Multi-period con guration of forward and reverse …scientiairanica.sharif.edu/article_21061_75fec80565c...and game theory problems. Di erent researchers have introduced various problems

Scientia Iranica E (2020) 27(2), 935{955

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringhttp://scientiairanica.sharif.edu

Multi-period con�guration of forward and reverseintegrated supply chain networks through transportmode

A. Eydia;�, S. Fazayelib, and H. Ghafourib

a. Faculty of Engineering, University of Kurdistan, Sanandaj, Iran.b. Department of Industrial Engineering, University of Kurdistan, Sanandaj, Iran.

Received 25 September 2017; received in revised form 25 August 2018; accepted 15 October 2018

KEYWORDSSupply chain networkdesign;Forward and reverselogistic;Multi-periodprogramming;Transportation mode;Genetic algorithm.

Abstract. In today's competitive business, buying and returning products have becomea common practice because of incompleteness of the products or the failure to meet thecustomer's satisfaction or reusing products. Before handling this cycle, companies needa proper logistics network because of its impact on the e�ciency and responsiveness ofthe supply chain. In this research, a forward and reverse logistics network was proposedfor product distribution and collection. The contribution of this paper is a multi-period, multi-echelon integrated forward and reverse supply chain network design problemwith transportation mode selection. Di�erent decisions including determination of theoptimum number and locations of facilities, facilities' opening time, and transportationmode selection were considered in this paper. Due to the multi-period nature of theproblem, the problem is exible for future periods. A mixed integer nonlinear programmingmodel was proposed for the introduced problem, considering di�erent levels of facilitycapacities. As another contribution, a genetic algorithm was developed to cope with theproblem's complexity, especially for large-sized instances. E�ectiveness and reliability ofthe algorithm, evaluated by solving several random instances with the obtained numericalresults and comparisons, con�rmed the capability of the proposed algorithm to �nd goodsolutions within an acceptable processing time period.

© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

Supply chain con�guration has a signi�cant role inthe e�ectiveness of a supply chain. A typical supplychain network is composed of forward and reverselogistics. The forward logistics is utilized to transform

*. Corresponding author. Tel.: +98 8733660073;Fax: +98 8733668513E-mail addresses: [email protected] (A. Eydi);[email protected] (S. Fazayeli);[email protected] (H. Ghafouri)

doi: 10.24200/sci.2018.5261.1175

raw material into �nished goods before delivering themto the end-users such as suppliers, production centers,and distributors. On the other hand, reverse logistics iscomposed of collection and inspection centers, recyclingcenters, disposal centers, and centers for the reproduc-tion of returned goods [1]. In recent years, reverselogistic strategies have been increasingly regarded byresearchers. This is because of the residual value ofimperfect, out of fashion, or unsold products at theend of the forward supply chain. The other reasonbehind the importance of such strategies is to reduceor eliminate the waste generation to comply with en-vironmental regulations or other social commitments.

936 A. Eydi et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 935{955

As such, the reverse logistics has been mainly focusedon the management, inspection, and arrangement ofthe waste in terms of material, part, and productcollection/delivery from/to processing or repair centersand tracing their return to other supply chains ormarkets [2]. An independent design of forward andreverse logistic networks results in suboptimal designsin terms of costs and level of service. Therefore, ithas been recommended that the forward and reverselogistic networks be simultaneously designed to achievea so-called integrated design [1].

In recent years, a great deal of development hasbeen realized within the �eld of Supply Chain NetworkDesign (SCND); although a great deal of research hasbeen focused on this topic, most studies reported sofar have been limited to a single period of time, makingthem less exible when dealing with periodic variations.However, supply chain networks are expected to bedesigned with a multi-period scheme to boost their exibility and provide a basis for taking advantage ofmulti-facility networks in multiple periods.

This research is an extension to the forward andreverse SCND problem, wherein a multi-period, multi-echelon model is considered. The multi-echelon modelhelps consider the entire supply chain, while the multi-period model contributes to better solutions in the longrun. The proposed problem can be applied to suchproducts as tires, electrical reusable devices, bottlecaps or virtually any product that may be returnedby customers for any reason. The forward and reverselogistics network design is essential for (a) cost reduc-tion by reusing products and decreasing environmentalpollutions by collecting deteriorated items and (b)for enhancing customers' satisfaction by returning theproducts that are either not satisfactory from thecustomers' point of view or no longer used. Theproposed model allows decision-makers to account forsuch parameters as transportation cost, purchase cost,etc. by considering their values in di�erent periods.Another feature highlighted in this research is thetransportation mode selection. In reality, there are dif-ferent transportation modes with di�erent associatedcosts, delivery time, and safety level, etc. Therefore, thechoice of transportation mode a�ects the performanceand responsiveness of a supply chain. In order tominimize the total transportation costs, one shouldestablish a balance between the �xed initial costsand variable costs of transportation among networknodes. Furthermore, owing to its comprehensivenessand generalizability, the proposed model can be appliedto many of those goods that are to be shipped tothe consumer once they are delivered and reproduced.As another contribution to the literature, the presentresearch formulates an appropriate solving algorithmfor the newly introduced problem, with its performanceveri�ed by examining numerical examples.

The rest of this paper is organized as follows.Section 2 presents a review of the existing literature.Section 3 focuses on the problem description andmathematical formulation of the model. Section 4 isdedicated to the model validation while setting outan approach to the solving algorithm that is furthertested on numerical examples on various sizes. Finally,Section 5 draws some conclusion and elaborates onfuture trends.

2. Literature review

As a strategic decision for supply chain management,SCND plays a signi�cant role in determining thee�ectiveness, associated costs, and responsiveness ofa supply chain. This problem involves facility loca-tion, ow allocation, product transportation, inventoryand storage management, vehicle routing, etc. Mostprevious researchers have addressed the designs offorward and reverse logistic networks independently,leading to sub-optimal results. Considering the factthat the reverse logistic network design imposes directe�ects on the corresponding forward logistic network,one may suggest the necessity of designing forwardand reverse logistic networks together [3]. In thisregard, various integrated forward and reverse logisticsproblems have been de�ned such as network design,inventory management, capacity management, pricing,and game theory problems.

Di�erent researchers have introduced variousproblems based on SCND. Lin et al. (2006) [4], Duand Evans (2008) [5], Mehdizadeh et al. (2013) [6],Mirmajlesi and Shafaei (2016) [7], Taleizadeh andSadeghi (2018) [8], and Fathollahi Fard and Hajaghaei-Keshteli (2018) [9] focused on the number of supplychain echelons. Some other researchers such as Badriet al. (2013) [10] and Nobari and Kheirkhah (2018) [11]based their investigations on the number of program-ming periods. Multi-product systems have also becomean eye-catching interest to researchers [4,10]. Cardosoet al. (2013) [12], Pedram et al. (2017) [13], andLieckens and Vandaele (2007) [14] worked on theuncertainties associated with the parameters of suchsystems. They considered the demand as a source ofsuch uncertainties and modeled it via a scenario tree-based approach.

Center capacities (i.e., capacity extension or con-traction, addition or elimination of storage facilities,etc.) represent another signi�cant aspect of SCND.For instance, Aghezzaf (2005) [15] and Shaharudin etal. (2017) [16] considered the possibility of expandingthe capacity of facilities, while Lowe et al. (2002) [17]proposed a model to design supply chain networkswith a reducible capacity. Furthermore, Martel etal. (2006) [18] developed a logistic model wherein thecenter capacity could be either increased or decreased.

