integrated production and distribution scheduling for...

Scientia Iranica E (2017) 24(4), 2105{2118

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

Integrated production and distribution scheduling forperishable products

F. Marandi and S.H. Zegordi�

Department of Industrial Engineering, Tarbiat Modares University, Tehran, Iran.

Received 20 October 2015; received in revised form 18 April 2016; accepted 5 December 2016

KEYWORDSProduction anddistribution;Permutation ow shopscheduling;Vehicle routingproblem;Integration;Mixed integerprogramming;Particle swarmoptimization.

Abstract. This study is concerned with how the quality of perishable products canbe improved by shortening the time interval between production and distribution. Sincespecial types of food, such as dairy products, decay fast, the Integration of Productionand Distribution Scheduling (IPDS), is investigated. This article deals with a variationof IPDS that contains a short shelf life product; hence, there is no inventory of theproduct in the process. Once a speci�c amount of the product is produced, it must betransported with the least transportation time directly to various customer positions withinits limited lifespans to minimize the delivery and tardy costs required to complete producingand distributing of the product to satisfy the demand of customers within the limiteddeadline. After developing a mixed-integer nonlinear programming model of the problem,because it is NP-hard, an Improved Particle Swarm Optimization (IPSO) is proposed.IPSO performance is compared with commercial optimization software for small-size andmoderate-size problems. For large-size ones, it is compared with the genetic algorithmexisting in the literature. Computational experiments show the e�ciency and e�ectivenessof the proposed IPSO in terms of both the quality of the solution and the time of achievingthe best solution.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Production and distribution operations are two impor-tant operational functions in a supply chain. To achieveoptimal operational performance in a supply chain, it issigni�cant to integrate these two functions and schedulethem jointly in a coordinated manner. However,most of the proposed integrated and synchronizedapproaches focus on the tactical decision level of supplychains [1,2]. In recent years, integrated schedulinghas attracted much interest among researchers. Incontrast to classical scheduling, this type of schedul-ing problem involves not only the production partbut also distribution. The objective of integratedscheduling is to obtain a simultaneous optimization

*. Corresponding author. Tel.: +98 21 82883394E-mail address: [email protected] (S.H. Zegordi)

of both parts. The obtained procedure will becomea detailed schedule that provides an e�cient solutionfor operation management. In this paper, a class ofintegrated scheduling problems is considered, whichincludes production and distribution.

In many applications involving make-to-order ortime-sensitive (e.g., perishable) products, �nished or-ders are often delivered to customers immediatelyor shortly after the production to restrict qualityreduction. In such a supply chain, product quality isdetermined not only by the production processes butalso through the coordination of the production anddistribution decisions. In this situation, the deliveryof the products must be done within a strictly limitedtime after their production. The non-inventory pro-duction and transportation problem is routine in manyindustries, in which a time-sensitive product cannot bein storage due to its short shelf life. The delivery of

2106 F. Marandi and S.H. Zegordi/Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 2105{2118

the product must be made within a tightly limited timeafter its production. Therefore, the production and dis-tribution operations must be highly integrated. Whenthe production plant has a limited production rateand the transportation time is not instantaneous, anyine�ciency in the integrated schedule may either causethe product to expire before it reaches the customers orcause the delivery not to satisfy a customer's deliverydeadline. Then, the production and distribution oper-ations must be highly linked and integrated becauseany ine�ciency in the integrated schedule causes adecrease in product quality, expiration before delivery,extra expenses, and lack of customer satisfaction. Toease this coordination, production sites are usuallydirectly connected to customers by a eet of vehicles [3].Consequently, there is little or no �nished productinventory in the supply chain such that production andoutbound distribution are very closely linked, whichmust be scheduled jointly in order to achieve a desiredon-time delivery performance at a minimum total cost.However, the analysis of practices in case studiesshow that, currently, the production and distributionoperations are done separately, which cause operationaland customer dissatisfaction [4]. Thus, this paperinvestigates integrating production and distributiondecisions at a trade-o� among customer satisfaction,quality of delivered products, and total costs. Researchon integrated scheduling models of production anddistribution is relatively new, but it is growing veryrapidly.

This study was motivated by a practical schedul-ing problem encountered by a leading manufacturerof various industries, in which a limited number ofvehicles were available, where the departure time afterproduction in owshop scheduling was not �xed andneeded to be determined in order to minimize tardy anddelivery costs to satisfy the customers' deadline. Thedi�erences and contributions of this paper comparedwith IPDS's literature can be summarized as follows:

1. A new problem is de�ned. The IntegratedProduction-Distribution Scheduling (IPDS) prob-lem is considered, of which the �rst stage con-tains permutation ow shop scheduling and thesecond stage involves the distribution problem thatrequires designing vehicle routes for picking up�nished goods and delivering them from the manu-facturer to customers;

2. A new mixed integer nonlinear programming modelis developed, which simultaneously considers bothproduction and distribution scheduling in an inte-grated manner. The exact solution to the problemis provided by solving the model. This model ispractical in large scales for real cases, such as thedairy product industry, which is motivated by a casestudy from a dairy manufacturer in Tehran;

3. An Improved Particle Swarm Optimization (IPSO)is proposed to deal with the problem. The improv-ing operator, 1-exchanged and 2-opt, is added toprevent premature convergence;

4. New test problems are created, which can be usedfor future studies.

This paper studies the integrated production and dis-tribution scheduling problem and extends features suchas non-negligible transportation time and delivery con-solidation. Comparison of this problem with previousworks shows the combination of the product's limitedlifespan, machine scheduling decisions, and the vehiclerouting decisions as critical features that make it di�er-ent, which leads to the possibility of expiration beforeit reaches a customer. The problem is complicated bylimited transportation capacity, a customer's demandsize and location, and the departure time that mustbe determined and are not �xed. On the other hand,these complications also make the resulting integratedscheduling problem challenging and interesting. Theparticular variation that is considered involves a singleproduction plant with multiple production machinesin owshop scheduling, a eet of delivery trucks, anda given set of customers at di�erent locations over ageographic region.

