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J Intell Manuf DOI 10.1007/s10845-015-1130-9 Bi-objective mixed-integer nonlinear programming for multi-commodity tri-echelon supply chain networks M. H. Alavidoost 1 · Mosahar Tarimoradi 1 · M. H. Fazel Zarandi 1,2 Received: 6 April 2015 / Accepted: 17 July 2015 © Springer Science+Business Media New York 2015 Abstract The competitive market and declined economy have increased the relevant importance of making supply chain network efficient. Up to now, this has resulted in great motivations to reduce the cost of services, and simul- taneously, to improve their quality. A mere network model, as a tri-echelon, consists of Suppliers, Warehouses or Dis- tribution Centers (DCs), and Retailer nodes. To bring it closer to reality, the majority of parameters in this network involve retailer demands, lead-time, warehouses holding and shipment costs, and also suppliers procuring and stocking costs which are all assumed to be stochastic. The aim is to determine the optimum service level so that total cost is minimized. Obtaining such conditions requires determin- ing which supplier nodes, and which DC nodes in network should be active to satisfy the retailers’ needs, an issue which is a network optimization problem per se. The pro- posed supply chain network for this paper is formulated as a mixed-integer nonlinear programming, and to solve this complicated problem, since the literature for the related benchmark is poor, three numbers of genetic algorithm called Non-dominated Sorting Genetic Algorithm (NSGA-II), Non- dominated Ranking Genetic Algorithm (NRGA), and Pareto B Mosahar Tarimoradi [email protected] M. H. Alavidoost [email protected] M. H. Fazel Zarandi [email protected] 1 Department of Industrial Engineering, Computational Intelligent Systems Laboratory, Amirkabir University of Technology, 424 Hafez Ave, P.O.Box 15875-4413, Tehran, Iran 2 Knowledge Intelligent Systems Laboratory, University of Toronto, Toronto, Canada Envelope-based Selection Algorithm (PESA-II) are applied and compared to validate the obtained results. The Taguchi method is also utilized for calibrating and controlling the parameters of the applied triple algorithms. Keywords Supply chain management · Tri-echelon network · Mixed-integer nonlinear programming · NRGA · NSGA-II · PESA-II · Taguchi method Introduction Supply Chain Management (SCM) is a comprehensive approach that contains the processes like retailer demand management, order fulfillment, manufacturing management, procurement, product commercialization, returns manage- ment, etc. (Rogers & Leuschner 2004). From another point of view, it might also involve both internal and external func- tions of an company that enables its value chain to make products and provide services for the retailers (Handfield and Nichols 1999). SC usually consists of retailers, distribution centers (DCs), plants, and suppliers. In common SCs theses are supposed that raw material should be primed, products are manufactured at one or more plants, commodities are sent to warehouses and finally they might be shipped for the retailers. SCM is faced with handling a network of inter-connected businesses involved in the ultimate provision of commod- ity so that packaging services could be carried by the end retailers. Thus with such an aspect, SCM or in better terms Supply Chain Network (SCN), envelopes all the require- ments for synchronizing activities like material priming, work in processing to final products, and distribution of the manufactured products to retailers. The goals of SCN are usually known such as minimizing the system costs and sat- 123

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Page 1: Bi-objective mixed-integer nonlinear programming for multi-commodity tri-echelon supply chain networks

J Intell ManufDOI 10.1007/s10845-015-1130-9

Bi-objective mixed-integer nonlinear programming formulti-commodity tri-echelon supply chain networks

M. H. Alavidoost1 · Mosahar Tarimoradi1 · M. H. Fazel Zarandi1,2

Received: 6 April 2015 / Accepted: 17 July 2015© Springer Science+Business Media New York 2015

Abstract The competitive market and declined economyhave increased the relevant importance of making supplychain network efficient. Up to now, this has resulted ingreat motivations to reduce the cost of services, and simul-taneously, to improve their quality. A mere network model,as a tri-echelon, consists of Suppliers, Warehouses or Dis-tribution Centers (DCs), and Retailer nodes. To bring itcloser to reality, the majority of parameters in this networkinvolve retailer demands, lead-time, warehouses holding andshipment costs, and also suppliers procuring and stockingcosts which are all assumed to be stochastic. The aim isto determine the optimum service level so that total costis minimized. Obtaining such conditions requires determin-ing which supplier nodes, and which DC nodes in networkshould be active to satisfy the retailers’ needs, an issuewhich is a network optimization problem per se. The pro-posed supply chain network for this paper is formulatedas a mixed-integer nonlinear programming, and to solvethis complicated problem, since the literature for the relatedbenchmark is poor, three numbers of genetic algorithm calledNon-dominatedSortingGeneticAlgorithm (NSGA-II),Non-dominated Ranking Genetic Algorithm (NRGA), and Pareto

B Mosahar [email protected]

M. H. [email protected]

M. H. Fazel [email protected]

1 Department of Industrial Engineering, ComputationalIntelligent Systems Laboratory, Amirkabir University ofTechnology, 424 Hafez Ave, P.O. Box 15875-4413, Tehran,Iran

2 Knowledge Intelligent Systems Laboratory, University ofToronto, Toronto, Canada

Envelope-based Selection Algorithm (PESA-II) are appliedand compared to validate the obtained results. The Taguchimethod is also utilized for calibrating and controlling theparameters of the applied triple algorithms.

Keywords Supply chain management · Tri-echelonnetwork · Mixed-integer nonlinear programming · NRGA ·NSGA-II · PESA-II · Taguchi method

Introduction

Supply Chain Management (SCM) is a comprehensiveapproach that contains the processes like retailer demandmanagement, order fulfillment, manufacturing management,procurement, product commercialization, returns manage-ment, etc. (Rogers & Leuschner 2004). From another pointof view, it might also involve both internal and external func-tions of an company that enables its value chain to makeproducts and provide services for the retailers (Handfield andNichols 1999). SC usually consists of retailers, distributioncenters (DCs), plants, and suppliers. In common SCs thesesare supposed that raw material should be primed, productsare manufactured at one or more plants, commodities aresent to warehouses and finally they might be shipped for theretailers.

SCM is faced with handling a network of inter-connectedbusinesses involved in the ultimate provision of commod-ity so that packaging services could be carried by the endretailers. Thus with such an aspect, SCM or in better termsSupply Chain Network (SCN), envelopes all the require-ments for synchronizing activities like material priming,work in processing to final products, and distribution of themanufactured products to retailers. The goals of SCN areusually known such as minimizing the system costs and sat-

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isfying the service level requirements. Such a comprehensivesystem is a draft that depicts the quantities of commodities,location ofDCs, and even the time for the production process.There are numerous autonomous identities each of whichtries to satisfy their own objective in an SCN. Thus, trying tosolve a real SC problemmight be very hard and requiresmorethan one objective to be satisfied. Such a problem is calleda multi-objective optimization problem that has numerousPareto solutions.Attaining tomatters like lower costs, shorterprocessing time and lead-time, lower stock, larger commod-ity diversities, better reliable delivery time, improved quality,and priming the coordination between demand, procurementand manufacturing that are all known as KPI1 for businessowners, need a proper and well-devised SCN.

