119868 119896 isin 119870 (18)

119909119894119895119896 isin 0 1 119894 isin 119878 119895 isin 119878 119896 isin 119870 (19)

119910119894119895 isin 0 1 119894 isin 119868 119895 isin 119869 (20)

119911119895 isin 0 1 119895 isin 119869 (21)

Constraints (12) state that each customer is exactly served byone vehicle Constraints (13) are the equilibrium constraintsof vehicle routes on nodes Constraints (14) ensure that eachvehicle can be used by at most one DC Constraints (15)guarantee that the capacity of vehicle cannot be exceededConstraints (16) are the capacity constraints of distributioncenters Constraints (17) make sure that a customer can onlybe served by the vehicle which originates from the DC itis assigned to Constraints (18) are the subtours eliminationconstraints of vehicle routes

Proposition 2 For any distribution center 119895 isin 119869 its orderquantities 119876

119898119895 and 119876

119903119895 can be represented as a function of

customersrsquo demands respectivelyWe have 119876

119898119895 = radic2119860119898sum119894isin119868119863119894 sdot 119910119894119895ℎ119895 and 119876

119903119895 =

radic2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895(119903 + 1)ℎ119895 for any 119895 isin 119869

Proof It can be observed that no matter case (1 M) orcase (R 1) decision variables 119876119898119895 and 119876

119903119895 appear only in the

objective function For the case (R 1) the optimal value of119876119898119895and 119876119903119895 can be obtained by checking the extreme conditions120597TCM120597119876

119898119895 = 0 and 120597TCM120597119876

119903119895 = 0

Since 1205972TCM120597(119876

119898119895 )2

= (2119860119898

sdot (1 minus 119906)sum119894isin119868119863119894 sdot

119910119894119895)(119876119898119895 )3ge 0 and 120597

2TCM120597(119876119903119895)2= (2119860

119903sdot 119906sum119894isin119868119863119894 sdot

119910119894119895)(119876119903119895)3ge 0 consequently the extreme point is a minimum

point We obtain the optimal order quantities 119876119898119895 =

radic(2119860119898sum119894isin119868119863119894 sdot (1 minus 119906)119910119894119895)ℎ119895(1 minus ((119903 minus 119906)(1 minus 119906)) sdot (119906119903))

and 119876119903119895 = radic(119860119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and the corresponding

order time 119877119895 = (119906 sdot 119876119898119895 )(1 minus 119906 sdot 119876

119903119895)

For the same reason for the case (1 119872) we can derivethat the order quantities 119876119898119895 = radic(2119860119898sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and

119876119903119895 = radic(2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)(119903 + 1)ℎ119895 and the correspond-

ing order time119872119895 = (1 minus 119906 sdot 119876119903119895)(119906 sdot 119876

119898119895 )

Observe that without considering the inventory strategyof these two cases Cases 1 and 2 the LIRP-FRL is reducedto the classical capacitated location routing problem (CLRP)which is NP-hard in the strong sense As a result the LIRP-FRL is also NP-hard As Belenguer et al [24] reported to

Discrete Dynamics in Nature and Society 5

21 23 19 16 15 14 8 11 6 22 4 1 12 18 20 13 5 7 3 24 25 2 17 9 10

Figure 2 An example of solution representation

find CLRP optimal solution even for the median size CLRPinstances (with 50 customers and 5 potential distributioncenters) significant research efforts need to be expensedThus in this paper we focus onfinding near optimal solutionsfor the LIRP-FRL

3 New Tabu Search

Using tabu list to avoid visited solutions to be revisited thetabu search proposed by Glover [26] is one of the mosteffective approaches for solving mixed integer programmingproblems which has been successfully applied to a varietyof distribution network design problems For the detailedintroduction and application of the tabu search we referreaders to Glover [27] and Habet [28]

In this section a tabu search (NTS) algorithm thatprobabilistically accepts the second best solution in searchprocess when a local optimum is reached is proposed toeffectively solve the LIRP-FRL The key components of theNTS including a solution representation technique an initialsolution generation method neighbourhood structures andthe general framework of the new tabu search are presentedin detail as follows

31 Solution Representation A good solution representationcan not only describe the problem solutions clearly butalso improve the performance of heuristics adopted Similarto that of Yu et al [29] in our implementation solution1199091 1199092 119909119899 119909119899+1 119909119899+119898 is represented by a permuta-tion of number 1 2 119899 119899 + 1 119899 + 2 119899 + 119898 where119909119895 isin 1 2 119899 indexes customer 119909119895 and 119909119894 isin 119899 + 1 119899 +

2 119899 + 119898 denotes potential DC 119909119894An example was given in Figure 2 In this example the

first number is distribution center 21 followed by distributioncenter 23 Since there are no customers between distributioncenter 21 and distribution center 23 distribution center 21is closed Customers (19 16 15 14 8 11 and 6) betweendistribution center 23 and distribution center 22 are realcustomers thus distribution center 23 is opened to servicethese customers The first route of distribution center 23services customers 19 16 15 and 14 Because adding customer8 exceeds the vehicle capacity the second route servicescustomers 8 11 and 6 Customer 6 is followed by distributioncenter 22 so the second route is terminated For a similarreason asmentioned above distribution centers 22 and 25 areopen with two routes and one route respectively

This representation has determined which distributioncenter is open and the customers on each route With thatdecided we can easily calculate the objective function valuebecause the inventory cost is related to LRP which meanswhen LRP is solved the inventory cost can be uniquelydetermined by Proposition 2 Note that the capacity ofdistribution center is not taken into consideration So during

the decoding process a per unit penalty cost 119872 with a bigvalue is added to the objective function value when the totaldemand serviced by a distribution center exceeds its capacity

To simplify the presentation of the algorithms proposedin following subsections we say that an element 119909119894 of thesolution is before another element 119909119895 or element 119909119895 is afterelement 119909119894 if and only if 119894 lt 119895 Further with the smallest value|119894 minus 119895| if DC 119909119894 is before DC 119909119895 we say that DC 119909119894 is nearestbefore DC 119909119895 or DC 119909119895 is nearest after DC 119909119894

32 Constructing Initial Solution A greedy nearest neighbormethod is proposed to construct an initial feasible solution ofthe LIRP-FRL which is used as an input for the tabu searchDetails of the greedy algorithm are given as follows

Algorithm 3 (greedy nearest neighbour method)

Step 0 InitializeΩ0 fl 119869 Ω1 fl Φ0 fl 119868 and Φ1 fl

Step 1 Assign each customer inΦ0 to its closest DCs inΩ0

Step 2 Open the DC in Ω0 with the largest number of

customers assigned to say DC119895lowast

Step 3 Sort the customers of set Φ0 in nondecreasing orderof the distance between it and the DC119895lowast

Step 4 With the order let Φ = 1199041 1199042 119904|Φ0| be the subsetof customers of Φ0 with the largest |Φ| and sum

|Φ|119894=1 119889119904119894 le 119881119895lowast

Renew Ω1 fl Ω

1cup 119895lowast Ω0 fl 119869Ω

1 Φ1 fl Φ1cup Φ and

Φ0 fl 119868Φ

1

Step 5 If Φ0 = then return to Step 1 Otherwise open theDCs inΩ1 and assign each customer to a DC according to thecustomer-DC assignments determined in Step 4

Step 6 For each opened DC sort the customers assignedto it into sequence by using a nearest neighbor heuristic(refer to Rosenkrantz et al [30]) Split the sequence intoseveral feasible routes so that vehicle capacity constraints aresatisfied

Step 7 Output feasible solution

In our implementation of Step 2 if more than oneDChasthe largest number of customers assigned to we select theDCwith the highest capacity

33 Neighborhoods Given a solution

119883 = (1199091 1199092 119909119894minus1 119909119894 119909119894+1 119909119895minus1 119909119895 119909119895+1 119909119899) (22)

a permutation of 1 2 119899 119899 + 1 119899 + 119898 three kinds ofneighbourhoods are considered in our tabu search

6 Discrete Dynamics in Nature and Society

331 Insertion Neighbourhood The insertion neighborhoodis defined as the set of solutions that can be reached byinsertion move which deletes one element of the solution119883 and then inserts it to another place For example deleteelement 119909119894 and insert it into the 119895th place we obtain a newsolution

1198831015840= (1199091 1199092 119909119894minus1 119909119894+1 119909119895minus1 119909119895 119909119894 119909119895+1 119909119899) (23)

According to the element selected (customer or depot)and the position it is inserted in there are four cases

(a) If 119909119894 and 119909119895 are both customers then customer 119909119894 isreassigned to the DC that customer 119909119895 is assigned to

(b) If 119909119894 and 119909119895 are bothDCs then reassign the customerspreviously assigned to the DC nearest before it CloseDC 119909119895 and assign its customers to DC 119909119894

(c) If 119909119894 is a customer and 119909119895 is a DC then customer 119909119894 isreassigned to DC 119909119895

(d) If 119909119894 is a DC and 119909119895 is a customer reassign customersthat are nearest after customer 119909119895 to DC 119909119894 Allcustomers assigned to DC 119909119894 are reassigned to the DCnearest before it

332 Swap Neighbourhood The swap neighbourhood a setof solutions contains the solutions that can be reached byperforming a swap move which exchanges the places oftwo different elements of solution 119883 For example exchangeelement 119909119894 and element 119909119895 of119883 the resulting new solution is

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119894+1 119909119895minus1 119909119894 119909119895+1 119909119899) (24)

Similar to the insert move the swap move has also fourcases

(e) If 119909119894 and 119909119895 are both customers then we exchange thetwoDCs they are currently assigned to and obtain twonew customer-DC assignments

(f) If 119909119894 and 119909119895 are both DCs then we exchange all thecustomers currently assigned to them

(g) If 119909119894 is a customer and 119909119895 is a DC then the customersafter 119909119894 assigned to the same DC are reassigned to DC119909119895 and customer 119909119894 is reassigned to the DC nearestbefore DC 119909119895 All customers previously assigned toDC 119909119895 will be reassigned to the DC nearest before it

(h) If 119909119894 is a DC and 119909119895 is a customer this case is similarto case g

333 2-Opt Neighbourhood With the consideration of vehi-cle capacity in our definition the 2-opt neighbourhood con-tains the solutions that can be obtained by selecting two cus-tomers of119883 (eg 119909119894 and 119909119895) and then reversing the substringin the solution representation between them the so-called 2-opt move For example applying 2-opt move to elements 119909119894and 119909119895 we obtain

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119895minus1 119909119894+1 119909119894 119909119895+1 119909119899) (25)

34 Tabu Search Framework To increase the probability ofobtaining high quality solutions a diversification strategyis implemented This strategy probabilistically accepts thesecond best solution in the search process while no bettersolution can be found in the neighborhoods of the currentsolution By doing so we can effectively diversify the searchdirection and thusmore solution spacewill be explored Con-sequently better solution may be found The computationresults in Section 4 prove this strategy The proposed tabusearch can be summarized as follows

Algorithm 4 (new tabu search)

Step 0 Input an initial feasible solution 1198830 (see Section 32)and initialize Iter fl 0119883curr fl 1198830 and Objlowast fl Obj(1198830)

Step 1 Explore the neighborhoods of 119883curr and denote thebest solution in the neighborhoods the best nontabu solu-tion and the second best nontabu solution as119883best119883tabu-bestand119883tabu-2nd respectively

Step 2 If Obj(119883best) lt Objlowast then set 119883curr fl 119883best and addthe corresponding move to tabu list otherwise

if Obj(119883tabu-best) lt Obj(119883curr) then119883curr fl 119883tabu-bestand add the corresponding move to tabu list

otherwise probabilistically move to 119883tabu-best or119883tabu-2nd 119883curr fl Prob119883tabu-best 119883tabu-2nd and addthe corresponding move to tabu list

Step 3 Iter fl Iter + 1 If no stopping criterion is reachedreturn to Step 1

Step 4 Output the best solution objective function valueObjlowast

In the proposed tabu search algorithm Iter counts thenumber of iterations and 1198830 represents the initial solutionobtained by greedy nearest neighbour method (Section 32)119883curr denotes the current solution Objlowast is the objectivefunction value corresponding to the best solution found sofar The function Obj(119883) represents the objective functionvalue of solution119883

In Step 2 comparing with the current solution119883curr if nobetter solution can be found in its neighbourhood we needto probabilistically move 119883curr to 119883tabu-best or 119883tabu-2nd Inimplementation we randomly generate a value 120573 in interval[0 1] If120573 ge 120582 then119883curr ismoved to119883tabu-best and otherwise119883curr is moved to119883tabu-2nd where 120582 is a given parameter

At each iteration of the tabu search procedure thebest admissible nontabu insertion swap or 2-opt move isperformed The two elements of solution 119883curr involved insuch move are declared as tabu-active which is forbiddento be removed in the next 119905 iterations 119905 is the tabu-tenureparameter selected randomly from a range [119879min 119879max]

The tabu search iteration is terminated if the maximumnumber of iterations maxIter is reached or if the bestsolution found so far has been improved in successive SucIteriteration

Discrete Dynamics in Nature and Society 7

Table 1 Computational results on CLRP benchmarks

Problem |119868| |119869| 119876 BKS TS NTS DEVUB Gap Time UB Gap Time

P1 20 5 70 54793 54793 000 012 54793 000 049 000P2 20 5 150 39104 39104 000 012 39104 000 069 000P3 20 5 70 48908 48908 000 050 48908 000 053 000P4 20 5 150 37542 37611 018 037 37611 018 074 000P5 50 5 70 90111 90806 077 258 90461 039 335 minus038P6 50 5 150 63242 65010 280 1043 63242 000 1173 minus280P7 50 5 70 88298 89688 157 2207 88643 039 2394 minus118P8 50 5 150 67340 68071 109 748 67340 000 1116 minus109P9 50 5 70 84055 84227 020 1413 84110 007 1696 minus013P10 50 5 150 51822 51902 015 1021 51883 012 1349 minus003P11 50 5 70 86203 86223 002 920 86203 000 1471 minus002P12 50 5 150 61830 61978 024 1954 61830 000 2149 minus024P13 100 5 70 275993 281229 190 1151 278926 105 3841 minus085P14 100 5 150 214392 218886 210 952 216802 111 2974 minus099P15 100 5 70 194598 196587 102 14525 196436 094 9516 minus008P16 100 5 150 157173 159244 132 18038 158720 097 12463 minus035P17 100 5 70 200246 203297 152 12612 202682 120 11085 minus032P18 100 5 150 152586 157859 346 12247 156049 222 10526 minus124P19 100 10 70 290429 334486 1517 9780 294774 147 1432 minus1370P20 100 10 150 234641 285303 2159 7587 236217 067 1136 minus2092P21 100 10 70 244265 257455 540 10349 248003 151 13985 minus389P22 100 10 150 203988 216253 601 6965 208455 214 107 minus387P23 100 10 70 253344 291153 1492 16998 258221 189 15353 minus1303P24 100 10 150 204597 220528 779 6706 209390 229 13152 minus550P25 200 10 70 479425 518815 822 34764 491097 243 55488 minus579P26 200 10 150 378773 408401 782 46961 384126 139 79975 minus643P27 200 10 70 450468 494247 972 49136 461810 246 11018 minus726P28 200 10 150 374435 396998 603 33917 380004 147 77562 minus456P29 200 10 70 472898 485672 270 39796 479708 142 77673 minus128P30 200 10 150 364178 374923 295 35823 372971 236 91982 minus059

Average 422 12266 100 21135 minus322

4 Computational Results

The proposed tabu search algorithm was coded in the Cprogramand ran on a computerwith Intel(R)Core(TM)CPU(32GHz) with 4GB of RAM under the Microsoft Windows7 operation system

41 Test Beds Two test beds were used to evaluate theperformance of the proposed tabu search algorithms

(1) The benchmarks of CLRP (Prins et al [31]) whichcontains 30 instances two potential distribution cen-ter sizes |119869| = 5 or 10 are considered The numberof customers is |119868| = 20 50 100 and 200 Thecustomers and DCs are uniformly distributed in a[1 50]times[1 50] squareThe fixed opening costs of DCsare randomly generated from interval [5000 125000]The traveling cost between two nodes equals theirEuclidean distance multiplied by 100 and rounded

up to the next integer The setup cost of a vehicleis 1000 The vehicle capacity 119876 equals 70 or 150Customersrsquo demands are randomly selected frominterval [11 20]

(2) Randomly generated LIRP-FRL instances based onthe data of CLRP benchmarks and using the settingof [25] for the parameters such as the order costof new products of DCs 119860119898 (=200) the order costof recovered products of DCs 119860119903 (=100) the cost ofmanufacturing 119888119898 (=60) recovering 119888119903 (=50) disposal119888119889 (minus10) and the return rate of product we randomlygenerate the LIRP-FRL instances as follows

(i) The holding cost for each distribution center isuniformly distributed in 119880 [5 10]

(ii) The transportation cost per unit product of thedistribution center at site 119895 119895 isin 119869 is uniformlydistributed in 119880 [1 10]

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

Chen et al [10] concluded that the consideration ofreverse logistics in distribution network design can greatlyreduce the logistics costs without interrupting the forwardflows Taking both the forward logistics and reverse logisticsinto account many researches on facility location vehiclerouting and inventory management had been presented

Vahdani et al [11] focused on a fuzzy facility location-allocation problem in closed loop logistics Differing from[11] Vahdani et al [12 13] proposed a fuzzy possibility pro-gramming method to solve multiobjective facility problemsin forward and reverse logistic

Under the background of forward and reverse logisticdifferent inventory models had also been developed in theliterature Richter [14] proposed the first inventory modelin a closed logistic using EOQ model It assumed thatcustomersrsquo demands can be satisfied by newly made productsor by repaired products For a given fixed waste disposalrate the optimal minimum cost expression was derived Shiet al [15] considered an inventory policy under uncertaincustomer demand A Lagrangian relaxation based approachwas proposed

Vehicle routing decision is another important researchsubject in forward and reverse logisticmanagement Zachari-adis and Kiranoudis [16] studied a VRP problem under thebackground of forward and reverse logistic A tabu searchmethod was proposed For a similar VRP problem Goksal etal [17] developed a particle swarm optimization (PSO) basedapproach for its resolution in which a variable neighborhooddescent (VND) algorithm was used as a local search toexplore solution space