The aim of this research is integrating schedulingin a mathematical model, which is developed to in-vestigate the e�ect of integrated production schedulingand distribution decisions on the total costs. Thisis similar to various real-world environments such asfood, dairy products, and chemical industries, whichare highly perishable. Since this problem has an NP-hard structure [5], a meta-heuristic method is used tosolve it. An Improved Particle Swarm Optimization(IPSO) is proposed in this study that improves theoperator, while 1-exchanged and 2-opt is added toprevent premature convergence.

The remainder of this paper is organized asfollows. Section 2 presents results from previous workson integrated supply chain scheduling. In Section 3,the problem is described and the mathematical modelof the problem is presented. In Section 4, an improvedparticle swarm optimization is proposed to solve theproblem. In Section 5, the experimental results ofthe proposed IPSO and managerial implications arepresented. At the end, Section 6 elaborates concludingremarks and some suggestions for future research.

2. Literature review

In this section, the literature that explicitly involvesboth production and transportation activities in in-tegrated supply chain scheduling is presented. Inte-grating production and outbound delivery scheduling isvery critical and common in the supply chain of time-

F. Marandi and S.H. Zegordi/Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 2105{2118 2107

sensitive products [6,7]. For example, Buer et al. [8]focus on the newspapers printing and distribution, andmail processing and distribution provided by Wanget al. [9], and industrial adhesive materials produc-tion and delivery provided by Devapriya et al. [10].Therefore, how to e�ectively integrate the productionand delivery stages at the operational level so as todecrease the operational costs and improve customerservice becomes very important for the success of acompany. However, most of the existing models onthe production-distribution scheduling problems onlystudy strategic or tactical levels of decisions, andvery few have addressed integrated decisions at theoperational level. Chandra and Fisher [11] emphasizethe need for studying these integrated scheduling issuesat the operational level. They consider an integratedscheduling problem where a plant produces and storesthe products until they are delivered to the customersby a eet of trucks. They provide two solutions.The �rst solution solves the production scheduling andvehicle routing problems, separately, but the secondone solves the problem in a coordinated manner. Theircomputational results show that the total operatingcost decreases in the coordinated one. Chen andVairaktarakis [12] and Pundoor and Chen [13] alsoshow that there is signi�cant bene�t by using theoptimally integrated production-distribution schedulecompared to the schedule generated by a separate andsequential scheduling approach in the context of themodels that are considered in their research. Theyemphasize that integration is a prior approach toimprove the total performance in decreasing costs thatis investigated in this paper.

Chen [6,14] reviews papers that deal with in-tegrated production scheduling and outbound distri-bution. He focuses on two main areas of directdelivery (one destination) and vehicle routing (multidestinations). He also provides a survey of modelsand results in the area of integrated scheduling ofproduction and distribution. He presents a uni�edmodel representation scheme, classi�es existing modelsinto several di�erent classes, and, for each class of themodels, gives an overview of the optimality properties,computational tractability, and solution algorithmsfor the various problems studied in the literature.In his survey on the relevant literature, he statesthat although there is a huge amount of research onintegrated production and distribution models, theneed for focusing on more studies in this realm isrequired. In his research, he notes a gap in theconsideration of a due date and distribution by vehiclerouting for practical situations. Lee and Chen [15]study the machine scheduling problems with explicittransportation considerations. They identify two typesof transportation situations in their models. The �rsttype involves transporting a semi-�nished job from one

machine to another for further processing. The secondtype involves transporting a �nished job to the cus-tomer or warehouse. Both transportation capacity andtransportation time are taken into account explicitly intheir model. They classify the computational complex-ity of various scheduling problems by either provingtheir NP-hardness or providing polynomial algorithms.Chang and Lee [5] consider an extension of Lee andChen's work where each job is assumed to occupy adi�erent amount of storage space in the vehicle duringdelivery. They show the problem that jointly considersproduction and delivery with the consideration thateach job may require a di�erent amount of spaceduring intractable transport and provide heuristics forsome cases of the problem. Zhong et al. [16] studythe similar problem as the one studied by [5] withthe objective of minimizing the makespan. In thepractical cases, focusing on the type of transportation,Hajiaghaei-Keshteli and Aminnayeri [17] present inte-grated production and rail transportation schedulingby emphasizing the application of rail distribution.Saidi-Mehrabad et al. [18] present a new integratedmodel in scheduling and routing by automated guidedvehicles that focus on new methods of transportation.In some research, the types of products are moreimportant than the types of transportation in practicalmodels investigated in integrated modelling. Chen etal. [19] and Farahani et al. [3] propose a production anddistribution scheduling for perishable food products.Liu et al. [20] present an integrated scheduling toimprove the operation of production and delivery inready-mixed concrete plants. Algorithms for solvingthese problems include approximation algorithms [21]and intelligent algorithms [4,22,23]. As mentioned, thetype of product and transportation is important inpractical cases and industries that is emphasized in thispaper, as follows.

To the best of our knowledge, it seems thatno research considers both permutation ow shopscheduling with due date and tardy cost in the pro-duction stage, and vehicle routing with considerationof deadline to satisfy customers. Few studies considerthe transportation eet in their models. This study isextended by assuming that the �rst stage of the supplychain is composed of m machines as the ow shopenvironment in the production stage, and v vehiclesinvolved in designing vehicle routes for picking up anddelivering �nished goods in the second stage. It can bepractical in real cases to serve a number of customersin various geographical zones.