A. Eydi et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 935{955 937

In another research, Karabakal et al. (2000) [19] ana-lyzed the supply chain across the American branch ofVolkswagen Company.

Some researchers have addressed the inventory inthe SCND. In the literature on inventory programming,the majority of decisions correspond to the properlocation of inventory storage facilities [15,20{22]. Someresearch studies have considered the inventory storageto be non-reducible to only one echelon [23{25], whilesome others have focused on the raw material supplyand �nal products storage, e.g., Jang et al. (2002) [23],Melo et al. (2006) [24], Syam (2002) [25], van Weele(2009) [26], and Cordeau et al. (2006) [27]. In anotherstudy, Yan et al. (2003) [28] proposed a multi-echelon,multi-product, single-time period model for SCND andraw material supply. Zhou and Chen (2017) [29]considered the inventory control problem for bothforward and reverse logistics.

Transportation mode selection has been widelyregarded in previous research studies. When it comesto the choice of transportation mode in SCND, eitherof two cases may arise: (1) a given pair of nodes can behandled by more than one transportation mode [30],(2) only one transportation mode is available betweena given pair of nodes [31].

Heydari et al. (2017) [32] and Taleizadeh andSadeghi (2018) [8] proposed a number of pricingstrategies for dealing with competitive reverse supplychains. Giri et al. (2017) [33] and Fathollahi Fardand Hajaghaei-Keshteli (2018) [9] investigated the im-pacts of cooperation and competition based on gametheory. Khakim Habibi et al. (2017) [34] presenteda collection-disassembly problem in the reverse supplychain. Butzer et al. (2017) [35] de�ned a performancemeasurement system to assess international reversesupply chains. Goh et al. (2007) [36] developed alocation programming model for international facilities.

Di�erent researchers have developed a variety ofapproaches to solving their proposed problems; thesecan be classi�ed into exact and approximate methods(heuristic and metaheuristic methods). Lu and Bostel(2007) [37] proposed a Lagrangian relaxation method.�Uster et al. (2007) [38] employed benders decomposi-tion. Min and Ko (2008) [39], Lee and Chan (2009) [40],Trappey et al. (2010) [41], and Fakhrzad and Moobed(2010) [42] used genetic algorithm. Mehdizadeh etal. (2013) [6] developed a hybrid priority-based ge-netic algorithm and simulated annealing algorithm.Zegordi et al. (2010) [43] considered a gender-basedgenetic algorithm, in which two di�erent types ofchromosomes with unequal structures were considered.More recently, a number of scholars have adoptednew algorithms for such a purpose. In this respect,Modiri-Delshad et al. (2016) [44] used back-trackingsearch algorithm; Kaboli et al. (2016) [45] proposedan arti�cial cooperative search algorithm; Ra�eerad

et al. (2017) [46] applied a multi-objective ParticleSwarm Optimization (PSO); Kaboli et al. (2017) [47]introduced rain-fall optimization algorithm; moreover,Kaboli et al. (2017) [48] proposed a gene expressionprogramming for electrical consumption forecasting;Sebtahmadi et al. (2018) [49] used PSO-DQ CurrentControl Scheme; Mansouri et al. (2012) [50] presenteda hybrid neuro-fuzzy- P.I. fed Controller for controllingthe rpm of brush-less D.C. motors; Modiri-Delshad etal. (2013) [51] proposed an iterative algorithm for aneconomic dispatch in a micro grid. They further usedthe algorithm to address the economic dispatch in apower system.

Given the focus of the present research, i.e.,integrated SCND, Amin and Zhang (2013) [1] proposeda multi-objective, multi-echelon, multi-product modelthat represented a consistent image of integrity acrossthe system. Mirmajlesi and Shafaei (2016) [7] con-sidered a multi-period, multi-product, multi-echeloncapacitated supply chain problem for products of ashort lifetime.

As a contribution to the forward and reverselogistics problem and to �ll in the existing researchgap in the literature on SCND, this study developsa deterministic multi-period, multi-echelon model tomaximize the overall pro�t. The model determines theoptimum number and locations of facilities, as well astheir establishment times. The considered logistics net-work structure had three echelons in the forward owand four echelons in the reverse ow. Furthermore, thechoice of transportation mode between di�erent nodes(facilities) was considered. The present research isamong the rare research studies that combines forwardand reverse logistics to address multi-period logisticnetwork design and scheduling simultaneously. Thisresearch further considers the choice of transportationmode (with di�erent �xed and variable costs associatedwith each transportation mode) between the associatednodes, i.e., mathematical programming for minimizingthe transportation costs leads to a proportional bal-ance between the transportation costs and the �xedand variable parameters, a�ecting the chosen trans-portation mode. Furthermore, given the numericalcomplexity of the problem, a metaheuristic algorithmwas developed to solve the model. Transportationproblem concept comprises a common idea withinSCND problems. Due to its superior performancefor solving similar problems, genetic algorithm wasadopted in this paper. Mixing the transportationproblem with the genetic algorithm, one can designan e�ective solver algorithm for problems of largersizes.

3. Problem description and modeling

In order to accurately demonstrate a schematic of the


Figure 1. Product distribution and collection diagram.

problem assumptions, Figure 1 provides a diagram viewof a forward and reverse integrated supply chain net-work. Beginning with the chain, following the forward ow, customers' required products are manufacturedin production centers before being dispatched to distri-bution centers from which they are to be delivered tothe customers. Such distribution centers are assignedto deliver all of the received products to the customerswithin the same period of time when the products arereceived. In terms of tasks and performances, there aretwo types of distribution centers: normal distributioncenters, which are only utilizable along the forward ow and their only task is to receive the productsfrom the distribution centers and forward them to theend-users, and hybrid distribution centers, which arenot only capable of accomplishing what normal centersdo, but also well set to collect the returned productsfrom the customers at the end of the period in areverse ow. As end customers, end-users or retailersare placed at the end of the supply chain with theirnumber and locations known within each period. Thedemand by these customers is known and the chainmust necessarily meet the demands in full.

In the reverse ow, products' end-users assumereceiving the products from the distribution centersand returning them to the supply chain once theproducts are consumed. Each customer has his/herspeci�c rate of return. This rate of return can bedi�erent in each period. Following the reverse ow,there are collection centers where the returned productsfrom the customers are collected. In this problem,all of returned products must be collected. Normalcollection centers can only be utilized along the reverse ow path to collect the returned products from thecustomers, while hybrid distribution-collection centerscan not only accomplish the collecting task, but also

perform the distribution task along the forward path ow. Once inspected at the collection centers, thecollected products will be sent to the recycling centers ifrecognized as recyclable; otherwise, the collected prod-ucts are sent to disposal centers. Di�erent recyclingcenters may exhibit a di�erent number of recyclableproducts within each period of time. Once recycled, thecollected products either turn into recycled productsthat can be resold to customers or provide raw materialfor the production centers. Therefore, the recycledproducts can be pro�table for the chain as they canbe sold.

In Figure 1, locating process is performed onthe manufacturing, distribution, hybrid distribution-collection, collection, recycling, and disposal facilitieswithin each period so that the demands by the cus-tomers can be ful�lled with minimum cost. Varioustransportation modes are provided for transporting theproducts between two related facilities. According tothe model assumptions, only one transportation modecan be chosen between each pair of nodes until the endof the current period; however, di�erent transportationmodes can be utilized for future periods. Each trans-portation mode has its own �xed and variable costs.Each facility opens at a limited initial capacity so thatit cannot serve beyond its initial capacity during theinitial period. In this research, expandable capacitywas considered, and each level of improvement wasassociated with some costs imposed on the system;for each facility, the capacity level enhancement wasallowed just once per period. The presented model wasdeveloped for a single-product system.

