A reverse logistics network for recovery systems and a robust metaheuristic solution approach

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The International Journal ofAdvanced Manufacturing Technology ISSN 0268-3768Volume 74Combined 9-12 Int J Adv Manuf Technol (2014)74:1393-1406DOI 10.1007/s00170-014-6045-7

A reverse logistics network for recoverysystems and a robust metaheuristic solutionapproach

Majid Eskandarpour, Ellips Masehian,Roya Soltani & Amirhossein Khosrojerdi

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ORIGINAL ARTICLE

A reverse logistics network for recovery systems and a robustmetaheuristic solution approach

Majid Eskandarpour & Ellips Masehian & Roya Soltani &Amirhossein Khosrojerdi

Received: 15 February 2012 /Accepted: 4 June 2014 /Published online: 1 July 2014# Springer-Verlag London 2014

Abstract One of the major concerns in a reverse supply chainmanagement (RSCM) system is dealing with returned products(due to being defective or obsolete) through a reverse logistics,such that the returned items reach their final destinations withminimum cost. In this paper, for managing returned products, acomprehensive seven-layer recovery network is designed, in-cluding primary customers, collection/redistribution centers, re-covery, recycling and disposal centers, and secondary cus-tomers. The network is mathematically modeled as a mixedinteger linear programming (MILP) model whose optimal solu-tion determines the proper collection and recycling centers forthe reverse and forward logistics of returned and recoveredproducts, such that the total cost is minimized. Since the prob-lem belongs to the network design class of problems which isNP-hard, the time for obtaining an optimal solution growsexponentially as the number of binary variables increases.Therefore, a new Tabu search-based heuristic method is devel-oped for computing optimal or near-optimal solutions for therecovery system. Also, the Taguchi experimental design tech-nique is employed for parameter tuning of the heuristic andcoming up with a robust design. The efficiency and effective-ness of the proposed heuristic method has been evaluatedthrough comparisons with a recently developed SA method, aswell as the global optimal solutions of the model. Experimentalresults showed that the new Tabu search-based approach

outperforms the SA method and has an average solution gapof 3.28 % with optimal solutions and a robustness of 2.18 %.

Keywords Closed-loop logistics . Reverse logistics .

Recovery system . Tabu searchmetaheuristic .

Taguchi robust design

1 Introduction

In today’s highly dynamic world, ongoing technological ad-vancements, scarcity of resources, high customer expectations,and competitive markets motivate and compel industries andorganizations to enhance their productivity and competitive-ness through improving their supply chain systems. One of themost important strategic decisions in a supply chain system—which directly affects the competitiveness of an enterprise—isdesigning its logistics network. A logistics network is the set ofsome entities with specific activities like producing, supplying,distributing, and consuming which have mutual connectionsthrough some “arcs” representing product flow channels.

The flow of products in a traditional logistic networkinitiates from producers or suppliers and goes to the end-customers through some (optional) intermediate facilities likewholesalers or retailers. However, in today’s highly interactivebusiness climate where marketing and customer feedback playa major role in the success of most producers, theabovementioned simplistic “unidirectional” model cannotsimulate most real-world requirements properly. Moreover,strict environmental regulations and savings caused byrecycling consumed products force many manufacturers torecollect their products and try to refurbish, recover, recycle,or dispose them safely and economically. Indeed, there is agrowing interest among companies in knowing about thefuture of their sold products. As a result, a secondary flow ofproducts can be considered which starts from the end-

M. Eskandarpour : E. Masehian (*) :R. Soltani :A. KhosrojerdiDepartment of Industrial Engineering, Tarbiat Modares University,Tehran, Irane-mail: [email protected]

M. Eskandarpoure-mail: [email protected]

R. Soltanie-mail: [email protected]

A. Khosrojerdie-mail: [email protected]

Int J Adv Manuf Technol (2014) 74:1393–1406DOI 10.1007/s00170-014-6045-7

Author's personal copy

customers and proceeds toward the suppliers through a num-ber of facilities such as collection/redistribution, recovery, orrecycling centers, which is known as “reverse logistics.”

Reverse logistics have been implemented in many indus-tries such as electronics, airplane, computers, and medicalequipments [1]. For instance, reverse logistics are extensivelyapplied in computer hardware industry: the IBM and Dellcompanies have implemented reverse logistics networks andtheir recovery systems make it easier for customers to refur-bish existing computers or buy new parts [2]. Grenchus et al.[3] reported that in order to support and enhance environmen-tal performance, some of the key components of reverselogistics network have been merged by the Global AssetRecovery Services (GARS) organization of IBM’s GlobalFinancing division [4]. Moreover, there are high turnoversand growing demands in consumer electronics (CE) industrysuch as in home entertainment segment. These types of indus-tries generate e-waste of the order of dozens of million tons,about one quarter of the world’s total waste [5].

In general, the following markets can be considered forimplementing reverse logistic systems and repair services [6]:

– Post-sale services and warranties– Collection and sorting of high-valued technological

products– Recovery of returned products– Recycling of hazardous materials for tests or

remanufacture

Products can be returned to manufacturers for variousreasons, such as commercial returns, returned warranties,and obsolescence of products [7]. Thus, to manage the supplychain system effectively, a recovery system is required forcontrolling the returned products and conducting their flowtoward appropriate facilities. The European Researchers’Group (Centre d’Études Européennes) who works on reverselogistics have defined the reverse logistic as “a process ofplanning, operating, and controlling the flow of raw materials,work in process materials, and final products, which streamfrom manufactures, distributors, or consumers towards suit-able recycle or disposal locations” [8].