3. Problem description

3.1. AssumptionsAs mentioned previously, the scheduling of productsand vehicles in a two-stage supply chain is investigated


here. The �rst stage contains permutation ow shopscheduling with m machines and n jobs, and thesecond stage is composed of v vehicles with di�erentspeeds and transportation capacities that transport njobs from the manufacturing company to c customersdistributed in various geographical zones. All vehiclesare available upon completion of all jobs (i.e., Cmax).

The problem is described as follows. There is a setof jobs (N = f1; 2; � � � ; ng) to be processed by a set ofmachines (M = f1; 2; � � � ;mg) at the production stage,which is permutation ow shop scheduling. All jobshave to be processed in an identical order on a given setof machines. The processing of each job is continuousonce the processing begins. After processing, the�nished jobs need to be delivered by a set of vehicles,V = f1; 2; � � � ; vg, which has a distinct capacity, Qk,to a set of customers, C = f1; 2; � � � ; cg, that aims to�nd a set of tours for several vehicles from a depot(i.e. manufacturer site) to a lot of customers. Theyreturn to the depot without exceeding capacity anddeadline constraints. Each customer is visited onlyonce by a single vehicle. Delay is not allowed, i.e.the jobs have to be delivered to the customers beforethe deadline. Moreover, when the last job terminates(i.e., makespan) after the due date, it will incur thetardy cost. The objective is to �nd an integratedproduction and distribution to minimize the sum ofjobs' delivery cost and tardy cost that refers to the duedate violation.

At the beginning of the horizon, customers re-quire a set of jobs and send the requirements to themanufacturer. At the subsequent delivery stage, a eet of capacitated vehicles delivers the �nished jobsto the pre-speci�ed customers. The capacity of thetransporter is measured by a certain volume. Eachcustomer is visited only once by a single vehicle. Onlyone vehicle is allowed to visit each customer.

3.2. Mathematical modelA mathematical formulation of the proposed IntegratedProduction-Distribution Scheduling (IPDS) problemis presented in this section. The parameters are asfollows:n Number of jobsm Number of machinesc Number of customersv Number of vehiclesi Job index, i = 1; 2; � � � ; np Job position index, p = 1; 2; � � � ; nr Machine index, r = 1; 2; � � � ;mj; l Customer indexk Vehicle indexvk Speed of vehicle k

Tri Operation time of job i on machine rdjl Distance between customer j and

customer leij Physical space of customer j's demand

iddk Deadline for vehicle k to deliver to

customersQk Capacity of vehicle k for transporting

job from manufacturing company tocustomers

du Due date for production stagecd Transportation cost per unit of

distancecp Violation due date cost in production

stageM A su�ciently large number

Variables of the model are as follows:

Brj Start time of job j on machine ruj Number of customers visited at

customer jrk Ready time of vehicle k representing

the latest completion timecmax Maximum completion time of all jobs

(makespan)zij 1 if job i in position j is processed;

zero otherwisexkjl 1 if customer l is served after customer

j by vehicle k; zero otherwisey 1 if cmax is after due date; zero

otherwise

Miller et al. [24] propose a mathematical pro-gramming formulation to prevent sub-tours in thevehicle routing problem. Kyparisis and Koulamas [25]propose a de�nition to consider the vehicle velocity anddistance instead of time. All the above assumptions areused to develop the following model for the IntegratedProduction-Distribution Scheduling (IPDS) problem:

Min cdvXk=1

cXl=0

cXj=0

dljxklj + cp:y(cmax � du);

nXp=1

Zip = 1 1 � i � n; (1)

nXi=1

Zip = 1 1 � p � n; (2)

B1p +nXi=1

T1iZip = B1;p+1 1 � p � n� 1; (3)

B11 = 0; (4)


Br1 +nXi=1

TriZi1 = Br+1;1 1 � r � m� 1; (5)

Brp +nXi=1

TriZip � Br+1;p

1 � r � m� 1 2 � p � n; (6)

Brp +nXi=1

TriZip � Br;p+1

1 � p � n� 1 2 � r � m; (7)

cmax = Bmn +nXi=1

TmiZin; (8)

vXk=1

cXl=1

xkjl = 1 j = 1; 2; � � � ; c; (9)

vXk=1

cXj=1

xkjl = 1 l = 1; 2; � � � ; c; (10)

cXl=0

xklh �cXj=0

xkhj = 0

k = 1; 2; � � � ; v; h = 1; 2; � � � ; c; (11)

cXj=1

xk0j = 1; k = 1; 2; � � � ; v; (12)

uj + 1 � ul + c(1� xkjl) k = 1; 2; � � � ; v;l = 1; 2; � � � ; c; j = 1; 2; � � � ; c; (13)

vXk=1

cXj=1

xk0j = v; (14)

vXk=1

cXl=1

xkl0 = v; (15)

nXi=1

cXj=0

cXl=0

eijxklj � Qk k = 1; 2; � � � ; v; (16)

rk � cmax k = 1; 2; � � � ; v; (17)

rk +cXl=1

cXj=0

djl=vkxkjl � ddk k = 1; 2; � � � ; v;(18)

cmax � du�M(1� y); (19)

cmax � du+My: (20)