3.1. Mathematical modelModel sets, indices, parameters, and decision variablesare as follows:


Setsv Set of �xed locations of outer

manufacturers, v = 1; 2; :::; Vi Set of potential locations for

manufacturing centers, i = 1; 2; :::; Ij Set of potential locations for

distribution centers, j = 1; 2; :::; Jk Set of �xed locations of customers,

k = 1; 2; :::;Kl Set of potential locations for collection

and inspection centers, l = 1; 2; :::; Lh Set of potential locations for hybrid

distribution-collection centers,h = 1; 2; :::; H

g Set of potential locations for recyclingcenters, g = 1; 2; :::; G

m Set of potential locations for disposalcenters, m = 1; 2; :::;M

n Set of available capacity levels forfacilities, n = 1; 2; :::; N

c Set of transport modes between a pairof nodes, c = 1; 2; :::; C

t Set of time periods, t = 1; 2; :::; TParametersDMkt The demand by the customer k in time

period tPRkt Product's unit price for the customer

k in time period tPRIit Recycled product's unit price for the

manufacturing center i in time period tPRVvt Recycled product's unit price for the

outer customer v in time period tPCit Unit production cost at the

manufacturing center i in timeperiod t

IClt Unit cost of inspection test at thecollection and inspection center l intime period t

ICht Unit cost of inspection test at thehybrid distribution-collection center hin time period t

RCgt Unit cost of product recycling atrecycling center g in time period t

DCmt Unit cost of returned product disposalat disposal center m in time period t

XOit Fixed cost to establish themanufacturing center i in timeperiod t, given it was closed at timeperiod t� 1

XCit Fixed cost to close the manufacturingcenter i in time period t, given it wasopen in time period t� 1

FEXint Fixed cost to expand the manufacturingcenter i with the capacity level of n intime period t

Y Ojt Fixed cost to establish the distributioncenter j in time period t, given it wasclosed in time period t� 1

Y Cjt Fixed cost to close the distributioncenter j in time period t, given it wasopen in time period t� 1

FEYjnt Fixed cost to expand the distributioncenter j with the capacity level of n intime period t

ZOlt Fixed cost to establish the collectionand inspection center l in time periodt, given it was closed in time periodt� 1

ZClt Fixed cost to close the collection andinspection center l in time period t,given it was open in time period t� 1

FEZl:nt Fixed cost to expand the collection andinspection center l with the capacitylevel of n in time period t

UOmt Fixed cost to establish the disposalcenter m in time period t, given it wasclosed in time period t� 1

UCmt Fixed cost to close the disposal centerm in time period t, given it was openin time period t� 1

FEUmnt Fixed cost to expand the disposalcenter m with the capacity level of nin time period t

WOht Fixed cost to establish the hybriddistribution-collection center h in timeperiod t, given it was closed in timeperiod t� 1

WCht Fixed cost to close the hybriddistribution-collection center h in timeperiod t, given it was open in timeperiod t� 1

FEWhnt Fixed cost to expand the hybriddistribution-collection center h withthe capacity level of n in time period t

QOgt Fixed cost to establish the recyclingcenter g in time period t, given it wasclosed in time period t� 1

QCgt Fixed cost to close the recycling centerg in time period t, given it was open intime period t� 1

FEQgnt Fixed cost to expand the recyclingcenter g with the capacity level of n intime period t

CaIit Capacity of the manufacturing center iin time period t


ECIint Development capacity of themanufacturing center i with thecapacity level of n in time period t

CaJjt Capacity of the distribution center j intime period t

ECJjnt Development capacity of thedistribution center j with the capacitylevel of n in time period t

CaLlt Capacity of the collection andinspection center l in time period t

ECLlnt Development capacity of the collectionand inspection center l with thecapacity level of n in time period t

CaHht Capacity of the hybrid distribution-collection center h in time periodt

ECHhnt Development capacity of the hybriddistribution-collection center h withthe capacity level of n in time period t

CaGgt Capacity of the recycling center g intime period t

ECGgnt Development capacity of the recyclingcenter g with the capacity level of n intime period t

CaMmt Capacity of the disposal center m intime period t

ECMmnt Development capacity of the disposalcenter m with the capacity level of nin time period t

CIJijtc Unit transportation cost from themanufacturing center i to thedistribution center j in time period tvia transport mode c

FIJijtc Fixed transportation cost fromthe manufacturing center i to thedistribution center j in time period tvia transport mode c

CJKjktc Unit transportation cost from thedistribution center j to the customer kin time period t via transport mode c

FJKjktc Fixed transportation cost from thedistribution center j to the customer kin time period t via transport mode c

CKLkltc Unit transportation cost from thecustomer k to the collection andinspection center l in time period t viatransport mode c

FKLkltc Fixed transportation cost from thecustomer k to the collection andinspection center l in time period t viatransport mode c

CHKhktc Unit transportation cost from thehybrid distribution-collection center hto the customer k in time period t viatransport mode c

FHKhktc Fixed transportation cost from thehybrid distribution-collection center hto the customer k in time period t viatransport mode c

CIHihtc Unit transportation cost from themanufacturing center i to the hybriddistribution-collection center h in timeperiod t via transport mode c

FIHihtc Fixed transportation cost from themanufacturing center i to the hybriddistribution-collection center h in timeperiod t via transport mode c

CLGlgtc Unit transportation cost from thecollection and inspection center l tothe recycling center g in time period tvia transport mode c

FLGlgtc Fixed transportation cost from thecollection and inspection center l tothe recycling center g in time period tvia transport mode c

CLMlmtc Unit transportation cost from thecollection and inspection center l tothe disposal center m in time period tvia transport mode c

FLMlmtc Fixed transportation cost from thecollection and inspection center l tothe disposal center m in time period tvia transport mode c

CHMhmtc Unit transportation cost from thehybrid distribution-collection center hto the disposal center m in time periodt via transport mode c

FHMhmtc Fixed transportation cost from thehybrid distribution-collection center hto the disposal center m in time periodt via transport mode c

CHGhgtc Unit transportation cost from thehybrid distribution-collection center hto the recycling center g in time periodt via transport mode c

FHGhgtc Fixed transportation cost from thehybrid distribution-collection center hto the recycling center g in time periodt via transport mode c

CGMgmtc Unit transportation cost from therecycling center g to the disposalcenter m in time period t via transportmode c


FGMgmtc Fixed transportation cost from therecycling center g to the disposalcenter m in time period t via transportmode c

CGIgitm Unit transportation cost from therecycling center g to the manufacturingcenter i in time period t via transportmode cgitc

FGIgitc Fixed transportation cost from therecycling center g to the manufacturingcenter i in time period t via transportmode c

CGVgvtc Unit transportation cost from therecycling center g to the outer customerv in time period t via transport mode c

FGVgvtc Fixed transportation cost from therecycling center g to the outer customerv in time period t via transport mode c

RKk Rate of return for the used productsby customer k

RI Rate of usable material in thecollection and hybrid distribution-collection centers

RG Rate of recyclable material at therecycling centers

Djt Unit distribution cost for thedistribution center j in time period t

Dht Unit distribution cost for the hybriddistribution-collection center h in timeperiod t

Non-negative variables

QIJijt The volume of products sent fromthe manufacturing center i to thedistribution center j in time period t

QJKjkt The volume of products sent from thedistribution center j to the customer kin time period t

QIHiht The volume of products sent from themanufacturing center i to the hybriddistribution-collection center h in timeperiod t