Major activities in recovery systems may include thefollowing [9]:

1. Collecting used products from customers2. Determining the states of the returned products after

inspecting and separating them, and deciding whether ornot they are repairable based on economic and technolog-ical viewpoints

3. Discarding unrecoverable products4. Recovering the returned products to restore their potential

value5. Redistributing the recovered products

Depending on the nature and context of the industry inwhich the reverse logistics is applied, there might be differentrecovery processes. For instance, Srivastava classified recov-ery processes into repair, refurbish, remanufacture, and recy-cle activities [10]. In Wadhwa et al. [11], based on the level ofquality and the degree of disassembly, the recovery processesare classified as repairing, refurbishing, remanufacturing,recycling, and cannibalization processes.

There are a wide range of studies in the past decade whichconsider recovery processes in diverse applications such as ininventory control [12, 13], vehicle routing problems [14, 15],facility location problems [16–18], outsourcing [19, 20], pricingof returned products [9, 21], and localizing inspection centers ofreturned products [22]. In [23] the connection between recoverycenters and customers in reverse logistics has been realizedthrough applying information technology. Some researchersproposed reverse logistics networks based on recovery process-es. Du and Evans [7] developed a biobjective reverse logisticsnetwork model for post-sales services. A recycle network hasparticularly been proposed for carpet [24], tires [25], and LPGtank [26] industries. In some studies, researchers consideredmore than one processes in their models [27–29].

Following a comprehensive literature review on the reverselogistic network design models, we classified some recentcontributions to the field based on seven attributes ofremanufacturing, reuse, post-sale, recycle, closed-loop,open-loop, and types of layers. As shown in Table 1, mostof the studied works were related to remanufacturing models,and less works were associated with reuse and recycle, andonly one work was dedicated to post-sales. Interestingly, anincreasing number of researches have dealt with closed-looplogistic network design in recent years. Since the majority oflogistics network design problems are in the category of NP-hard problems [1], developing powerful heuristics andmetaheuristics to solve the proposed models are needed.

In this paper, a mixed integer linear programming (MILP)model is presented for mathematically expressing the reverselogistics network for recovery systems. The proposed modelcan be applied for various industries such as electronics,computer, automotive, and airplanes. Also, a robust Tabusearch (TS) metaheuristic method is presented to solve thismodel. The main contributions of this study are as follows:

– Developing a comprehensive seven-layer reverse closed-loop logistic network based on activities of a recoverysystemmentioned in [9], and with considering both directand indirect shipments from recovery centers to cus-tomers of recovered products

– Developing a mixed integer linear programming (MILP)model for the reverse logistics network

– Proposing a robust and efficient metaheuristic solutionmethod which outperformed the closest method in theliterature and also produced near-optimal solutions

1394 Int J Adv Manuf Technol (2014) 74:1393–1406


The research problem is described and the developed MILPformulation is presented in Section 2, and the proposed heuristicmethod for solving the model is described in Section 3. For

making the solutionmethod robust and efficient, an experimentaldesign is proposed in Section 4. Finally, conclusions and direc-tions for future research are presented in Section 5.

Table 1 Reviewed articles in logistics network design

Reference articles Remanufacture Reuse Post-sales Recycle Closed-loop Open-loop Types of layers

Demiral and Gokcen[30]

✓ ✓ Customer, distribution, manufacture, collection,disposal

Du and Evans [7] ✓ ✓ Collection, repair, supplier

Figueiredo andMayerle [25]

✓ ✓ Customer, collection, recycling

Jayaraman et al. [29] ✓ ✓ ✓ Manufacture, refurbishing center, recycler, andretailer

Kim et al. [31] ✓ ✓ Disassembly, disposal, refurbish, externalsupplier, collection

Ko and Evans [20] ✓ ✓ Customer, factory, distribution, collection, repair

Lee and Dong [32] ✓ ✓ Customer, factory, distribution/collection

Lee and Dong [33] ✓ ✓ Customer, factory, distribution/collection

Lee et al. [18] ✓ ✓ Collection, disassembly, processing, supplier

Listes [34] ✓ ✓ Manufacture, customer, disposal, recovery

Lu and Bostel [35] ✓ ✓ Customer, collection, recovery, factory

Min and Ko [36] ✓ ✓ Manufacture, warehouse, customer, repair

Mutha and Pokharel[27]

✓ ✓ Retailer, disposal, recycling, reprocessing,supplier, factory, spare markets

Pati et al. [37] ✓ ✓ Customer, dealer, warehouse, supplier andmanufacture

Pishvaee et al. [1] ✓ ✓ Customer, distribution, disposal, collection/distribution, manufacture/remanufacture

Pishvaee and Torabi[38]

✓ ✓ Plant, distribution center, customer, recovery,collection, recycling, material customer

Pishvaee et al. [39] ✓ ✓ Customer, collection, disposal, recovery

Qin and Ji [40] ✓ ✓ Customer, collection, recovery

Salema et al. [41] ✓ ✓ Manufacture, distribution, disassembly, customer

Sayed et al. [42] ✓ ✓ ✓ Supplier, manufacture, redistribution, disassembly,customer

Wang and Hsu [43] ✓ ✓ Supplier, customer, recycle, manufacture/remanufacture

Youngshen andShouyang [26]