The aim is to minimize the sum of delivery and tardy

costs. Constraint sets (1) and (2) represent the assign-ment that guarantees to assign every job to one positionand every position to one job. Constraint sets (3)-(5)ensure the permutation ow shop scheduling propertyin which there is no idle time on the �rst machineand the �rst job is processed on all machines withoutdelay. Constraint set (6) guarantees that the start ofeach job on machine r + 1 is not earlier than its �nishtime on machine r, which prevents the simultaneousprocessing of jobs on one machine. Constraint set (7)ensures that the job processes in position p + 1 in thesequence do not start until the processing of the job inposition p on that machine is completed. Constraintset (8) represents makespan. Constraint sets (9) and(10) are common constraints for the vehicle routingproblem that causes every customer to be servicedby only one vehicle. Constraint set (11) is the owconservation constraint for each customer and whenthe vehicle serves the customer, it must depart thecurrent customer to service the next ones. Constraintset (12) ensures that every vehicle must serve thecustomer once. Constraint set (13) prevents the subtour that is disconnected to the depot. Constraintsets (14) and (15) con�ne the number of vehicles toserve the customers to v vehicles. Constraint set(16) guarantees that the maximum ow in any arcleaving the root is equal to Qk and the maximumvehicle k capacity is con�ned to Qk. The relation ofthe production and distribution stages is importantin the IPDS problem. Constraint set (17) focuses onintegration of the production and distribution stages,which emphasizes the ready time of vehicles uponcompletion of all jobs. Constraint set (18) focuseson satisfying the deadline to serve the customer inorder to increase customer satisfaction. Constraintsets (19) and (20) determine variable y to get 1 or 0that is dependent on makespan violation of the duedate.

It should be mentioned that as the objectivefunction contains non-linear terms, the mathematicalmodel is a Mixed-Integer Non-Linear Programming(MINLP) model. The model could be solved withcommercial software packages such as LINGO.

Lemma 1. The Integrated Production-DistributionScheduling (IPDS) problem is NP-hard.

Proof. Chang and Lee [5] proved a problem withonly one geographical zone, one manufacturer, and onevehicle as NP-hard. According to [26,27], the ow shopscheduling problem with Cmax minimization objectiveis also NP-hard. As the IPDS problem is an extensionof two NP-hard problems, i.e. the ow shop schedulingand vehicle routing problem, the integrated problemmust be NP-hard in the strong sense. �


NP is not equal to P problems, so there is noalgorithm to �nd global optima in polynomial compu-tational time. Therefore, meta-heuristics or heuristicsmust be applied to solve large-scale problems withinreasonable computation time. After introducing theparticle swarm optimization algorithm, the improvedalgorithm, IPSO, is proposed in Section 4. In Section 5,it is validated by the generated test problems.

4. Meta-heuristic algorithms

Each of the integrated production and distributionscheduling problems is strongly NP-hard [4,5]. There-fore, an Improved Particle Swarm Optimization isproposed to deal with the proposed problem in thisresearch. For validation of IPSO, a genetic algorithmemanated by [4] is applied.

4.1. Improved Particle Swarm Optimization(IPSO)

Various approaches have been proposed to solvescheduling problems. Among them, PSOs have beengreatly adopted during recent years in practical do-mains [28,29]. The Particle Swarm Optimization(PSO) is a population-based optimization algorithmbased on the benchmarking of social interactions suchas bird ocking or �sh school. It is developed by [30]in which the members of the whole population aremaintained through the search procedure, so thatinformation is socially shared among individuals toguide the search towards the best position in the searchspace. PSO is an evolutionary algorithm, which is ini-tialized with a population (swarm) of random solutions(particles) that y in the search space. It searchesfor optima by updating generations. Individuals orpotential solutions are named particles. They y inthe problem space with a velocity dynamically updatedaccording to the ying experiences of every particle andthe whole population.

The PSO algorithm has been successfully used forvarious applications such as neural network training,task assignment, supplier selection, ordering problem,permutation ow shop sequencing problems, lot sizingproblem, and single machine total weighted tardinessproblems [31,32]. The advantages of a PSO algorithmare observable not only in the wide applications, butalso in the simple structures, immediate accessibil-ity for practical applications, ease of implementation,speed to get the solutions, and robustness.

The basic elements of the PSO algorithm aresummarized as follows:

� Particle: Xti is the ith particle at iteration t with

a D-dimensional vector, which is located at Xti =

(xti1; xti2; � � � ; xtiD) in the searching space;� Population: Consists of P particles with D dimen-

sions in the swarm, iteratively;

� Particle velocity: The ith particle's velocityis also a D-dimensional vector as V ti = vti1; vti2;� � � ; vtiD, which is updated iteratively;

� Inertia weight and acceleration coe�cients:! is a parameter to control the impact of previousvelocities on the current velocity. c1 and c2 areconstant parameters called acceleration coe�cientsto control the maximum step size that the particlecan carry out;

� Personal best: The best position of the ithparticle with the best �tness value until iteration t isP ti = (pti1; pti2; � � � ; ptiD) that means the best positionassociated with the best �tness value of the particlesso far. It is updated iteratively for each particle;

� Global best: The best position of the popula-tion obtained so far in the whole swarm is P tg =(ptg; ptg; � � � ; ptg) at iteration t;

� Termination criterion: It de�nes a condition tostop searching the solution space. It is optionalaccording to the max generation or CPU time.

The velocity of each particle based on the currentvelocity, the best experience of the particle, and thatof the entire population are updated as follows:

vti =!:vt�1id +c1:r1(ptid�xtid)+c2:r2(ptgd�xtid): (21)

Eq. (1) describes the velocity of a particle at iterationt. It is determined by the previous velocity of theparticle, the cognition part, and the social part. c1is the cognition factor, c2 is the social learning factor,and ! is inertia weight to control the impact ofprevious velocities on the current velocity, calculatedin Eq. (2). The position of each particle is updated ineach iteration in Eq. (3):

! =2��2� '�p'2 � 4'

�� ;' = '1 + '2 > 4;

c1 = !'1;

c2 = !'2;

xtid = xt�1id + vtid: (22)

In order to control excessive ying of particles outsidethe search space, the velocity values are con�ned to therange [�vmax; vmax].