SCM could be summarized into three main processes:SC structuring, SC programming, and SC control and mon-itoring. In SC structuring, we make strategic plan such asplant location, capacity of plants and the quantity of materi-als that are required in producing operations or distributionamong facilities. The focus of structuring in traditional SCMis mainly devoted to a single objective, such as minimiz-ing the cost or maximizing the profit, whilst a real SC hasoften to be optimized considering more than one. In fact,real SC problems can usually be formulated as a case of amulti-objective problem requiring an algorithm capable forsearching the space of objectives in a short run-time.

In this paper, an attempt is made to optimize a bi-objectivetri-echelonmulti-commodity SC problem. The proposed net-work would of some suppliers, DCs, and retailers nodes.Putting the existing models into practice and bring themto reality is the contribution of this paper. This is attainedusing more realistic and applied assumptions in terms ofuncertainties involved in all of the three strategic, tactical,and structuring the proposed SCN levels. Depicting it in amore specific manner, the fixed and variable costs, retailersdemand, total available production time for plants, setup andproduction time of producing products, are all assumed to bestochastic internal parameters following the uniform distrib-utions; a common probability model that is suitable for manynatural stochastic processes based on the central limit theo-rem. Moreover, the goal is to determine the active suppliersand DCs assumed as Boolean variables so that the optimumpaths for retailers’ demands satisfaction could be achieved.In other words, this paper aims to determine the optimumnetwork for satisfying retailers’ demands subject to the twogoals of minimum cost and maximum service levels.

The problem is formulated to obtain the deterministicmodel of a bi-objective MINLP.2 The proposed mathemati-cal model of this work is hard to be resolved by the commonanalytical or exact approaches, thus three of MOGAs are

1 Key Performance Indicator.2 Mixed-Integer Nonlinear Programming.

utilized to find Pareto Fronts; and since the literature forbenchmarks to validate the obtained solutions is poor, theseapplied algorithms called NSGA-II,3 NRGA,4 and PESA-II5

are compared together via six numbers of the cited indexes.The Taguchi DOE6 method is used for controlling and cal-ibrating these applied algorithms parameters. Finally, thenumerical example is presented and detailed comparisonresults are exposed and discussed.

The rest of this paper aims to explain the problem back-ground in section “Problem background”. Afterward insection “Formulating the proposed supply chain problem”the proposed problem has been formulated. Then the solvingprocedure consisting of the current approaches for dealingwith SCN problems, applied algorithms and their character-istics are considered in section “Solving procedure”. Indexesfor multi-objective evolutionary algorithm provided in sec-tion “Comparison measures”. After that, the experimentalresults and a comparison between triplex calibrated algo-rithms based upon the defined indexes are considered insection “The experiments”. Finally, the conclusions and someguidelines for future studies are provided in section “Con-cluding remarks and implications for future works”.

Problem background

In the past few years, it has become obvious that many com-panies have reduced operational costs as much as possible.They are discovering that effective SCM is the next neces-sary step to take in order to increase profits and market share(Simchi-Levi et al. 2003).

In many of the classical SCN structuring, the goal issending/receiving merchandise from/to a layer to/from theother(s) so that procuring costs for both strategic and opera-tional functions are minimized. As an instance, Amiri (2006)structured SC model for making the best strategic decisionson locating the plants and DCs for dispatching commodi-ties from manifesting site to the retailers side, coincide withthe goal of minimum total costs of the DCs in the net-work. In another work, Gebennini et al. (2009) offered athree-layered manufacturing–dispatching system for mini-mum costs. Network designing faces with relations betweenvarious SCportions together, aremutually under the risks anduncertainties through the whole chain; an issue that createda controversial problem for the SC decision-making process,so that the recent goals are propounded. The uncertaintiesinvolved in SC networks could be depicted into three divi-sions based on the supplier layer, the receiver layer, and inthe DC layer. Since the reversible logistic decisions and their

3 Non-Dominated Sorting Genetic Algorithm.4 Non-Dominated Ranking Genetic Algorithm.5 Pareto Envelope-Based Selection Algorithm.6 Design of Experiments.

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relation to the SCN scaffolding is very difficult and costly,the momentous of the interactions between these decisionsis vastly enhanced under uncertainty. Mohammadi BidhandiandMohd Yusuff (2011) formulated a stochastic SCNmodelas a two-level program considering both strategic and tacticaldecisions. In their model, the retailer demands, the opera-tion cost, and the capacity of facilities can be uncertain asthey can all have tardiness effects on the strategic decisions.For the strategic level, Snyder (2006) considered an RFLP7

for locating DCs level of an SC under uncertainty when thefacilities might have random failures. Murthy et al. (2004)mentioned that uncertainty for the strategic level is the mostdifficult and important issue to be considered. For the tacticallevel, Van Landeghem and Vanmaele (2002) considered anSC structuring problem that consists of the merchandise andraw material dispatching. Moreover, Jamshidi et al. (2012)proposed a multi-echelon bi-objective SCN structure involv-ing several transportation options for each level with variablecosts and restrictions on capacity.

Some other approaches in literature which are noticeablefor SC problems could be taken into account as Moncayo-Martínez and Zhang (2011) proposed an algorithm based ona Pareto AC8 optimization for minimizing both the SC cur-rent cost and the total lead-time for a family of commodities.In another work, Cardona-Valdés et al. (2014) studied thestructure of a two-echelon SC with an uncertain demand.An important contribution in this work is the deploymentof TS9 within the multi-objective adaptive memory pro-gramming architecture to prepare optimal Pareto Fronts fora two-stage stochastic bi-objective programming problem.While Shankar et al. (2013) considered the optimization ofthe strategic structure and DCs decisions for a tri-echelonSC simultaneously, for solving the problem a MOHPSO10

has also been proposed in their work. In addition, Maru-fuzzaman et al. (2014) considered a two-stage stochasticmodel used for scaffolding and handling the biodiesel SC.Their model catches the effects of biomass supply anduncertainties in technology on SC related decisions. Chesh-mehgaz et al. (2013) provided a flexible three-level logisticnetwork design considering time and cost criteria with amulti-objective evolutionary algorithm.Mousavi et al. (2014)considered a modified particle swarm optimization for solv-ing the integrated location and inventory control problems ina two-echelon supply chain network. Hajipour et al. (2015)primed a Pareto-based meta-heuristics for solving multi-objective multi-item capacitated lot-sizing problems. Heprovided amulti-objective harmony search algorithm to opti-mize multi-server location–allocation problem in congested

7 Reliable Facility Location Problem.8 Ant Colony.9 Tabu Search.10 Multi-Objective Hybrid Particle Swarm Optimization.

systems (Hajipour et al. 2014). Considering redundancyqueuing-location-allocation problem (Hajipour et al. 2014)is the other noteworthy work. Fattahi et al. (2015) proposed abi-objective continuous review inventory control model andused Pareto-based meta-heuristic algorithms. Also Rahmatiet al. (2013) offered soft-computing Pareto-based meta-heuristic algorithm for a multi-objective multi-server facilitylocation problem. And for multi-objective works in the pro-duction fields the proposed fuzzy adaptive multi-objectivegenetic algorithm provided by Alavidoost et al. (2015) isnoteworthy.