✓ ✓ ✓ Manufacture, distribution, customer, centralizedreturn center, disposal

Primarycustomer l

Collection / Redistributioncenter i

Recovery center j

Recycling center h

Disposalcenter k

Customer s ofrecycled parts

Customer m of recovered products

1

2

2

2

3

3

4

3

Xli

Wik

Vih

Zij

Rim

Qjm

Oji

Ths

Fig. 1 Structure of the proposedseven-layer reverse network. Thenumbers indicate the order ofproducts’ flows, and dashedarrows represent reverse flows.The symbols represent decisionvariables as defined in Table 4

Int J Adv Manuf Technol (2014) 74:1393–1406 1395


2 Problem definition and mathematical model

We study a multilayer closed-loop supply chain network.Customers, collection/redistribution centers, disposal centers,recycling centers, customers of recycled parts, recovery cen-ters, and costumers of recovered products are all differentlayers of the network. The proposed reverse closed-loopseven-layer network is sketched in Fig. 1.

The figure shows that once used products reach acollection/redistribution center from primary customers, theyare sent for disposal, recycling, or recovery. Recycled prod-ucts are then sent to new customers, and recovered parts aresent to new customers either directly or indirectly via thecollection/redistribution center.

There are many advantages in applying multipurpose fa-cilities as they contribute to lower costs and higher utilizationof facilities for both forward and reverse flows [32]. Thus, amultipurpose facility in the form of a collection/redistributioncenter is also considered.

The function of a collection center is to collect returnedproducts and make decisions about assigning subsequent ap-plicable processes to these parts mainly based on their quality.A subsequent applicable process can be a disposal, recycling,or a recovery process. In fact, all returned products areinspected and divided into three groups based on their quality.

1. The set of products that are no more usable and must betransferred to disposal centers

2. The set of products comprised of parts still usable in otherproducts, which must be transferred to recycling centers

3. The set of products which can still be used after somerepairs, and must be transferred to recovery centers

After modifying products in the recovery centers, the re-covered products are delivered to customers in two ways:either through direct flows between recovery centers andcustomers, or through indirect flows so that the recoveredproducts are sent from recovery centers to redistribution cen-ters and afterwards to end-customers (as depicted in Fig. 1).Although the direct flow is more costly than the indirect flowdue to increased drop shipping costs realized by multiplepackage carriers (compared to truckload carriers), it has ad-vantages such as less delivery time for the network [44].

There are a number of assumptions in this model:

– The demands are deterministic.– The number, location, and capacity of recovery and dis-

posal centers are definite and fixed.– The percentages of returned products being sent to recov-

ery, disposal, or recycling centers are definite and fixed.

Before presenting the developed mixed integer linearprogramming model of the network, its parameters,

decision variables, objective function, and constraintsare introduced in Tables 2, 3, and 4, and in Eqs. (1)through (19).

Table 2 Indices used in the model

Index Description

L The set of primary customers

I The set of potential nodes ofcollection/redistribution centers

J The set of defined nodes of recovery centers

K The set of defined nodes of disposal centers

H The set of defined nodes of recycling centers

S The set of customers of recycled parts

M The set of customers of recovered products

Table 3 Parameters of the model

Parameter Description

d Average percentage of disposable products

e Average percentage of recyclable products

rl Number of returned products from the primary customer l

fi Fixed cost of setting up the collection/redistribution center i

gh Fixed cost of setting up the recycling center h

ρj Fixed cost of setting up the recovery center j

θk Fixed cost of setting up the disposal center k

cfli Transportation cost for a unit of returned product from theprimary customer l to the collection/redistribution center i

csij Transportation cost for a unit of recoverable product from thecollection/redistribution center i to the recovery center j(csji is for reverse flow)

ctik Transportation cost for a unit of disposable product from thecollection/redistribution center i to the disposal center k

crih Transportation cost for a unit of recyclable product from thecollection/redistribution center i to the recycling center h

cqjm Transportation cost for a unit of recovered product from therecovery center j to the customer m of recovered products

cuim Transportation cost for a unit of recovered product from thecollection/redistribution center i to the customer m ofrecovered products

cmhs Transportation cost for a unit of recycled part from therecycling center h to the customer s of recycled parts

cafi Capacity of the collection center i

casj Capacity of the recovery center j

catk Capacity of the disposal center k

carh Capacity of the recycling center h

carfi Capacity of the redistribution center i

PSs Penalty cost for a unit of unsatisfied recycled part of thecustomer s

PMm Penalty cost for a unit of unsatisfied recovered product of thecustomer m

SDm Demand of customer m for recovered products

MDs Demand of customer s for recycle parts



The proposed mathematical model of the network isas follows:

Min C ¼Xi∈I

f iY i þXh∈H

ghUhþXj∈ J

ρ jP jþXk∈K

θkDk

þXl∈L

Xi∈I

c f lirlX li þXi∈I

Xj∈ J

csijZij

þXi∈I

Xk∈K

ctikW ikþXi∈I

Xh∈H

crihV ih

þXj∈ J

Xi∈I

csjiOjiþXj∈ J

Xm∈M

cqjmQjm

þXi∈I

Xm∈M

cuimRimþXh∈H

Xs∈S

cmhsThs

þXs∈S

PSsTRs þXm∈M

PMmT Fm ð1Þ

subject to:Xi∈I

X li ¼ 1; ∀l∈L ð2Þ

Xk∈K

Wik ¼ dXl∈L

rlX li; ∀i∈I ð3Þ

Xh∈H

V ih ¼ eXl∈L


Xj∈ J

Zij ¼ 1−d−eð ÞXl∈L


Table 4 Decision variables of themodel, some of which are shownin Fig. 1

Variable Description

Zij Number of recoverable products transferred from thecollection/redistribution center i to the recovery center j