4.1.1. Proposed IPSO algorithmIn this section, a particle swarm optimization algorithmnamed Improved Particle Swarm Optimization (IPSO)algorithm is proposed. The procedure of the IPSO is


Figure 1. Process ow of proposed IPSO.

shown in Figure 1. For this purpose, among manydi�erent kinds of neighbourhood structures in theliterature, a local improvement procedure, namely (1-exchange, 2-opt), is added to improve the solution'squality. It is randomly selected for each iterationto extend solution search. The following two typesare considered to improve solution quality and widensearch space:

� 1-exchange: Select two tours; then, omit onecustomer in one of two tours and add it to the othertour;

� 2-opt: Select two tours; then, interchange twocustomers of one tour with two of the other.

When a particle is going to stagnate, the improvingoperators are used to search its neighbourhood to pre-vent premature convergence. The improving structurescan in uence quality of the solution. Numerical resultsshow how these procedures prevent the algorithm fromgetting stuck in a local optimum.

4.2. Computational procedureThe complete computational procedure of the IPSO issummarized as follows:

- Step 1: Initialization:� Initialize P particles randomly as a population

with (D = n + c + v � 1)-dimensions; n, c,v, respectively refer to the numbers of jobs,

customers, and vehicles. In the production stage,customers are randomly assigned to a de�nitenumber of v vehicles. There is a capacity violationcost in order to guarantee the vehicle capacityconstraint and time violation cost to satisfy thedeadline constraint by minimizing violation costs.In the distribution stage, the permutation of jobsis considered in the initial population with con-sideration of better �tness value (lower objectivefunction). First, calculate each particle's �tnessvalue of the initialization population and, then,rank them. Choose the smaller one that is P ti .Choose the particle with the best �tness value ofthe whole population as P tg ;

� Set t = 1; generate the position of particlei, X1

i (i = 1; 2; � � � ; p) randomly, where X1i =

(x1i1; x1

i2; � � � ; x1iD) for the whole population;

� Generate velocity of particle i, V 1i (i = 1; 2; � � � ; p)

randomly, where V 1i = v1

i1; v1i2; � � � ; v1

iD for thewhole population;

� Compute the performance measurement, i.e. thetotal delivery and tardy cost, and set it as the�tness value f1

i of x1i (i = 1; 2; � � � ; p);

� For each particle in the swarm, P 1i = (p1

i1 =x1i1; p1

i2 = x1i2; � � � ; p1

iD = x1iD) for i = 1; 2; � � � ; p;

� Find the best �tness value in the population P 1g =

min(f1i ) for i = 1; 2; � � � ; p.

- Step 2: Update iteration:

� t = t+ 1.

- Step 3: Apply improving operator:

� Apply one of two improving operators randomlyat each iteration to search the solution spacewidely and achieve various solutions.

- Step 4: Update P ti and P tg :

� For i = 1; 2; � � � ; p calculate the �tness value; iff ti < P t�1

i , put P ti = f ti ; otherwise, P ti = P t�1i ;

� For i = 1; 2; � � � ; p calculate �tness value; �nd theminimum �tness value in the swarm; if min(f ti ) <P (t�1)g , put P tg = min(f ti ); otherwise, P tg = P t�1

g .

- Step 5: Update velocity:

� vti = !:vt�1id + c1:r1(ptid � xtid) + c2:r2(ptgd � xtid);

'1 and '2 are equal to 2.05; r1 and r2 are randomnumbers uniformly distributed in [0,1], and c1 andc2 are random numbers in [0,2].

- Step 6: Update position

� xtid = xt�1id + vtid.

- Step 7: Termination criterion

� The termination criterion is determined whenthe number of iterations is achieved by the maxgeneration; otherwise, go to Step 2.


Figure 2. The particle encoded as the number of jobs and customers.

4.3. Solution representation in an illustrativeexample

In an integrated production and distribution schedulingproblem, the information of both problems is stored inone particle instead of de�ning two separate particles.Hence, the encoding scheme contains real numbers.Each particle is consisted to have (n + c + v � 1)dimensions, in which n, c, and v are, respectively, thenumbers of jobs, customers, and vehicles.

In this section, an illustrative example has beenincluded to show the e�ciency of the proposed ap-proach. After completion of jobs (Cmax) in the pro-duction stage, which contains permutation ow shopscheduling, vehicles deliver customers' orders with re-strictions of vehicles capacity and deadline constraint.An instance of the problem with three jobs, eightcustomers, and three vehicles is given in Figure 2 as anexample of the encoding integrated problem. A particlein this encoding system is a feasible sequence of jobsthat are assigned to machines in spite of the sequenceof customers that are assigned to vehicles with respectto vehicle capacity.

5. Computational results

In this section, the computational experiments areperformed to test the performance of the proposedalgorithm. The IPSO algorithm is tested on di�erentscale instances and compared with the recently pro-posed e�cient algorithms. As there are no benchmarksolutions to compare with the solution of the proposedalgorithm, for small-size and moderate-size instances,the IPSO algorithm is compared with the optimalsolution calculated by a commercial optimization soft-ware, LINGO 13.0. For large-size instances, it iscompared with the genetic algorithm of the similarproblem [4]. All the algorithms in this paper arecoded in MATLAB 8.3. They are implemented on acomputer with Intel Core 2Duo 2.5 GHZ and 3 GBRAM.