Different combination of facility location decision inven-torymanagement decision and vehicle routing decision givesrise to different research problems in forward and reverselogistics For example Liu et al [18] proposed a hybridparticle swarm optimization to solve a location routingproblem using grey systems theory in reverse logistics Yuand Lin [19] presented a location routing problem withsimultaneous pickup and delivery which was solved via amultistart simulated annealing algorithm with multistart hillclimbing Modeling the problem as a bilevel programmingproblemWang et al [20] studied realistic location-inventoryproblem in reverse logistic of a China B2C company Li etal [21] solved a location and inventory control problem bya two-stage heuristic algorithm that combined Lagrangianrelaxation method with an ant colony algorithm

Due to the intrinsic complexity of the problem inmathematics to our best knowledge only a few researchesexcept that of Li et al [22] simultaneously considered thedecisions of the facility location the inventory managementand vehicle routing in forward and reverse logistics in theliterature In their model they consider retrieving thoseproducts that customers dislike to theDCs As these productsare newor unused they can be sold to customers directlywithno need to be returned to manufacturer for repair

However as the work of Adler et al [23] indicated used(old or damaged) products (eg the engines of CaterpillarCompany) can also be retrieved recovered and returnedto the market to satisfy customersrsquo demands as new ones

This realistic new feature motivates us to focus on goodsproduction distribution and recycling

A location-inventory-routing problem in forward andreverse logistic (LIRP-FRL) network design is studied in thispaper The LIRP-FRL simultaneously integrates the locationdecisions of distribution centers (DCs) the inventory policiesof opened DCs and the vehicle routing decision in servingcustomers in which new goods are produced and damagedgoods are repaired by a manufacturer and then transportedto opened DCs Vehicles that start from and end in thesame DC distribute new or recovered goods to satisfy thedemands of customers and retrieve damaged goods Ourobjective is to minimize the total costs of manufacturingand remanufacturing goods building DCs shipping goods(new or recovered) between the manufacturer and openedDCs distributing new or recovered goods to customers andretrieving damaged goods from customers to DCs and theinventory costs of DCs including ordering new or recoveredgoods and holding these goods

Without taking the inventory strategy of DCs intoaccount our problem is simplified to the classical capacitatedlocation routing problem (CLRP) [24] which is NP-hardin strong sense Consequently our problem is also NP-hard Further the LIRP-FRL is a nonlinear problem as theEOQ inventory policy is adopted Due to its complexitywe focus on finding near optimal solutions for the LIRP-FRL A new tabu search (NTS) algorithm is proposed tofind approximately optimal solutions This algorithm prob-abilistically accepts a second best solution to change searchdirection to achieve this goal Numerical experiments of theCLRP benchmarks demonstrate the effectiveness and theefficiency of the proposed NTS method in obtaining highquality solutions in a very reasonable time Computationaland sensitivity analysis results on randomly generated LIRP-FRL instances are also reported

The rest of this paper is organized as follows In Section 2a mathematical model is developed to describe the studiedLIRP-FRL A tabu search that seeks to search near optimalsolution for the LIRP-FRL is proposed in Section 3 Algo-rithm performance evaluations are conducted in Section 4Finally we conclude the paper in Section 5

2 Mathematical Model

Given a set of potential sites of distribution centers (DCs) thestudied location-inventory-routing problem in forward andreverse logistic (LIRP-FRL) network design is to optimallydetermine the locations of DCs and their order quantitiesfrom amanufacturer the product flows (forward and reverse)between the manufacturer and the opened DCs and thepaths of vehicles in serving customers We assume that eachroute starts from and ends in the same DC by which newor recovered products are delivered to satisfy the demandsof customers for each cycle and damaged products areretrieved The objective of the LIRP-FRL is to minimize thesum of the opening costs and the inventory costs of DCs thesetup costs of vehicles and the transportation costs of (newrepaired and damaged) products An example of the studiedLIRP-FRL is illustrated in Figure 1

Discrete Dynamics in Nature and Society 3

Nonselected DCSelected DCManufacturer

CustomerForward logisticsReverse logistics

Figure 1 Location-inventory-routing problem in forward andreverse logistic

21 Notations The notations are described as follows

119868 the index set of customers119869 the index set of potential distribution centers119870 the index set of vehicles119878 the merged set of customers and potential distribu-tion centers (119868 cup 119869)119881119895 the maximum capacity of the distribution center119895 119895 isin 119869FC119895 the fixed opening cost of establishing distribu-tion center 119895 119895 isin 119869119876 the maximum capacity of homogeneous vehicle119888119894119895 the transportation cost of a vehicle on arc(119894 119895) 119894119895 isin 119878 119894 = 119895119891119895 the setup cost of using a vehicle at distributioncenter 119895 119895 isin 119869

119888119895 the product unit transportation cost from themanufacturer to distribution center 119895 119895 isin 119869119860119898 the order cost of new products of DCs per cycle

119860119903 the order cost of recovered or repaired products

of DCs per cycle119888119898 the cost of manufacturing a new product119888119903 the cost of repairing a damaged product to a newproduct119863119894 the demand of customers per cycleℎ119895 the unit holding cost of distribution center 119895 119895 isin 119869119903 the rate of return in percentage

119888119889 the unit cost of disposal of useless retrievedproducts119906 the percentage of reuse of retrieved products

22 Decision Variables The decision variables are describedas follows

119876119898119895 is the order quantity of new products of DC 119895 isin 119869

for each period119876119903119895 is the order quantity of recovered products of DC

119895 isin 119869 for each period119872119895 is the order times of new products from DC 119895 isin 119869

during each period119877119895 is the order times of recovered products from DC119895 isin 119869 during each period119909119894119895119896 = 1 if arc(119894 119895) is travelled by vehicle 119896 0otherwise (119894 119895 isin 119878 119896 isin 119870)119910119894119895 = 1 if customer 119894 isin 119868 is served by distributioncenter 119895 isin 119869 0 otherwise119885119895 = 1 if distribution center 119895 isin 119869 is established 0otherwise

23 Cost Analysis To simplify the presentation we firstabbreviate the costs related to the LIRP-FRL as follows

(1) The total fixed cost of opening distribution centers is

FIXCOST = sum

119895isin119869

FC119895 sdot 119911119895 (1)

(2) The total cost of ordering new and recovered productsis

OC = sum

119895isin119869

119860119898sdotsum119894isin119868119863119894 (1 minus 119906) 119910119894119895

119876119898119895+ 119860119903sdotsum119894isin119868119863119894119906119910119894119895

119876119903119895 (2)

(3) The total cost of producing new products and recov-ering retrieved products is

MR = sum

119895isin119869

sum

119894isin119868

119863119894 sdot ((1 minus 119906) sdot 119888119898+ 119906 sdot 119888

119903) sdot 119910119894119895 (3)

(4) The total cost of disposal of useless retrieved productsis

DISP = sum

119895isin119869

sum

119894isin119868

119863119894 sdot (119903 minus 119906) sdot 119888119889sdot 119910119894119895 (4)

(5) The transportation cost between the manufacturerand distribution centers is

TRAN = sum

119895isin119869

119888119895sum

119894isin119868

119863119894 sdot (1 + 119906) sdot 119910119894119895 (5)

(6) The transportation cost between distribution centersand customers is

TRC = sum

119896isin119870

sum

119895isin119878

sum

119894isin119878

119891119895 sdot 119909119894119895119896 + sum

119896isin119870

sum

119895isin119878

sum

119894isin119868

119888119894119895 sdot 119909119894119895119896 (6)

4 Discrete Dynamics in Nature and Society

Given the locations of DCs and the routes of vehicles theLIRP-FRL is reduced to the inventory model of Teunter [25]Some of their valuable conclusions can be directly applied toour LIRP-FRL

Proposition 1 (Teunter [25]) ldquoIt is reasonable to restrict ourattention to those policies with either119872 = 1 or 119877 = 1rdquo

Teunter [25] concluded that the policies with either119872 =

1 or 119877 = 1 can obtain near optimal solution with lesscomputational time by comparing with that of achievingoptimal solution Since our problem is more complicatedthan that of Teunter [25] to simplify the resolution of LIRP-FRL we consider only the two cases in our implementationThe two cases are as follows

Case 1 (1M) That means 119877 = 1 in this case the order timesof recovered products for each DC are equal to one

Case 2 (R 1) That means119872 = 1 in this case the order timesof new products for each DC are equal to one

For Case 1 (1M) the holding cost is

HM = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895 + 119876119903119895 sdot

119906

119903) sdot 119911119895 (7)

Consequently the total cost is denoted as

TCM = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HM(8)

For Case 2 (R 1) the holding cost is

HR = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895

+ (119903 (119876119898119895 + 119876

119903119895) minus (119903 minus 119906) sdot (119876

119898119895 +

119906

1 minus 119906119876119898119895 ))

119906

119903)

sdot 119911119895

(9)

The corresponding total cost is

TCR = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HR(10)

24 Mathematical Model The LIRP-FRL can be formulatedas the following nonlinearmixed integer programming prob-lemmin TCMTCR (11)

st sum

119896isin119870

sum

119894isin119878

119909119894119895119896 = 1 119895 isin 119868 (12)

sum

119895isin119878

119909119894119895119896 minussum

119895isin119878

119909119895119894119896 = 0 119894 isin 119878 119896 isin 119870 (13)

sum

119895isin119869

sum

119894isin119868

119909119894119895119896 le 1 119896 isin 119870 (14)

sum

119895isin119869

sum

119894isin119868

119863119894119909119894119895119896 le 119876 119896 isin 119870 (15)

sum

119894isin119868

119863119894119910119894119895 minus 119881119895 sdot 119911119895 le 0 119895 isin 119869 (16)

119910119894119895 + 1 ge sum

119892isin119868

119909119894119892119896 + sum

119892isin119878119895

119909119892119895119896

119894 isin 119869 119895 isin 119868 119896 isin 119870

(17)

sum

119895isin119882

sum

119894isin119882

119909119894119895119896 le |119882| minus 1 119882 sube 119868 119896 isin 119870 (18)

119909119894119895119896 isin 0 1 119894 isin 119878 119895 isin 119878 119896 isin 119870 (19)

119910119894119895 isin 0 1 119894 isin 119868 119895 isin 119869 (20)

119911119895 isin 0 1 119895 isin 119869 (21)

Constraints (12) state that each customer is exactly served byone vehicle Constraints (13) are the equilibrium constraintsof vehicle routes on nodes Constraints (14) ensure that eachvehicle can be used by at most one DC Constraints (15)guarantee that the capacity of vehicle cannot be exceededConstraints (16) are the capacity constraints of distributioncenters Constraints (17) make sure that a customer can onlybe served by the vehicle which originates from the DC itis assigned to Constraints (18) are the subtours eliminationconstraints of vehicle routes

Proposition 2 For any distribution center 119895 isin 119869 its orderquantities 119876

119898119895 and 119876

119903119895 can be represented as a function of

customersrsquo demands respectivelyWe have 119876

119898119895 = radic2119860119898sum119894isin119868119863119894 sdot 119910119894119895ℎ119895 and 119876

119903119895 =

radic2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895(119903 + 1)ℎ119895 for any 119895 isin 119869

Proof It can be observed that no matter case (1 M) orcase (R 1) decision variables 119876119898119895 and 119876

119903119895 appear only in the

objective function For the case (R 1) the optimal value of119876119898119895and 119876119903119895 can be obtained by checking the extreme conditions120597TCM120597119876

119898119895 = 0 and 120597TCM120597119876

119903119895 = 0

Since 1205972TCM120597(119876

119898119895 )2

= (2119860119898

sdot (1 minus 119906)sum119894isin119868119863119894 sdot

119910119894119895)(119876119898119895 )3ge 0 and 120597

2TCM120597(119876119903119895)2= (2119860

119903sdot 119906sum119894isin119868119863119894 sdot

119910119894119895)(119876119903119895)3ge 0 consequently the extreme point is a minimum

point We obtain the optimal order quantities 119876119898119895 =

radic(2119860119898sum119894isin119868119863119894 sdot (1 minus 119906)119910119894119895)ℎ119895(1 minus ((119903 minus 119906)(1 minus 119906)) sdot (119906119903))

and 119876119903119895 = radic(119860119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and the corresponding

order time 119877119895 = (119906 sdot 119876119898119895 )(1 minus 119906 sdot 119876

119903119895)

For the same reason for the case (1 119872) we can derivethat the order quantities 119876119898119895 = radic(2119860119898sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and

119876119903119895 = radic(2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)(119903 + 1)ℎ119895 and the correspond-

ing order time119872119895 = (1 minus 119906 sdot 119876119903119895)(119906 sdot 119876

119898119895 )

Observe that without considering the inventory strategyof these two cases Cases 1 and 2 the LIRP-FRL is reducedto the classical capacitated location routing problem (CLRP)which is NP-hard in the strong sense As a result the LIRP-FRL is also NP-hard As Belenguer et al [24] reported to

Discrete Dynamics in Nature and Society 5

21 23 19 16 15 14 8 11 6 22 4 1 12 18 20 13 5 7 3 24 25 2 17 9 10

Figure 2 An example of solution representation

find CLRP optimal solution even for the median size CLRPinstances (with 50 customers and 5 potential distributioncenters) significant research efforts need to be expensedThus in this paper we focus onfinding near optimal solutionsfor the LIRP-FRL

3 New Tabu Search

Using tabu list to avoid visited solutions to be revisited thetabu search proposed by Glover [26] is one of the mosteffective approaches for solving mixed integer programmingproblems which has been successfully applied to a varietyof distribution network design problems For the detailedintroduction and application of the tabu search we referreaders to Glover [27] and Habet [28]

In this section a tabu search (NTS) algorithm thatprobabilistically accepts the second best solution in searchprocess when a local optimum is reached is proposed toeffectively solve the LIRP-FRL The key components of theNTS including a solution representation technique an initialsolution generation method neighbourhood structures andthe general framework of the new tabu search are presentedin detail as follows

31 Solution Representation A good solution representationcan not only describe the problem solutions clearly butalso improve the performance of heuristics adopted Similarto that of Yu et al [29] in our implementation solution1199091 1199092 119909119899 119909119899+1 119909119899+119898 is represented by a permuta-tion of number 1 2 119899 119899 + 1 119899 + 2 119899 + 119898 where119909119895 isin 1 2 119899 indexes customer 119909119895 and 119909119894 isin 119899 + 1 119899 +

2 119899 + 119898 denotes potential DC 119909119894An example was given in Figure 2 In this example the

first number is distribution center 21 followed by distributioncenter 23 Since there are no customers between distributioncenter 21 and distribution center 23 distribution center 21is closed Customers (19 16 15 14 8 11 and 6) betweendistribution center 23 and distribution center 22 are realcustomers thus distribution center 23 is opened to servicethese customers The first route of distribution center 23services customers 19 16 15 and 14 Because adding customer8 exceeds the vehicle capacity the second route servicescustomers 8 11 and 6 Customer 6 is followed by distributioncenter 22 so the second route is terminated For a similarreason asmentioned above distribution centers 22 and 25 areopen with two routes and one route respectively

This representation has determined which distributioncenter is open and the customers on each route With thatdecided we can easily calculate the objective function valuebecause the inventory cost is related to LRP which meanswhen LRP is solved the inventory cost can be uniquelydetermined by Proposition 2 Note that the capacity ofdistribution center is not taken into consideration So during

the decoding process a per unit penalty cost 119872 with a bigvalue is added to the objective function value when the totaldemand serviced by a distribution center exceeds its capacity

To simplify the presentation of the algorithms proposedin following subsections we say that an element 119909119894 of thesolution is before another element 119909119895 or element 119909119895 is afterelement 119909119894 if and only if 119894 lt 119895 Further with the smallest value|119894 minus 119895| if DC 119909119894 is before DC 119909119895 we say that DC 119909119894 is nearestbefore DC 119909119895 or DC 119909119895 is nearest after DC 119909119894

32 Constructing Initial Solution A greedy nearest neighbormethod is proposed to construct an initial feasible solution ofthe LIRP-FRL which is used as an input for the tabu searchDetails of the greedy algorithm are given as follows

Algorithm 3 (greedy nearest neighbour method)

Step 0 InitializeΩ0 fl 119869 Ω1 fl Φ0 fl 119868 and Φ1 fl

Step 1 Assign each customer inΦ0 to its closest DCs inΩ0

Step 2 Open the DC in Ω0 with the largest number of

customers assigned to say DC119895lowast

Step 3 Sort the customers of set Φ0 in nondecreasing orderof the distance between it and the DC119895lowast

Step 4 With the order let Φ = 1199041 1199042 119904|Φ0| be the subsetof customers of Φ0 with the largest |Φ| and sum

|Φ|119894=1 119889119904119894 le 119881119895lowast

Renew Ω1 fl Ω

1cup 119895lowast Ω0 fl 119869Ω

1 Φ1 fl Φ1cup Φ and

Φ0 fl 119868Φ

1

Step 5 If Φ0 = then return to Step 1 Otherwise open theDCs inΩ1 and assign each customer to a DC according to thecustomer-DC assignments determined in Step 4

Step 6 For each opened DC sort the customers assignedto it into sequence by using a nearest neighbor heuristic(refer to Rosenkrantz et al [30]) Split the sequence intoseveral feasible routes so that vehicle capacity constraints aresatisfied

Step 7 Output feasible solution

In our implementation of Step 2 if more than oneDChasthe largest number of customers assigned to we select theDCwith the highest capacity

33 Neighborhoods Given a solution

119883 = (1199091 1199092 119909119894minus1 119909119894 119909119894+1 119909119895minus1 119909119895 119909119895+1 119909119899) (22)

a permutation of 1 2 119899 119899 + 1 119899 + 119898 three kinds ofneighbourhoods are considered in our tabu search

6 Discrete Dynamics in Nature and Society

331 Insertion Neighbourhood The insertion neighborhoodis defined as the set of solutions that can be reached byinsertion move which deletes one element of the solution119883 and then inserts it to another place For example deleteelement 119909119894 and insert it into the 119895th place we obtain a newsolution

1198831015840= (1199091 1199092 119909119894minus1 119909119894+1 119909119895minus1 119909119895 119909119894 119909119895+1 119909119899) (23)