Wik Number of disposable products transferred from thecollection/redistribution center i to the disposal center k

Vih Number of recyclable products transferred from thecollection/redistribution center i to the recycling center h

Oji Number of recovered products transferred from the recoverycenter j to the collection/redistribution center i

Qjm Number of recovered products transferred from the recoverycenter j to the customer m of recovered products

Rim Number of recovered products transferred from thecollection/redistribution center i to the customer mof recovered products

Ths Number of recycled parts transferred from the recyclingcenter h to the customer s of recycled parts

TRs Number of unsatisfied recycled parts at the secondary customer s

TFm Number of unsatisfied recovered products at the secondarycustomer m

Yi 1 if a collection=redistribution center is opened at location i0 otherwise

�Uh 1 if a recycling center is opened at location h

0 otherwise

�Dk 1 if a disposal center is opened at location k

0 otherwise

�Pj 1 if a recovery center is opened at location j

0 otherwise

�Xli 1 if customer l is allocated to the collection center i

0 otherwise

�

1 … L + I 1 … I + J 1 … I + K 1 … I + H

1st segment: Xli , Yi 2nd segment: Zij 3rd segment: Wik 4th segment: Vjh , Uh

1 … H + S 1 … J + I + M 1 … I + M

5th segment: Ths , TRs 6th segment: Oji , Qjm , Rim 7th segment: TFm

Fig. 2 The encoding scheme of the proposed Tabu search approach



Xj∈ J

Oji ¼Xm∈M

Rim; ∀i∈I ð6Þ

Xj∈ J

Qjm þXi∈I

Rim þ T Fm ≥ SDm; ∀m∈M ð7Þ

Xh∈H

Ths þ TRs≥MDs; ∀s∈S ð8Þ

Xl∈L

X li≤ caf iY i; ∀i∈I ð9Þ

Xj∈ J

Oji≤car f i; ∀i∈I ð10Þ

Xi∈I

Zij≤ρ jcas j; ∀ j∈J ð11Þ

Xi∈I

W ik ≤θkcatk ; ∀k∈K ð12Þ

Xi∈I

V ih≤carhUh; ∀h∈H ð13Þ

Y i∈ 0; 1f g; ∀i∈I ð14Þ

Before: Segment 1 Segment 2 1 2 3 4 5 6 Segment 4 Segment 5 Segment 6 Segment 7

After: Segment 1 Segment 2 4 6 5 1 3 2 Segment 4 Segment 5 Segment 6 Segment 7

Fig. 3 The three-swap neighborhood procedure is applied to the third segment, in which the following elements are swapped consecutively: (1↔4),(2↔6), and (3↔5)

Tabu Search Procedure

1. Begin2. Initialize Max_iteration, Tabu_list_size, and N_size3. Define a tabu list for each segment4. Randomly generate a set of initial solutions of size |L| + |I| + |J| + |K| + |H| + |M| +|S|5. Consider the best solution as the initial solution6. Set the initial solution as the current solution and also the best solution7. Set NUM = 08. While (stopping condition is not met) do9. Stochastically search the neighborhoods of the current solution:10. Do the following steps N_size times

a. Randomly select a segment to be changedb. For each selected segment employ iterative swap neighborhood structure and generate a new

solutionEnd

11. If the best neighboring solution is tabu but smaller than the aspiration level12. choose it as the current solution13. Else choose the best non-tabu solution as the current solution14. End15. Update the tabu list16. If current fitness < best fitness17. best solution = current solution18. NUM = 019. Else NUM = NUM +1

End20. If NUM > pn21. Apply diversification22. NUM = 0

End23. Empty the tabu list

End

Fig. 4 Pseudocode of the proposed Tabu search method



Uh∈ 0; 1f g; ∀h∈H ð15Þ

ρ j∈ 0; 1f g; ∀ j∈J ð16Þ

θk∈ 0; 1f g; ∀k∈K ð17Þ

X li∈ 0; 1f g; ∀l∈L; i∈I ð18Þ

∀ Zij; Wik ; V ih; Oji; Qjm; Rim; Ths; TRs; T Fm≥0∀ i∈I ; j∈J ; k∈K; m∈M

ð19Þ

The objective function (1) is the sum of fixed costs(consisted of costs for opening collection and recycling cen-ters) and variable costs. Constraints (2) impose a single sourcepolicy from customers to collection centers in such a way thateach customer must be assigned to only one collection center.Constraints (3−5) secure the percentage of products allocatedto disposal, recycling, and recovery centers, respectively.Constraints (6) demonstrate the flow equilibrium betweenrecovery and redistribution centers. Constraints (7−8) ensurethat the maximum demands of recovered products or recycledparts are satisfied. The stockout situation is also consideredwhen recovery or recycling centers cannot respond to de-mands. Constraints (9−13) enforce the capacity of recovery,recycling, and disposal centers, and finally, constraints(14−19) impose binary and non-negativity restrictions oncorresponding decision variables.