5.1. Test problems generationTest problems are randomly generated as follows.25 random instances for each problem size are createdbased on the following parameter settings:

� The job processing time on the machine is randomlygenerated from the uniform distribution in the rangeof [1; 99];

� The volumes of di�erent jobs are randomly gener-ated from the uniform distribution in the range of[1; 10];

� The distance between di�erent customers is ran-domly generated from the uniform distribution inthe range of [1; 100];

� The vehicle capacity is randomly generated fromthe uniform distribution in the range of [

Pcj=1Pn

i=1 eij=v,Pcj=1

Pni=1 eij ];

� In this paper, a new and practical range is ap-plied with consideration of velocity and distance,the deadlines of which are randomly generatedfrom the uniform distribution in the range of[�Pmr=1

Pni=1 Tri, �(Tmin + Tmax)=2 + (dmin +

dmax)=2v], where Tmin and Tmax are minimum andmaximum job processing times, respectively, anddmin and dmax are the minimum and maximumdistances, respectively in which � = 0:2 and � =1:5 by numerical experiments. It is di�erent fromPundoor and Chen's [13] research;

� The due dates are randomly generated from theuniform distribution in range of [du(1�R=2), du(1+R=2)] that du =

Pmr=1

Pni=1 Tri(1 � �), R = 0:6,

� = 0:2 [33].

Problems are classi�ed into three groups of small,moderate, and large sizes in order to compare exactprocedures, IPSO, and GA, for each test problem thatis described in the next section.


Table 1. Results of random instances with small sizes.

Instanceno.

Size IPSO LINGOPD

Machine Job Vehicle Customer Value CPUT(s)

Value CPUT(s)

1 2 5 2 6 29493 22 29493 4 02 2 10 3 6 38736 19 38736 17 03 2 15 3 6 39937 20 39937 42 04 5 5 2 6 32967 19 33517 12 05 5 15 2 6 42337 20 42234 60 0.0026 2 5 3 7 30043 21 29993 38 0.0017 2 10 3 7 32586 20 32586 27 08 2 15 3 7 32837 21 32837 51 09 5 5 2 7 29467 19 29467 15 010 5 15 2 7 38101 22 38184 52 011 2 5 3 8 29943 17 32443 23 012 2 10 3 8 39886 17 39886 160 013 2 15 3 8 40154 21 40137 129 0.000414 5 5 2 8 34467 21 34467 107 015 5 15 2 8 43984 17 43834 136 0.00316 2 5 3 9 30143 17 33843 116 017 2 10 3 9 40236 19 40236 160 018 2 15 3 9 40637 20 40487 92 0.00419 5 5 2 9 33417 19 33417 72 020 2 5 3 10 24983 19 24943 59 0.00121 2 10 2 10 34236 16 33936 67 0.00922 2 15 2 10 37187 17 37237 153 023 5 5 2 10 28417 21 28317 37 0.00324 5 10 2 10 34759 20 34759 90 025 5 15 2 10 43034 16 43147 548 0

Average 19.2 83 0.0009

5.2. Validation and veri�cation of the IPSOIn this section, the outcomes of the model in the exactprocedure, IPSO, and GA for each test problem, in theway of randomized block design [34], are reported. Thetables consist of three parts: the objective function,running time (CPUT), and percentage deviation. Theindicator PD (Percentage Deviation) is de�ned asPD = 100 � (IPSOs � LINGOs)=LINGOs to assessthe quality of the obtained solution. It measuresthe amount of improvement in terms of the objectivefunction. In the PD formula, LINGOs and IPSOsrepresent the LINGO solution and IPSO solution,respectively. It should be mentioned that LINGO �ndsthe optimal solution for all small-size instances.

For comparison, the MINLP model is solved withthe mathematical software LINGO 13.0. It is usedto exactly solve the model with small-size instances.Since the complexity of the problem is categorized inthe NP-hard class, only small-size problems can be

solved. All PD percentages and the total average ofthe small-size problems shown in Table 1 are less than0.9%. The average running time of the IPSO is lessthan that of the optimal approach. For moderate-sizeproblems, LINGO returns a local optimal solution ina limitation of 2000 seconds for running time. Table 2shows the solutions of the proposed IPSO algorithm,which are better than the local optimal solutions foundby LINGO. The gap of the local solution is 9.5% onaverage for all moderate-size instances. For moderate-size problems, the PD formula is rewritten as PD =100�max(0; IPSOs� LINGOs)=LINGOs.

It is shown that the IPSO runs much faster thanthe LINGO solver. Although the LINGO solver �ndsthe optimal solution, the computational time of LINGOgrows exponentially as the instance size increases.Moreover, the IPSO cannot obtain optimal solutionsfor moderate sizes, but can �nd locally optimal solu-tions in the limitation time of 2000 seconds.


Table 2. Results of random instances with moderate sizes.

Instanceno.