According to the element selected (customer or depot)and the position it is inserted in there are four cases

(a) If 119909119894 and 119909119895 are both customers then customer 119909119894 isreassigned to the DC that customer 119909119895 is assigned to

(b) If 119909119894 and 119909119895 are bothDCs then reassign the customerspreviously assigned to the DC nearest before it CloseDC 119909119895 and assign its customers to DC 119909119894

(c) If 119909119894 is a customer and 119909119895 is a DC then customer 119909119894 isreassigned to DC 119909119895

(d) If 119909119894 is a DC and 119909119895 is a customer reassign customersthat are nearest after customer 119909119895 to DC 119909119894 Allcustomers assigned to DC 119909119894 are reassigned to the DCnearest before it

332 Swap Neighbourhood The swap neighbourhood a setof solutions contains the solutions that can be reached byperforming a swap move which exchanges the places oftwo different elements of solution 119883 For example exchangeelement 119909119894 and element 119909119895 of119883 the resulting new solution is

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119894+1 119909119895minus1 119909119894 119909119895+1 119909119899) (24)

Similar to the insert move the swap move has also fourcases

(e) If 119909119894 and 119909119895 are both customers then we exchange thetwoDCs they are currently assigned to and obtain twonew customer-DC assignments

(f) If 119909119894 and 119909119895 are both DCs then we exchange all thecustomers currently assigned to them

(g) If 119909119894 is a customer and 119909119895 is a DC then the customersafter 119909119894 assigned to the same DC are reassigned to DC119909119895 and customer 119909119894 is reassigned to the DC nearestbefore DC 119909119895 All customers previously assigned toDC 119909119895 will be reassigned to the DC nearest before it

(h) If 119909119894 is a DC and 119909119895 is a customer this case is similarto case g

333 2-Opt Neighbourhood With the consideration of vehi-cle capacity in our definition the 2-opt neighbourhood con-tains the solutions that can be obtained by selecting two cus-tomers of119883 (eg 119909119894 and 119909119895) and then reversing the substringin the solution representation between them the so-called 2-opt move For example applying 2-opt move to elements 119909119894and 119909119895 we obtain

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119895minus1 119909119894+1 119909119894 119909119895+1 119909119899) (25)

34 Tabu Search Framework To increase the probability ofobtaining high quality solutions a diversification strategyis implemented This strategy probabilistically accepts thesecond best solution in the search process while no bettersolution can be found in the neighborhoods of the currentsolution By doing so we can effectively diversify the searchdirection and thusmore solution spacewill be explored Con-sequently better solution may be found The computationresults in Section 4 prove this strategy The proposed tabusearch can be summarized as follows

Algorithm 4 (new tabu search)

Step 0 Input an initial feasible solution 1198830 (see Section 32)and initialize Iter fl 0119883curr fl 1198830 and Objlowast fl Obj(1198830)

Step 1 Explore the neighborhoods of 119883curr and denote thebest solution in the neighborhoods the best nontabu solu-tion and the second best nontabu solution as119883best119883tabu-bestand119883tabu-2nd respectively

Step 2 If Obj(119883best) lt Objlowast then set 119883curr fl 119883best and addthe corresponding move to tabu list otherwise

if Obj(119883tabu-best) lt Obj(119883curr) then119883curr fl 119883tabu-bestand add the corresponding move to tabu list

otherwise probabilistically move to 119883tabu-best or119883tabu-2nd 119883curr fl Prob119883tabu-best 119883tabu-2nd and addthe corresponding move to tabu list

Step 3 Iter fl Iter + 1 If no stopping criterion is reachedreturn to Step 1

Step 4 Output the best solution objective function valueObjlowast

In the proposed tabu search algorithm Iter counts thenumber of iterations and 1198830 represents the initial solutionobtained by greedy nearest neighbour method (Section 32)119883curr denotes the current solution Objlowast is the objectivefunction value corresponding to the best solution found sofar The function Obj(119883) represents the objective functionvalue of solution119883

In Step 2 comparing with the current solution119883curr if nobetter solution can be found in its neighbourhood we needto probabilistically move 119883curr to 119883tabu-best or 119883tabu-2nd Inimplementation we randomly generate a value 120573 in interval[0 1] If120573 ge 120582 then119883curr ismoved to119883tabu-best and otherwise119883curr is moved to119883tabu-2nd where 120582 is a given parameter

At each iteration of the tabu search procedure thebest admissible nontabu insertion swap or 2-opt move isperformed The two elements of solution 119883curr involved insuch move are declared as tabu-active which is forbiddento be removed in the next 119905 iterations 119905 is the tabu-tenureparameter selected randomly from a range [119879min 119879max]

The tabu search iteration is terminated if the maximumnumber of iterations maxIter is reached or if the bestsolution found so far has been improved in successive SucIteriteration

Discrete Dynamics in Nature and Society 7

Table 1 Computational results on CLRP benchmarks

Problem |119868| |119869| 119876 BKS TS NTS DEVUB Gap Time UB Gap Time

P1 20 5 70 54793 54793 000 012 54793 000 049 000P2 20 5 150 39104 39104 000 012 39104 000 069 000P3 20 5 70 48908 48908 000 050 48908 000 053 000P4 20 5 150 37542 37611 018 037 37611 018 074 000P5 50 5 70 90111 90806 077 258 90461 039 335 minus038P6 50 5 150 63242 65010 280 1043 63242 000 1173 minus280P7 50 5 70 88298 89688 157 2207 88643 039 2394 minus118P8 50 5 150 67340 68071 109 748 67340 000 1116 minus109P9 50 5 70 84055 84227 020 1413 84110 007 1696 minus013P10 50 5 150 51822 51902 015 1021 51883 012 1349 minus003P11 50 5 70 86203 86223 002 920 86203 000 1471 minus002P12 50 5 150 61830 61978 024 1954 61830 000 2149 minus024P13 100 5 70 275993 281229 190 1151 278926 105 3841 minus085P14 100 5 150 214392 218886 210 952 216802 111 2974 minus099P15 100 5 70 194598 196587 102 14525 196436 094 9516 minus008P16 100 5 150 157173 159244 132 18038 158720 097 12463 minus035P17 100 5 70 200246 203297 152 12612 202682 120 11085 minus032P18 100 5 150 152586 157859 346 12247 156049 222 10526 minus124P19 100 10 70 290429 334486 1517 9780 294774 147 1432 minus1370P20 100 10 150 234641 285303 2159 7587 236217 067 1136 minus2092P21 100 10 70 244265 257455 540 10349 248003 151 13985 minus389P22 100 10 150 203988 216253 601 6965 208455 214 107 minus387P23 100 10 70 253344 291153 1492 16998 258221 189 15353 minus1303P24 100 10 150 204597 220528 779 6706 209390 229 13152 minus550P25 200 10 70 479425 518815 822 34764 491097 243 55488 minus579P26 200 10 150 378773 408401 782 46961 384126 139 79975 minus643P27 200 10 70 450468 494247 972 49136 461810 246 11018 minus726P28 200 10 150 374435 396998 603 33917 380004 147 77562 minus456P29 200 10 70 472898 485672 270 39796 479708 142 77673 minus128P30 200 10 150 364178 374923 295 35823 372971 236 91982 minus059

Average 422 12266 100 21135 minus322

4 Computational Results

The proposed tabu search algorithm was coded in the Cprogramand ran on a computerwith Intel(R)Core(TM)CPU(32GHz) with 4GB of RAM under the Microsoft Windows7 operation system

41 Test Beds Two test beds were used to evaluate theperformance of the proposed tabu search algorithms

(1) The benchmarks of CLRP (Prins et al [31]) whichcontains 30 instances two potential distribution cen-ter sizes |119869| = 5 or 10 are considered The numberof customers is |119868| = 20 50 100 and 200 Thecustomers and DCs are uniformly distributed in a[1 50]times[1 50] squareThe fixed opening costs of DCsare randomly generated from interval [5000 125000]The traveling cost between two nodes equals theirEuclidean distance multiplied by 100 and rounded

up to the next integer The setup cost of a vehicleis 1000 The vehicle capacity 119876 equals 70 or 150Customersrsquo demands are randomly selected frominterval [11 20]

(2) Randomly generated LIRP-FRL instances based onthe data of CLRP benchmarks and using the settingof [25] for the parameters such as the order costof new products of DCs 119860119898 (=200) the order costof recovered products of DCs 119860119903 (=100) the cost ofmanufacturing 119888119898 (=60) recovering 119888119903 (=50) disposal119888119889 (minus10) and the return rate of product we randomlygenerate the LIRP-FRL instances as follows

(i) The holding cost for each distribution center isuniformly distributed in 119880 [5 10]

(ii) The transportation cost per unit product of thedistribution center at site 119895 119895 isin 119869 is uniformlydistributed in 119880 [1 10]

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

Nonselected DCSelected DCManufacturer

CustomerForward logisticsReverse logistics

Figure 1 Location-inventory-routing problem in forward andreverse logistic

21 Notations The notations are described as follows

119868 the index set of customers119869 the index set of potential distribution centers119870 the index set of vehicles119878 the merged set of customers and potential distribu-tion centers (119868 cup 119869)119881119895 the maximum capacity of the distribution center119895 119895 isin 119869FC119895 the fixed opening cost of establishing distribu-tion center 119895 119895 isin 119869119876 the maximum capacity of homogeneous vehicle119888119894119895 the transportation cost of a vehicle on arc(119894 119895) 119894119895 isin 119878 119894 = 119895119891119895 the setup cost of using a vehicle at distributioncenter 119895 119895 isin 119869

119888119895 the product unit transportation cost from themanufacturer to distribution center 119895 119895 isin 119869119860119898 the order cost of new products of DCs per cycle

119860119903 the order cost of recovered or repaired products

of DCs per cycle119888119898 the cost of manufacturing a new product119888119903 the cost of repairing a damaged product to a newproduct119863119894 the demand of customers per cycleℎ119895 the unit holding cost of distribution center 119895 119895 isin 119869119903 the rate of return in percentage

119888119889 the unit cost of disposal of useless retrievedproducts119906 the percentage of reuse of retrieved products

22 Decision Variables The decision variables are describedas follows

119876119898119895 is the order quantity of new products of DC 119895 isin 119869

for each period119876119903119895 is the order quantity of recovered products of DC

119895 isin 119869 for each period119872119895 is the order times of new products from DC 119895 isin 119869

during each period119877119895 is the order times of recovered products from DC119895 isin 119869 during each period119909119894119895119896 = 1 if arc(119894 119895) is travelled by vehicle 119896 0otherwise (119894 119895 isin 119878 119896 isin 119870)119910119894119895 = 1 if customer 119894 isin 119868 is served by distributioncenter 119895 isin 119869 0 otherwise119885119895 = 1 if distribution center 119895 isin 119869 is established 0otherwise

23 Cost Analysis To simplify the presentation we firstabbreviate the costs related to the LIRP-FRL as follows

(1) The total fixed cost of opening distribution centers is

FIXCOST = sum

119895isin119869

FC119895 sdot 119911119895 (1)

(2) The total cost of ordering new and recovered productsis

OC = sum

119895isin119869

119860119898sdotsum119894isin119868119863119894 (1 minus 119906) 119910119894119895

119876119898119895+ 119860119903sdotsum119894isin119868119863119894119906119910119894119895

119876119903119895 (2)

(3) The total cost of producing new products and recov-ering retrieved products is

MR = sum

119895isin119869

sum

119894isin119868

119863119894 sdot ((1 minus 119906) sdot 119888119898+ 119906 sdot 119888

119903) sdot 119910119894119895 (3)

(4) The total cost of disposal of useless retrieved productsis

DISP = sum

119895isin119869

sum

119894isin119868

119863119894 sdot (119903 minus 119906) sdot 119888119889sdot 119910119894119895 (4)

(5) The transportation cost between the manufacturerand distribution centers is

TRAN = sum

119895isin119869

119888119895sum

119894isin119868

119863119894 sdot (1 + 119906) sdot 119910119894119895 (5)

(6) The transportation cost between distribution centersand customers is

TRC = sum

119896isin119870

sum

119895isin119878

sum

119894isin119878

119891119895 sdot 119909119894119895119896 + sum

119896isin119870

sum

119895isin119878

sum

119894isin119868

119888119894119895 sdot 119909119894119895119896 (6)

4 Discrete Dynamics in Nature and Society

Given the locations of DCs and the routes of vehicles theLIRP-FRL is reduced to the inventory model of Teunter [25]Some of their valuable conclusions can be directly applied toour LIRP-FRL

Proposition 1 (Teunter [25]) ldquoIt is reasonable to restrict ourattention to those policies with either119872 = 1 or 119877 = 1rdquo

Teunter [25] concluded that the policies with either119872 =

1 or 119877 = 1 can obtain near optimal solution with lesscomputational time by comparing with that of achievingoptimal solution Since our problem is more complicatedthan that of Teunter [25] to simplify the resolution of LIRP-FRL we consider only the two cases in our implementationThe two cases are as follows

Case 1 (1M) That means 119877 = 1 in this case the order timesof recovered products for each DC are equal to one

Case 2 (R 1) That means119872 = 1 in this case the order timesof new products for each DC are equal to one

For Case 1 (1M) the holding cost is

HM = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895 + 119876119903119895 sdot

119906

119903) sdot 119911119895 (7)

Consequently the total cost is denoted as

TCM = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HM(8)

For Case 2 (R 1) the holding cost is

HR = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895

+ (119903 (119876119898119895 + 119876

119903119895) minus (119903 minus 119906) sdot (119876

119898119895 +

119906

1 minus 119906119876119898119895 ))

119906

119903)

sdot 119911119895

(9)

The corresponding total cost is

TCR = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HR(10)

24 Mathematical Model The LIRP-FRL can be formulatedas the following nonlinearmixed integer programming prob-lemmin TCMTCR (11)

st sum

119896isin119870

sum

119894isin119878

119909119894119895119896 = 1 119895 isin 119868 (12)

sum

119895isin119878

119909119894119895119896 minussum

119895isin119878

119909119895119894119896 = 0 119894 isin 119878 119896 isin 119870 (13)

sum

119895isin119869

sum

119894isin119868

119909119894119895119896 le 1 119896 isin 119870 (14)

sum

119895isin119869

sum

119894isin119868

119863119894119909119894119895119896 le 119876 119896 isin 119870 (15)

sum

119894isin119868

119863119894119910119894119895 minus 119881119895 sdot 119911119895 le 0 119895 isin 119869 (16)

119910119894119895 + 1 ge sum

119892isin119868

119909119894119892119896 + sum

119892isin119878119895

119909119892119895119896

119894 isin 119869 119895 isin 119868 119896 isin 119870

(17)

sum

119895isin119882

sum

119894isin119882

119909119894119895119896 le |119882| minus 1 119882 sube 119868 119896 isin 119870 (18)

119909119894119895119896 isin 0 1 119894 isin 119878 119895 isin 119878 119896 isin 119870 (19)

119910119894119895 isin 0 1 119894 isin 119868 119895 isin 119869 (20)

119911119895 isin 0 1 119895 isin 119869 (21)

Constraints (12) state that each customer is exactly served byone vehicle Constraints (13) are the equilibrium constraintsof vehicle routes on nodes Constraints (14) ensure that eachvehicle can be used by at most one DC Constraints (15)guarantee that the capacity of vehicle cannot be exceededConstraints (16) are the capacity constraints of distributioncenters Constraints (17) make sure that a customer can onlybe served by the vehicle which originates from the DC itis assigned to Constraints (18) are the subtours eliminationconstraints of vehicle routes

Proposition 2 For any distribution center 119895 isin 119869 its orderquantities 119876

119898119895 and 119876

119903119895 can be represented as a function of

customersrsquo demands respectivelyWe have 119876

119898119895 = radic2119860119898sum119894isin119868119863119894 sdot 119910119894119895ℎ119895 and 119876

119903119895 =

radic2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895(119903 + 1)ℎ119895 for any 119895 isin 119869

Proof It can be observed that no matter case (1 M) orcase (R 1) decision variables 119876119898119895 and 119876

119903119895 appear only in the

objective function For the case (R 1) the optimal value of119876119898119895and 119876119903119895 can be obtained by checking the extreme conditions120597TCM120597119876

119898119895 = 0 and 120597TCM120597119876

119903119895 = 0

Since 1205972TCM120597(119876

119898119895 )2

= (2119860119898

sdot (1 minus 119906)sum119894isin119868119863119894 sdot

119910119894119895)(119876119898119895 )3ge 0 and 120597

2TCM120597(119876119903119895)2= (2119860

119903sdot 119906sum119894isin119868119863119894 sdot

119910119894119895)(119876119903119895)3ge 0 consequently the extreme point is a minimum

point We obtain the optimal order quantities 119876119898119895 =

radic(2119860119898sum119894isin119868119863119894 sdot (1 minus 119906)119910119894119895)ℎ119895(1 minus ((119903 minus 119906)(1 minus 119906)) sdot (119906119903))

and 119876119903119895 = radic(119860119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and the corresponding

order time 119877119895 = (119906 sdot 119876119898119895 )(1 minus 119906 sdot 119876

119903119895)

For the same reason for the case (1 119872) we can derivethat the order quantities 119876119898119895 = radic(2119860119898sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and

119876119903119895 = radic(2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)(119903 + 1)ℎ119895 and the correspond-

ing order time119872119895 = (1 minus 119906 sdot 119876119903119895)(119906 sdot 119876

119898119895 )

Observe that without considering the inventory strategyof these two cases Cases 1 and 2 the LIRP-FRL is reducedto the classical capacitated location routing problem (CLRP)which is NP-hard in the strong sense As a result the LIRP-FRL is also NP-hard As Belenguer et al [24] reported to

Discrete Dynamics in Nature and Society 5

21 23 19 16 15 14 8 11 6 22 4 1 12 18 20 13 5 7 3 24 25 2 17 9 10

Figure 2 An example of solution representation

find CLRP optimal solution even for the median size CLRPinstances (with 50 customers and 5 potential distributioncenters) significant research efforts need to be expensedThus in this paper we focus onfinding near optimal solutionsfor the LIRP-FRL