3 The solution approach

Since reverse logistics network models are NP-hard [1], theircomputational complexity increases by a growth in the num-ber of customers, collection centers, disposal centers, recoverycenters, recycling centers, and secondary customers. As aresult, exact optimization tools and software are no longereffective, while metaheuristic approaches have been quitesuccessful. In this paper, the well-known Tabu search (TS)metaheuristic method is utilized to solve the model. TS wasinitially developed by Fred Glover [45, 46] for solving variouscombinatorial optimization problems, and its main idea is toprevent the search process from getting trapped in local opti-ma by avoiding cycling, that is, repeating the last few moves(hence, “tabu” moves) stored in a short-term memory calledTabu list. The main elements of the proposed tabu search aredescribed in the following subsections.

3.1 Solution encoding and evaluation

In order for a solution to be compatible with the tabu searchrepresentation, an encoding scheme is needed. Here, we usethe encoding scheme proposed by Gen et al. [47], in whicheach segment is composed of a random permutationrepresenting the priorities of the sources and depots. As shownin Fig. 2, our encoding scheme is composed of 7 segments,which control the 12 decision variables presented in Table 4,by locating Xli and Yi in the first segment, Zij in the secondsegment, Wik in the third segment, Vjh and Uh in the fourth

Table 5 Problem scenarios with different number of facilities

Scenario Collectionsites (I)

Recoverysites (J)

Recyclingsites (H)

Disposal sites(K)

Customers(L)

Secondarycustomers (m)

Secondarycustomersof parts (s)

1 10 3 3 4 15 15 6

2 30 4 4 5 40 40 8

3 50 5 5 6 70 70 10

4 80 6 6 7 100 100 12

5 100 7 7 8 120 120 14

Table 6 Transportation costs fora unit of flow between the net-work’s facilities

Flow route Parameter Uniformdistribution range

Customer center → collection/redistribution center cfli (8, 16)

Collection/redistribution center → recovery center csij (5, 12)

Collection/redistribution center → recycling center crih (3, 8)

Collection/redistribution center → disposal center ctik (3, 8)

Recycling center → secondary customer of material cmhs (3, 8)

Recovery center → secondary customer cqjm (10, 20)

Collection/redistribution center → secondary customer cuim (8, 16)



segment, Ths and TRs in the fifth segment, Oji, Qjm, and Rim inthe sixth segment, and finally TFm in the seventh segment.

The solutions are evaluated according to their objectivefunction value which is the minimization of both transporta-tion and fixed costs (Eq. (1)). The used evaluation procedureis taken from [47], in which a solution is decoded by deter-mining the values of variables and already open centers ineach segment. However, as the structure of the encoding inFig. 2 implies, for decoding the second, third, and forthsegments, the first segment must first be decoded, and fordecoding the fifth and seventh segments, the fourth and sixthsegments must first be decoded, respectively.

3.2 Initial design

To start with, the algorithm needs an initial network design asa seed for representing a point in the search space. This initialdesign can be either a current design or the best-known design.In the absence of an existing design, a set of random designsare initially generated and the one with the highest quality isselected. Afterwards, a set of “neighboring” designs(solutions) are generated from the current one to search aroundit for a better design.

3.3 Moves

Once an initial solution is created, for moving from that pointin the search space to another point (solution), a neighborhoodstructure is needed, through which a new set of solutions isgenerated by applying a perturbation to the current solution(design). For this purpose, we use a stochastic descent methodso that a set of neighboring solutions (with the size of N_size)are stochastically generated according to the multiswap neigh-borhood procedure. In this procedure, for each segment, arandom number in the [0, 1] interval is generated: if it isgreater than or equal to 0.5, then that segment is selected foremploying the multiswap neighborhood structure, in whichtwo or more priorities are changed stochastically.

To preserve diversity, the number of swaps is set to be highat the beginning of the algorithm, but decreases as the iterationnumber increases. Figure 3 illustrates the three-swap neigh-borhood structure employed to the third segment. After gen-erating the neighboring solutions, the best non-tabu and

admissible solution is selected from the neighboring solutionsand is set as the current solution.

3.4 Tabu list

Being an essential element of the Tabu search method, a tabulist of size k stores the last kmoves of the search and preventsrevisiting them. In fact, the tabu list is defined as a dynamicmemory which keeps the attributes of new solutions andblocks them to be repeated until a number of iterations(Tabu_list_size) have passed. After each iteration, the tabuattributes are released on a first-in-first-out basis, and the mostrecent move is appended to the tabu list. Since our encodingconsists of seven segments, seven tabu lists are consideredsimultaneously, and once a segment is chosen for beingswapped, its corresponding tabu list is checked and a moveis permitted only if it is not included in the tabu list.

3.5 Aspiration criterion

Aspiration criterion is a measure that provides more flexibilityto the search by overriding the tabu status of a move if it issatisfied. Whenever a better solution (move) is produced bythe attributes that already exist in the tabu list, that move isaccepted despite its tabu status, and its attributes are excludedfrom the tabu list.

3.6 Diversification

Through the diversification mechanism, the whole feasibleregion is widely explored with the purpose of finding a betternew solution. For this reason, any new solution is comparedwith the best solution found so far and replaces it as long as itsquality is better than the best existing solution. If for a

Table 7 Capacities of thefacilities Facility Parameter Uniform distribution range

Collection center cafi (70, 120)

Recovery center casj 1:5; 2:2½ � � Sum of demandsNumber of recovery centers

Disposal center catk 0:4; 0:7½ � � Sum of demandsNumber of disposal centers

Recycling center carh 0:4; 0:7½ � � Sum of demandsNumber of recycling centers

Redistribution center carfi (60, 110)

Table 8 Tabu searchparameters and theirlevels

Parameters Levels

1 2 3

Max_iteration 300 150 100

Tabu_list_size 3 5 8

N_size 100 200 300



specified number of iterations (e.g., pn) the best solution is notimproved, the search process is directed to another area of thesearch space by randomly generating a new solution. Asmentioned earlier, the multiswap structure is adopted as theneighborhood structure, and the number of swaps is high atearly iterations and decreases as the search proceeds.Whenever a diversification occurs, the number of swaps in-creases again to generate more diverse solutions.