QHKhkt The volume of products sent from thehybrid distribution-collection center hto the customer k in time period t

QKLklt The volume of products sent fromthe customer k to the collection andinspection center l in time period t

QKHkht The volume of products sent fromthe customer k to the hybriddistribution-collection center h in timeperiod t

QLGlgt The volume of products sent from thecollection and inspection center l tothe recycling center g in time period t

QLMlmt The volume of products sent from thecollection and inspection center l tothe disposal center m in time period t

QHMhmt The volume of products sent from thehybrid distribution-collection centerh to the disposal center m in timeperiod t

QHGhgt The volume of products sent from thehybrid distribution-collection centerh to the recycling center g in timeperiod t

QGMgmt The volume of products sent fromthe recycling center g to the disposalcenter m in time period t

QGIgit The volume of products sent from therecycling center g to the manufacturingcenter i in time period t

QGVgvt The volume of products sent fromthe recycling center g to the outercustomer v in time period t

Binary variables

Xit

8><>:1 : If the manufacturing center iis established in time period t

0 : Otherwise

EXi:nt

8>>><>>>:1 : If the manufacturing center i

is developed with the capacity level ofn in time period t

0 : Otherwise

Yjt

8><>:1 : If the distribution center jis established in time period t

0 : Otherwise

EYjnt

8>>><>>>:1 : If the distribution center j


0 : Otherwise

Zlt

8><>:1 : If the collection and inspection center lis established in time period t

0 : Otherwise


EZl:nt

8>>><>>>:1 : If the collection and inspection center l


0 : Otherwise

Umt

8><>:1 : If the disposal center mis established in time period t

0 : Otherwise

EUmnt

8>>><>>>:1 : If the disposal center m

is developed with the capacity level of nin time period t

0 : Otherwise

Wht

8><>:1 : If the hybrid distribution-collection centerh is established in time period t

0 : Otherwise

EWhnt

8>>><>>>:1 : If the hybrid distribution-collection center

h is developed with the capacity level ofn in time period t

0 : Otherwise

Qgt

8><>:1 : If the recycling center gis established in time period t

0 : Otherwise

EQgnt

8>>><>>>:1 : If the recycling center g


0 : Otherwise

IJijtc

8>>><>>>:1 : If the transport mode c

is chosen for the route from ito j in time period t

0 : Otherwise

KLkltc


is chosen for the route from kto l in time period t

0 : Otherwise

JKjktc


is chosen for the route from jto k in time period t

0 : Otherwise

Hkhktc


is chosen for the route from hto k in time period t

0 : Otherwise

IHihtc


is chosen for the route from i to hin time period t

0 : Otherwise

LGlgtc


is chosen for the route from lto g in time period t

0 : Otherwise

LMlmtc


is chosen for the route from lto m in time period t

0 : Otherwise

HMhmtc


is chosen for the route from hto m in time period t

0 : Otherwise

HGhgtc


is chosen for the route from hto g in time period t

0 : Otherwise

GMgmtc


is chosen for the route from gto m in time period t

0 : Otherwise

GIgitc


is chosen for the route from gto i in time period t

0 : Otherwise

GVgvtc


is chosen for the route from gto v in time period t

0 : Otherwise

KHkhtc


is chosen for the route from kto h in time period t

0 : Otherwise


where M is a large number, and RI and RK denoterates of return, can take values between 0 and 1, andexpress the fraction of original products that were re-turned. Based on the assumptions, the parameters andvariables of the model are de�ned, and the objectivefunction, aimed at pro�t maximization, is formulatedas follows:

Objective function and its components

Maximise Profit =Xt

(Incomet � Costt): (1)

Incomet =Xj

Xk

(QJK�jktPRkt)

+Xh

Xk

(QHK�hktPRkt)

+Xg

Xi

(QGI�gitPRIit)

+Xg

Xv

(QGV �gvtPRVvt) 8t (2)

Costt =Xi

XOit(1�Xi;t�1)Xit

+Xi

XC�itXi;t�1(1�Xit)

+Xj

Y Ojt(1� Yj;t�1)Yjt

+Xj

Y C�jtYj;t�1(1� Yjt)

+Xl

ZOlt(1� Zl;t�1)Zlt

+Xl

ZC�ltZl;t�1(1� Zlt)

+Xm

UOmt(1� Um;t�1)Umt

+Xm

UC�mtUm;t�1(1� Umt)

+Xh

WOht(1�Wh;t�1)Wit

+Xh

WC�htWh;t�1(1�Wht)

+Xg

QOgt(1�Qg;t�1)Qgt

+Xg

QC�gtQg;t�1(1�Qgt) (3.1)

+Xi

Xn

FEXint(EXint � EXin;t�1)

+Xj

Xn

FEYjnt(EYjnt � EYjn;t�1)

+Xl

Xn

FEZlnt(EZlnt � EZln;t�1)

+Xm

Xn

FEUmnt(EUmnt � EUmn;t�1)

+Xh

Xn

FEWhnt(EWhnt � EWih;t�1)

+Xg

Xn

FEQgnt(EQgnt�EQgn;t�1) (3.2)

+Xi

Xj

(QIJ�ijtPCit)+Xi

Xh

(QIH�ihtPCit) (3.3)

+Xj

Xk

(QJK�jktDjt)+Xh

Xk

(QHK�hktDht) (3.4)

+Xk

Xl

(QKL�kltIClt)+Xk

Xh

(QKH�khtICht) (3.5)

+Xl

Xm

(QLM�lmtDCmt)

+Xh

Xm

(QHM�hmtDCmt)

+Xg

Xm

(QGM�gmtDCmt) (3.6)

+Xl

Xg

(QLG�lg tRCgt)

+Xh

Xg

(QHG�hgtRCgt) (3.7)

+Xi

Xj

QIJ�ijt

Xc

IJ�ijtcCIJijtc!!

+Xj

Xk

QJK�jkt

Xc

JK�jktcCJKjktc

!!+Xk

Xl

(QKL�klt(Xc

KL�kltcCKLkltc))


+Xh

Xk

QHK�hkt

Xc

HK�hktcCHKhktc

!!Xk

Xh

QKH�kht

Xc

KH�khtcCKHkhtc

!!+Xi

Xh

QIH�iht

Xc

IH�ihtcCIHihtc

!!+Xl

Xg

QLG�lg t

Xc

LG�lg tcCLGlg tc

!!

+Xl

Xm

QLM�lmt

Xc

LM�lmtcCLMlmtc

!!+Xh

Xm

QHM�hmt

Xc

HM�hmtcCHMhmtc

!!+Xh

Xg

QHG�hgt

Xc

HG�hgtcCHGhgtc!!

+Xg

Xm

QGM�gmt

Xc

GM�gmtcCGMgmtc

!!

+Xg

Xi

QGI�git

Xc

GI�gitcCGIgitc!!

+Xg

Xv

QGV �gvt

Xc

GV �gvtcCVgvtc!!

(3.8)

+Xi

Xj

Xc

IJ�ijtcFIJijtc

+Xj

Xk

Xc

JK�jktcFJKjktc

+Xk

Xl

Xc

KL�kltcFKLkltc

+Xh

Xk

Xc

HK�hktcFHKhktc

+Xi

Xh

Xc

IH�tcFIHihtc

+Xl

Xg

Xc

LG�lgtcFLGlg tc

+Xl

Xm

Xc

LM�lmtcFLMlmtc

+Xh

Xm

Xc

HM�hmtcFHMhmtc

+Xh

Xg

Xc

HG�hgtcFHGhgtc

+Xg

Xm

Xc

GM�gmtcFGMgmtc

+Xg

Xi

Xc

GI�gitcFGIgitc

+Xg

Xv

Xc

GV �gvtcFGVgvtc: (3.9)

The �rst term of the expression sets out a gen-eral form of the objective function. Based on thisgeneral form, it is clear that the pro�t function isobtained to be the di�erence between the sum ofrevenues generated along the chain and associatedcosts. For the sake of simplicity and to obtain a simplemathematical expression, the income function and costfunction were expressed as the second and third terms,respectively.