3 New Tabu Search

Using tabu list to avoid visited solutions to be revisited thetabu search proposed by Glover [26] is one of the mosteffective approaches for solving mixed integer programmingproblems which has been successfully applied to a varietyof distribution network design problems For the detailedintroduction and application of the tabu search we referreaders to Glover [27] and Habet [28]

In this section a tabu search (NTS) algorithm thatprobabilistically accepts the second best solution in searchprocess when a local optimum is reached is proposed toeffectively solve the LIRP-FRL The key components of theNTS including a solution representation technique an initialsolution generation method neighbourhood structures andthe general framework of the new tabu search are presentedin detail as follows

31 Solution Representation A good solution representationcan not only describe the problem solutions clearly butalso improve the performance of heuristics adopted Similarto that of Yu et al [29] in our implementation solution1199091 1199092 119909119899 119909119899+1 119909119899+119898 is represented by a permuta-tion of number 1 2 119899 119899 + 1 119899 + 2 119899 + 119898 where119909119895 isin 1 2 119899 indexes customer 119909119895 and 119909119894 isin 119899 + 1 119899 +

2 119899 + 119898 denotes potential DC 119909119894An example was given in Figure 2 In this example the

first number is distribution center 21 followed by distributioncenter 23 Since there are no customers between distributioncenter 21 and distribution center 23 distribution center 21is closed Customers (19 16 15 14 8 11 and 6) betweendistribution center 23 and distribution center 22 are realcustomers thus distribution center 23 is opened to servicethese customers The first route of distribution center 23services customers 19 16 15 and 14 Because adding customer8 exceeds the vehicle capacity the second route servicescustomers 8 11 and 6 Customer 6 is followed by distributioncenter 22 so the second route is terminated For a similarreason asmentioned above distribution centers 22 and 25 areopen with two routes and one route respectively

This representation has determined which distributioncenter is open and the customers on each route With thatdecided we can easily calculate the objective function valuebecause the inventory cost is related to LRP which meanswhen LRP is solved the inventory cost can be uniquelydetermined by Proposition 2 Note that the capacity ofdistribution center is not taken into consideration So during

the decoding process a per unit penalty cost 119872 with a bigvalue is added to the objective function value when the totaldemand serviced by a distribution center exceeds its capacity

To simplify the presentation of the algorithms proposedin following subsections we say that an element 119909119894 of thesolution is before another element 119909119895 or element 119909119895 is afterelement 119909119894 if and only if 119894 lt 119895 Further with the smallest value|119894 minus 119895| if DC 119909119894 is before DC 119909119895 we say that DC 119909119894 is nearestbefore DC 119909119895 or DC 119909119895 is nearest after DC 119909119894

32 Constructing Initial Solution A greedy nearest neighbormethod is proposed to construct an initial feasible solution ofthe LIRP-FRL which is used as an input for the tabu searchDetails of the greedy algorithm are given as follows

Algorithm 3 (greedy nearest neighbour method)

Step 0 InitializeΩ0 fl 119869 Ω1 fl Φ0 fl 119868 and Φ1 fl

Step 1 Assign each customer inΦ0 to its closest DCs inΩ0

Step 2 Open the DC in Ω0 with the largest number of

customers assigned to say DC119895lowast

Step 3 Sort the customers of set Φ0 in nondecreasing orderof the distance between it and the DC119895lowast

Step 4 With the order let Φ = 1199041 1199042 119904|Φ0| be the subsetof customers of Φ0 with the largest |Φ| and sum

|Φ|119894=1 119889119904119894 le 119881119895lowast

Renew Ω1 fl Ω

1cup 119895lowast Ω0 fl 119869Ω

1 Φ1 fl Φ1cup Φ and

Φ0 fl 119868Φ

1

Step 5 If Φ0 = then return to Step 1 Otherwise open theDCs inΩ1 and assign each customer to a DC according to thecustomer-DC assignments determined in Step 4

Step 6 For each opened DC sort the customers assignedto it into sequence by using a nearest neighbor heuristic(refer to Rosenkrantz et al [30]) Split the sequence intoseveral feasible routes so that vehicle capacity constraints aresatisfied

Step 7 Output feasible solution

In our implementation of Step 2 if more than oneDChasthe largest number of customers assigned to we select theDCwith the highest capacity

33 Neighborhoods Given a solution

119883 = (1199091 1199092 119909119894minus1 119909119894 119909119894+1 119909119895minus1 119909119895 119909119895+1 119909119899) (22)

a permutation of 1 2 119899 119899 + 1 119899 + 119898 three kinds ofneighbourhoods are considered in our tabu search

6 Discrete Dynamics in Nature and Society

331 Insertion Neighbourhood The insertion neighborhoodis defined as the set of solutions that can be reached byinsertion move which deletes one element of the solution119883 and then inserts it to another place For example deleteelement 119909119894 and insert it into the 119895th place we obtain a newsolution

1198831015840= (1199091 1199092 119909119894minus1 119909119894+1 119909119895minus1 119909119895 119909119894 119909119895+1 119909119899) (23)

According to the element selected (customer or depot)and the position it is inserted in there are four cases

(a) If 119909119894 and 119909119895 are both customers then customer 119909119894 isreassigned to the DC that customer 119909119895 is assigned to

(b) If 119909119894 and 119909119895 are bothDCs then reassign the customerspreviously assigned to the DC nearest before it CloseDC 119909119895 and assign its customers to DC 119909119894

(c) If 119909119894 is a customer and 119909119895 is a DC then customer 119909119894 isreassigned to DC 119909119895

(d) If 119909119894 is a DC and 119909119895 is a customer reassign customersthat are nearest after customer 119909119895 to DC 119909119894 Allcustomers assigned to DC 119909119894 are reassigned to the DCnearest before it

332 Swap Neighbourhood The swap neighbourhood a setof solutions contains the solutions that can be reached byperforming a swap move which exchanges the places oftwo different elements of solution 119883 For example exchangeelement 119909119894 and element 119909119895 of119883 the resulting new solution is

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119894+1 119909119895minus1 119909119894 119909119895+1 119909119899) (24)

Similar to the insert move the swap move has also fourcases

(e) If 119909119894 and 119909119895 are both customers then we exchange thetwoDCs they are currently assigned to and obtain twonew customer-DC assignments

(f) If 119909119894 and 119909119895 are both DCs then we exchange all thecustomers currently assigned to them

(g) If 119909119894 is a customer and 119909119895 is a DC then the customersafter 119909119894 assigned to the same DC are reassigned to DC119909119895 and customer 119909119894 is reassigned to the DC nearestbefore DC 119909119895 All customers previously assigned toDC 119909119895 will be reassigned to the DC nearest before it

(h) If 119909119894 is a DC and 119909119895 is a customer this case is similarto case g

333 2-Opt Neighbourhood With the consideration of vehi-cle capacity in our definition the 2-opt neighbourhood con-tains the solutions that can be obtained by selecting two cus-tomers of119883 (eg 119909119894 and 119909119895) and then reversing the substringin the solution representation between them the so-called 2-opt move For example applying 2-opt move to elements 119909119894and 119909119895 we obtain

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119895minus1 119909119894+1 119909119894 119909119895+1 119909119899) (25)

34 Tabu Search Framework To increase the probability ofobtaining high quality solutions a diversification strategyis implemented This strategy probabilistically accepts thesecond best solution in the search process while no bettersolution can be found in the neighborhoods of the currentsolution By doing so we can effectively diversify the searchdirection and thusmore solution spacewill be explored Con-sequently better solution may be found The computationresults in Section 4 prove this strategy The proposed tabusearch can be summarized as follows

Algorithm 4 (new tabu search)

Step 0 Input an initial feasible solution 1198830 (see Section 32)and initialize Iter fl 0119883curr fl 1198830 and Objlowast fl Obj(1198830)

Step 1 Explore the neighborhoods of 119883curr and denote thebest solution in the neighborhoods the best nontabu solu-tion and the second best nontabu solution as119883best119883tabu-bestand119883tabu-2nd respectively

Step 2 If Obj(119883best) lt Objlowast then set 119883curr fl 119883best and addthe corresponding move to tabu list otherwise

if Obj(119883tabu-best) lt Obj(119883curr) then119883curr fl 119883tabu-bestand add the corresponding move to tabu list

otherwise probabilistically move to 119883tabu-best or119883tabu-2nd 119883curr fl Prob119883tabu-best 119883tabu-2nd and addthe corresponding move to tabu list

Step 3 Iter fl Iter + 1 If no stopping criterion is reachedreturn to Step 1

Step 4 Output the best solution objective function valueObjlowast

In the proposed tabu search algorithm Iter counts thenumber of iterations and 1198830 represents the initial solutionobtained by greedy nearest neighbour method (Section 32)119883curr denotes the current solution Objlowast is the objectivefunction value corresponding to the best solution found sofar The function Obj(119883) represents the objective functionvalue of solution119883

In Step 2 comparing with the current solution119883curr if nobetter solution can be found in its neighbourhood we needto probabilistically move 119883curr to 119883tabu-best or 119883tabu-2nd Inimplementation we randomly generate a value 120573 in interval[0 1] If120573 ge 120582 then119883curr ismoved to119883tabu-best and otherwise119883curr is moved to119883tabu-2nd where 120582 is a given parameter

At each iteration of the tabu search procedure thebest admissible nontabu insertion swap or 2-opt move isperformed The two elements of solution 119883curr involved insuch move are declared as tabu-active which is forbiddento be removed in the next 119905 iterations 119905 is the tabu-tenureparameter selected randomly from a range [119879min 119879max]

The tabu search iteration is terminated if the maximumnumber of iterations maxIter is reached or if the bestsolution found so far has been improved in successive SucIteriteration

Discrete Dynamics in Nature and Society 7

Table 1 Computational results on CLRP benchmarks

Problem |119868| |119869| 119876 BKS TS NTS DEVUB Gap Time UB Gap Time

P1 20 5 70 54793 54793 000 012 54793 000 049 000P2 20 5 150 39104 39104 000 012 39104 000 069 000P3 20 5 70 48908 48908 000 050 48908 000 053 000P4 20 5 150 37542 37611 018 037 37611 018 074 000P5 50 5 70 90111 90806 077 258 90461 039 335 minus038P6 50 5 150 63242 65010 280 1043 63242 000 1173 minus280P7 50 5 70 88298 89688 157 2207 88643 039 2394 minus118P8 50 5 150 67340 68071 109 748 67340 000 1116 minus109P9 50 5 70 84055 84227 020 1413 84110 007 1696 minus013P10 50 5 150 51822 51902 015 1021 51883 012 1349 minus003P11 50 5 70 86203 86223 002 920 86203 000 1471 minus002P12 50 5 150 61830 61978 024 1954 61830 000 2149 minus024P13 100 5 70 275993 281229 190 1151 278926 105 3841 minus085P14 100 5 150 214392 218886 210 952 216802 111 2974 minus099P15 100 5 70 194598 196587 102 14525 196436 094 9516 minus008P16 100 5 150 157173 159244 132 18038 158720 097 12463 minus035P17 100 5 70 200246 203297 152 12612 202682 120 11085 minus032P18 100 5 150 152586 157859 346 12247 156049 222 10526 minus124P19 100 10 70 290429 334486 1517 9780 294774 147 1432 minus1370P20 100 10 150 234641 285303 2159 7587 236217 067 1136 minus2092P21 100 10 70 244265 257455 540 10349 248003 151 13985 minus389P22 100 10 150 203988 216253 601 6965 208455 214 107 minus387P23 100 10 70 253344 291153 1492 16998 258221 189 15353 minus1303P24 100 10 150 204597 220528 779 6706 209390 229 13152 minus550P25 200 10 70 479425 518815 822 34764 491097 243 55488 minus579P26 200 10 150 378773 408401 782 46961 384126 139 79975 minus643P27 200 10 70 450468 494247 972 49136 461810 246 11018 minus726P28 200 10 150 374435 396998 603 33917 380004 147 77562 minus456P29 200 10 70 472898 485672 270 39796 479708 142 77673 minus128P30 200 10 150 364178 374923 295 35823 372971 236 91982 minus059

Average 422 12266 100 21135 minus322

4 Computational Results

The proposed tabu search algorithm was coded in the Cprogramand ran on a computerwith Intel(R)Core(TM)CPU(32GHz) with 4GB of RAM under the Microsoft Windows7 operation system

41 Test Beds Two test beds were used to evaluate theperformance of the proposed tabu search algorithms

(1) The benchmarks of CLRP (Prins et al [31]) whichcontains 30 instances two potential distribution cen-ter sizes |119869| = 5 or 10 are considered The numberof customers is |119868| = 20 50 100 and 200 Thecustomers and DCs are uniformly distributed in a[1 50]times[1 50] squareThe fixed opening costs of DCsare randomly generated from interval [5000 125000]The traveling cost between two nodes equals theirEuclidean distance multiplied by 100 and rounded

up to the next integer The setup cost of a vehicleis 1000 The vehicle capacity 119876 equals 70 or 150Customersrsquo demands are randomly selected frominterval [11 20]

(2) Randomly generated LIRP-FRL instances based onthe data of CLRP benchmarks and using the settingof [25] for the parameters such as the order costof new products of DCs 119860119898 (=200) the order costof recovered products of DCs 119860119903 (=100) the cost ofmanufacturing 119888119898 (=60) recovering 119888119903 (=50) disposal119888119889 (minus10) and the return rate of product we randomlygenerate the LIRP-FRL instances as follows

(i) The holding cost for each distribution center isuniformly distributed in 119880 [5 10]

(ii) The transportation cost per unit product of thedistribution center at site 119895 119895 isin 119869 is uniformlydistributed in 119880 [1 10]

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Given the locations of DCs and the routes of vehicles theLIRP-FRL is reduced to the inventory model of Teunter [25]Some of their valuable conclusions can be directly applied toour LIRP-FRL

Proposition 1 (Teunter [25]) ldquoIt is reasonable to restrict ourattention to those policies with either119872 = 1 or 119877 = 1rdquo

Teunter [25] concluded that the policies with either119872 =

1 or 119877 = 1 can obtain near optimal solution with lesscomputational time by comparing with that of achievingoptimal solution Since our problem is more complicatedthan that of Teunter [25] to simplify the resolution of LIRP-FRL we consider only the two cases in our implementationThe two cases are as follows

Case 1 (1M) That means 119877 = 1 in this case the order timesof recovered products for each DC are equal to one

Case 2 (R 1) That means119872 = 1 in this case the order timesof new products for each DC are equal to one

For Case 1 (1M) the holding cost is

HM = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895 + 119876119903119895 sdot

119906

119903) sdot 119911119895 (7)

Consequently the total cost is denoted as

TCM = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HM(8)

For Case 2 (R 1) the holding cost is

HR = sum

119895isin119869

1

2ℎ119895 ((1 minus 119906) sdot 119876

119898119895 + 119906 sdot 119876

119903119895

+ (119903 (119876119898119895 + 119876

119903119895) minus (119903 minus 119906) sdot (119876

119898119895 +

119906

1 minus 119906119876119898119895 ))

119906

119903)

sdot 119911119895

(9)

The corresponding total cost is

TCR = FIXCOST +OC +MR + DISP + TRAN

+ TRC +HR(10)

24 Mathematical Model The LIRP-FRL can be formulatedas the following nonlinearmixed integer programming prob-lemmin TCMTCR (11)

st sum

119896isin119870

sum

119894isin119878

119909119894119895119896 = 1 119895 isin 119868 (12)

sum

119895isin119878

119909119894119895119896 minussum

119895isin119878

119909119895119894119896 = 0 119894 isin 119878 119896 isin 119870 (13)

sum

119895isin119869

sum

119894isin119868

119909119894119895119896 le 1 119896 isin 119870 (14)

sum

119895isin119869

sum

119894isin119868

119863119894119909119894119895119896 le 119876 119896 isin 119870 (15)

sum

119894isin119868

119863119894119910119894119895 minus 119881119895 sdot 119911119895 le 0 119895 isin 119869 (16)

119910119894119895 + 1 ge sum

119892isin119868

119909119894119892119896 + sum

119892isin119878119895

119909119892119895119896

119894 isin 119869 119895 isin 119868 119896 isin 119870

(17)

sum

119895isin119882

sum

119894isin119882

119909119894119895119896 le |119882| minus 1 119882 sube 119868 119896 isin 119870 (18)

119909119894119895119896 isin 0 1 119894 isin 119878 119895 isin 119878 119896 isin 119870 (19)

119910119894119895 isin 0 1 119894 isin 119868 119895 isin 119869 (20)

119911119895 isin 0 1 119895 isin 119869 (21)

Constraints (12) state that each customer is exactly served byone vehicle Constraints (13) are the equilibrium constraintsof vehicle routes on nodes Constraints (14) ensure that eachvehicle can be used by at most one DC Constraints (15)guarantee that the capacity of vehicle cannot be exceededConstraints (16) are the capacity constraints of distributioncenters Constraints (17) make sure that a customer can onlybe served by the vehicle which originates from the DC itis assigned to Constraints (18) are the subtours eliminationconstraints of vehicle routes

Proposition 2 For any distribution center 119895 isin 119869 its orderquantities 119876

119898119895 and 119876

119903119895 can be represented as a function of

customersrsquo demands respectivelyWe have 119876

119898119895 = radic2119860119898sum119894isin119868119863119894 sdot 119910119894119895ℎ119895 and 119876

119903119895 =

radic2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895(119903 + 1)ℎ119895 for any 119895 isin 119869

Proof It can be observed that no matter case (1 M) orcase (R 1) decision variables 119876119898119895 and 119876

119903119895 appear only in the

objective function For the case (R 1) the optimal value of119876119898119895and 119876119903119895 can be obtained by checking the extreme conditions120597TCM120597119876

119898119895 = 0 and 120597TCM120597119876

119903119895 = 0

Since 1205972TCM120597(119876

119898119895 )2

= (2119860119898

sdot (1 minus 119906)sum119894isin119868119863119894 sdot

119910119894119895)(119876119898119895 )3ge 0 and 120597

2TCM120597(119876119903119895)2= (2119860

119903sdot 119906sum119894isin119868119863119894 sdot

119910119894119895)(119876119903119895)3ge 0 consequently the extreme point is a minimum

point We obtain the optimal order quantities 119876119898119895 =

radic(2119860119898sum119894isin119868119863119894 sdot (1 minus 119906)119910119894119895)ℎ119895(1 minus ((119903 minus 119906)(1 minus 119906)) sdot (119906119903))

and 119876119903119895 = radic(119860119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and the corresponding

order time 119877119895 = (119906 sdot 119876119898119895 )(1 minus 119906 sdot 119876