3.7 Stopping criterion

To terminate the search process, one or more criteria areneeded. Here, we consider two termination criteria: the max-imum number of iterations searched by the tabu search (max_-iteration), and the variance of the solutions which should besmaller than a threshold. The whole search procedure is pre-sented in Fig. 4.

4 Experimental design

In order to evaluate the performance of the proposed Tabusearch (TS) method, several network instances were designedand solved. For this purpose, two sets of experiments wereconducted. In the first set, the proposed TS was comparedwith an existing simulated annealing (SA) method taken fromPishvaee et al. [39], and in the second set, the TS was com-pared with the global optimums of the MILP model exactlysolved by the CPLEX 12.4 optimization package. The TS andSA methods were coded in C++ and run using the generatedinstances described below. All experiments were executed ona 2 GHz Intel CPU with 2 GB of RAM.

4.1 Data generation

In order to test the developed method, 5 different scenarioswere considered (as shown in Table 5), and 4 random in-stances were designed for each scenario, resulting in 20 testproblems. As mentioned earlier, the proposed reverse logistics

network model contains seven layers, and the products flowbetween each two layers with a certain cost obtained from auniform distribution, as presented in Table 6.

For the fifth and seventh segments of the encoding, penal-ties of each unsatisfied demand (PSs and PMm) take theirvalues from a uniform distribution over integers between 10and 20. Customer demands for products returns (rl) are pickedfrom the interval [10, 50], demands for recycled parts (MDs)are uniformly distributed in the interval [50, 100], and de-mands for recovered products (SDm) are distributed over theinterval [10, 40]. Opening costs for each collection/redistribution center (fi) and for each recycling center (gh)are taken from uniform distributions over the intervals of[5,000, 10,000] and [1,000, 2,000], respectively. Also, collec-tion, recovery, recycling, disposal, and redistribution centersare capacitated with limits generated from uniform distribu-tions presented in Table 7.

4.2 Parameter setting

Prior to implementing the proposed TS and solving the devel-oped networkmodel, the algorithm’s parameters are calibratedthrough the Taguchi experimental design method to achievebetter performance. We investigated the best values of threemain parameters of the Tabu search, namely Max_iteration,Tabu_list_size, and N_size, by considering different probablescenarios for each one. Table 8 presents the algorithm’s pa-rameters (i.e., control factors) and their possible levels.

Table 9 The orthogonal arrayL9(3

3) Experiment Max_iteration Tabu_list_size N_size S/N ratio Mean

1 1 1 1 −106.680 191,046

2 1 2 2 −106.578 189,110

3 1 3 3 −106.523 188,082

4 2 1 2 −106.711 191,620

5 2 2 3 −106.652 190,349

6 2 3 1 −106.807 193,495

7 3 1 3 −106.706 191,571

8 3 2 1 −106.843 194,265

9 3 3 2 −106.769 192,700

Table 10 Response table for signal-to-noise (S/N) ratios (smaller isbetter, shown in boldface)

Level Max_iteration Tabu_list_size N_size

1 −106.594 −106.699 −106.7772 −106.723 −106.691 −106.6863 −106.773 −106.700 −106.627Delta 0.179 0.009 0.150

Rank 1 3 2



An advantage of the Taguchi method over otherexperimental design methods is that it can investigatea large number of parameters with just a small numberof experiments by arranging the noise and design fac-tors in an orthogonal array. The appropriate orthogonalarray is determined by considering the overall degree offreedom (DOF), which in our case is 2×3+1=7,consisting two degrees of freedom for each three-levelfactor, and one for the total mean. As a result, the mostfit standard Taguchi orthogonal array would be L9(3

3),as shown in Table 9, where the control factors appear inthe columns of the orthogonal array and each rowrepresents a unique combination of their levels. Notethat the Taguchi method has filtered 18 out of the 33=27 full factorial combinations.

Nine experiments were conducted according to the Taguchischeme. For each experiment, the 20 test problems eachwith 5replications were run and the results were transformed intosignal-to-noise (S/N) ratios using (20), in which yi is the valueof the parameter in the ith test of a combination. The resultingratios are also presented in Table 9.

S=N ratio : η j ¼ −10⋅log1

N

Xi¼1

N

y2i

!decibels½ � ð20Þ

Table 10 demonstrates the value of S/N ratio for each levelof the parameters. In this case, the level that has the smallestS/N ratio is selected for each parameter (shown in boldface).The results are also depicted against each parameter in Fig. 5.Consequently, the analysis reveals that the best levels for the

parametersMax_iteration, Tabu_list_size, andN_size are 300,5, and 300, respectively.

To examine the statistical significance of the factors, ananalysis of variance (ANOVA) is carried out at α=0.05 sig-nificance level as shown in Table 11. The results show thatMax_iteration and N_size have significant impacts on therobustness of the proposed tabu search method since theyhave very small p values.

