a tabu search heuristic for redesigning a multi-echelon supply chain network over a planning horizon

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A tabu search heuristic for redesigning a multi-echelon supply chain network over a planning horizon M.T. Melo a,b , S. Nickel c,d , F. Saldanha-da-Gama b,e,n a Business School, Saarland University of Applied Sciences, D 66123 Saarbr¨ ucken, Germany b Operations Research Center, University of Lisbon, P 1749-016 Lisbon, Portugal c Karlsruhe Institute of Technology, D 76128 Karlsruhe, Germany d Fraunhofer Institute for Industrial Mathematics, D 67663 Kaiserslautern, Germany e Department of Statistics and Operations Research, University of Lisbon, P 1749-016 Lisbon, Portugal article info Article history: Received 8 July 2010 Accepted 19 November 2011 Available online 26 November 2011 Keywords: Supply chain network redesign Facility relocation Metaheuristics Tabu search Strategic oscillation abstract This paper addresses the problem of redesigning a supply chain network with multiple echelons and commodities. The problem belongs to a comprehensive class of network redesign problems previously introduced in the literature. Redesign decisions comprise the relocation of existing facilities to new sites under an available budget over a finite time horizon, the supply of commodities by upstream facilities, the inventory levels at storage facilities, and the flow of commodities through the network. The problem is modeled as a large-scale mixed-integer linear program. Feasible solutions are obtained by using a tabu search procedure that explores the space of the facility location variables. The latter prescribe the time periods in which changes in the network configuration occur. They are triggered by the setup of new facilities, which operate with capacity transferred from the existing facilities, and by closing the latter upon their entire relocation. As the problem is highly constrained, infeasible solutions with excess budget are allowed during the course of the search process. However, such solutions are penalized for their infeasibility. Computational experiments on realistically sized randomly generated instances indicate that this strategic oscillation scheme used in conjunction with tabu search performs very well. & 2011 Elsevier B.V. All rights reserved. 1. Introduction The redesign of a supply chain network is a complex decision- making process that arises in supply chain management. In this context, facility relocation is a strategic planning problem of vital concern to many companies: it provides the infrastructure of the supply chain and has a significant impact on other subsequent managerial decisions. According to Ballou (2001), the reconfigura- tion process can result in a 5–15% reduction of the logistics costs. Chandra and Grabis (2007) and Hammami et al. (2008) identify the key triggers for supply chain network redesign. The former authors also review case studies in the automotive and retail industries. Mergers, acquisitions, strategic alliances, and the removal of trade barriers between nations are among the most common factors leading to network reconfiguration. In this paper, we address the problem of redesigning a multi- echelon supply chain network. The decisions to be made concern: (i) the relocation of existing facilities to new sites through the gradual transfer of capacities over a multi-period horizon; (ii) the investment of an available budget for facility relocation, estab- lishing new facilities and closing existing facilities; (iii) the amount of commodities to be supplied by upstream facilities; (iv) the inventory levels at storage facilities; and (v) the flows of multiple commodities through the network. These decisions must be taken so as to satisfy customer demands and other supply chain specific constraints over a time horizon while minimizing the total net supply chain cost. The latter includes fixed and variable costs associated with supply, inventory holding, trans- portation, and facility operating costs. In all periods of the planning horizon, revenue is generated by the fraction of the available budget that is not invested in facility relocation. Our problem belongs to a comprehensive class of network redesign problems introduced by Melo et al. (2006). This class generalizes classical multi-period facility location models and in addition, it captures various essential aspects of supply chain network design (SCND) under a relocation scenario, such as: Facilities may be categorized into echelons, although this is not mandatory. Moreover, any network configuration can be modeled as no restrictions on the number of echelons and on the type of facilities (e.g. plants, warehouses, cross-docks, etc.) are imposed. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics 0925-5273/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2011.11.022 n Corresponding author at: University of Lisbon, P 1749-016 Lisbon, Portugal. Tel.: þ351 217 500 010. E-mail address: [email protected] (F. Saldanha-da-Gama). Int. J. Production Economics 136 (2012) 218–230

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Page 1: A tabu search heuristic for redesigning a multi-echelon supply chain network over a planning horizon

Int. J. Production Economics 136 (2012) 218–230

Contents lists available at SciVerse ScienceDirect

Int. J. Production Economics

0925-52

doi:10.1

n Corr

Tel.: þ3

E-m

journal homepage: www.elsevier.com/locate/ijpe

A tabu search heuristic for redesigning a multi-echelon supply chain networkover a planning horizon