The multi-commodity problem considers fixed locationcosts, linear transportation costs, and assumes that eachware-house can be assigned at most one commodity which arestudied by Warszawski and Peer (1973). After that, Geof-frion and Graves (1974) considered the extended version ofmulti-commodity location problemas capacitated, anddevel-oped amodel to solve the problem of designing a distributionsystem with optimal location of the intermediate distributionfacilities between plants and customers. They also explainedthe risk of using heuristic models in distribution planning.The plant location problem has two derivatives as capaci-tated and incapacitated per se. These two types of problemsare studied by Mirchandani and Francis (1990), Church andReVelle (1976), Shen et al. (2003), Sridharan (1995).

One momentous issue in the SCM study is overcomingmore than one objective such as minimizing costs, maxi-mizing profits and improving customer services (Mosaharet al. 2014). Different methodologies were developed forsolving multi-objective optimization problems such as theweighted-sum method, the ε-constraint method, the goal-programming method and fuzzy method (Azapagic and Clift1999; Chen and Lee 2004; Zhou et al. 2000). In this context,Sabri and Beamon (2000) presented a multi-objective tech-nique for simultaneous strategic and operational planningin SC design. The model considered production, delivery,demand uncertainty, and a multi-objective performance vec-tor for the entire SC network. In line with the mathematicalmodel of this paper, Nozick and Turnquist (2001) proposed amodel that minimizes the costs and maximizes the services.

For further study, themulti-objective locationmodels pub-lishedbyShen et al. (2003) canbe referred.On the other hand,Chen (2004) proposed a model for optimizing the conflictingobjectives such as participant’s profits, the average customerservice levels and the average safe inventory level. Kopanoset al. (2009) also presented a multi-objective stochasticmixed-integer linear programming model for SCM. Theysolved their model using the standard ε-constraint methodand branch and bound techniques. Graves and Willems(2005) applied an optimization algorithm to find the bestinventory levels of all sites on the SC. Nowadays, the GA isconsidered as one of the most used optimization tools whichis applied in the resolution of several types of linear and non-

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.

.

.

.

.

.

Supplier-2-DC Lead-time

Suppliers DCs

Retailers

.

.

.

Fig. 1 A tri-echelon SCN

linear optimization problems (Goldberg 1989). However, inthe real problem conception of SC decisions, one is encoun-teredwithmultiple choices. Themain difference between theabove mentioned issues that were obtained through literaturesurveying, with the proposed model of this paper is that thispaper considers supplier and inventory location (as DCs) bydetermining their Boolean value (i.e. null or active) in theproposed network and also material flow decisions, whilethe pre-mentioned works consider other minded issues thathave referred.

Formulating the proposed supply chain problem

As shown in Fig. 1, the assumed tri-echelon SCN for thispaper consists of suppliers to the left, distribution centers(DCs) in the middle, and retailer nodes to the right. It issupposed that the proposed SC Network:i. Has an integrated structure consisting both of potential

supplier and potential DCs to procure retailer demandsfor multitude commodities.

ii. Has predefinednumbers of suppliers andDCswith iden-tified capacities.

iii. The number of its retailers and their demands distribu-tion are identified.

iv. Operates in an uncertain circumstance, i.e. its maininterior parameters as demands, lead-time, procure andtransportation costs, and also holding costs of inventoryfor commodities are all supposed to be uniform randomvariables with identified average and variance.

v. Its DCs and suppliers are all supposed to be poten-tially operational at the beginning of the constructingnetwork.

vi. Its suppliers and DCs do their procuring, shipment, andholding duties perfectly.

vii. Any retailer receives its demand for a specific merchan-dize only from one of the DCs.

viii. Shortage cannot occur at the retailer nodes in any form.ix. More than one supplier can replete the demand of a

specific DC.x. More than one DC can replete the demand of each

retailer.

Table 1 Notation used in the mathematical formulation

Parameters’notation

Donates

μki Average daily demand from i th retailer for kth

commodity

V ki Variance daily demand from i th retailer for kth

commodity

Fj j th Warehouse opening fixed cost

hkj j th Warehouse holding cost for kth commodity

AkJ j th Warehouse ordering cost for kth commodity

w j Potential capacity of jth warehouse

tckji Unit cost of kth commodity shipping from j th

warehouse to i th retailer

gm mth Supplier fixed cost to be selected/accept toprocure

rckmj Cost of procuring, stocking and shipping kth

commodity from mth supplier to j th warehouse

sm Potential capacity of mth supplier

lkmj Lead-time for kth commodity from mth supplier to

j th warehouse

R The maximum possible number of supplier

N The maximum possible number of warehouses

β The number of working-day per year

In other words, it is a BSCN11 in which the total capacityof all suppliers is greater or equal to the total consumptionof all commodities, and the total procurement capacity of allcommodities are equal to the total consumption of all end cus-tomers, and the total consumption of all commodities is equalto the total procurement of all commodities by suppliers.Used indices :

i: Number of Retailersj: Number of Warehouses (DCs)m: Number of Suppliersk: Number of Commodities

Considered sets :

SI : Set of Retailers SI = {i |i = 1, 2, . . . , I }SJ : Set of Potential DCs SJ = { j | j = 1, 2, . . . , J }SM : Set of Suppliers SM = {m|m = 1, 2, . . . , M}SK : Set of Commodities SK = {k|k = 1, 2, . . . , K }

Tables 1 and 2 depict the used parameters’ notations anddecision variables respectively.

Basedon the stock theory, the j thwarehouse daily demand

distribution for the kth commodity follows N(

Dkj , θ

kj

).

Where Dkj delegates the average daily demand of the j th

11 Balanced Supply Chain Network.

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Table 2 Decision variables used in the mathematical formulation

Decision variables Definition

xkji ∈ [0, 1] 1, If the demand of kth commodity for the

i th retailer is satisfied by the j th ware-house, Else 0

ykmj ∈ [0, 1] 1, If the stock of kth commodity for the j th

warehouse is procured by the mth supplier,Else 0.

u j ∈ [0, 1] 1, If the j th warehouse is open/active,Else0.

pm ∈ [0, 1] 1, If the mth Supplier is selectedfor/accepted procuring, Else 0.