By showing income sources of the chain, thesecond term of the objective function is composed oftwo components: (1) the sum of incomes providedby the sale of the �nished products to the end-usersin the forward ow during various periods, and (2)total incomes raised by selling recycled products tomanufacturing centers or external customers during allprogramming periods.

By re ecting the incurred costs, the third termof the objective function consists of 9 components.For the sake of simplicity, these 9 components areformulated as follows. Component (3.1) denotes to-tal �xed costs associated with the establishment andtermination of centers during all periods. Component(3.2) refers to total �xed costs spent on capacityexpansion of the centers from the initial capacity toincreased capacity levels. Component (3.3) indicatestotal manufacturing cost spent at the manufacturingcenters to ful�ll customers' demands within the pro-gramming periods. Component (3.4) evaluates thecosts incurred upon distributing the products from thedistribution or hybrid collection-distribution centersto the end-users. Component (3.5) denotes the totalcost incurred to inspect and test the collected returnedproducts; it can be seen as a collection-associated cost.Component (3.6) represents the sum of disposal costsincurred at the disposal centers. Component (3.7)indicates total product recycling costs incurred at therecycling centers. Component (3.8) shows the sum ofvariable costs of transportation across associated nodesand, �nally, Component (3.9) represents the total �xedtransportation cost.


The model constraints:Xi

QIJijt =Xk

QJKjkt 8t; j; (4)Xj

QJKjkt +Xh

QHKhkt = DMkt 8t; k; (5)

Xi

QIHiht =Xk

QHKhkt 8t; h; (6)

Xh

QKHkht +Xl

QKLklt = DMkt�RKk 8t; k;

(7)

RI�Xk

QKHkht =Xg

QHGhgt 8t; h; (8)

(1�RI)Xk

QKHkht =Xm

QHMhmt 8t; h; (9)

RI�Xk

QKLklt =Xg

QLGlgt 8t; i; (10)

(1�RI)Xk

QKLklt =Xm

QLMlmt 8t; l; (11)

RG� X

h

QHGhgt +Xl

QLGlg t

!=Xv

QGVgvt +Xi

QGIgit 8t; g; (12)

(1�RG)

Xh

QHGhgt +Xl

QLGlg t

!=Xm

QGMgmt 8t; g; (13)Xj

QIJijt +Xh

QIHiht � CaI�itXit

+Xn

(ECI�intEXint) 8t; i; (14)Xi

QIJijt � CaJ�jtYjt +Xn

(ECJ�jntEYjnt)

8t; j; (15)Xk

QKLklt � CaL�ltZlt +Xn

(ECL�lntEZlnt)

8t; l; (16)Xi

QIHiht +Xk

QKHkht � CaH�htWht

+Xn

(ECH�hntEWhnt) 8t; h; (17)

Xl

QLGlg t +Xh

QHGhgt � CaG�gtQgt

+Xn

(ECG�gntEQgnt) 8t; g; (18)

Xl

QLMlmt +Xh

QHMhmt +Xg

QGMgmt

� CaM�mtUmt

+

Xn

ECM�mntEUmnt!8t;m; (19)

Xn

EXint � Xit 8t; i; (20)

Xn

EYJnt � Yjt 8t; j; (21)

Xn

EZlnt � Zlt 8t; l; (22)

Xn

EUmnt � Umt 8t;m; (23)

Xn

EQgnt � Qgt 8t; g; (24)

Xn

EWhnt �Wht 8t; h; (25)

EXin;t�1 � EXint 8t; n; i; (26)

EYjn;t�1 � EYjnt 8t; n; j; (27)

EZln;t�1 � EZlnt 8t; n; l; (28)

EUmn;t�1 � EUmnt 8t; n;m; (29)

EQgn;t�1 � EQgnt 8t; n; g; (30)

EWhn;t�1 � EWhnt 8t; n; h; (31)Xc

IJijtc � 1 8t; i; j; (32)

Xc

JKjktc � 1 8t; j; k; (33)

Xc

KLkltc � 1 8t; k; l; (34)

Xc

HKhktc � 1 8t; h; k; (35)

Xc

IHihtc � 1 8t; i; h; (36)


Xc

LGlg tc � 1 8t; l; g; (37)Xc

LMlmtc � 1 8t; l;m; (38)Xc

HMhmtc � 1 8t; h;m; (39)Xc

GMgmtc � 1 8t; g;m; (40)Xc

HGhgtc � 1 8t; h; g; (41)Xc

GIgitc � 1 8t; g; i; (42)Xc

GVgvtc � 1 8t; g; v; (43)Xc

KHkhtc � 1 8t; h; k; (44)

QIJijt �MXc

IJijtc 8t; i; j; (45)

QJKjkt �MXc

JKjktc 8t; j; k; (46)

QKLklt �MXc

KLkltc 8t; k; l; (47)

QHKhkt �MXc

HKhktc 8t; h; k; (48)

QIHiht �MXc

IHihtc 8t; i; h; (49)

QLGlg t �MXc

LGlg tc 8t; l; g; (50)

QLMlmt �MXc

LMlmtc 8t; l;m; (51)

QHMhmt �MXc

HMhmtc 8t; h;m; (52)

QHGhgt �MXc

HGhgtc 8t; h; g; (53)

QGMgmt �MXc

GMgmtc 8t; g;m; (54)

QGIgit �MXc

GIgitc 8t; g; i; (55)

QGVgvt �MXc

GVgvtc 8t; g; v; (56)

QKHkht �MXc

KHkhtc 8t; k; h; (57)

Xit; EXint; Yjt; EYjnt; Zlt; EZln t; Umt; EUmnt;

Qgt; EQgnt;Wht; EWhnt; IJijtc; JKjktc;

KLkltc;HKhktc; IHihtc; LGlg tc; LMlmtc;

HMhmtc;HGhgtc; GMgmtc; GIgitc; GVgvtc;

KHkhtc 2 f0; 1g8i; j; k; l; h; g;m; v; n; c; t; (58)

QIJijt; QJKjkt; QIHiht; QHKhkt; QKLklt;

QKHkht; QLGlg t; QLMlmt; QHMhmt;

QHGhgt; QGMgmt; QGIgit; QGVgvt � 0

8i; j; k; l; h; g;m; v; t: (59)

Constraints (4){(6) ensure ow equilibriumwithin the related nodes to the forward ow path, whileConstraints (7){(13) ensure ow equilibrium withinthose along the reverse ow path. For the purposeof this paper, ow equilibrium implies the state ofequality between the sum of products received anddispatched by a particular node. Constraints (14){(19) de�ne the capacity limitations at di�erent centers.Once opened, each center can operate at its initialcapacity only unless a capacity expansion process isundertaken for that center. Constraints (20){(25)ensure that a capacity expansion can solely be realizedfor the centers established in the corresponding period.Constraints (26){(31) introduce the impossibility ofa capacity reduction to occur in capacity-expandedfacilities, i.e., once the capacity of a facility expandsin a given period of time, it cannot be either reducedor returned back to the initial capacity within thesame period of time. Constraints (32){(44) prove thatthere is only one transportation mode between eachpair of nodes during each period of time. Constraints(45){(57) ensure that no product can be transportedbetween a pair of nodes unless a transportation modeis chosen. Finally, Constraints (58) and (59) implythat the variables shall be non-negative and binary,respectively.