119903119895)

For the same reason for the case (1 119872) we can derivethat the order quantities 119876119898119895 = radic(2119860119898sum119894isin119868119863119894 sdot 119910119894119895)ℎ119895 and

119876119903119895 = radic(2119860119903 sdot 119903 sdot sum119894isin119868119863119894 sdot 119910119894119895)(119903 + 1)ℎ119895 and the correspond-

ing order time119872119895 = (1 minus 119906 sdot 119876119903119895)(119906 sdot 119876

119898119895 )

Observe that without considering the inventory strategyof these two cases Cases 1 and 2 the LIRP-FRL is reducedto the classical capacitated location routing problem (CLRP)which is NP-hard in the strong sense As a result the LIRP-FRL is also NP-hard As Belenguer et al [24] reported to

Discrete Dynamics in Nature and Society 5

21 23 19 16 15 14 8 11 6 22 4 1 12 18 20 13 5 7 3 24 25 2 17 9 10

Figure 2 An example of solution representation

find CLRP optimal solution even for the median size CLRPinstances (with 50 customers and 5 potential distributioncenters) significant research efforts need to be expensedThus in this paper we focus onfinding near optimal solutionsfor the LIRP-FRL

3 New Tabu Search

Using tabu list to avoid visited solutions to be revisited thetabu search proposed by Glover [26] is one of the mosteffective approaches for solving mixed integer programmingproblems which has been successfully applied to a varietyof distribution network design problems For the detailedintroduction and application of the tabu search we referreaders to Glover [27] and Habet [28]

In this section a tabu search (NTS) algorithm thatprobabilistically accepts the second best solution in searchprocess when a local optimum is reached is proposed toeffectively solve the LIRP-FRL The key components of theNTS including a solution representation technique an initialsolution generation method neighbourhood structures andthe general framework of the new tabu search are presentedin detail as follows

31 Solution Representation A good solution representationcan not only describe the problem solutions clearly butalso improve the performance of heuristics adopted Similarto that of Yu et al [29] in our implementation solution1199091 1199092 119909119899 119909119899+1 119909119899+119898 is represented by a permuta-tion of number 1 2 119899 119899 + 1 119899 + 2 119899 + 119898 where119909119895 isin 1 2 119899 indexes customer 119909119895 and 119909119894 isin 119899 + 1 119899 +

2 119899 + 119898 denotes potential DC 119909119894An example was given in Figure 2 In this example the

first number is distribution center 21 followed by distributioncenter 23 Since there are no customers between distributioncenter 21 and distribution center 23 distribution center 21is closed Customers (19 16 15 14 8 11 and 6) betweendistribution center 23 and distribution center 22 are realcustomers thus distribution center 23 is opened to servicethese customers The first route of distribution center 23services customers 19 16 15 and 14 Because adding customer8 exceeds the vehicle capacity the second route servicescustomers 8 11 and 6 Customer 6 is followed by distributioncenter 22 so the second route is terminated For a similarreason asmentioned above distribution centers 22 and 25 areopen with two routes and one route respectively

This representation has determined which distributioncenter is open and the customers on each route With thatdecided we can easily calculate the objective function valuebecause the inventory cost is related to LRP which meanswhen LRP is solved the inventory cost can be uniquelydetermined by Proposition 2 Note that the capacity ofdistribution center is not taken into consideration So during

the decoding process a per unit penalty cost 119872 with a bigvalue is added to the objective function value when the totaldemand serviced by a distribution center exceeds its capacity

To simplify the presentation of the algorithms proposedin following subsections we say that an element 119909119894 of thesolution is before another element 119909119895 or element 119909119895 is afterelement 119909119894 if and only if 119894 lt 119895 Further with the smallest value|119894 minus 119895| if DC 119909119894 is before DC 119909119895 we say that DC 119909119894 is nearestbefore DC 119909119895 or DC 119909119895 is nearest after DC 119909119894

32 Constructing Initial Solution A greedy nearest neighbormethod is proposed to construct an initial feasible solution ofthe LIRP-FRL which is used as an input for the tabu searchDetails of the greedy algorithm are given as follows

Algorithm 3 (greedy nearest neighbour method)

Step 0 InitializeΩ0 fl 119869 Ω1 fl Φ0 fl 119868 and Φ1 fl

Step 1 Assign each customer inΦ0 to its closest DCs inΩ0

Step 2 Open the DC in Ω0 with the largest number of

customers assigned to say DC119895lowast

Step 3 Sort the customers of set Φ0 in nondecreasing orderof the distance between it and the DC119895lowast

Step 4 With the order let Φ = 1199041 1199042 119904|Φ0| be the subsetof customers of Φ0 with the largest |Φ| and sum

|Φ|119894=1 119889119904119894 le 119881119895lowast

Renew Ω1 fl Ω

1cup 119895lowast Ω0 fl 119869Ω

1 Φ1 fl Φ1cup Φ and

Φ0 fl 119868Φ

1

Step 5 If Φ0 = then return to Step 1 Otherwise open theDCs inΩ1 and assign each customer to a DC according to thecustomer-DC assignments determined in Step 4

Step 6 For each opened DC sort the customers assignedto it into sequence by using a nearest neighbor heuristic(refer to Rosenkrantz et al [30]) Split the sequence intoseveral feasible routes so that vehicle capacity constraints aresatisfied

Step 7 Output feasible solution

In our implementation of Step 2 if more than oneDChasthe largest number of customers assigned to we select theDCwith the highest capacity

33 Neighborhoods Given a solution

119883 = (1199091 1199092 119909119894minus1 119909119894 119909119894+1 119909119895minus1 119909119895 119909119895+1 119909119899) (22)

a permutation of 1 2 119899 119899 + 1 119899 + 119898 three kinds ofneighbourhoods are considered in our tabu search

6 Discrete Dynamics in Nature and Society

331 Insertion Neighbourhood The insertion neighborhoodis defined as the set of solutions that can be reached byinsertion move which deletes one element of the solution119883 and then inserts it to another place For example deleteelement 119909119894 and insert it into the 119895th place we obtain a newsolution

1198831015840= (1199091 1199092 119909119894minus1 119909119894+1 119909119895minus1 119909119895 119909119894 119909119895+1 119909119899) (23)

According to the element selected (customer or depot)and the position it is inserted in there are four cases

(a) If 119909119894 and 119909119895 are both customers then customer 119909119894 isreassigned to the DC that customer 119909119895 is assigned to

(b) If 119909119894 and 119909119895 are bothDCs then reassign the customerspreviously assigned to the DC nearest before it CloseDC 119909119895 and assign its customers to DC 119909119894

(c) If 119909119894 is a customer and 119909119895 is a DC then customer 119909119894 isreassigned to DC 119909119895

(d) If 119909119894 is a DC and 119909119895 is a customer reassign customersthat are nearest after customer 119909119895 to DC 119909119894 Allcustomers assigned to DC 119909119894 are reassigned to the DCnearest before it

332 Swap Neighbourhood The swap neighbourhood a setof solutions contains the solutions that can be reached byperforming a swap move which exchanges the places oftwo different elements of solution 119883 For example exchangeelement 119909119894 and element 119909119895 of119883 the resulting new solution is

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119894+1 119909119895minus1 119909119894 119909119895+1 119909119899) (24)

Similar to the insert move the swap move has also fourcases

(e) If 119909119894 and 119909119895 are both customers then we exchange thetwoDCs they are currently assigned to and obtain twonew customer-DC assignments

(f) If 119909119894 and 119909119895 are both DCs then we exchange all thecustomers currently assigned to them

(g) If 119909119894 is a customer and 119909119895 is a DC then the customersafter 119909119894 assigned to the same DC are reassigned to DC119909119895 and customer 119909119894 is reassigned to the DC nearestbefore DC 119909119895 All customers previously assigned toDC 119909119895 will be reassigned to the DC nearest before it

(h) If 119909119894 is a DC and 119909119895 is a customer this case is similarto case g

333 2-Opt Neighbourhood With the consideration of vehi-cle capacity in our definition the 2-opt neighbourhood con-tains the solutions that can be obtained by selecting two cus-tomers of119883 (eg 119909119894 and 119909119895) and then reversing the substringin the solution representation between them the so-called 2-opt move For example applying 2-opt move to elements 119909119894and 119909119895 we obtain

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119895minus1 119909119894+1 119909119894 119909119895+1 119909119899) (25)

34 Tabu Search Framework To increase the probability ofobtaining high quality solutions a diversification strategyis implemented This strategy probabilistically accepts thesecond best solution in the search process while no bettersolution can be found in the neighborhoods of the currentsolution By doing so we can effectively diversify the searchdirection and thusmore solution spacewill be explored Con-sequently better solution may be found The computationresults in Section 4 prove this strategy The proposed tabusearch can be summarized as follows

Algorithm 4 (new tabu search)

Step 0 Input an initial feasible solution 1198830 (see Section 32)and initialize Iter fl 0119883curr fl 1198830 and Objlowast fl Obj(1198830)

Step 1 Explore the neighborhoods of 119883curr and denote thebest solution in the neighborhoods the best nontabu solu-tion and the second best nontabu solution as119883best119883tabu-bestand119883tabu-2nd respectively

Step 2 If Obj(119883best) lt Objlowast then set 119883curr fl 119883best and addthe corresponding move to tabu list otherwise

if Obj(119883tabu-best) lt Obj(119883curr) then119883curr fl 119883tabu-bestand add the corresponding move to tabu list

otherwise probabilistically move to 119883tabu-best or119883tabu-2nd 119883curr fl Prob119883tabu-best 119883tabu-2nd and addthe corresponding move to tabu list

Step 3 Iter fl Iter + 1 If no stopping criterion is reachedreturn to Step 1

Step 4 Output the best solution objective function valueObjlowast

In the proposed tabu search algorithm Iter counts thenumber of iterations and 1198830 represents the initial solutionobtained by greedy nearest neighbour method (Section 32)119883curr denotes the current solution Objlowast is the objectivefunction value corresponding to the best solution found sofar The function Obj(119883) represents the objective functionvalue of solution119883

In Step 2 comparing with the current solution119883curr if nobetter solution can be found in its neighbourhood we needto probabilistically move 119883curr to 119883tabu-best or 119883tabu-2nd Inimplementation we randomly generate a value 120573 in interval[0 1] If120573 ge 120582 then119883curr ismoved to119883tabu-best and otherwise119883curr is moved to119883tabu-2nd where 120582 is a given parameter

At each iteration of the tabu search procedure thebest admissible nontabu insertion swap or 2-opt move isperformed The two elements of solution 119883curr involved insuch move are declared as tabu-active which is forbiddento be removed in the next 119905 iterations 119905 is the tabu-tenureparameter selected randomly from a range [119879min 119879max]

The tabu search iteration is terminated if the maximumnumber of iterations maxIter is reached or if the bestsolution found so far has been improved in successive SucIteriteration

Discrete Dynamics in Nature and Society 7

Table 1 Computational results on CLRP benchmarks

Problem |119868| |119869| 119876 BKS TS NTS DEVUB Gap Time UB Gap Time

P1 20 5 70 54793 54793 000 012 54793 000 049 000P2 20 5 150 39104 39104 000 012 39104 000 069 000P3 20 5 70 48908 48908 000 050 48908 000 053 000P4 20 5 150 37542 37611 018 037 37611 018 074 000P5 50 5 70 90111 90806 077 258 90461 039 335 minus038P6 50 5 150 63242 65010 280 1043 63242 000 1173 minus280P7 50 5 70 88298 89688 157 2207 88643 039 2394 minus118P8 50 5 150 67340 68071 109 748 67340 000 1116 minus109P9 50 5 70 84055 84227 020 1413 84110 007 1696 minus013P10 50 5 150 51822 51902 015 1021 51883 012 1349 minus003P11 50 5 70 86203 86223 002 920 86203 000 1471 minus002P12 50 5 150 61830 61978 024 1954 61830 000 2149 minus024P13 100 5 70 275993 281229 190 1151 278926 105 3841 minus085P14 100 5 150 214392 218886 210 952 216802 111 2974 minus099P15 100 5 70 194598 196587 102 14525 196436 094 9516 minus008P16 100 5 150 157173 159244 132 18038 158720 097 12463 minus035P17 100 5 70 200246 203297 152 12612 202682 120 11085 minus032P18 100 5 150 152586 157859 346 12247 156049 222 10526 minus124P19 100 10 70 290429 334486 1517 9780 294774 147 1432 minus1370P20 100 10 150 234641 285303 2159 7587 236217 067 1136 minus2092P21 100 10 70 244265 257455 540 10349 248003 151 13985 minus389P22 100 10 150 203988 216253 601 6965 208455 214 107 minus387P23 100 10 70 253344 291153 1492 16998 258221 189 15353 minus1303P24 100 10 150 204597 220528 779 6706 209390 229 13152 minus550P25 200 10 70 479425 518815 822 34764 491097 243 55488 minus579P26 200 10 150 378773 408401 782 46961 384126 139 79975 minus643P27 200 10 70 450468 494247 972 49136 461810 246 11018 minus726P28 200 10 150 374435 396998 603 33917 380004 147 77562 minus456P29 200 10 70 472898 485672 270 39796 479708 142 77673 minus128P30 200 10 150 364178 374923 295 35823 372971 236 91982 minus059

Average 422 12266 100 21135 minus322

4 Computational Results

The proposed tabu search algorithm was coded in the Cprogramand ran on a computerwith Intel(R)Core(TM)CPU(32GHz) with 4GB of RAM under the Microsoft Windows7 operation system

41 Test Beds Two test beds were used to evaluate theperformance of the proposed tabu search algorithms

(1) The benchmarks of CLRP (Prins et al [31]) whichcontains 30 instances two potential distribution cen-ter sizes |119869| = 5 or 10 are considered The numberof customers is |119868| = 20 50 100 and 200 Thecustomers and DCs are uniformly distributed in a[1 50]times[1 50] squareThe fixed opening costs of DCsare randomly generated from interval [5000 125000]The traveling cost between two nodes equals theirEuclidean distance multiplied by 100 and rounded

up to the next integer The setup cost of a vehicleis 1000 The vehicle capacity 119876 equals 70 or 150Customersrsquo demands are randomly selected frominterval [11 20]

(2) Randomly generated LIRP-FRL instances based onthe data of CLRP benchmarks and using the settingof [25] for the parameters such as the order costof new products of DCs 119860119898 (=200) the order costof recovered products of DCs 119860119903 (=100) the cost ofmanufacturing 119888119898 (=60) recovering 119888119903 (=50) disposal119888119889 (minus10) and the return rate of product we randomlygenerate the LIRP-FRL instances as follows

(i) The holding cost for each distribution center isuniformly distributed in 119880 [5 10]

(ii) The transportation cost per unit product of thedistribution center at site 119895 119895 isin 119869 is uniformlydistributed in 119880 [1 10]

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

21 23 19 16 15 14 8 11 6 22 4 1 12 18 20 13 5 7 3 24 25 2 17 9 10

Figure 2 An example of solution representation

find CLRP optimal solution even for the median size CLRPinstances (with 50 customers and 5 potential distributioncenters) significant research efforts need to be expensedThus in this paper we focus onfinding near optimal solutionsfor the LIRP-FRL

3 New Tabu Search

Using tabu list to avoid visited solutions to be revisited thetabu search proposed by Glover [26] is one of the mosteffective approaches for solving mixed integer programmingproblems which has been successfully applied to a varietyof distribution network design problems For the detailedintroduction and application of the tabu search we referreaders to Glover [27] and Habet [28]

In this section a tabu search (NTS) algorithm thatprobabilistically accepts the second best solution in searchprocess when a local optimum is reached is proposed toeffectively solve the LIRP-FRL The key components of theNTS including a solution representation technique an initialsolution generation method neighbourhood structures andthe general framework of the new tabu search are presentedin detail as follows

31 Solution Representation A good solution representationcan not only describe the problem solutions clearly butalso improve the performance of heuristics adopted Similarto that of Yu et al [29] in our implementation solution1199091 1199092 119909119899 119909119899+1 119909119899+119898 is represented by a permuta-tion of number 1 2 119899 119899 + 1 119899 + 2 119899 + 119898 where119909119895 isin 1 2 119899 indexes customer 119909119895 and 119909119894 isin 119899 + 1 119899 +

2 119899 + 119898 denotes potential DC 119909119894An example was given in Figure 2 In this example the

first number is distribution center 21 followed by distributioncenter 23 Since there are no customers between distributioncenter 21 and distribution center 23 distribution center 21is closed Customers (19 16 15 14 8 11 and 6) betweendistribution center 23 and distribution center 22 are realcustomers thus distribution center 23 is opened to servicethese customers The first route of distribution center 23services customers 19 16 15 and 14 Because adding customer8 exceeds the vehicle capacity the second route servicescustomers 8 11 and 6 Customer 6 is followed by distributioncenter 22 so the second route is terminated For a similarreason asmentioned above distribution centers 22 and 25 areopen with two routes and one route respectively

This representation has determined which distributioncenter is open and the customers on each route With thatdecided we can easily calculate the objective function valuebecause the inventory cost is related to LRP which meanswhen LRP is solved the inventory cost can be uniquelydetermined by Proposition 2 Note that the capacity ofdistribution center is not taken into consideration So during

the decoding process a per unit penalty cost 119872 with a bigvalue is added to the objective function value when the totaldemand serviced by a distribution center exceeds its capacity

To simplify the presentation of the algorithms proposedin following subsections we say that an element 119909119894 of thesolution is before another element 119909119895 or element 119909119895 is afterelement 119909119894 if and only if 119894 lt 119895 Further with the smallest value|119894 minus 119895| if DC 119909119894 is before DC 119909119895 we say that DC 119909119894 is nearestbefore DC 119909119895 or DC 119909119895 is nearest after DC 119909119894

32 Constructing Initial Solution A greedy nearest neighbormethod is proposed to construct an initial feasible solution ofthe LIRP-FRL which is used as an input for the tabu searchDetails of the greedy algorithm are given as follows