4.3 Computational results

For testing the efficiency and effectiveness of the proposedrobust TS against recent related methods, we compared it witha simulated annealing method proposed by Pishvaee et al.[39]. Their method was developed to solve a four-layer re-verse logistic model (consisting customers, and collection,disposal and recovery centers), which was the closest modelin the literature to our seven-layer model. Actually, we coded

Me

an

of

SN

rati

os

321

-106.60

-106.65

-106.70

-106.75

-106.80321

321

-106.60

-106.65

-106.70

-106.75

-106.80

Maxiteration Tabu list

n_size

Main Effects Plot (data means) for SN ratios

Signal-to-noise: Smaller is better

Fig. 5 Average S/N ratio plots

Table 11 Analysis of variance for S/N ratios

Source DOF Sum ofsquares (SS)

Mean squares(MS)

F P value

Max_iteration 2 0.051153 0.025576 532.83 0.00187

Tabu_list_size 2 0.000141 0.000071 1.48 0.40323

N_size 2 0.034060 0.017030 354.79 0.00281

Error 2 0.000096 0.000048

Total 8 0.085449

F0.05(2, 2)=19.2 (threshold in corresponding F distribution)



their method, and solved 20 four-layer test problems (4 ran-domly generated problems for each scenario) generated ac-cording to the scheme described in Section 4.1, by bothmethods. Table 12 shows the experimental results, which areillustrated in Fig. 6.

The two methods were compared in terms of the gap ofobjective values as defined in (21) and reported in Table 12,which shows that in all instances the Gap was negative,meaning that the proposed TS found better values for theobjective function Z than the SA method.

Gap %ð Þ ¼ min ZTSð Þ−min ZSAð Þmin ZSAð Þ � 100 ð21Þ

The robustness of the TS method reflects the variations inobtained objective function values in different runs and iscalculated by (22) and reported in Table 12. Since the averagerobustness percentage is about 3.15±2.12, it can be concludedthat the method is sufficiently robust.

Robustness %ð Þ ¼ max ZTSð Þ−min ZTSð Þmin ZTSð Þ � 100 ð22Þ

We further tested the efficiency of the TS by solving20 new instances of seven-layer reverse logistics problems

(4 random instances for each of the 5 scenarios inTable 5) and compared the results with those obtainedfrom solving their respective mixed integer linear

Table 12 Objective function values and runtimes of the proposed TS and the SA in [39]

Scenario Test no. TS (proposed) SA ([39]) TS–SA Gap (%) TS Robustness (%)

Z Time Z Time

Min Avg. Max Min Avg. Max Min Min

1 1 36,119 36,119 36,119 0.86 0.92 1.03 36,132 0.25 −0.04 0.00

2 38,916 38,924 38,928 0.86 0.91 1.03 38,966 0.24 −0.13 0.03

3 47,821 47,838 47,862 0.92 0.97 1.09 47,984 0.25 −0.34 0.09

4 53,213 53,222 53,230 0.89 0.98 1.09 53,232 0.24 −0.04 0.03

2 5 88,642 89,049 89,977 4.52 4.53 4.55 88,897 2.20 −0.29 1.51

6 82,664 83,900 84,354 4.41 4.43 4.45 83,980 2.19 −1.57 2.05

7 111,904 113,163 115,014 4.53 4.74 5.33 113,824 2.27 −1.69 2.78

8 93,061 95,166 96,963 4.64 4.97 6.17 93,526 2.22 −0.50 4.19

3 9 158,694 162,139 167,301 13.19 14.09 17.48 160,601 9.39 −1.19 5.42

10 158,880 162,780 167,104 15.33 15.37 15.44 160,167 11.77 −0.80 5.18

11 154,713 157,586 162,446 13.07 14.45 17.45 156,558 9.42 −1.18 5.00

12 138,627 139,593 143,244 13.00 13.05 13.09 139,711 9.27 −0.78 3.33

4 13 232,531 238,046 242,370 31.84 38.15 52.47 235,805 31.58 −1.39 4.23

14 212,595 218,016 223,206 28.08 31.00 42.09 215,265 31.28 −1.24 4.99

15 216,957 219,733 226,089 28.20 29.36 33.19 218,686 31.52 −0.79 4.21

16 257,335 261,949 267,663 28.99 35.05 43.83 259,437 31.70 −0.81 4.01

5 17 253,900 260,415 272,080 42.45 52.16 69.69 257,422 56.41 −1.37 7.16

18 278,670 282,224 286,049 42.72 45.62 49.78 278,969 57.70 −0.11 2.65

19 262,783 269,504 275,497 42.45 43.22 43.74 263,030 56.70 −0.09 4.84

20 258,238 259,973 261,795 42.80 50.03 70.28 260,003 56.55 −0.68 1.38

(a)

(b)Fig. 6 a Percentage of Gaps between the objective function valuesobtained by TS and SA. b Comparison of runtimes of the two methods



programming models to their optimality using the CPLEXsolver. The outputs are presented in Table 13 and show that thesolution gap of the TS with optimal solutions was averagely2.67 %, which is significantly tight, and the average robust-ness percentage was about 2.06±1.68, which is within satis-factory limits. The results of gap and robustness percentagesare plotted in Fig. 7.

It should be noted that for large inputs (e.g., scenarios 4and 5), the optimal solutions were computed in much longertime than the maximum computation time of the TS, which wasabout 337 s. Also, the memory requirement for exact optimiza-tion was far beyond than that of the metaheuristic method.