4. Model validation and solution approach

In this research, in order to solve small- to medium-sized problems, optimization software called GAMSv.24.2.2 was used without con�guring a particularalgorithm. The results obtained by this software wereused as a reference to compare other solution methodsand algorithms. All case studies were processed ona laptop powered by an Intel® CoreTM 2 Dou CPU


operating at 2.4 GHz and further equipped with 4 GBof RAM.

By utilizing GAMS software, two important issueswere analyzed: the value of M (an adequately largenumber) and the choice of the solver. Due to thenon-linear nature of the mathematical model, BARONsolver was employed to solve the illustrative problems.Furthermore, in order to investigate the sensitivity ofthis solver to the value of M , a set of experimentswas designed, whose results were analyzed. Accordingto the analyses, the minimum value of M before theproblem turns infeasible or unjusti�ed is the maximumvalue of total demands in di�erent periods. Followingthis approach, the variable ranges were limited tocontract the solution space for the solver so as toaccelerate the process of �nding the optimum solution.

Having introduced various local techniques toanalyze supply chain networks, Krarup and Pruzan(1983) investigated the issues associated with the be-havior of such techniques for large-sized problems [52],and proved that the SCND problems were NP-hard.Therefore, for each mathematical model developed forsuch problems, it is required to present an appropriatealgorithm �tted to the problem structure. In order toprove that the processing time increases with the prob-lem size, several purposive cases were designed withdi�erent sizes and, then, solved by GAMS software.Table 1 reports the obtained processing times. Thetable makes it clear that an increase in the number ofperiods adds to the processing time dramatically.

In Table 1, the �rst 4-digit part of the codeindicates, from left to right, the number of manufac-turing centers, the number of distribution centers, thenumber of hybrid collection-distribution centers, andthe number of customers, respectively. The second 4-digit part shows, again from left to right, the number ofcollection centers, the number of recycling centers, thenumber of disposal centers, and the number of externalcustomers, respectively. Finally, the third part denotes,from left to right, the number of capacity levels, thenumber of transportation modes between two relatednodes, and the number of programming periods. It is

worth mentioning that the parameters in the designedillustrative cases have all been set randomly.

4.1. Solving approachGenetic algorithm is among the widely used algorithmsfor solving NP-hard problems. It is applicable toproblems with large feasible spaces. Furthermore, itis particularly helpful for complex problems where thein uences of constraints and parameters are unknown.This method enjoys a lower probability of gettingtrapped in local optimum, as compared to similartechniques. The capability of GA to reach near-optimum solutions has widened the scope of its appli-cation to large-sized problems. It is a nature-inspiredand population-based method with a capability tocope with the problems of special structure and isconsidered in this research because of its ability to �ndnear-optimum solutions and its exibility to solve awide range of problems. This is the reason why thisalgorithm has been selected from a pool of varioussolving approaches. Furthermore, new chromosomeswere proposed here to encode the problem. GA hasbeen previously used to solve reverse logistics problemsby some researchers, e.g., Min and Ko (2008) [39],Lee and Chan (2009) [40], Trappey et al. (2010) [41],and Fakhrzad and Moobed (2010) [42]. Accordingly,genetic algorithm was used to solve the problem in thisresearch.

4.2. Algorithm componentsComponents of the algorithm used in this study aredescribed in this subsection.

4.2.1. Initiator operator and solution demonstrationIn this research, common methods in transportationproblems were used to present the solution. Thenumber of products transported from manufacturers tocustomers was organized into a matrix. As mentionedpreviously, there is no direct transportation betweenmanufacturers and customers, and the matrix merelyshows the origin of the received products. Anotherfeature of the model is the time periods. For eachtime period, two matrices were used: one for forward

Table 1. Results of solving the illustrative cases.

Problem Problem

Number ofperiods

Group codeProcessing

time(seconds)

Objectivefunction

value

Number ofperiods

Group codeProcessing

time(seconds)

Objectivefunction

value

T = 12112-2222-121 2 193246

T = 32112-2222-123 3220 735079

2212-2223-121 6 187956 2113-2223-123 323 1629623

T = 22112-2222-122 70 489659 T = 4 2112-2222-124 14400 933059

3112-2222-222 158 484262


logistics and another for reverse logistics. In terms ofcapacities, if the sum of demands in one period exceedsthe capacity of facilities, the capacity increases until theentire volume of demands is addressed.

In each period, the number of products shippedfrom the manufacturing centers to the customers isexpressed in terms of a transportation matrix shownbelow; therefore, each period corresponds to a trans-portation matrix. Transportation matrices can beseen as chromosomes in this algorithm. In otherwords, the solution of each period can be seen as twotransportation matrices: one for the forward ow pathand another for the reverse one:

Xp =

2664x11 x12 : : : x1nx21 x22 : : : x2n: : : : : : : : : : : :xm1 xm2 : : : xmn

3775 ;in which rows and columns represent the manufactur-ing centers and customers, respectively.

The following pseudo-code was used to form theabove matrix:

1. Form a set, T , including the set of numbers from 1to mn.

2. Randomly choose a number, k, from the set T .3. Through the following relationship, extract the row

number and column number corresponding to k:

Row number (i) = [((k � 1) =n) + 1] ;

Column number (j) = ((k � 1) =mod n) + 1:

4. Choose the minimum value across the productionvalues in the ith row and the demand in the jthcolumn and name it xij .

5. Subtract xij from the production in the ith row andthe demand in the jth column.

6. Eliminate k from the set T .7. Repeat the process until all members in T are

eliminated.

Once the above matrix was formed, in order to de-termine the volume of products shipped to customers,each xij value was randomly decomposed into twoparts: the volume of products sent to the distributioncenters and that to hybrid distribution-collection cen-ters. Following the process of chromosome formation,one can determine the required quantity of products tobe redistributed among the facilities along the reverse ow to reach a complete solution after applying the rateof product return by customers. After applying therate of product recycling at collection and hybrid cen-ters and determining the number of pushed productstoward the recycling and disposal centers (similar to

what was done before), the transportation matrix wasrandomly allocated between two nodes; �nally, volumesof the products sent to recycling centers, manufactur-ers, and external customers were determined by thesame approach.

4.2.2. Crossover operatorThe following pseudocode was used to apply thecrossover operator once the parents (matrices) wereselected as X1 = (x1

ij) and X2 = (x2ij):

1. Use the following method to de�ne the matricesD = (dij) and R = (rij).

dij = [(x1ij + x2

ij)=2]; rij = (x1ij + x2

ij)mod 2:

2. Divide the matrix R into two matrices of R1 = (r1ij)

and R2 = (r2ij):

R = R1 +R2;

nXj=1

r1ij =

nXj=1

r2ij =

12

nXj=1

rij i = 1; 2; : : : ;m;

mXi=1

r1ij =

mXi=1

r2ij =

12

mXi=1

rij j = 1; 2; : : : ; n:

3. Form the o�spring X 01 and X 02 as follows:

X 01 = D +R1; X 02 = D +R2:

4.2.3. Mutation operatorIn order to apply this operator when a transportationmatrix is selected, �rst, a number of rows and columnsare randomly selected within the matrix. In the devel-oped submatrix, the existing values are manipulated,such that the sum of values within each row and columnremains unchanged. For instance, assume the followingmatrix to be subjected to the mutation operator:

Parent X0 0 5 0 30 4 0 0 00 0 5 7 03 1 0 0 2

Suppose that the 2nd and 4th rows and the2nd, 3rd, and 5th columns are selected randomly; thissubmatrix can be manipulated in the following to meetthe mentioned condition:

The selectedmatrix

4 0 01 0 2

The manipulatedmatric

2 0 23 0 0

Consequently, the following o�spring will resultfrom the mutation operator:


O�spring0 0 5 0 30 2 0 0 20 0 5 7 02 3 0 0 0

4.2.4. Fitness functionFitness function is exactly the same as the problem'sobjective function, i.e., to maximize the overall pro�tgenerated along the chain. This function calculates theincurred costs and gained income along the chain andreturns the �tness value for each chromosome. As amaximization function, there is no need to modify theobjective function.