Algorithm 3 (greedy nearest neighbour method)

Step 0 InitializeΩ0 fl 119869 Ω1 fl Φ0 fl 119868 and Φ1 fl

Step 1 Assign each customer inΦ0 to its closest DCs inΩ0

Step 2 Open the DC in Ω0 with the largest number of

customers assigned to say DC119895lowast

Step 3 Sort the customers of set Φ0 in nondecreasing orderof the distance between it and the DC119895lowast

Step 4 With the order let Φ = 1199041 1199042 119904|Φ0| be the subsetof customers of Φ0 with the largest |Φ| and sum

|Φ|119894=1 119889119904119894 le 119881119895lowast

Renew Ω1 fl Ω

1cup 119895lowast Ω0 fl 119869Ω

1 Φ1 fl Φ1cup Φ and

Φ0 fl 119868Φ

1

Step 5 If Φ0 = then return to Step 1 Otherwise open theDCs inΩ1 and assign each customer to a DC according to thecustomer-DC assignments determined in Step 4

Step 6 For each opened DC sort the customers assignedto it into sequence by using a nearest neighbor heuristic(refer to Rosenkrantz et al [30]) Split the sequence intoseveral feasible routes so that vehicle capacity constraints aresatisfied

Step 7 Output feasible solution

In our implementation of Step 2 if more than oneDChasthe largest number of customers assigned to we select theDCwith the highest capacity

33 Neighborhoods Given a solution

119883 = (1199091 1199092 119909119894minus1 119909119894 119909119894+1 119909119895minus1 119909119895 119909119895+1 119909119899) (22)

a permutation of 1 2 119899 119899 + 1 119899 + 119898 three kinds ofneighbourhoods are considered in our tabu search

6 Discrete Dynamics in Nature and Society

331 Insertion Neighbourhood The insertion neighborhoodis defined as the set of solutions that can be reached byinsertion move which deletes one element of the solution119883 and then inserts it to another place For example deleteelement 119909119894 and insert it into the 119895th place we obtain a newsolution

1198831015840= (1199091 1199092 119909119894minus1 119909119894+1 119909119895minus1 119909119895 119909119894 119909119895+1 119909119899) (23)

According to the element selected (customer or depot)and the position it is inserted in there are four cases

(a) If 119909119894 and 119909119895 are both customers then customer 119909119894 isreassigned to the DC that customer 119909119895 is assigned to

(b) If 119909119894 and 119909119895 are bothDCs then reassign the customerspreviously assigned to the DC nearest before it CloseDC 119909119895 and assign its customers to DC 119909119894

(c) If 119909119894 is a customer and 119909119895 is a DC then customer 119909119894 isreassigned to DC 119909119895

(d) If 119909119894 is a DC and 119909119895 is a customer reassign customersthat are nearest after customer 119909119895 to DC 119909119894 Allcustomers assigned to DC 119909119894 are reassigned to the DCnearest before it

332 Swap Neighbourhood The swap neighbourhood a setof solutions contains the solutions that can be reached byperforming a swap move which exchanges the places oftwo different elements of solution 119883 For example exchangeelement 119909119894 and element 119909119895 of119883 the resulting new solution is

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119894+1 119909119895minus1 119909119894 119909119895+1 119909119899) (24)

Similar to the insert move the swap move has also fourcases

(e) If 119909119894 and 119909119895 are both customers then we exchange thetwoDCs they are currently assigned to and obtain twonew customer-DC assignments

(f) If 119909119894 and 119909119895 are both DCs then we exchange all thecustomers currently assigned to them

(g) If 119909119894 is a customer and 119909119895 is a DC then the customersafter 119909119894 assigned to the same DC are reassigned to DC119909119895 and customer 119909119894 is reassigned to the DC nearestbefore DC 119909119895 All customers previously assigned toDC 119909119895 will be reassigned to the DC nearest before it

(h) If 119909119894 is a DC and 119909119895 is a customer this case is similarto case g

333 2-Opt Neighbourhood With the consideration of vehi-cle capacity in our definition the 2-opt neighbourhood con-tains the solutions that can be obtained by selecting two cus-tomers of119883 (eg 119909119894 and 119909119895) and then reversing the substringin the solution representation between them the so-called 2-opt move For example applying 2-opt move to elements 119909119894and 119909119895 we obtain

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119895minus1 119909119894+1 119909119894 119909119895+1 119909119899) (25)

34 Tabu Search Framework To increase the probability ofobtaining high quality solutions a diversification strategyis implemented This strategy probabilistically accepts thesecond best solution in the search process while no bettersolution can be found in the neighborhoods of the currentsolution By doing so we can effectively diversify the searchdirection and thusmore solution spacewill be explored Con-sequently better solution may be found The computationresults in Section 4 prove this strategy The proposed tabusearch can be summarized as follows

Algorithm 4 (new tabu search)

Step 0 Input an initial feasible solution 1198830 (see Section 32)and initialize Iter fl 0119883curr fl 1198830 and Objlowast fl Obj(1198830)

Step 1 Explore the neighborhoods of 119883curr and denote thebest solution in the neighborhoods the best nontabu solu-tion and the second best nontabu solution as119883best119883tabu-bestand119883tabu-2nd respectively

Step 2 If Obj(119883best) lt Objlowast then set 119883curr fl 119883best and addthe corresponding move to tabu list otherwise

if Obj(119883tabu-best) lt Obj(119883curr) then119883curr fl 119883tabu-bestand add the corresponding move to tabu list

otherwise probabilistically move to 119883tabu-best or119883tabu-2nd 119883curr fl Prob119883tabu-best 119883tabu-2nd and addthe corresponding move to tabu list

Step 3 Iter fl Iter + 1 If no stopping criterion is reachedreturn to Step 1

Step 4 Output the best solution objective function valueObjlowast

In the proposed tabu search algorithm Iter counts thenumber of iterations and 1198830 represents the initial solutionobtained by greedy nearest neighbour method (Section 32)119883curr denotes the current solution Objlowast is the objectivefunction value corresponding to the best solution found sofar The function Obj(119883) represents the objective functionvalue of solution119883

In Step 2 comparing with the current solution119883curr if nobetter solution can be found in its neighbourhood we needto probabilistically move 119883curr to 119883tabu-best or 119883tabu-2nd Inimplementation we randomly generate a value 120573 in interval[0 1] If120573 ge 120582 then119883curr ismoved to119883tabu-best and otherwise119883curr is moved to119883tabu-2nd where 120582 is a given parameter

At each iteration of the tabu search procedure thebest admissible nontabu insertion swap or 2-opt move isperformed The two elements of solution 119883curr involved insuch move are declared as tabu-active which is forbiddento be removed in the next 119905 iterations 119905 is the tabu-tenureparameter selected randomly from a range [119879min 119879max]

The tabu search iteration is terminated if the maximumnumber of iterations maxIter is reached or if the bestsolution found so far has been improved in successive SucIteriteration

Discrete Dynamics in Nature and Society 7

Table 1 Computational results on CLRP benchmarks

Problem |119868| |119869| 119876 BKS TS NTS DEVUB Gap Time UB Gap Time

P1 20 5 70 54793 54793 000 012 54793 000 049 000P2 20 5 150 39104 39104 000 012 39104 000 069 000P3 20 5 70 48908 48908 000 050 48908 000 053 000P4 20 5 150 37542 37611 018 037 37611 018 074 000P5 50 5 70 90111 90806 077 258 90461 039 335 minus038P6 50 5 150 63242 65010 280 1043 63242 000 1173 minus280P7 50 5 70 88298 89688 157 2207 88643 039 2394 minus118P8 50 5 150 67340 68071 109 748 67340 000 1116 minus109P9 50 5 70 84055 84227 020 1413 84110 007 1696 minus013P10 50 5 150 51822 51902 015 1021 51883 012 1349 minus003P11 50 5 70 86203 86223 002 920 86203 000 1471 minus002P12 50 5 150 61830 61978 024 1954 61830 000 2149 minus024P13 100 5 70 275993 281229 190 1151 278926 105 3841 minus085P14 100 5 150 214392 218886 210 952 216802 111 2974 minus099P15 100 5 70 194598 196587 102 14525 196436 094 9516 minus008P16 100 5 150 157173 159244 132 18038 158720 097 12463 minus035P17 100 5 70 200246 203297 152 12612 202682 120 11085 minus032P18 100 5 150 152586 157859 346 12247 156049 222 10526 minus124P19 100 10 70 290429 334486 1517 9780 294774 147 1432 minus1370P20 100 10 150 234641 285303 2159 7587 236217 067 1136 minus2092P21 100 10 70 244265 257455 540 10349 248003 151 13985 minus389P22 100 10 150 203988 216253 601 6965 208455 214 107 minus387P23 100 10 70 253344 291153 1492 16998 258221 189 15353 minus1303P24 100 10 150 204597 220528 779 6706 209390 229 13152 minus550P25 200 10 70 479425 518815 822 34764 491097 243 55488 minus579P26 200 10 150 378773 408401 782 46961 384126 139 79975 minus643P27 200 10 70 450468 494247 972 49136 461810 246 11018 minus726P28 200 10 150 374435 396998 603 33917 380004 147 77562 minus456P29 200 10 70 472898 485672 270 39796 479708 142 77673 minus128P30 200 10 150 364178 374923 295 35823 372971 236 91982 minus059

Average 422 12266 100 21135 minus322

4 Computational Results

The proposed tabu search algorithm was coded in the Cprogramand ran on a computerwith Intel(R)Core(TM)CPU(32GHz) with 4GB of RAM under the Microsoft Windows7 operation system

41 Test Beds Two test beds were used to evaluate theperformance of the proposed tabu search algorithms

(1) The benchmarks of CLRP (Prins et al [31]) whichcontains 30 instances two potential distribution cen-ter sizes |119869| = 5 or 10 are considered The numberof customers is |119868| = 20 50 100 and 200 Thecustomers and DCs are uniformly distributed in a[1 50]times[1 50] squareThe fixed opening costs of DCsare randomly generated from interval [5000 125000]The traveling cost between two nodes equals theirEuclidean distance multiplied by 100 and rounded

up to the next integer The setup cost of a vehicleis 1000 The vehicle capacity 119876 equals 70 or 150Customersrsquo demands are randomly selected frominterval [11 20]

(2) Randomly generated LIRP-FRL instances based onthe data of CLRP benchmarks and using the settingof [25] for the parameters such as the order costof new products of DCs 119860119898 (=200) the order costof recovered products of DCs 119860119903 (=100) the cost ofmanufacturing 119888119898 (=60) recovering 119888119903 (=50) disposal119888119889 (minus10) and the return rate of product we randomlygenerate the LIRP-FRL instances as follows

(i) The holding cost for each distribution center isuniformly distributed in 119880 [5 10]

(ii) The transportation cost per unit product of thedistribution center at site 119895 119895 isin 119869 is uniformlydistributed in 119880 [1 10]

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

331 Insertion Neighbourhood The insertion neighborhoodis defined as the set of solutions that can be reached byinsertion move which deletes one element of the solution119883 and then inserts it to another place For example deleteelement 119909119894 and insert it into the 119895th place we obtain a newsolution

1198831015840= (1199091 1199092 119909119894minus1 119909119894+1 119909119895minus1 119909119895 119909119894 119909119895+1 119909119899) (23)

According to the element selected (customer or depot)and the position it is inserted in there are four cases

(a) If 119909119894 and 119909119895 are both customers then customer 119909119894 isreassigned to the DC that customer 119909119895 is assigned to

(b) If 119909119894 and 119909119895 are bothDCs then reassign the customerspreviously assigned to the DC nearest before it CloseDC 119909119895 and assign its customers to DC 119909119894

(c) If 119909119894 is a customer and 119909119895 is a DC then customer 119909119894 isreassigned to DC 119909119895

(d) If 119909119894 is a DC and 119909119895 is a customer reassign customersthat are nearest after customer 119909119895 to DC 119909119894 Allcustomers assigned to DC 119909119894 are reassigned to the DCnearest before it

332 Swap Neighbourhood The swap neighbourhood a setof solutions contains the solutions that can be reached byperforming a swap move which exchanges the places oftwo different elements of solution 119883 For example exchangeelement 119909119894 and element 119909119895 of119883 the resulting new solution is

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119894+1 119909119895minus1 119909119894 119909119895+1 119909119899) (24)

Similar to the insert move the swap move has also fourcases

(e) If 119909119894 and 119909119895 are both customers then we exchange thetwoDCs they are currently assigned to and obtain twonew customer-DC assignments

(f) If 119909119894 and 119909119895 are both DCs then we exchange all thecustomers currently assigned to them

(g) If 119909119894 is a customer and 119909119895 is a DC then the customersafter 119909119894 assigned to the same DC are reassigned to DC119909119895 and customer 119909119894 is reassigned to the DC nearestbefore DC 119909119895 All customers previously assigned toDC 119909119895 will be reassigned to the DC nearest before it

(h) If 119909119894 is a DC and 119909119895 is a customer this case is similarto case g

333 2-Opt Neighbourhood With the consideration of vehi-cle capacity in our definition the 2-opt neighbourhood con-tains the solutions that can be obtained by selecting two cus-tomers of119883 (eg 119909119894 and 119909119895) and then reversing the substringin the solution representation between them the so-called 2-opt move For example applying 2-opt move to elements 119909119894and 119909119895 we obtain

1198831015840= (1199091 1199092 119909119894minus1 119909119895 119909119895minus1 119909119894+1 119909119894 119909119895+1 119909119899) (25)

34 Tabu Search Framework To increase the probability ofobtaining high quality solutions a diversification strategyis implemented This strategy probabilistically accepts thesecond best solution in the search process while no bettersolution can be found in the neighborhoods of the currentsolution By doing so we can effectively diversify the searchdirection and thusmore solution spacewill be explored Con-sequently better solution may be found The computationresults in Section 4 prove this strategy The proposed tabusearch can be summarized as follows

Algorithm 4 (new tabu search)

Step 0 Input an initial feasible solution 1198830 (see Section 32)and initialize Iter fl 0119883curr fl 1198830 and Objlowast fl Obj(1198830)

Step 1 Explore the neighborhoods of 119883curr and denote thebest solution in the neighborhoods the best nontabu solu-tion and the second best nontabu solution as119883best119883tabu-bestand119883tabu-2nd respectively

Step 2 If Obj(119883best) lt Objlowast then set 119883curr fl 119883best and addthe corresponding move to tabu list otherwise

if Obj(119883tabu-best) lt Obj(119883curr) then119883curr fl 119883tabu-bestand add the corresponding move to tabu list

otherwise probabilistically move to 119883tabu-best or119883tabu-2nd 119883curr fl Prob119883tabu-best 119883tabu-2nd and addthe corresponding move to tabu list

Step 3 Iter fl Iter + 1 If no stopping criterion is reachedreturn to Step 1

Step 4 Output the best solution objective function valueObjlowast

In the proposed tabu search algorithm Iter counts thenumber of iterations and 1198830 represents the initial solutionobtained by greedy nearest neighbour method (Section 32)119883curr denotes the current solution Objlowast is the objectivefunction value corresponding to the best solution found sofar The function Obj(119883) represents the objective functionvalue of solution119883

In Step 2 comparing with the current solution119883curr if nobetter solution can be found in its neighbourhood we needto probabilistically move 119883curr to 119883tabu-best or 119883tabu-2nd Inimplementation we randomly generate a value 120573 in interval[0 1] If120573 ge 120582 then119883curr ismoved to119883tabu-best and otherwise119883curr is moved to119883tabu-2nd where 120582 is a given parameter

At each iteration of the tabu search procedure thebest admissible nontabu insertion swap or 2-opt move isperformed The two elements of solution 119883curr involved insuch move are declared as tabu-active which is forbiddento be removed in the next 119905 iterations 119905 is the tabu-tenureparameter selected randomly from a range [119879min 119879max]

The tabu search iteration is terminated if the maximumnumber of iterations maxIter is reached or if the bestsolution found so far has been improved in successive SucIteriteration

Discrete Dynamics in Nature and Society 7

Table 1 Computational results on CLRP benchmarks

Problem |119868| |119869| 119876 BKS TS NTS DEVUB Gap Time UB Gap Time

P1 20 5 70 54793 54793 000 012 54793 000 049 000P2 20 5 150 39104 39104 000 012 39104 000 069 000P3 20 5 70 48908 48908 000 050 48908 000 053 000P4 20 5 150 37542 37611 018 037 37611 018 074 000P5 50 5 70 90111 90806 077 258 90461 039 335 minus038P6 50 5 150 63242 65010 280 1043 63242 000 1173 minus280P7 50 5 70 88298 89688 157 2207 88643 039 2394 minus118P8 50 5 150 67340 68071 109 748 67340 000 1116 minus109P9 50 5 70 84055 84227 020 1413 84110 007 1696 minus013P10 50 5 150 51822 51902 015 1021 51883 012 1349 minus003P11 50 5 70 86203 86223 002 920 86203 000 1471 minus002P12 50 5 150 61830 61978 024 1954 61830 000 2149 minus024P13 100 5 70 275993 281229 190 1151 278926 105 3841 minus085P14 100 5 150 214392 218886 210 952 216802 111 2974 minus099P15 100 5 70 194598 196587 102 14525 196436 094 9516 minus008P16 100 5 150 157173 159244 132 18038 158720 097 12463 minus035P17 100 5 70 200246 203297 152 12612 202682 120 11085 minus032P18 100 5 150 152586 157859 346 12247 156049 222 10526 minus124P19 100 10 70 290429 334486 1517 9780 294774 147 1432 minus1370P20 100 10 150 234641 285303 2159 7587 236217 067 1136 minus2092P21 100 10 70 244265 257455 540 10349 248003 151 13985 minus389P22 100 10 150 203988 216253 601 6965 208455 214 107 minus387P23 100 10 70 253344 291153 1492 16998 258221 189 15353 minus1303P24 100 10 150 204597 220528 779 6706 209390 229 13152 minus550P25 200 10 70 479425 518815 822 34764 491097 243 55488 minus579P26 200 10 150 378773 408401 782 46961 384126 139 79975 minus643P27 200 10 70 450468 494247 972 49136 461810 246 11018 minus726P28 200 10 150 374435 396998 603 33917 380004 147 77562 minus456P29 200 10 70 472898 485672 270 39796 479708 142 77673 minus128P30 200 10 150 364178 374923 295 35823 372971 236 91982 minus059