Qkj ≥ 0 Optimum quantity of the kth commodity

for the j th warehouse

SSkj ≥ 0 Buffer quantity of the kth commodity for

the j th warehouse

rkj ≥ 0 The kth Commodity order point for the j th

warehouse

warehouse for the kth commodity and θkj is the daily demand

variance of the j th warehouse for the kth commodity. Theformulas for calculating Dk

j and θkj are as Eq. (1):

Dkj =

I∑i=1

μki · xk

ji , θkj =

I∑i=1

V ki · xk

ji ,

∀ j = 1, 2, . . . , J, k = 1, 2, . . . , K (1)

The expected value of the kth commodity lead-time deliveryin the j th warehouse could be calculated by Eq. (2):

Ekj =

M∑m=1

Lkmj · yk

mj , ∀ j = 1, 2, . . . , J, k = 1, 2, . . . , K

(2)

The mean and variance of the specific kth commoditydemand in lead-time for the j th warehouse are as given inEqs. (3) and (4), while ∀ j = 1, 2, . . . , J, k = 1, 2, . . . , K :

D′kj = Ek

j · Dkj = Ek

j ·I∑

i=1

μki · xk

ji , ∀ j = 1, 2, . . . , J,

k = 1, 2, . . . , K (3)

θ ′kj = Ek

j · θkj = Ek

j ·I∑

i=1

V ki · xk

ji , ∀ j = 1, 2, . . . , J,

k = 1, 2, . . . , K (4)

Then SSkj , the buffer quantity of the kth commodity for the

j th warehouse could be calculated by Eq. (5). Note that α

donates the interval confidence and z1−α represents the α

critical value of the normal standard distribution:

SSkj = z1−α ·

[√θ ′k

j

], ∀ j = 1, 2, . . . , J,

k = 1, 2, . . . , K (5)

The order point and optimum quantity of the j th warehouse(Q∗k

j ) are as Eqs. (6) and (7):

rkj = D′k

j + SSkj , ∀ j = 1, 2, . . . , J,

k = 1, 2, . . . , K (6)

Q∗kj =

√√√√2 · Akj · β

∑Ii=1 μk

i · xkji

hkj

, ∀ j = 1, 2, . . . , J,

k = 1, 2, . . . , K (7)

The mathematical model for the mentioned SC networkis described as follows:

Objective 1 : f1

= Min

⎧⎨⎩

M∑m=1

gm · pm +J∑

j=1

Fj · u j

K∑k=1

M∑m=1

J∑j=1

I∑i=1

μki · rck

mj · xkji · yk

mj

K∑k=1

J∑j=1

I∑i=1

μki · tck

ji · xkji

+K∑

k=1

J∑j=1

√√√√2.Akj · hk

j ·[β

I∑i=1

μki · xk

ji

]

+K∑

k=1

J∑j=1

hkj · z1−α ·

√√√√ M∑m=1

I∑i=1

Lkmj · V k

i · xkji · yk

mj

⎫⎬⎭(8)

Objective 2 : f2

= Max

{∑Kk=1∑M

m=1∑J

j=1∑I

i=1 μki .x

kji .y

kmj∑K

k=1∑I

i=1 μki

}(9)

Subject to:

J∑j=1

xkji ≤ 1, ∀i ∈ SI , k ∈ SK (10)

xkij ≤ uj, ∀ i ∈ SI , j ∈ SJ , k ∈ SK (11)

M∑m=1

ykmj ≤ 1, ∀ j ∈ SJ , k ∈ SK (12)

ykmj ≤ pm, ∀ m ∈ SM , j ∈ SJ , k ∈ SK (13)

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J∑j=1

u j ≤ N (14)

M∑m=1

pm ≤ R (15)

I∑i=1

μki .x

kji + z1−α.

⎡⎣√√√√ M∑

m=1

I∑i=1

Lkmj .V

ki .xk

ji .ykmj

⎤⎦

≤ w j .u j , ∀ j ∈ SJ , k ∈ SK (16)J∑

j=1

[I∑

i=1

μki · xk

ji

]· ykmj ≤ sm · pm,

∀ m ∈ SM , k ∈ SK (17)

xkji ∈ [0, 1] , yk

mj ∈ [0, 1], u j ∈ [0, 1], pm ∈ [0, 1] (18)

The objective function 1 [Eq. (8)] minimizes the total cost ofsetting up and operating the network, and objective function2 [Eq. (9)] maximizes the replenishing rate or service level.The constraint in Eq. (10) states that the i th retailer receivesthe kth commodity that could be satisfied just from one ware-house. The constraint in Eq. (11) specifies that the variablesare bounded. The constraint in Eq. (12) enforces the kth com-modity demand of the j th warehouse that could be procuredjust by one supplier. The constraint in Eq. (13) states that ifthe mth supplier is open, the j th warehouse will receive itsdemand from the mth supplier. The constraint in (14) indi-cates the maximum number of warehouses. The constraintin Eq. (15) specifies the maximum number of suppliers. Theconstraint in Eq. (16) ensures that the j th warehouse capac-ity is greater than the i th retailer demand and its buffer. Theconstraint in Eq. (17) enforces that the supplier capacitymustbe greater than the warehouse capacity. The constraint in Eq.(18) donates that the variables are binary.

Solving procedure

Overall, to solve the complicated multi-objective optimiza-tion problems, there are two approaches. In one, the prob-lem is converted to a single-objective optimization usingMCDM12 methods [proposed by Hwang et al. (1979)]. Thenan SOEA13 such as GA,14 PSO,15 SA,16 HAS,17 or ICA18

could be deployed to solve the single-objective problem in

12 Some Multi-Criteria Decision Making.13 Single-Objective Evolutionary Algorithm.14 Genetic Algorithm.15 Particle Swarm Optimization.16 Simulated Annealing.17 Harmony Search Algorithm.18 Imperialist Competition Algorithm.

one single run (Deb et al. 2002). In another one, an MOEA19

such as NSGA-II, NRGA, or PESA-II is directly used to findan optimal set named Pareto Optimal Front in a single run(Al Jadaan et al. 2006). SinceMOEAs are usually fast to findPareto Fronts in a single run and also SOEAs need multituderuns for obtaining a Front, anMOEAmight be utilized in thissection as well to solve a complex bi-objective optimizationproblem at hand.

All in all, solving SCM problems using GA is a preva-lent between the practitioners of this context. As Tsai andChao (2009) applied an adaptive GA by a chromosomes’repairing procedure tomake genes’ ordinal structure adapted.In another work, Wang et al. (2011) considered a facilitylocation and task allocation problem of a two-echelon SCwith stochastic demands for gainingmaximization. They pre-sented a GA with efficient greedy heuristics to solve theirproblem. Prakash et al. (2012), primed a KBGA20 for opti-mizing an SCN. Altiparmak et al. (2006) used a GA to findthe Pareto optimal set of a multi-objective four-echelon SCusing a different weighting method. Bandyopadhyay andBhattacharya (2014) proposed a tri-objective problem fora two-echelon serial SC. They considered modification ofNSGA-II with an embedded mutation algorithm. In anotherwork, Sourirajan et al. (2009) studied a two-stage SC witha single product replenished in a production facility andapplied a GA to solve it. LHA21 was also deployed in theirwork for the comparison of the obtained results. Zegordi et al.(2010) used a GA for solving a mixed-integer programmingfor a two-stage SC problem containing scheduling of mer-chandizes and vehicles.

Among MOEAs, the NSGA-II is selected for the sakeof its popularity, capability and ease of use. Furthermore,as previously mentioned, since the literature for validate theobtained results is poor benchmark to, a couple of multi-objective evolutionary algorithm called NRGA and PESA-IIare utilized as well. For all applied algorithms, Taguchi DOEis used for controlling and calibrating algorithms parameters.Finally, a numerical example and comparison results betweenthese calibrated algorithms are presented and discussed.