4.2.5. Selection mechanismsTwo mechanisms were employed to perform the selec-tion task in the proposed algorithm: the roulette wheeland tournament mechanisms. The designed algorithmwas con�gured in such a way that, for each roundof selection, 50% of the parents were selected by theroulette wheel mechanism, while the other 50% wereselected according to the tournament approach.

4.2.6. Termination criterionFor the considered algorithm in this research, thetermination criterion was set as reaching a limitednumber of iterations and generation number.

4.3. Setting the GA parametersIn order to set the parameters of the genetic algorithm,a sample problem was designed and, then, solved usingthe considered algorithm in MATLAB software. In thenext step, for each parameter within the GA, a set ofprede�ned discrete values (reported in Table 2) wasassumed and the problem was solved. Figure 2 showsthe contributions of crossover parameter variations inthe objective function value when other parameters arekept unchanged. As is observed, the objective functionis maximal when the crossover parameter takes a valueof 0.8. Further, Figure 3 depicts the contribution ofvariations in the mutation parameter at another levelof mutation parameter; �nally, Figure 4 considers thecontributions of mutation parameter variations in theobjective function value when other parameters arekept unchanged.

The proposed genetic algorithm encompasses �veoperating parameters. Considering the number and

Figure 2. Plot of variations in mutation level 1.

Figure 3. Plot of variations in mutation level 2.

Figure 4. Plot of variations in crossover parameter.

levels of parameters at which each parameter is tobe analyzed, one may conclude that it is impossibleto consider the contributions of variations in eachand any parameter, as well as interactions betweenthe parameters. Therefore, based on the principlesof the design of experiments, instead of performingeach and every experimental e�ort, only 80 tests wereconsidered (including single and coupled ones) to lookinto the e�ect of such variations on the �tness function.

Table 2. Di�erent levels of Genetic Algorithm (GA) parameters.

Parameter Number of levels Value

Maximum number of iterations 10 400Population size 10 60Crossover rate 9 0.8Mutation rate 9 0.4

Tournament selection size 5 4


Reported in Table 2 are the GA parameters, thenumber of investigated levels, and the values obtained.

4.4. Generating cases studies and setting theparameters

Most of the problems used for numerical tests investi-gated in the literature on SCND have been randomlygenerated. Accordingly, in this research, problem datawere randomly generated using uniform distribution incertain domains. Considering the number of param-eters in the model, no comprehensive reference couldbe found for benchmarking the parameters; thus, someof the parameters were generated randomly. There-fore, the parameters were generated with reference toOlivares-Benitez et al. (2010) as a valid reference [31].Furthermore, the problem size was assessed based ontwo criteria: number of periods and programmingnumber for each problem (Table 3).

For each problem, the programming number isequal to the sum of indices, i.e., the sum of digits inthe corresponding group code, which can be calculatedvia the following formula:

PN = i+ j + h+ k + l + g +m+ v + n+ c+ t:

In Table 4, all details of the problem can be observed.Furthermore, the information regarding the exactmathematical solution of GAMS software is presentedin this table.

Presumably, an increase in the time period of theproblem adds to the problem complexity and process-ing time, and the number of recycling centers a�ects thecomplexity of the problem. The designed numerical ex-amples were used to analyze these assumptions. Each

problem was designated by a two-digit number, withthe �rst digit from the left indicating the number ofperiods in the problem and the other digit re ecting thenumber of problems within the corresponding period.Regarding the problem type, an attempt was made toincorporate as many combinations as possible in eachperiod. Average processing times for the two-periodand three-period problems were found to be 136 and2863 seconds, respectively. As expected, the processingtime increased by 21 folds when the value of T changedfrom 2 to 3. Comparing the processing times obtainedfor problems 3.1 and 3.4 shown in Table 4, one caneasily �nd that processing time became doubled whenthe value of m index (number of recycling centers)changed from 1 to 2. These two comparisons revealthe e�ect of the parameter T on the sample problemprocessing time. Furthermore, this table may con�rmthe performance of the introduced sizing criterion forthe sample problems.

In problem 3.1, the processing time was 7200 sec-onds, i.e., the GAMS was not able to solve the problemin 2 hours and the reported objective value was merelythe best integer solution found after 2 hours of runningthe software. This problem number shows that, inparticular numerical examples, especially large ones,the software cannot solve the problem at an acceptabletime interval; therefore, alternative approaches areneeded.

Following the research, some sample problemswere solved with genetic algorithm. To do so, �rst ofall, the parameters of each exactly solved problem were�tted into the structure of metaheuristic algorithm soas to establish the same conditions. Subsequently, the

Table 3. Sample problem size assessment.

Problem Problem

Size Number ofperiods

Programmingnumber

Size Number ofperiods

Programmingnumber

Small 1 Any number Large 2 Greater than or equal 18Medium 2 Lower than 18 Large 3 Any number

Table 4. Solutions of two-period and three-period sample problems as obtained by GAMS software.

Examplenumber

Group code Solvetime

Objectivefunction

value

Examplenumber

Groupcode

Solvetime

Objectivefunction

value3.1 2113-1221-123 7200 556462 2.1 2113-1221-122 74 11796403.2 1113-1221-123 950 520647 2.2 3113-1221-122 133 10578653.3 2113-1121-123 2157 726932 2.3 3113-1231-122 282 6911553.4 2113-1211-123 3850 916650 2.4 3113-1331-122 86 3868333.5 1113-1111-123 160 1022315 2.5 3212-1331-122 105 538735

Average solution time fora three-period case

2863 Average solution time fora two-period case

136


sample problems were tested in both GAMS (as the ex-act solution) and MATLAB (as the heuristic solution).In order to reduce the randomness e�ect of the solvingalgorithm and prevent it from getting trapped withinpoor solutions, each sample was launched in MATLAB�ve times and the obtained results were averaged beforebeing reported as the �nal result. Table 5 reports theinformation about each sample.

The processing times were reported in seconds.Table 6 represents a summary of the above tabletogether with analytical results. Shown in this table arethe average values for each sample problem, standard

deviation within the results obtained for each sample,and the gap between the average solution and exactsolution for each sample. The gap was calculated bydividing the di�erence between the exact and approx-imate solutions by the exact one. Taking a quick lookat this table, one may �nd that the gap values weregenerally lower for two-period problems rather than thethree-period ones. Considering the gap values for thetwo-period and three-period problems, it is observedthat the value of the gap slightly increases by just 2%when the parameter T changes from 2 to 3. This iswhile, as mentioned before, such a change in parameter

Table 5. Values of the objective function and processing times for sample problems used with the solving algorithm.