Average 422 12266 100 21135 minus322

4 Computational Results

The proposed tabu search algorithm was coded in the Cprogramand ran on a computerwith Intel(R)Core(TM)CPU(32GHz) with 4GB of RAM under the Microsoft Windows7 operation system

41 Test Beds Two test beds were used to evaluate theperformance of the proposed tabu search algorithms

(1) The benchmarks of CLRP (Prins et al [31]) whichcontains 30 instances two potential distribution cen-ter sizes |119869| = 5 or 10 are considered The numberof customers is |119868| = 20 50 100 and 200 Thecustomers and DCs are uniformly distributed in a[1 50]times[1 50] squareThe fixed opening costs of DCsare randomly generated from interval [5000 125000]The traveling cost between two nodes equals theirEuclidean distance multiplied by 100 and rounded

up to the next integer The setup cost of a vehicleis 1000 The vehicle capacity 119876 equals 70 or 150Customersrsquo demands are randomly selected frominterval [11 20]

(2) Randomly generated LIRP-FRL instances based onthe data of CLRP benchmarks and using the settingof [25] for the parameters such as the order costof new products of DCs 119860119898 (=200) the order costof recovered products of DCs 119860119903 (=100) the cost ofmanufacturing 119888119898 (=60) recovering 119888119903 (=50) disposal119888119889 (minus10) and the return rate of product we randomlygenerate the LIRP-FRL instances as follows

(i) The holding cost for each distribution center isuniformly distributed in 119880 [5 10]

(ii) The transportation cost per unit product of thedistribution center at site 119895 119895 isin 119869 is uniformlydistributed in 119880 [1 10]

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society 7

Table 1 Computational results on CLRP benchmarks

Problem |119868| |119869| 119876 BKS TS NTS DEVUB Gap Time UB Gap Time

P1 20 5 70 54793 54793 000 012 54793 000 049 000P2 20 5 150 39104 39104 000 012 39104 000 069 000P3 20 5 70 48908 48908 000 050 48908 000 053 000P4 20 5 150 37542 37611 018 037 37611 018 074 000P5 50 5 70 90111 90806 077 258 90461 039 335 minus038P6 50 5 150 63242 65010 280 1043 63242 000 1173 minus280P7 50 5 70 88298 89688 157 2207 88643 039 2394 minus118P8 50 5 150 67340 68071 109 748 67340 000 1116 minus109P9 50 5 70 84055 84227 020 1413 84110 007 1696 minus013P10 50 5 150 51822 51902 015 1021 51883 012 1349 minus003P11 50 5 70 86203 86223 002 920 86203 000 1471 minus002P12 50 5 150 61830 61978 024 1954 61830 000 2149 minus024P13 100 5 70 275993 281229 190 1151 278926 105 3841 minus085P14 100 5 150 214392 218886 210 952 216802 111 2974 minus099P15 100 5 70 194598 196587 102 14525 196436 094 9516 minus008P16 100 5 150 157173 159244 132 18038 158720 097 12463 minus035P17 100 5 70 200246 203297 152 12612 202682 120 11085 minus032P18 100 5 150 152586 157859 346 12247 156049 222 10526 minus124P19 100 10 70 290429 334486 1517 9780 294774 147 1432 minus1370P20 100 10 150 234641 285303 2159 7587 236217 067 1136 minus2092P21 100 10 70 244265 257455 540 10349 248003 151 13985 minus389P22 100 10 150 203988 216253 601 6965 208455 214 107 minus387P23 100 10 70 253344 291153 1492 16998 258221 189 15353 minus1303P24 100 10 150 204597 220528 779 6706 209390 229 13152 minus550P25 200 10 70 479425 518815 822 34764 491097 243 55488 minus579P26 200 10 150 378773 408401 782 46961 384126 139 79975 minus643P27 200 10 70 450468 494247 972 49136 461810 246 11018 minus726P28 200 10 150 374435 396998 603 33917 380004 147 77562 minus456P29 200 10 70 472898 485672 270 39796 479708 142 77673 minus128P30 200 10 150 364178 374923 295 35823 372971 236 91982 minus059

Average 422 12266 100 21135 minus322

4 Computational Results

The proposed tabu search algorithm was coded in the Cprogramand ran on a computerwith Intel(R)Core(TM)CPU(32GHz) with 4GB of RAM under the Microsoft Windows7 operation system

41 Test Beds Two test beds were used to evaluate theperformance of the proposed tabu search algorithms

(1) The benchmarks of CLRP (Prins et al [31]) whichcontains 30 instances two potential distribution cen-ter sizes |119869| = 5 or 10 are considered The numberof customers is |119868| = 20 50 100 and 200 Thecustomers and DCs are uniformly distributed in a[1 50]times[1 50] squareThe fixed opening costs of DCsare randomly generated from interval [5000 125000]The traveling cost between two nodes equals theirEuclidean distance multiplied by 100 and rounded

up to the next integer The setup cost of a vehicleis 1000 The vehicle capacity 119876 equals 70 or 150Customersrsquo demands are randomly selected frominterval [11 20]

(2) Randomly generated LIRP-FRL instances based onthe data of CLRP benchmarks and using the settingof [25] for the parameters such as the order costof new products of DCs 119860119898 (=200) the order costof recovered products of DCs 119860119903 (=100) the cost ofmanufacturing 119888119898 (=60) recovering 119888119903 (=50) disposal119888119889 (minus10) and the return rate of product we randomlygenerate the LIRP-FRL instances as follows

(i) The holding cost for each distribution center isuniformly distributed in 119880 [5 10]

(ii) The transportation cost per unit product of thedistribution center at site 119895 119895 isin 119869 is uniformlydistributed in 119880 [1 10]

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

42 Parameter Settings The maximum number of iterationsfor NTS maxIter is set to 10000 and the search iteration isterminated if the best upper bound has not been improvedin successive 800 iteration (SusIter = 800) Parameter 120582 isuniformly distributed in 119880 [001 08] The lower and upperbound of tabu-tenure 119879min and 119879max are set to 5 and 12respectively

43 Numerical Results To evaluate the performance of theproposed new tabu search (NTS) we compare it with theclassical tabu search algorithm (denoted as TS in followingdiscussion) in terms of the best upper bound (UB) foundso far and the corresponding computational time in CPUseconds (time) Moreover the parameter settings of TS isthe same as NTSrsquos except parameter 120582 that TS did not haveComputational results on CLRP benchmarks are listed inTable 1The column BKS is the best known solution objectivefunction value reported in the literature Gap is calculated bythe following formula Gap = 100 lowast (UB minus BKS)UB DEVis the deviation of the gap of NTS from the gap of TS that isDEV = GapNTS minus GapTS

Using the diversification strategy of probabilisticallyaccepting the second best solution the NTS increases theprobability of exploring more solution space As a result theNTS takes more computational time than the TS Howeverfrom Table 1 we can observe that although the averagecomputational time of NTS is about two times that of TSit is still in a very acceptable range The NTS providesmore competitive upper bound than the TS The average gapbetween the best upper bound found so far by NTS and thebest known upper bound in the literature is about 100compared to the 422 of the TS The NTS outperformsthe TS in the quality of solutions obtained with the averagedeviation of the gap DEV = minus322

Table 2 reports the computational results on test bed ofrandomly generated LIRP-FRL instances From the table wecan also find that the NTS performs better than the classic TSwith an average gap between the best upper bound of NTSand that of TS about minus279 The average computationaltime of NTS is slightly bigger than that of TS

44 Sensitivity Analysis Sensitivity analysis is also conductedto investigate the effects on the total costs when the inputparameters such as the return rate(119903) or the rate between theunit producing cost of new products and recovered products(119888119898 119888119903) changes

Table 3 reports the computational results when theproduct return rate is changed for each randomly generatedinstance To manifest the trend more clearly for each returnrate value we calculate the arithmetic average objectivefunction values of all tested instances The results are shownin Figure 3 From the figure we can find that the total costsdecrease with the increase of return rate 119903 Focusing on thoseproducts with high return rate can effectively reduce thesystem costs

Table 4 shows the numerical results when the ratebetween the cost of producing a new product and that ofrecovering a damaged or old product varies For each rate we

Table 2 Computational results on LIRP-FRL instances

Prob ID |119868| |119869| 119876TS NTS lowastGap 1 ()

UB Time UB TimeP1 20 5 70 67528 050 67528 053 000P2 20 5 150 42867 055 42867 061 000P3 20 5 70 60021 046 60021 065 000P4 20 5 150 40255 057 40255 062 000P5 50 5 70 115984 2256 115534 2295 minus039P6 50 5 150 84468 3643 84401 3418 minus008P7 50 5 70 117828 2256 117530 2406 minus025P8 50 5 150 96633 2161 96161 2164 minus049P9 50 5 70 112493 2431 111625 2451 minus078P10 50 5 150 82667 2301 82392 2466 minus033P11 50 5 70 108639 2128 108568 2032 minus007P12 50 5 150 82938 2158 82804 2207 minus016P13 100 5 70 342786 13924 342309 13997 minus014P14 100 5 150 276104 13764 273409 13872 minus137P15 100 5 70 239544 14375 238937 15807 minus025P16 100 5 150 201533 12558 200708 13506 minus041P17 100 5 70 250239 12128 249565 12488 minus027P18 100 5 150 198842 12684 197310 12830 minus078P19 100 10 70 406342 13830 396469 16150 minus249P20 100 10 150 368668 13884 339644 15786 minus855P21 100 10 70 328174 12159 320363 15115 minus244P22 100 10 150 278729 12850 274026 13476 minus172P23 100 10 70 388259 11347 320651 14394 minus2108P24 100 10 150 339851 12077 283008 13613 minus2009P25 200 10 70 645014 39375 629604 52543 minus245P26 200 10 150 558189 36891 528721 51052 minus557P27 200 10 70 627737 42791 611151 54292 minus271P28 200 10 150 543988 37941 517132 43250 minus519P29 200 10 70 607894 41925 598889 42833 minus150P30 200 10 150 530675 42272 509796 56314 minus410

Average 13877 16367 minus279lowastGap 1 () = 100 lowast (H2 solution value minus H1 solution value)H2 solutionvalue

average the best upper boundobtained for all tested instancesThe results are depicted in Figure 4 From the figure wecan observe that the total costs decrease with the increaseof the rate That means reducing the unit recovering cost ofproduct can effectively reduce the total costs which indicatesus to retrieve those subnew products or to improve repairingtechnique so as to reduce recovering cost

5 Conclusions and Future Works

We study a location-inventory-routing problem in forwardand reverse logistics (LIRP-FRL) in this paper A nonlinearmixed integer programming model is proposed to formulatethe LIRP-FRL Based on this model a new tabu searchthat probabilistically accepts the second best solution in itsneighborhoods is proposed to find near optimal solution forthe problem Numerical experiments on CLRP benchmarks

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

Table 3 Results of different return rates

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P1

01

20 5 70

67617 322 67617 266 00002 67522 338 67522 348 00003 67423 364 67423 314 00004 67568 273 67320 328 minus03705 67293 469 67211 322 minus01206 67094 326 67094 295 00007 66966 319 66966 277 00008 66904 315 66822 367 minus01209 66646 268 66646 315 000

P2

01

20 5 150

42972 254 42972 287 00002 42969 243 42969 296 00003 42965 307 42965 257 00004 42961 340 42961 266 00005 42955 300 42955 318 00006 42948 263 42948 292 00007 42939 268 42939 308 00008 42927 266 42927 302 00009 42908 247 42908 304 000

P3

01

20 5 70

61544 270 61296 268 minus04002 61175 314 61175 340 00003 61050 268 61050 299 00004 61011 316 60921 291 minus01505 60786 262 60786 298 00006 60733 255 60644 281 minus01507 60492 299 60492 323 00008 60325 287 60325 310 00009 60308 318 60127 319 minus030

P4

01

20 5 150

40396 256 40396 271 00002 40395 258 40395 263 00003 40393 264 40393 294 00004 40390 268 40390 241 00005 40386 266 40386 312 00006 40381 253 40381 246 00007 40373 268 40373 266 00008 40363 246 40363 263 00009 40346 288 40346 261 000

P5

01

50 5 70

117680 3619 116409 3462 minus10802 115813 3265 115115 3158 minus06003 114965 3064 114542 3143 minus03704 115279 307 114927 3134 minus03105 116446 2622 115078 3092 minus11706 114151 3048 114039 3118 minus01007 114504 3134 114190 3204 minus02708 115578 2866 114239 2656 minus11609 112326 3131 112310 3362 minus001

P6

01

50 5 150

87885 2030 87678 2404 minus02402 87289 2382 86748 2787 minus06203 88191 2395 87140 2459 minus11904 89497 2047 86970 2387 minus28205 87240 2181 86955 2484 minus03306 87233 2434 86283 2426 minus10907 86366 2417 86173 2105 minus02208 85494 2281 84771 2478 minus08509 84673 2393 84358 2424 minus037

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P7

01

50 5 70

122232 2684 121605 2223 minus05102 121038 2614 120826 2616 minus01803 121147 2617 120530 2666 minus05104 120239 2641 119945 2686 minus02405 120223 2611 119788 2603 minus03606 119636 2605 119545 2881 minus00807 119795 2550 119440 3204 minus03008 119994 2605 119100 2780 minus07509 119728 2572 118222 2731 minus126

P8

01

50 5 150

99470 2638 99178 2775 minus02902 99455 2521 99319 2561 minus01403 100163 2865 98682 2178 minus14804 98722 2514 98642 2532 minus00805 98930 2527 98617 2710 minus03206 98131 2490 98056 2623 minus00807 98723 2560 97815 2355 minus09208 97633 2578 97296 3313 minus03509 97752 2543 96882 2607 minus089

P9

01

50 5 70

86438 2522 86398 2974 minus00502 86031 2220 85926 2847 minus01203 86300 2501 85635 2150 minus07704 85292 2161 85283 2287 minus00105 85084 2446 84834 2805 minus02906 84628 2310 84495 2872 minus01607 84358 2498 84116 2287 minus02908 83239 2188 83103 2164 minus01609 83386 2248 82806 2597 minus070

P10

01

50 5 150

115321 2576 115143 2826 minus01502 114968 2616 114609 2155 minus03103 114628 2712 114347 2459 minus02504 114620 2662 113993 2786 minus05505 114243 2593 113821 2795 minus03706 113720 2605 113461 2929 minus02307 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P11

01

50 5 70

112376 2682 111843 3235 minus04702 111582 2594 111286 2718 minus02703 111920 2313 111054 2951 minus07704 111642 2491 110986 2805 minus05905 111345 2308 110891 2775 minus04106 110712 2611 110203 2893 minus04607 110789 2579 109949 2903 minus07608 110232 2579 109923 2818 minus02809 109797 2480 109042 3289 minus069

P12

01

50 5 150

85505 2469 85246 2745 minus03002 85187 2425 85019 2392 minus02003 84879 2473 84757 2478 minus01404 84797 2833 84337 3083 minus05405 84312 2584 83955 2838 minus04206 84053 2665 83933 2182 minus01407 83933 2631 83358 3133 minus06908 83651 2692 82897 2992 minus09009 82655 2322 82439 2695 minus026

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P13

01

100 5 70

343359 18912 341812 14689 minus04502 346569 18744 343870 18775 minus07803 345210 16949 342124 14165 minus08904 348100 18008 345668 1909 minus07005 349269 15925 343318 18488 minus17006 347719 18223 343891 16219 minus11007 349525 18284 342643 16988 minus19708 349171 16017 341151 15868 minus23009 351251 18208 349382 18692 minus053

P14

01

100 5 150

282054 13584 281389 19002 minus02402 281199 12894 279541 15139 minus05903 279903 15598 279153 16032 minus02704 278748 16152 277988 13087 minus02705 279806 15232 278460 16229 minus04806 280003 18832 279155 12799 minus03007 280823 12666 278542 17161 minus08108 276206 13694 275974 13812 minus00809 276528 15765 275917 16314 minus022

P15

01

100 5 70

246010 15628 245337 14854 minus02702 245120 17584 244537 17242 minus02403 244563 15452 243754 15432 minus03304 243537 15453 243124 13388 minus01705 242570 12021 241529 19288 minus04306 241778 17175 241435 15070 minus01407 240726 15304 240237 13858 minus02008 239819 14901 239716 15379 minus00409 238793 14575 238624 14434 minus007

P16

01

100 5 150

209027 13128 208711 15317 minus01502 208060 14978 207229 15739 minus04003 206328 16799 205454 20044 minus04204 207220 15439 205877 19570 minus06505 206676 16329 205654 17225 minus04906 204833 13778 204547 16887 minus01407 204402 13786 204284 15395 minus00608 203509 17304 203068 19172 minus02209 203695 15395 201648 14662 minus100

P17

01

100 5 70

259816 19504 258025 14948 minus06902 258004 16499 256937 19331 minus04103 256120 12451 256057 12360 minus00204 254658 13021 254594 15079 minus00305 254764 13342 253722 16532 minus04106 255047 15520 252587 15101 minus09607 250732 15372 249812 14922 minus03708 253732 18223 249612 16772 minus16209 252906 16849 249337 15779 minus141

P18

01

100 5 150

204826 14528 204678 19150 minus00702 205474 17275 204621 19328 minus04203 203254 16753 202691 15511 minus02804 203120 14882 201380 12134 minus08605 202003 12200 201329 14570 minus03306 200401 14562 199858 15180 minus02707 199032 14758 198618 15551 minus02108 198858 15376 197776 14595 minus05409 199502 13457 197513 15085 minus100

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P19

01

100 10 70

414613 17101 401365 1985 minus32002 413680 16418 400546 20434 minus31703 420538 16747 399896 19542 minus49104 414460 15411 394054 20268 minus49205 411688 15896 395828 21136 minus38506 412339 16217 394750 21373 minus42707 410641 16397 394617 21072 minus39008 409561 18148 394513 21827 minus36709 408967 1781 394294 20113 minus359