Size IPSO LINGOPD


Value CPUT(s)

1 2 100 10 10 118481 21 122976 2000 02 2 50 20 10 96522 23 96706 2000 03 2 50 10 20 89117 22 96535 2000 04 5 10 10 100 169617 29 141113 2000 0.205 4 20 10 50 98768 25 135901 2000 06 4 200 10 10 214359 29 218871 2000 07 4 50 40 10 125039 26 122600 2000 0.028 5 100 10 20 150885 26 133666 2000 0.129 5 200 5 20 220857 28 211838 2000 0.0410 6 100 5 40 159283 20 121982 2000 0.3011 7 100 4 50 173507 29 131489 2000 0.2112 7 20 20 50 127811 26 137781 2000 013 7 80 5 50 149301 22 113730 2000 0.3114 7 20 10 100 19727 23 73188 2000 015 6 60 10 50 164775 22 174470 2000 016 5 100 10 30 167996 21 181159 2000 017 6 30 10 100 216093 24 171509 2000 0.2518 5 50 10 60 164422 22 158756 2000 0.0319 5 60 10 50 158729 22 164456 2000 020 6 30 20 50 117182 22 171450 2000 021 3 50 40 15 128993 22 143768 2000 022 4 150 10 20 188416 22 162631 2000 0.1623 5 30 10 100 211993 23 178145 2000 0.1924 6 40 10 75 174108 22 141551 2000 0.2325 6 20 20 75 159411 24 120766 2000 0.32

Average 23.76 2000 0.095

The superiority of the suggested IPSO is ex-amined by analysis of variance and Tukey pairwisecomparisons [34]. The statistical model presented inEq. (4) is the same for both response variables, i.e.objective function and running time. Consequently,Yij can be the objective function or time durationof the ith test problem solved by the jth algorithm(LINGO, IPSO, or GA). The size of the test problemis considered as the block e�ect (pi) in I levels and thealgorithm is determined as the main factor (aj) in Jlevels. The statistical model is as follows:

Yij = �+ pi + aj + "ij ;

i = 1; 2; � � � ; I; j = 1; 2; � � � ; J: (23)

In order to show that the proposed IPSO converges onthe optimal solution in a polynomial time for solvingsmall-size and moderate-size test problems, the resultsof Tables 1 and 2 are utilized. As mentioned before,

the executions are based on a randomized block designwith I = 50 and J = 2.

Tables 3 and 4 report the analysis results ofvariance obtained by a statistical software package,MINITAB 17. Moreover, the results of the Tukeytest are reported in Tables 5 and 6. According toTable 3, the equivalence hypothesis of LINGO andIPSO solutions in terms of objective function is notrejected. Furthermore, based on Table 5, both LINGOand IPSO algorithms are categorized in the samegroup. As a result, the convergence of IPSO onthe optimal solution, i.e., the veri�cation of IPSO, isstatistically proven.

Tables 4 and 6 represent results of the ANOVAand Tukey test for CPU times, respectively. As theP -value of Table 4 for the algorithm is about zero,LINGO and IPSO solutions are di�erent in termsof running times. With di�erent categories of theTukey test in Table 5, it is proven that the running


Table 3. Results of comparing exact solutions and IPSO solutions with respect to objective function.

Source DF SS MS F -value P -valueAlgorithm 1 174952471 174952471 0.88 0.354Problem 49 387824000000 7914775129 39.64 0.000Error 49 9784033151 199674146Total 99 397783000000

Table 4. Results of comparing exact solutions and IPSO solutions with respect to running time.


Table 5. Tukey results of comparing exact solutions andIPSO solutions with respect to objective function.

Tukey grouping Mean N Algorithm

A 92947.6 50 LINGO

A 90302.2 50 IPSO

Table 6. Tukey results of comparing exact solutions andIPSO solutions with respect to running time.


A 1045.34 50 LINGO

B 21.45 50 IPSO

time of IPSO is much less than that of the LINGOsolver.

As LINGO cannot solve large-size problems, forthese instances, the proposed IPSO algorithm is com-pared with the genetic algorithm proposed by [4] forthe similar integrated scheduling problem. This is doneto assess the e�ciency of the proposed algorithm insolving real cases. The PD indicator is not de�nedhere because the optimal solutions are not available.Table 7 presents the results of IPSO and GA in termsof the objective function and the CPU time. The IPSOalgorithm is capable of generating solutions with goodquality and within a reasonable amount of CPU time,which is less than that in the GA algorithm on average.

Using the ANOVA procedure and Tukey test,both IPSO and GA are compared with each other.Results of the statistical model with MINITAB 17,where I = 25 and J = 2, are reported in Tables 8-11. According to Tables 7 and 9, the algorithms aresigni�cantly di�erent in terms of the solution's quality.Therefore, IPSO is greatly superior to GA.

In addition to the objective function, the com-parison made between the two algorithms in terms ofCPU time is reported in Tables 9 and 11. It showsthat running times of both IPSO and GA do not

di�er signi�cantly because the P -value is larger than5% and they are in the same category of the Tukeytest. However, this comparison is not worth mentioningbecause IPSO and GA are both meta-heuristics whichsolve problems in polynomial time.

From the above test results, it is found that theproposed IPSO algorithm can solve IPDS e�ciently,providing reliable solutions. It is more e�ective andbetter than the other compared algorithms, i.e. GA andLINGO software. The e�ciency of the IPSO makes itsuitable for solving real cases, which are normally largein scale.

5.3. Managerial implicationThis paper discusses the practical implications of man-agerial decisions to integrate production and distribu-tion scheduling. The focus of this study is to showthe practical implication of the integrated model toconsider simultaneous owshop scheduling and vehiclerouting decisions. The integrated scheduling problemsstudied in this paper are based on the models forproduction and distribution that are applied to a widerange of practical applications. The integrated ap-proach prevents decrease in product quality, expirationbefore delivery, extra expenses, and lack of customersatisfaction. It is practical in many industries involvingmake-to-order or time-sensitive (e.g., perishable, dairy)products, where �nished orders are often delivered tocustomers immediately or shortly after the productionto restrict quality reduction.

6. Conclusion and future research

This paper has dealt with practice-oriented integratedproduction and distribution scheduling. IPDS focuseson products with a short lifespan that include owshopscheduling decisions in a single plant, a eet of limited-capacity trucks for delivering customers' demand withconsideration of vehicle routing, and a number of


Table 7. Results of random instances with large sizes.