Instance Iterationnumber

1 2 3 4 5 Instance Iterationnumber

1 2 3 4 5

3.1Solution 475375 476684 487625 489297 488839

2.1Solution 934420 930074 950265 944300 948471

Time 182 311 232 252 211 Time 19 16 17 24 18

3.2Solution 441857 451536 450551 439688 446 045

2.2Solution 920390 924900 914044 937830 918110

Time 201 179 175 185 197 Time 18 19 18 18 21

3.3Solution 649945 653505 644582 654642 646 721

2.3Solution 621060 631960 616550 609310 624950

Time 173 194 245 188 189 Time 19 19 16 20 20

3.4Solution 765027 791046 762100 787237 781 247

2.4Solution 328980 336650 340090 345040 322150

Time 173 213 175 263 214 Time 19 21 18 22 18

3.5Solution 793794 771966 805189 805266 799 156

2.5Solution 510835 502625 520040 503920 511390

Time 176 183 182 185 222 Time 18 22 18 20 19

Table 6. A comparison between the results of exact and metaheuristic solutions.

Instancenumber

GAMSoutput

Average ofMATLABoutputs

Standarddeviation of

MATLAB outputs

MATLAB andGAMS outputs

gap

Average gap(%)

3.1 556462 483564 6027 13.10

15.093.2 520647 445935 4130 14.343.3 726932 649879 3382 10.593.4 916650 777331 11014 15.193.5 1022315 795074 9755 22.22

2.1 1179640 941506 7407 20.18

12.402.2 1057865 923054 6648 12.742.3 691155 620766 6268 10.182.4 386833 334582 7213 13.502.5 538735 509762 5191 5.37

Average total gap13.74


Table 7. Results of robustness analysis of the proposed metaheuristic algorithm.

Examplenumber

Minimumvalue

Maximumvalue

Averageoutput

Standarddeviation

Standard deviationto average

output ratio(%)

3.1 475375 489297 483564 6027 1.2%3.2 439688 451536 445935 4130 0.9%3.3 646721 654642 649879 3382 0.5%3.4 765027 791046 777331 11014 1.4%3.5 771966 805266 795074 9755 1.2%2.1 930074 950265 941506 7407 0.7%2.2 918110 937830 923054 6648 0.7%2.3 609310 631960 620766 6268 1%2.4 322150 345040 334582 7213 2.1%2.5 520040 511390 509762 5191 1%

Average 1.1%

Figure 5. Curve of comparison of the objective functionbetween the exact and metaheuristic solutions.

T increased the processing time by about 2100% (21order of magnitudes). Hence, this obvious di�erence(2% versus 2100%) is well justi�able and a�ordable;indeed, it con�rms the signi�cant e�ect of the proposedmetaheuristic algorithm.

A comparison between the results of the proposedalgorithm and GAMS revealed that the algorithmfound better solutions within a reasonable time, whileGAMS failed to �nd the optimum solution in somecases. Further, GAMS required longer processing timesthan GA. Figures 5 and 6 show that GA outperformsGAMS.

Considering all comparisons and according toFigures 5 and 6, it can be suggested that the advantageso�ered by the proposed GA for the mathematicalmodel formulated in this research dominate outnumberits disadvantages, proving the e�ectiveness and capa-bility of the algorithm to solve not only two-periodand three-period problems, but also problems of higherprogramming periods. It is worth mentioning that,based on the criteria set by the decision-makers and thenature of the logistic network, di�erent measurementunits can be used to measure a period.

Figure 6. Logarithmic curve of comparison of theprocessing time between the exact and metaheuristicsolutions.

In Figures 5 and 6, the comparative resultsand processing times of the exact and GA algorithmoutputs are shown. The results con�rm the superiorperformance of the proposed algorithm, such that thepresented GA provides an e�cient approach to thesolution of the introduced problem.

By analyzing the algorithm robustness, the resultsof di�erent runs of the algorithm are presented inTable 7.

The results show that, in the worst case (i.e.,problem number 2.4), the standard deviation to aver-age output ratio was 2.1%, while the average ratio forall cases was as low as 1.1%, indicating the robustnessof the proposed algorithm.

5. Conclusion and recommendations

Today, organizations need to design integrateddistribution-collection networks because of various rea-sons imposed by governments, society, and competi-tors. Multi-period programming, multi-echelon struc-ture, and other design variables must be appropriately


taken into consideration before an applicable and close-to-reality model can be achieved. Furthermore, multi-period programming enhances the level of exibility.Every transpiration network needs to use di�erenttransportation modes, and the resent research consid-ers the choice of transportation mode in various pro-gramming periods. Designing such a network entailsmore e�orts from the researchers.

Considering these needs, this research startedby proposing a mathematical model under severalreasonable assumptions taken from the real world.The applicability of the proposed problem and thevalidity of the mathematical model were analyzed bysolving the problem in GAMS software. Solutionstructure and complexity of the mathematical modelwere further studied, concluding that the mathematicalmodel, especially medium- and large-sized cases, waspractically impossible to solve via exact approachesas it took an extremely long time to be solved.Accordingly, there is a need for a solving algorithmwith the capability of returning correct, rational, well-�tted, reliable solutions on an accepted time scale.By studying di�erent solution approaches and thestructure of the mathematical models and by meansof available resources and references, metaheuristicgenetic algorithm was found to be a proper approachto solving the research problem. GA was designedand coded in MATLAB software. Finally, based onthe comparison of the results obtained via the exactapproach and the metaheuristic algorithm outputs, themetaheuristic algorithm was found to be an e�ectiveand e�cient approach to solving the problem.

In order to enhance supply chain e�ciency, alogistics manager should consider various parameterswhen forming a logistics network. In order to meetparticular regulations, reuse products, or decreasewastes, some supply chains need reverse logistics; thispaper provides the logistics managers with a propertool for such supply chains. The results validated themodel and the performance of the proposed algorithm.

Based on the investigations and studies performedso far and considering the demands arisen in the realworld, the following recommendations can be proposedfor extending the mathematical model and getting itas close to the real world as possible:

1. Considering the idea of multi-objectivity within themodel, i.e., interactions among such objectives asmaximization of system responsiveness, minimiza-tion of negative environmental e�ects and deliverytime, etc., can provide subjects for future researchworks;

2. Investigation of the uncertainties associated withsuch parameters as customer demands within dif-ferent periods, transportation costs, and volume

of returned products may be performed in futurestudies;

3. Performance of other metaheuristic algorithms suchas TS and SA in this context can be compared tothat of GA in order to further validate it.

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Biographies

Alireza Eydi is an Associate Professor of IndustrialEngineering in the Faculty of Engineering at Universityof Kurdistan, Iran. He received his PhD from the De-partment of Industrial Engineering at Tarbiat ModaresUniversity, Tehran, Iran in 2009. His main areas ofteaching and research interests include supply chainand transportation planning and network optimizationproblems include routing and location problems onnetworks.

Saeed Fazayeli is a PhD student of Industrial En-gineering in the Faculty of Engineering at Universityof Kurdistan, Sanandaj, Iran. He received his MScdegree from Iran University of Science and Technology,Tehran, Iran. His main areas of research interestsare Location-Routing problems and multimodal trans-portation planning.

Hossein Ghafouri is an MSc graduate of IndustrialEngineering in the faculty of Engineering at Universityof Kurdistan, Sanandaj, Iran. He received his BSc fromthe Department of Engineering at University of PayamNour Tabriz, Tabriz, Iran in 2015. His thesis subjectrevolves around supply chain con�guration problem.

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Multi-period con guration of forward and reverse …scientiairanica.sharif.edu/article_21061_75fec80565c...and game theory problems. Di erent researchers have introduced various problems