P20

01

100 10 150

365982 11365 362144 11433 minus10502 365164 11179 361157 14229 minus11003 365868 11797 349578 11322 minus44504 365307 14530 349277 14462 minus43905 366266 11501 348378 15855 minus48806 363989 15105 348198 11737 minus43407 113367 2295 113204 2836 minus01408 113157 2673 112657 2841 minus04409 112518 3252 111908 2690 minus054

P21

01

100 10 70

330001 13205 323564 13191 minus19502 329657 15575 322848 17839 minus20703 329303 16630 322162 17631 minus21704 328939 16091 321756 16707 minus21805 328561 16771 321589 15201 minus21206 328166 16606 321183 18285 minus21307 327744 13212 321102 17483 minus20308 327281 13330 320160 17993 minus21809 326742 13459 319244 18847 minus229

P22

01

100 10 150

281700 14071 278464 12913 minus11502 285498 14001 278181 17995 minus25603 285035 12887 277760 18458 minus25504 282821 15709 277349 15942 minus19305 284563 12931 277133 18107 minus26106 284057 15207 276776 18334 minus25607 280960 12957 276530 17201 minus15808 282955 15732 276208 18127 minus23809 278306 13202 273261 17852 minus181

P23

01

100 10 70

399113 12510 375576 16477 minus59002 396761 14684 371533 16097 minus63603 397280 16897 360348 15558 minus93004 396341 13925 347788 17599 minus122505 393942 14391 346723 18396 minus119906 392957 15330 344273 14558 minus123907 393378 15910 325508 17295 minus172508 390858 13043 325392 17215 minus167509 391111 14030 322910 16781 minus1744

P24

01

100 10 150

297873 12628 296530 13804 minus04502 295735 13968 294394 16450 minus04503 334584 13860 292118 17375 minus126904 334011 15264 288251 18357 minus137005 345871 13023 287245 14265 minus169506 343622 15007 286258 14921 minus166907 344063 13218 285833 13230 minus169208 343388 15100 285505 14531 minus168609 342624 16102 276622 16518 minus1926

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 13

Table 3 Continued

Prob ID 119903 |119868| |119869| 119876TS NTS Gap 1 ()UB Time UB Time

P25

01

200 10 70

665405 100337 646837 135856 minus27902 663368 102224 646011 127581 minus26203 662357 84078 645933 105216 minus24804 660795 81973 643790 90059 minus25705 660155 81955 641200 117031 minus28706 657577 83574 640619 117254 minus25807 655423 83123 640582 85198 minus22608 654409 83095 637338 85812 minus26109 653286 79233 622908 83645 minus465

P26

01

200 10 150

573667 75347 553421 106924 minus35302 570023 76506 551891 129782 minus31803 569666 77615 537807 106980 minus55904 567939 79795 535782 116339 minus56605 564777 90056 535225 110899 minus52306 564413 88718 535083 119513 minus52007 562167 84913 531505 123666 minus54508 560098 79995 514584 115703 minus81309 557554 86490 513652 102390 minus787

P27

01

200 10 70

661785 138445 631700 122842 minus45502 662602 131790 630853 96939 minus47903 647333 114441 630036 111912 minus26704 647377 123822 625782 106019 minus33405 647837 95581 625659 115976 minus34206 651234 97211 624175 110426 minus41607 643657 99418 623635 120308 minus31108 648507 100751 618926 125166 minus45609 640631 97455 617253 135217 minus365

P28

01

200 10 150

560151 103968 554838 90112 minus09502 567066 80596 552250 120561 minus26103 565177 110727 551150 142966 minus24804 551554 115046 546455 128346 minus09205 550006 84509 546211 90112 minus06906 548151 67959 543161 85063 minus09107 546181 83508 539990 96550 minus11308 545049 86930 538293 144422 minus12409 543203 93776 522946 106527 minus373

P29

01

200 10 70

621423 103198 619749 81058 minus02702 624921 75535 618833 81709 minus09703 620040 71157 617782 99129 minus03604 617601 86788 615738 95845 minus03005 623408 113524 614301 116853 minus14606 614480 79956 612048 133713 minus04007 612960 109573 611365 94858 minus02608 611261 112674 609791 96975 minus02409 609581 84660 608927 71233 minus011

P30

01

200 10 150

522946 93556 520709 121420 minus04302 521952 69692 518710 87110 minus06203 520505 73031 516812 118222 minus07104 519262 78089 516352 128981 minus05605 517999 70729 516019 133597 minus03806 516930 105288 515388 121271 minus03007 515604 95461 513938 105447 minus03208 514221 68726 511705 107318 minus04909 497335 102075 492080 111348 minus106

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Discrete Dynamics in Nature and Society

Table 4 Results of different rate between the cost of recovered products and the cost of new products

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P1

1

20 5 70

68142 300 68142 278 000125 67402 268 67402 239 00015 67211 214 66964 261 minus037175 66898 315 66649 254 minus0372 66496 276 66414 279 minus012

P2

1

20 5 150

43008 333 43008 235 000125 42964 311 42964 254 00015 42934 249 42934 222 000175 42913 240 42913 315 0002 42897 355 42897 319 000

P3

1

20 5 70

61680 272 61680 282 000125 61028 246 61028 281 00015 60595 247 60595 254 000175 60367 259 60285 271 minus0142 60132 252 60052 250 minus013

P4

1

20 5 150

40458 245 40458 205 000125 40391 218 40391 236 00015 40346 286 40346 296 000175 40314 269 40314 232 0002 40290 192 40290 228 000

P5

1

50 5 70

117832 2639 116378 2598 minus123125 117029 2568 114692 2663 minus20015 115617 2614 114374 2588 minus108175 115126 2670 114022 2630 minus0962 113413 2584 112590 2575 minus073

P6

1

50 5 150

90053 2294 87484 2019 minus285125 88307 2422 86991 2015 minus14915 87796 2382 86910 2321 minus101175 86976 2081 86535 2224 minus0512 86703 2288 86394 1723 minus036

P7

1

50 5 70

123939 1823 122295 2188 minus133125 121637 1925 120958 1691 minus05615 120474 1875 119532 2084 minus078175 119790 2188 118308 2161 minus1242 118183 1695 117858 2177 minus027

P8

1

50 5 150

100524 1741 100439 2551 minus008125 99371 2859 99067 2973 minus03115 98231 2437 97807 2953 minus043175 97514 2448 96608 2280 minus0932 96942 2515 96560 1795 minus039

P9

1

50 5 70

117880 2700 115956 2217 minus163125 116053 3039 114876 3522 minus10115 114690 2403 113420 2806 minus111175 113128 1920 112769 2895 minus0322 112006 2862 111756 2696 minus022

P10

1

50 5 150

88057 2326 87883 3121 minus020125 86485 2583 86222 3148 minus03015 85367 3162 84589 1958 minus091175 83386 2610 83478 2237 0112 83455 2545 83235 2190 minus026

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 15

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P11

1

50 5 70

114793 2004 112544 2568 minus196125 112136 1999 111395 2760 minus06615 111087 2016 111062 2996 minus002175 110640 2713 109472 2212 minus1062 110405 2260 108934 2643 minus133

P12

1

50 5 150

87232 2463 86433 2955 minus004125 86410 2209 86199 1889 minus05715 85173 2798 83868 1914 minus194175 84523 2153 83656 1563 minus2182 83027 1838 82912 2022 minus146

P13

1

100 5 70

364895 15568 361755 15932 minus086125 363309 16069 360230 16093 minus08515 362096 16384 359999 16878 minus058175 359974 15796 356857 14178 minus0872 355624 15657 339404 16189 minus456

P14

1

100 5 150

305053 16411 300033 13962 minus165125 303135 14189 296364 12195 minus22315 301834 13924 295118 12168 minus223175 298777 13976 294442 13388 minus1452 291740 13399 276706 13566 minus515

P15

1

100 5 70

296619 12499 258199 12521 minus1295125 260317 13747 254461 12882 minus22515 249922 11505 247152 18334 minus111175 249195 13222 244535 15443 minus1872 247529 14944 241351 17113 minus250

P16

1

100 5 150

233346 12031 223398 12611 minus426125 229687 12975 220760 11491 minus38915 226932 12763 218861 11739 minus356175 216624 13742 214174 11509 minus1132 211223 12879 205238 12239 minus283

P17

1

100 5 70

267713 13493 265309 11060 minus090125 266067 13217 262547 12234 minus13215 263994 13092 260267 11049 minus141175 261711 11678 259735 15061 minus0762 260804 17099 252673 12440 minus312

P18

1

100 5 150

226178 12486 224348 13970 minus081125 223053 12708 219578 13479 minus15615 220970 11528 218823 18485 minus097175 219154 15819 216341 12900 minus1282 217751 15937 209257 18273 minus390

P19

1

100 10 70

412240 13546 409643 14111 minus063125 406518 13913 403969 17398 minus06315 404199 13706 401006 16712 minus079175 403475 13664 400046 17888 minus0852 401608 13654 394231 14981 minus184

P20

1

100 10 150

366576 10510 364188 16650 minus065125 364342 12315 362564 10025 minus04915 360397 15327 358732 14308 minus046175 357502 15804 356178 11459 minus0372 350419 10043 345948 15421 minus128

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Discrete Dynamics in Nature and Society

Table 4 Continued

Prob ID 119888119898 119888119903

|119868| |119869| 119876TS NTS Gap 1 ()

UB Time UB Time

P21

1

100 10 70

337749 11331 334240 10011 minus104125 333248 11445 330768 11204 minus07415 330981 12766 329900 13301 minus033175 329982 14346 325218 13932 minus1442 327911 18902 322382 15034 minus169

P22

1

100 10 150

300245 11222 297406 9424 minus095125 293642 12766 291212 11995 minus08315 290938 18605 287471 19295 minus119175 288716 18372 285885 12707 minus0982 286473 13229 281933 12507 minus158

P23

1

100 10 70

369343 12738 364871 13222 minus121125 364871 15270 359677 14196 minus14215 359613 14770 355286 15278 minus120175 358094 15505 354980 14297 minus0872 341610 14696 320343 17226 minus623

P24

1

100 10 150

328663 12491 299018 13271 minus902125 325042 12513 295553 13729 minus90715 323847 15683 291860 13543 minus988175 299214 13560 288433 15716 minus3602 294706 12122 281938 16867 minus433

P25

1

200 10 70

665281 82911 653633 102607 minus175125 657935 83427 652114 106126 minus08815 655818 82379 651574 98423 minus065175 649604 82010 646761 105581 minus0442 642489 82087 641864 113961 minus010

P26

1

200 10 150

581623 61152 580270 61826 minus023125 579961 60825 569283 62620 minus18415 574684 60460 536008 61379 minus673175 556955 62166 535570 62355 minus3842 554052 62192 528999 62253 minus452

P27

1

200 10 70

773781 79309 662523 77808 minus1438125 713012 81964 655573 80325 minus80615 681260 80427 642081 73951 minus575175 635834 77017 632602 72311 minus0512 632061 73914 623493 70485 minus136

P28

1

200 10 150

576983 66669 555923 98724 minus365125 568736 67582 546433 85983 minus39215 564064 67470 543330 85983 minus368175 560726 91460 535818 76102 minus4442 534830 97703 522356 79576 minus233

P29

1

200 10 70

666957 79082 636582 82080 minus455125 658832 83247 633813 111970 minus38015 653454 114963 621988 96582 minus482175 622078 80189 619907 82517 minus0352 616051 86347 608644 108037 minus120

P30

1

200 10 150

542071 73401 540384 70536 minus031125 525159 72755 524172 78656 minus01915 521758 71637 510933 76303 minus207175 518474 78704 506479 75231 minus2312 515142 75566 502884 69885 minus238

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 17

254000256000258000260000262000264000266000268000270000272000274000

01 02 03 04 05 06 07 08 09

Tota

l cos

t

Return rate

Figure 3 Results of different return rates

255000

260000

265000

270000

275000

280000

285000

1 125 15 175 2

Tota

l cos

t

cm cr

Figure 4 Results of different rate between the cost of recoveredproducts and the cost of new products

and randomly generated instances of LIRP-FRL with variousproblem sizes demonstrate the effectiveness and the efficiencyof the new tabu search which can find high quality solutionwith a reasonable time Sensitivity analyses are also con-ducted to investigate the intrinsicmanagement insights of theproblems

For future research to be more close to the reality thestochastic demand of customer should also be considered inthe LIRP-FRL

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research presented in this paper is supported by theNational Natural Science Foundation Project of China(71390331 71572138 71271168 and 71371150) the NationalSocial Science Foundation Project of China (12ampZD070) theNational Soft Science Project of China (2014GXS4D151) theSoft Science Project of Shaanxi Province (2014KRZ04) theProgram for New Century Excellent Talents in University(NCET-13-0460) and the Fundamental Research Funds forthe Central Universities

References

[1] A A Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[2] S C Liu and S B Lee ldquoA two-phase heuristic method for themulti-depot location routing problem taking inventory controldecisions into considerationrdquo International Journal of AdvancedManufacturing Technology vol 22 no 11-12 pp 941ndash950 2003

[3] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[4] Y Zhang M Qi L Miao and E Liu ldquoHybrid metaheuristicsolutions to inventory location routing problemrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 305ndash323 2014

[5] D Ambrosino and M G Scutella ldquoDistribution networkdesign new problems and related modelsrdquo European Journal ofOperational Research vol 165 no 3 pp 610ndash624 2005

[6] Z-J M Shen and L Qi ldquoIncorporating inventory and routingcosts in strategic location modelsrdquo European Journal of Opera-tional Research vol 179 no 2 pp 372ndash389 2007

[7] W J Guerrero C Prodhon N Velasco and C A AmayaldquoA relax-and-price heuristic for the inventory-location-routingproblemrdquo International Transactions in Operational Researchvol 22 no 1 pp 129ndash148 2015

[8] V Ravi R Shankar andM K Tiwari ldquoAnalyzing alternatives inreverse logistics for end-of-life computers ANP and balancedscorecard approachrdquo Computers amp Industrial Engineering vol48 no 2 pp 327ndash356 2005

[9] BVahdani andMNaderi-Beni ldquoAmathematical programmingmodel for recycling network design under uncertainty aninterval-stochastic robust optimization modelrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 73no 5ndash8 pp 1057ndash1071 2014

[10] C-T Chen P-F Pai and W-Z Hung ldquoAn integrated method-ology using linguistic PROMETHEE and maximum deviationmethod for third-party logistics supplier selectionrdquo Interna-tional Journal of Computational Intelligence Systems vol 3 no4 pp 438ndash451 2010

[11] B Vahdani J Razmi and R Tavakkoli-Moghaddam ldquoFuzzypossibilistic modeling for closed loop recycling collection net-worksrdquo Environmental Modeling amp Assessment vol 17 no 6 pp623ndash637 2012

[12] B Vahdani R Tavakkoli-Moghaddam and F Jolai ldquoReliabledesign of a logistics network under uncertainty a fuzzypossibilistic-queuing modelrdquo Applied Mathematical Modellingvol 37 no 5 pp 3254ndash3268 2013

[13] B Vahdani and M Mohammadi ldquoA bi-objective interval-stochastic robust optimization model for designing closed loopsupply chain network with multi-priority queuing systemrdquoInternational Journal of Production Economics vol 170 pp 67ndash87 2015

[14] K Richter ldquoThe EOQ repair and waste disposal model withvariable setup numbersrdquo European Journal of OperationalResearch vol 95 no 2 pp 313ndash324 1996

[15] J Shi G Zhang and J Sha ldquoOptimal production planning for amulti-product closed loop system with uncertain demand andreturnrdquo Computers amp Operations Research vol 38 no 3 pp641ndash650 2011

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

18 Discrete Dynamics in Nature and Society

[16] E E Zachariadis and C T Kiranoudis ldquoA local searchmetaheuristic algorithm for the vehicle routing problem withsimultaneous pick-ups and deliveriesrdquo Expert Systems withApplications vol 38 no 3 pp 2717ndash2726 2011

[17] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

[18] H Liu Q Zhang andWWang ldquoResearch on locationminusroutingproblem of reverse logistics with grey recycling demands basedon PSOrdquo Grey Systems Theory and Application vol 1 no 1 pp97ndash104 2011

[19] V F Yu and S-W Lin ldquoMulti-start simulated annealing heuris-tic for the location routing problem with simultaneous pickupand deliveryrdquo Applied Soft Computing vol 24 pp 284ndash2902014

[20] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[21] Y Li M Lu and B Liu ldquoA two-stage algorithm for the closed-loop location-inventory problem model considering returns inE-commercerdquoMathematical Problems in Engineering vol 2014Article ID 260869 9 pages 2014

[22] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory- routing prob-lem considering returns under E-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[23] D P Adler V Kumar P A Ludewig and J W SutherlandldquoComparing energy and other measures of environmental per-formance in the original manufacturing and remanufacturingof engine componentsrdquo in Proceedings of the ASME Interna-tional Conference onManufacturing Science andEngineering pp851ndash860 October 2007

[24] J-M Belenguer E Benavent C Prins C Prodhon and RWolfler Calvo ldquoA branch-and-cut method for the capacitatedlocation-routing problemrdquo Computers amp Operations Researchvol 38 no 6 pp 931ndash941 2011

[25] R H Teunter ldquoEconomic ordering quantities for recoverableitem inventory systemsrdquoNaval Research Logistics vol 48 no 6pp 484ndash495 2001

[26] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquoComputers ampOperations Research vol 13no 5 pp 533ndash549 1986

[27] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[28] D Habet ldquoTabu search to solve real-life combinatorial opti-mization problems a case of studyrdquo Foundations of Computa-tional Intelligence vol 3 pp 129ndash151 2009

[29] V F Yu S-W Lin W Lee and C-J Ting ldquoA simulated anneal-ing heuristic for the capacitated location routing problemrdquoComputers and Industrial Engineering vol 58 no 2 pp 288ndash299 2010

[30] D J Rosenkrantz R E Stearns and P M Lewis ldquoApproximatealgorithms for the traveling salesperson problemrdquo in Proceed-ings of the IEEE Conference Record of 15th Annual Symposiumon Switching and Automata Theory pp 33ndash42 October 1974

[31] C Prins C Prodhon and R Wolfler-Calvo ldquoNouveaux algo-rithmes pour le probleme de localisation et routage souscontraintes de capaciterdquo MOSIM vol 4 no 2 pp 1115ndash11222004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of