Non-dominated sorting genetic algorithm (NSGA-II)

NSGA-II, introduced by Deb et al. (2000), is one of mostused and propounded GA-based algorithms for solvingmulti-objective problems (Alavidoost and Nayeri 2014). Itcommences by a randomly generated population with sizenPop (as one of the algorithm parameters). During theiterations, the objectives value for each individual of the pop-ulation would be assessed via an evaluator function. After

19 Multi-Objective Evolutionary Algorithm.20 Knowledge-Based Genetic Algorithm.21 Lagrangian Heuristic Approach.

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that, the population individuals would be ranked based onthe non-dominated sorting process. The individuals of thepopulation label a rank equal to their non-dominated levelso that the first front contains individuals with the smallestrank; the second front corresponds to the individuals withthe second rank; and so on. In the next stage, the CrowdingDistance between members on each front would be calcu-lated. As a Boolean tournament selection operator based ona crowded-comparison operation is used, it is necessary to

reckon both the rank and the crowding distance for each pop-ulation individual. Thus, two members would first be caughtbetween the populations by this operator. In what follows,

the member with a larger Crowding Distance is selected ifthey share an equal rank. Otherwise, the member with thelower rank would be chosen. Then, a new offspring popula-tion with a size of n would be created through the selectionand the crossover, and then the mutation operators are goingto be run to create a population consisting of the existing andthe new (nPop + n) population size. Finally, a population of

an exact size of nPop would be attained by the sorting proce-dure. In this procedure, the solutions could be sorted in twosteps: one based upon their Crowding Distances in descend-ing order, and other according to their ascending order ranks.The new population is used to generate the next new gener-ation by iterating the mentioned stages respectively. Such aprocedure would be continued till the termination conditionis reached. The pseudo code forNon-dominated Sorting (NS)algorithm is supposed as follow:

Procedure: Typical Non-dominated Sorting (NS) AlgorithmBeginStep 1. Put and also for all the population’s individuals , & .

Where:l is the counter of front.Sp donates set of population dominated by individual number p (pth member).Np the number of the pth member is dominated by the others.

Step 2. for each two members of the population (like p and q):IF q is dominated by p, THEN add q to the set of Sp and Nq=Nq+1IF p is dominated by q, THEN add p to the set of Sq and Np=Np+1

Step 3. Put members valued with Np=0 in first front (F1).Step 4. Put as a default for the front of .Step 5. For each member of (like p), if q is member of Sp, subtract a unit from Nq and if Nq=0, add q to the Q (i.e. ).Step 6. IF the Q value is null, THEN end up the sorting process,

ELSE and beck to step 4.END

Likewise the pseudo code for NSGA-II which deploysNon-dominated Sorting (NS) algorithm as an embedded stepused as follow:

Procedure: Typical NSGA-IIBeginStep 1. Determine population size (nPop), crossover rate ( ), mutation rate ( ), and iteration numbers (Itr).Step 2. Randomly generate first population.Step 3. Sort individuals using Non-dominated Sorting (NS) Algorithm and calculate Crowding Distance for each of them.

3.1. Select a couple of individuals from population randomly.3.2. IF they are from different fronts, THEN select one with less front number,

ELSE select one with higher Crowding Distance.Step 4. Deploy crossover and mutation operators on parents for generating offspring.Step 5. Sort parents and offspring using Non-dominated Sorting (NS) Algorithm and calculate Crowding Distance for each of them.Step 6. Select best individuals (in rank and crowding distance) from existing population and offspring and generate new generation.Step 7. Consider termination condition and back to step 3 if the condition is not satisfied.END

Non-dominated ranking genetic algorithms (NRGA)

In this part, a second popular MOEA called NRGA has beenused to obtain Pareto Fronts.Al Jadaan et al. (2009) presentedNRGA by transforming the NSGA-II selection strategy fromthe Tournament selection to the Roulette Wheel selection.As it evident, NRGA works similar to NSGA-II, except in

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their selection mechanism to choose the parents and copyingthem in the mating pool. More specifically, it combines anRBRW,22 selection operatorwith a PBPRA23 inwhich one ofthe fronts is first selected applying the based Roulette Wheelselection operator. Then, one solution within the candidatefront set would be selected by the same procedure. Thus, thehighest possibility to be chosen is for the set of first front, thesolutions within a set of the second front could be selectedwith lower possibility, and so on. The difference betweenthis algorithm and NSGA-II is in their member selection.According to this modification, the possibility of selecting amember like i from the population is equal to Pi and couldbe calculated by Formula (19).

Pi = 2 ∗ Ranki

n Pop ∗ (n Pop + 1)(19)

Note that N is the population size, and Ranki is the i th mem-ber rank in the population. Thus the pseudo code for NRGAis similar to the NSGA-II except in member selection whichfollows Formula (19).

Pareto envelope-based selection algorithm (PESA-II)

To make NSGA-II faster and to mitigate its complexity,Corne et al. (2001) presented an algorithm called PESA-II.To use it as one of the benchmarks, an extra memory whichsaves the iterations’ best solutions as an archive ought tobe predicted in addition to the main population. Reachingsuch an issue, it makes objective function space reticulatedand devotes a number to every container place that is equalwith the existing population size in that place. Thus, thereis no calculate to Rank and Crowding Distance for everymember in the sort and only a number devoted to places iscontained in the archive population. This reduces calculationload and makes the algorithm operation faster. The pseudocode for PESA-II is as follow (Note that Popt donates maingeneration in iteration t, and Arch represents the archivepopulation):

Procedure: Typical PESA-IIBegin

Step 1. Randomly generate Pop0 and put Arch = {}.Step 2. Segregate the non-dominated individuals andsave them in Arch.Step 3. Regionize the explored goal space and devote aprobability value to each region.Step 4. Eliminate redundant individuals according to thereagioning and result state.Step 5. Consider termination condition.

22 Ranked-Based Roulette Wheel23 Pareto-Based Population-Ranking Algorithm.

Step 6. Select parents from Arch.Step 7. Generate offspring by mutation and crossoveroperators.Step 8. Assess new population.Step 9. Return to Step 2.

END

Characteristics of the algorithms

In this section, common characteristics for deployed algo-rithms as their Parameters Calibration, Initial PopulationGeneration, Selection, Crossover, Mutation, and Termina-tion Condition are going to be considered.

Putting the algorithms into practice needs to generate astochastic vector. This vector renders the problem chromo-someswith amaximum length equal to the problemvariables.These stochastic numbers have four portions. The 1st and 2ndare Boolean variables each of which defines the active DCsand suppliers between existing ones. The 3rd and 4th partswould be generated by preliminary parts (1st and 2nd) anddefine X and Y variables. Note that each X and Y are threedimensional Boolean variables. (X dimensions = number ofDCs× number of retailers× number of commodities). Thusit is necessary to generate a stochastic number series withthe size of “number of retailers×number of commodities”from active DCs set (exposing X) and also a stochastic num-ber series with size of “number of active DCs×number ofcommodities” from the active suppliers set (exposing Y), torepresent them by the vector.