4.4 Managerial implications

One of the main motivations for this study has been theincreasing importance of reverse logistics and product recov-ery systems in the current competitive world. Along the samelines, because of legal requirements, environmental protec-tion, and also related economic benefits, an increasing numberof companies such as General Motors, Kodak, and Xeroxfocused on managing their reverse logistics and recoveryactivities and achieved significant successes in this area.By incorporating such activities as collection, disposal,recycling, recovery, and redistribution, reverse logistics

Table 13 Results of solving the sample problems by the proposed TS and the CPLEX optimization package

Scenario Test no. TS CPLEX Gap (%) Robustness (%)

C Time

Min Avg. Max Min Avg. Max C Time

1 1 41,737.5 41,762.8 41,788.7 3 3 3 41,427.5 1 0.75 0.12

2 41,476.0 41,486.6 41,515.6 3 3 3 41,476.0 2 0.00 0.10

3 45,539.2 45,571.1 45,614.5 3 3 3 45,389.2 1 0.33 0.17

4 45,551.9 45,555.9 45,572.0 3 3 3 45,551.9 1 0.00 0.04

2 5 100,507.0 100,863.9 101,464.7 77 80 89 98,507.0 18,355 2.03 0.95

6 103,739.0 103,990.7 104,542.4 68 73 76 102,239.0 773 1.47 0.77

7 105,225.0 105,424.6 105,739.9 71 72 76 102,825.0 145 2.33 0.49

8 100,088.5 100,610.8 101,383.7 76 82 88 99,138.5 588 0.96 1.29

3 9 166,988.0 167,715.2 173,524.0 147 149 154 163,988.0 18,079 1.83 3.91

10 170,658.0 171,951.3 174,529.6 146 166 170 163,758.0 18,117 4.21 2.27

11 173,213.0 175,273.7 176,563.8 142 159 171 167,813.0 18,079 3.22 1.93

12 165,247.0 165,912.4 169,337.4 139 154 166 159,847.0 18,122 3.38 2.48

4 13 253,961.0 258,853.0 261,370.0 321 341 354 242,385.0 18,056 4.78 2.92

14 256,577.0 258,538.0 263,040.0 375 391 395 247,553.0 18,036 3.65 2.52

15 252,531.0 254,050.0 260,359.0 297 303 331 243,505.0 18,052 3.71 3.10

16 254,770.0 257,040 263,458.0 386 399 407 243,583.0 18,025 4.59 3.41

5 17 315,862.0 317,419.0 330,348.0 467 482 489 300,434.0 18,002 5.14 4.59

18 306,303.0 309,733.0 314,836.0 398 406 431 296,185.0 18,001 3.42 2.79

19 316,070.0 319,620.0 324,707.0 436 454 457 303,072.0 18,002 4.29 2.73

20 306,270.0 308,959.0 320,170.0 439 457 462 296,414.0 18,001 3.33 4.54

Fig. 7 Percentage of Gaps andRobustness of the objectivefunction values obtained by theTS



systems offer a more comprehensive and realistic view towardsupply chain management than conventional direct-flow net-works. Used products can be recovered in a variety of ways indealing with reverse supply chain management.

The outcomes of this paper are applicable to manufacturersor distributors who are involved in handling returned productsfrom their customers. Such an after-sales service is increas-ingly gaining popularity, and nowadays is widely practiced inindustries related to electronics, computers, carpets, tires, LPGtanks, etc.

From the applied point of view, the proposedmodel is well-matched with the characteristics of automobile, petroleum,and high-tech electronic manufacturing industries and theirdecision-making environments. On one hand, the proposedmathematical model can help supply chain managers anddecision makers to design a comprehensive reverse logisticnetwork which accommodates numerous variables and con-straints, and on the other hand the proposed robust solutionapproach can help them in yielding optimal or near-optimalsolutions in a reasonable amount of time, while handling alarge number of integer variables. In short, through the out-comes of this research, designers can easily deal with thereverse logistic network design considerations of their supplychain, while decreasing high computational costs.

5 Conclusions

Dealing with returned products through a reverse logistics isthe main concern in a reverse supply chain managementsystem. There are different recovery options, each of whichaffects the configuration of a reverse logistic network. In thispaper, it has been tried to incorporate various recovery optionsfor managing recovery systems through a seven-layer reverselogistics network, which considers primary customers, sec-ondary customers, collection/redistribution, recovery,recycling, and disposal centers. In order to design an opti-mized reverse logistic network and to choose an efficientrecovery option, a novel mixed integer linear programming(MILP) mathematical model is proposed. Due to its NP-hardness, the model cannot be solved in polynomial time forlarge instances. Therefore, as a remedy, a priority-based Tabusearch (TS) metaheuristic approach is proposed to solve themodel. The results were compared to the optimal solutionsobtained from solving the MILP model by the CPLEX solver.Comparisons showed that the divergence (gap) between thesolutions of the TS and optimal approaches were about2.67 %, which is quite minor. Also, the variation in obtainedobjective function values in different runs of the TS was about2.05 % in average, which confirm the robustness of themethod. The parameters of the proposed TS method weretuned using the Taguchi’s robust design methodology, and

the solution approach was validated by comparing its resultswith a simulated annealing (SA) method proposed byPishvaee et al. [39] in solving a four-layer recovery system,which outperformed it.

A typical issue in reverse logistics is the uncertainties withrespect to quality and quantities of the returned products, andthus, incorporating such an assumption can be a subject offuture research. Furthermore, one may be interested in con-sidering multiobjective models, such as a combination ofresponsiveness and reliability of the network as the objectivefunction.

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A reverse logistics network for recovery systems and a robust metaheuristic solution approach

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