Instanceno.

Size IPSO GA


Value CPUT(s)

1 200 20 20 200 648781 69 937723 732 200 30 30 200 697987 79 1052365 683 200 50 50 200 773656 121 1151122 1504 200 20 50 200 815652 120 1283420 1095 200 30 50 200 761005 110 1353470 886 200 50 50 300 841468 126 1064971 1397 200 50 30 500 1058764 129 1557706 1168 200 40 50 400 1006698 152 1186959 1819 300 20 20 300 964474 90 1363511 15310 300 30 30 300 1080526 106 1279986 20011 300 50 50 200 968308 111 1649322 8612 300 20 50 300 1039654 138 1299667 17713 300 30 50 300 1146759 138 1533116 12114 300 50 50 300 1151689 136 1476345 15015 300 50 30 500 1255513 137 1484446 10316 300 40 50 400 1250807 164 1537098 9917 500 20 20 500 1700525 130 1932005 16018 500 30 30 500 1715614 157 2033918 18819 500 50 50 400 1688019 183 2245663 18020 500 20 50 500 1980044 210 2663903 19921 500 30 50 500 1809463 201,4 2483166 19022 500 50 50 500 1788781 211 2311044 20123 500 50 30 500 1755100 157 2250077 19024 500 40 50 400 1733551 174 2818420 18925 400 50 50 400 1378299 174 2913846 178

Average 132.8 147.5

Table 8. Results of comparing exact solutions and IPSO solutions with respect to the objective function.


Table 9. Results of comparing exact solutions and IPSO solutions with respect to the CPU time.

Source DF SS MS F -value P -valueAlgorithm 1 545.2 545.2 1.00 0.326Problem 24 67577.0 2815.7 5.19 0.000Error 24 13023.5 542.6Total 49 81145.7

customers with de�ned locations and demands. Thegoal was to design routes that serve all customerswithin each trip and to schedule production in orderto minimize the total tardy and delivery costs whileful�lling the vehicle capacity and deadline for each tripin order to meet lifespan constraints. In the perishable

products industry, costs are perhaps the most promi-nent feature. They are strongly in uenced by the twointerrelated stages of production and distribution. Inthis paper, a scheduling problem in a two-stage sup-ply chain environment is proposed with the objectivefunction of minimizing the sum of delivery and tardy


Table 10. Tukey results of IPSO and GA with respect tothe objective function.


A 1714531 25 GA

B 1240445 25 IPSO

Table 11. Tukey results of IPSO and GA with respect tothe CPU time.


A 147.536 25 GA

B 140.932 25 IPSO

costs. The production and distribution schedulingis integrated into ow shop environment and vehiclerouting. This model is still a good representation ofthe real-world cases such as the dairy product industry.After presenting the IPDS problem as a mixed integernonlinear programming model, since the problem isNP-hard in the strong sense, an Improved ParticleSwarm Optimization (IPSO) algorithm was proposedto schedule the integrated problem. 1-exchange and2-opt improving operators enhanced quality of thesolution in IPSO. Since this is the �rst study of thisproblem, a new data set was generated.

Results of the numerical survey indicated de-creasing costs and, as a result, increasing customersatisfaction and product quality. All of the 25 small-size instances were optimally solved by the proposedIPSO. The mean relative deviation from the optimumof 0.9% was close to zero. The gap of the localsolution was 9.5% on average for all moderate-sizeinstances. The superiority of the suggested IPSO wasexamined by analysis of variance and Tukey pairwisecomparisons. As a result, the convergence of IPSO onthe optimal solution, i.e. the veri�cation of IPSO, wasstatistically proven. Moreover, for large-size instances,the proposed IPSO algorithm was compared withthe genetic algorithm proposed by [4] for the similarintegrated scheduling problem. Using the ANOVAprocedure and Tukey test, both IPSO and GA werecompared. Results of the statistical model reportedthat the e�ciency of the IPSO made it suitable forsolving real cases, which were normally large in scale.An investigation of large-size instances indicated thatthe IPSO was able to provide relatively good resultswithin an acceptable computation time.

Future research is needed to investigate an inte-grated scheduling problem where the machine con�g-urations and distribution are more complex, such asbatch scheduling, two-stage vehicle routing, or multiplants which are distributed in various geographicregions. Integration of di�erent decision levels, liketactical or strategic, helps in optimizing the wholesupply chain. Tightening the gap between theory and

practical applications would be highly worthwhile formore research.

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Biographies

Fateme Marandi is a PhD student in the Departmentof Industrial Engineering at Amirkabir University, Iran.She received her MSc degree from the Department ofIndustrial Engineering at Tarbiat Modares Universityin 2012. Her main areas of research include productionplanning and scheduling, optimization problems, andmeta-heuristics.

Seyed Hesameddin Zegordi is an Associate Pro-fessor of Industrial Engineering in the School of En-gineering at Tarbiat Modares University, Iran. Hereceived his PhD degree from the Department ofIndustrial Engineering and management at Tokyo In-stitute of Technology, Japan, in 1994. He holds anMSc degree in Industrial Engineering and Systemsfrom Sharif University of Technology, Iran, and aBSc degree in Industrial Engineering from IsfahanUniversity of Technology, Iran. His main areas ofteaching and research interests include productionplanning and scheduling, multi-objective optimizationproblems, meta-heuristics, quality management, andproductivity. He has published several articles ininternational conferences and academic journals in-cluding European Journal of Operational Research,International Journal of Production Research, Journalof Operational Research Society of Japan, Computers& Industrial Engineering, Amirkabir Journal of Scienceand Engineering, and Scientia Iranica InternationalJournal of Science and Technology.

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