Representation of the chromosomes—In order toembody each solution as a chromosome, one binary vectoris used for integer-valued variables. Let’s suppose that thenumber of potential DCs, the number of potential suppliers,the number of retailers, and the number of commodities are3, 4, 2, and 2, respectively. Figure 2 presents a generatedchromosome with the mentioned method.

As Fig. 2 shows, the generated chromosome contains 4parts. In first part the DCs 1 and 3 are both active and DC2 is null. The second part also indicates that suppliers 1, 2,and 4 are a set of active suppliers. The 3rd portion of thechromosome contains a stochastic chain between active DCs(1 and 3) the length of which is equal to “number of retail-ers×number of commodities”. This section is divided intosubsets (number of retailers) per se, and the length of eachoneis equal to the number of commodities. This section showsthat the 1st retailer delivers the 1st commodity from the 3rdDC and the 2nd commodity from the 1st DC. Also, the 2ndretailer delivers both the 1st and 2nd commodities from the1st DC. The last portion of the chromosome has a stochasticchain between active suppliers (i.e. 1, 2, and 4) the lengthof which is equal to “active DCs×number of commodities”.This section is also divided into subsets (coincide with num-

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Fig. 2 A case in chromosome representation

ber of DCs) the length of each one being equal to the numberof commodities. This section shows that the 1st active DC(number 1) procures the 1st commodity from the 1st supplierand the 2nd commodity from the 4th supplier, and also the2nd active DC (number 3) procures both commodities fromthe 2nd supplier (Note that the length of this section variesaccording to the number of active DCs).

Initial population generation—The first populationwould be generated according to the described representa-tion procedure.

Selection operator—This operator acts as Tournament(Haupt and Haupt 2004) for NSGA-II and PESA-II, andRoulette Wheel (Haupt and Haupt 2004) for NRGA.

Crossover operator—For this operator, three methodsof Single-Point Crossover, Two-point Crossover and Uni-form Crossover are considered as the possible operations foroptimizer algorithms. In crossover each part of the parentchromosome is combined with the same one in another par-ent chromosome. Note that it is possible to be extended asthough in a specific portion, multiple portions, or even allparts of the parents’ chromosome. It is also possible thatan offspring chromosome be an unfeasible one, since its 3rdand 4th parts are generated by 1st and 2nd parts. Overcomingsuch a problem, the repairing procedure comes in handy. Inother words, the genes in the 3rd and 4th parts each of whichmay cause infeasible offspring chromosomes could be sub-stituted by stochastic numbers between active DCs (in 3rd)and between active suppliers (in 4th) (Fig. 3). The repairingmechanism to make them feasible is the path that GAs over-come to face this problem. But repairing does not necessarilyproduce a unique sequence

Mutation operator—This operator like crossover mayaffect one or more portions of a chromosome. For the 1stand 2nd which are Boolean, stochastically a specific per-centage of the genes would be selected and changed (nullto active or active to null) (Haupt and Haupt 2004). For the3rd and 4th portion presumably a specific percentage mightbe changed but the stochastically selected gene(s) could besubstituted with an active set of DCs (for 3rd or active set ofsuppliers (for 4th). Note that like crossover, after operation,the produced chromosome might be an infeasible one thatrequires the mentioned repairing process.

Termination condition—As it is clear by its name, it deter-mines the circumstance for ceasing iteration. For appliedalgorithms, the termination condition in this paper is a spe-cific number of iterations dictated by the calibration methodexplained in section “The experiments”.

Comparison measures

The non-dominated results could be compared throughdifferent criteria as CPU Time, Ratio of Non-dominatedIndividuals (RNI), Uniformly Distribution (UD), Diversity,Coverage of Two Set (C), and Quality Metric (QM) each ofwhich are explained beneath:

CPU time—It induces the processing time of each algo-rithm, and the lower value for this criterion is known to bethe better.

Ratio of non-dominated individuals (RNI)—The RNIcriterion (Tan et al. 2002), determines the ratio of non-dominatedmember numbers to the total population [Eq. (20)](n =number of identified points in Pareto Front (P FK nown),

n Pop=number of population).

RN I = n

n Pop(20)

Clearly for this criterion, the greater RNI value approachingthe unit (i.e. 1), the better.

Uniformly Distribution (UD) of Pareto Front—The Uni-formly Distribution of Pareto Front could be calculated bySchott’s Spacing (SS) Metric (Knowles and Corne 2002).[Formula (21)]

SS =√√√√ 1

n − 1

n∑i=1

(d̄ − di

)2

di = minj

⎛⎝

nObj∑k=1

∣∣∣ f ik − f j

k

∣∣∣⎞⎠ ; ∀i, j = 1, 2, . . . n, i �= j

d̄ = 1

n

n∑i=1

di (21)

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Fig. 3 Crossover operators and repair procedure for the presented chromosome

where:

f ik = kth value of objective function in ith PFK nown Point

nObj =number of objective

TheSchott’s Spacing (SS)Metric is an inverse relationshipwith Uniformly Distribution [Formula (22)]. Clearly, for thiscriterion, the higher Uniformly Distribution value, the moreutility.

U D = 1

1 + SS(22)

Diversity of Pareto Front—The Diversity of Pareto Frontsolutions could be calculated using Formula (23) and ofcourse the bigger value for this criterion, the better the utility.

Diversity =nObj∑k=1

maxi, j

∣∣∣ f ik − f j

k

∣∣∣ ; ∀i, j = 1, 2, . . . , n

(23)

Coverage of Two Set (C)—Coverage of Two Set (Zitzler andThiele 1999) identified the domination of two sets pointsagainst each other. In other words, it measures how many ofY set is dominated by X set (C(X, Y)) and vice versa.

Quality Metric (QM)—The quality of caught solutionscould be considerable by this criterion (Rabiee et al. 2012).Measuring requires combining all obtained results by triplealgorithms, so that with a complete comparison betweenthem, the global non-dominated points (signed as set of PT ∗)could to be determined. The QMvalue is equal to the num-ber of each algorithm’s non-dominated results (entitled to

be members of PT ∗), divided by the total number of non-dominated results of this specific algorithm as Formula (24):

QMl = |PTl ∈ PT ∗||PTl | (24)

The experiments

As an instance and for the experiment, the problem size isconsidered though that the number of the potential suppli-ers is 7, the number of DCs is 15, the number of retailers is20, and the diversity of commodities is 6. Likewise for theproblem parameters, the average daily demand from the i thretailer for the kth commodity, the variance daily demandfrom the i th retailer for the kth commodity, the j th ware-house holding cost for the kth commodity, the unit cost ofthe kth commodity shipping from the j thwarehouse to the i thretailer, the cost of procuring and stocking and shipping thekth commodity from the mth supplier to the j th warehouse,and the lead-time for the kth commodity from the mth sup-plier to the j th warehouse are randomly generated based onuniform distributions in their corresponding ranges for theproblem instance and are given in Table 3. Whilst the otherparameters as the j th warehouse opening fixed cost, the j thwarehouse ordering cost for the kth commodity, the potentialcapacity of the jthwarehouse, the mth supplier fixed cost tobe selected/accept to procure, themaximum possible numberof supplier, the maximum possible number of warehouses,and the number of working-day per year are supposed to beconstantly valued as provided in Table 4. Likewise in thispaper α is considered 0.95, (hence Z1−α is 1.96).

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Table 3 Variant parameters for the experiment

Parameters Distribution Dimension

μki U[70–120] Unit

Vki U[10–25] Unit

hkj U[70–90] $

tckji U[10–15] $

rckmj U[65–80] $

lkmj U[2–3] Day

Table 4 Constant parameters for the experiments

Parameters Constant value Dimension

Fj 650 $

Akj 5$ $

wj 750 Unit

sm 500 Monthly

R 50 Unit

N 25 Unit

β 220 Day

Parameters control using Taguchi Method

There are various methods to calibrate the meta-heuristicalgorithmparameters, someofwhich are full factorial design,i.e. they examine all possible combinations (Ruiz et al. 2006;Montgomery 2008), which is intrinsically time and cost con-suming. The Taguchi method (Taguchi 1986) uses a specialdesign of orthogonal arrays to study the entire parametersspace with a small number of experiments. The Taguchimethod clusters the factors into two main groups: control-lable and noise factors (uncontrollable). Since noise factorsare uncontrollable and their elimination is unpractical andalmost impossible, the Taguchi method tries to reach the bestcontrollable factors level from the robustness point of view.Furthermore, Taguchi establishes the relative importance foreach factor with respect to its main impacts on the objectivefunction (Alavidoost et al. 2014; Naderi et al. 2009). In otherwords, the Taguchi relates controllable factors and responsevariables and decides accordingly.

In contrast with single-objective problems that the objec-tive function should be selected as a response variable, inmulti-objective problems the relation between “algorithmassessment indexes” and “response variable” have to be iden-tified. The Weighted Sum is a simple method for attainingsuch an issue. In this paper calculation of the response vari-able would be achieved through Formula (25). Note that inthis paper, for each algorithm, a couple values of C (C1, C2)

are devoted since each algorithm is compared with two oth-ers.

Response = w1 × CPU Time + w2 × RN I + w3

×U D + w4 × QM + w5 × C1 + w6 × C2

where:7∑

q=1

wq = 1, w1 = w2 = w3 = w4 = w5 = w6

(25)

Note that each index should be changed into a digit with nodimension and after that, it could be used in Formula (25).For changing indexes to a digitwith no dimension theRelatedDeviation Index (RDI) (Cardona-Valdés et al. 2014) has beenused [Formula (26)].

RDI = |Algsol − Bestsol |Maxsol − Minsol

(26)

To analyze the experimental data and find the optimal factorcombination, the Taguchi method uses two criteria entitledS/N 24 ratio and Means. The S/N ratio should be maximizedregardless of the types of objective functions, whilst for themeans, the type of objective function is important and RDIhave been taken for undimensioning and clearly, the lowervalue for RDI, the better. Summing up, a level for the para-meters should be selected and in this level, the S/N ratio hasthe maximum value while means has the minimum valuein comparison with the other levels, and just in case thesecriteria are not satisfied for a level simultaneously, anotherexperiment for that specific parameter should be designed(Alavidoost et al. 2014). For each algorithm, according totheir characteristics and attributes, the controllable parame-ters and their levels are identified and presented in Table 5.Note that for PESA-II, the numbers of archives are supposedto be equal to the number of population, and also mutationrate and crossover rate in this algorithm are correlated andtheir summation is equal to 1.

Implementing the Taguchi method has come in actionusing Minitab17. The Taguchi experiments for each of thesethree algorithms give S/N ratio and mean criteria for NSGA-II,NRGA, andPESA-II separately and are exposed in Figs. 4,5, and 6, respectively.

As Fig. 5 shows, the 3rd level for nPopB and βB, and1st level for ItrB might be selected as well, but supplemen-tary experiments would be needed for determining γ B. Thesame analysis for Figs. 4 and 6 could be carried out andthus relying these exposed diagrams and after complimen-tary experiments, the offered levels by the Taguchi methodare presented as Table 6.

24 Signal-to-Noise.

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Table 5 Parameters and theirlevels

Algorithm Parameter Symbol Level

NSGA-II Population size nPopA nPopA(1):100, nPopA(2):200, nPopA(3):300

Number of iteration ItrA ItrA(1):50, ItrA(2):100, ItrA(3):200

Crossover rate β A β A(1):0.7, β A(2):0.8, β A(3):0.9

Number of grid γ A γ A(1):0.1, γ A(2): 0.2, γ A(3):0.3

NRGA Population size nPopB nPopB(1):100, nPopB(2):200, nPopB(3):300

Number of iteration ItrB ItrB(1):50, ItrB(2):100, ItrB(3):200

Crossover rate β B β B(1):0.7, β B(2):0.8, β B(3):0.9

Number of grid γ B γ B(1):0.1, γ B(2): 0.2, γ B(3):0.3

PESA-II Population size nPopC nPopC(1):100, nPopC(2):200, nPopC(3):300

Number of iteration ItrC ItrC(1):50, ItrC(2):100, ItrC(3):200

Crossover rate βC βC(1):0.7, βC(2):0.8, βC(3):0.9

Number of grid nGrid nGrid(1):10, nGrid(2):20, nGrid(3):30

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nPopA

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n of

Mea

ns

ItrA A A

Main Effects Plot for MeansData Means

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-31,0

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nPopA

Mea

n of

SN

ratio

s

ItrA A A

Main Effects Plot for SN ratiosData Means

Fig. 4 Mean of means and S/N ratios for each parameter in NSGA-II

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SN

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s

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Main Effects Plot for SN ratiosData Means

Fig. 5 Mean of means and S/N ratios for each parameter in NRGA

Results, comparisons, and discussion

The obtained results by codedMO-algorithms are going to beexplained and exposed as follows. First the emerged ParetoFronts by triple algorithm would be presented and comparedtogether and after that they would be assessed using formerly

mentioned indexes. The produced results by the triple algo-rithms are reported in Table 7.

Emerged Pareto Fronts in a specific run for three appliedalgorithms are also exposed as Fig. 7 for comparison alto-gether.

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321

60

58

56

54

52

50

48

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Mea

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Mea

ns

ItrC C nGrid

Main Effects Plot for MeansData Means

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-34,5

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-35,5321 321 321

nPopC

Mea

n of

SN

ratio

s

ItrC C nGrid

Main Effects Plot for SN ratiosData Means

Fig. 6 Mean of means and S/N ratios for each parameter in PESA-II

Table 6 Selected level for the parameters

Algorithm Population size Iteration max. number Crossover rate Mutation rate Number of grids

NSGA-II nPopA(2): 200 ItrA(3): 200 β A(1) : 0.7 γ A(2) : 0.2 –

NRGA nPopB(3): 300 ItrB(1): 50 β B(3) : 0.9 γ B(1) : 0.1 –

PESA-II nPopC(1): 100 ItrC(2): 100 βC(3) : 0.9 – nGrid(2) : 20

Table 7 Generated Pareto Frontby NSGA-II, NRGA, andPESA-II

NSGA-II NRGA PESA-II

Objective 1 Objective 2 Objective 1 Objective 2 Objective 1 Objective 2

1806700277 0.893723435 1798404908 0.862162458 1836537237 0.903223488

1817700143 0.91895891 1812190804 0.907168669 1832596523 0.888383488

1815086446 0.916845219 1809455489 0.904796991 1836042362 0.901883488

1812962041 0.91507546 1807367822 0.903177609 1832915524 0.890110154

1809167459 0.906855776 1803871662 0.894375386 1834310488 0.896596821

1807309547 0.898491178 1798623826 0.869228559 1833880556 0.894983488

1809134830 0.902356196 1811142390 0.90520572 1833952602 0.895016821

1806947832 0.894517142 1805429881 0.898086796 1833066466 0.890550154

1808751217 0.901368073 1806596071 0.901390496 1835534444 0.900410154

1812025359 0.912787881 1800059506 0.882668827 1835127435 0.899216821

1810989221 0.910794976 1799502702 0.879991122 1835974427 0.901783488

1812208928 0.913016385 1801589556 0.887161726 1836721074 0.903796821

1808484181 0.898686334 1805692368 0.899253552 1832769512 0.889410154

1810597436 0.909261244 1798664967 0.873815143 1837205008 0.904750154

1809912639 0.907711339 1799359033 0.876954044 1835911693 0.901563488

1810259319 0.908028198 1802425467 0.887885272 1832560943 0.888376828

1808163432 0.898634074 1802851699 0.889332768 1836069655 0.901976821

1810952468 0.910383505 1803094518 0.893059102 1836059749 0.901943488

1806700277 0.893723435 1798404908 0.862162458 1836537237 0.903223488

As Fig. 7 presents, for NRGA and NSGA-II, producebetter results rather than PESA-II. In continue, they wouldbe compared together using indexes mentioned in section“Comparison measures”.

The comparison results for the three applied algorithmsare presented in Figs. 8, 9, 10, 11, 12, and 13 as follow:

Now and after applying the triple evolutionary calibratedalgorithms, it is the time to have an analysis and explanation

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0.85

0.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

1.79E+09 1.8E+09 1.81E+09 1.82E+09 1.83E+09 1.84E+09

Obj

ectiv

e 02

Objective 01

PESAII NSGAII NRGA

Fig. 7 Generated Pareto Front by triplex algorithms

on the experimental results. Thus, a comparison on the algo-rithm run time is exposed in Fig. 8. NSGA-II and NRGACPU Time are almost the same, whilst PESA-II needs lessprocessing time rather than the others (almost two-thirds incomparison with the others). Thus PESA-II in comparisonwith the others caught better rank with this index.

Invoking RNI criterion, Fig. 9 reveals the same values forNSGA-II and PESA-II each of which are much more than

what was reported for NRGA. Accordingly by this index,NRGA operates weaker than the others since provide fewernumber off non-dominated results. In another comparisonpresented in Fig. 10, based upon the Uniform Distributionof the Pareto Front, one can clearly find that PESA-II hasthe highest value that is followed by NRGA, and the last isNSGA-II of course. Also, based upon this measure, PESA-IIand NRGA are almost near and they have significantly betterposition than NSGA-II.

Figure 11 considers theSolutionDiversities for the appliedalgorithm and as exposed there is no significant differentia-tion between them.

QM for non-dominated solutions is presented in Fig. 12for each one. It indicates that in this criterion, NRGA pos-sesses the highest value, and then comesNSGA-II after it thatcatches the 2nd order; thus it is clear that PESA-II comes last,that proofs better position for NRGA.

Last but not least, the Coverage of Two Set (C) index ispresented in Fig. 13. As a case, the upper corner graph on theleft hand side shows C(NSGA-II, NSGA-II), C(NSGA-II,NRGA), and C(NSGA-II, PESA-II). Note that this criterion

0

100

200

300

400

500

600

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CPU

TIM

E (S

EC

)

RUN NO.

NSGA-II NRGA PESA-II

0

100

200

300

400

500

600

NSGA-II NRGA PESA-II

Outlier

Fig. 8 Line plot and box plot for CPU time

0

0.2

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1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RA

TIO

OF

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-DO

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AT

ED

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IVID

UA

LS

(RN

I)

RUN NO.

NSGA-II NRGA PESA-II

0

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0.6

0.8

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1.2

NSGA-II NRGA PESA-II

Outlier

Fig. 9 Line plot and box plot for RNI

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IFO

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D)

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1

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NSGA-II NRGA PESA-II

Outlier

Fig. 10 Line plot and box plot for UD

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Fig. 11 Line plot and box plot for diversity

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Fig. 12 Line plot and box plot for QM

in a case that an algorithm is compared by itself turns nullvalue (C(NSGA-II, NSGA-II)=0). All in all, one could judgeand extract this statement that in terms of a criterion like C,NRGA has the best condition, the circumstance that is fol-lowed by NSGA-II, while PESA-II catches the lowest rank.Summing up, in general it could be alleged that PESA-II havelover convergence rate and faster than the two others, whilstNRGA turns better results from quality point of view.

Concluding remarks and implications for futureworks

In this innovative paper, amathematical formulation has beendeveloped for a supposed tri-echelon supply chain networkas a MINLP. Since the proposed model of this paper washard to be resolved analytically or with the exact methods,three ones of MOGAs called the Non-dominated Sort-

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Fig. 13 Box plots of coverage of two set

ing Genetic Algorithm (NSGA-II), Non-dominated RankingGenetic Algorithm (NRGA), and Pareto Envelope-basedSelection Algorithm (PESA-II) have been applied and com-pared to validate the obtained results. In each applied triplemulti-objective algorithms, the Taguchi method has been uti-lized for calibrating and controlling their parameters. Afterthe parameter calibration process, they have been deployedto solve the problem. The comparison results in the end haveshown that although NSGA-II and NRGA algorithms havealmost the samemeasured results by somecriteria, thePESA-II does not significantly act as well as the two others butclearly it has better CPU time than the others.

Several recommendations could be kept inmind for futurestudies as follow. Four main issues to be mentioned are:

1. Considering the problem under extra constraint such asshortage costs, discounts or inflation, non-perfect suppli-ers and DCs, suppliers and DCs set-up time.

2. Utilizing other meta-heuristics such as MOGA, MOSA,MOPSO, andMOHS to solve the problemand comparingtheir performances.

3. Using other GA operators for mutation and crossover.4. Invoking queuing models as a hybridized portion for the

network and also considering some of the intake parame-ters as fuzzy numbers.

Acknowledgments The authors are thankful for the time and consid-eration the anonymous reviewer spent in this manuscript. Taking careof the comments significantly improved the presentation.

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