M.T. Melo a,b, S. Nickel c,d, F. Saldanha-da-Gama b,e,n

a Business School, Saarland University of Applied Sciences, D 66123 Saarbrucken, Germanyb Operations Research Center, University of Lisbon, P 1749-016 Lisbon, Portugalc Karlsruhe Institute of Technology, D 76128 Karlsruhe, Germanyd Fraunhofer Institute for Industrial Mathematics, D 67663 Kaiserslautern, Germanye Department of Statistics and Operations Research, University of Lisbon, P 1749-016 Lisbon, Portugal

a r t i c l e i n f o

Article history:

Received 8 July 2010

Accepted 19 November 2011Available online 26 November 2011

Keywords:

Supply chain network redesign

Facility relocation

Metaheuristics

Tabu search

Strategic oscillation

73/$ - see front matter & 2011 Elsevier B.V. A

016/j.ijpe.2011.11.022

esponding author at: University of Lisbon, P

51 217 500 010.

ail address: [email protected] (F. Saldanha-da-G

a b s t r a c t

This paper addresses the problem of redesigning a supply chain network with multiple echelons and

commodities. The problem belongs to a comprehensive class of network redesign problems previously

introduced in the literature. Redesign decisions comprise the relocation of existing facilities to new sites

under an available budget over a finite time horizon, the supply of commodities by upstream facilities, the

inventory levels at storage facilities, and the flow of commodities through the network. The problem is

modeled as a large-scale mixed-integer linear program. Feasible solutions are obtained by using a tabu

search procedure that explores the space of the facility location variables. The latter prescribe the time

periods in which changes in the network configuration occur. They are triggered by the setup of new

facilities, which operate with capacity transferred from the existing facilities, and by closing the latter upon

their entire relocation. As the problem is highly constrained, infeasible solutions with excess budget are

allowed during the course of the search process. However, such solutions are penalized for their

infeasibility. Computational experiments on realistically sized randomly generated instances indicate that

this strategic oscillation scheme used in conjunction with tabu search performs very well.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The redesign of a supply chain network is a complex decision-making process that arises in supply chain management. In thiscontext, facility relocation is a strategic planning problem of vitalconcern to many companies: it provides the infrastructure of thesupply chain and has a significant impact on other subsequentmanagerial decisions. According to Ballou (2001), the reconfigura-tion process can result in a 5–15% reduction of the logistics costs.Chandra and Grabis (2007) and Hammami et al. (2008) identifythe key triggers for supply chain network redesign. The formerauthors also review case studies in the automotive and retailindustries. Mergers, acquisitions, strategic alliances, and theremoval of trade barriers between nations are among the mostcommon factors leading to network reconfiguration.

In this paper, we address the problem of redesigning a multi-echelon supply chain network. The decisions to be made concern:(i) the relocation of existing facilities to new sites through thegradual transfer of capacities over a multi-period horizon; (ii) the

ll rights reserved.

1749-016 Lisbon, Portugal.

ama).

investment of an available budget for facility relocation, estab-lishing new facilities and closing existing facilities; (iii) theamount of commodities to be supplied by upstream facilities;(iv) the inventory levels at storage facilities; and (v) the flows ofmultiple commodities through the network. These decisions mustbe taken so as to satisfy customer demands and other supplychain specific constraints over a time horizon while minimizingthe total net supply chain cost. The latter includes fixed andvariable costs associated with supply, inventory holding, trans-portation, and facility operating costs. In all periods of theplanning horizon, revenue is generated by the fraction of theavailable budget that is not invested in facility relocation.

Our problem belongs to a comprehensive class of networkredesign problems introduced by Melo et al. (2006). This classgeneralizes classical multi-period facility location models and inaddition, it captures various essential aspects of supply chainnetwork design (SCND) under a relocation scenario, such as:

Facilities may be categorized into echelons, although this isnot mandatory. Moreover, any network configuration can bemodeled as no restrictions on the number of echelons and onthe type of facilities (e.g. plants, warehouses, cross-docks, etc.)are imposed.
Page 2: A tabu search heuristic for redesigning a multi-echelon supply chain network over a planning horizon

M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230 219

Facility relocation can be planned under fairly general condi-tions including opening new facilities and closing existingfacilities. � Non-invested financial funds on network reconfiguration gain

interest and can be spent later in the time horizon.

� Commodities can flow through the network between any pair

of facilities (e.g. directly from upstream sources to customers;among facilities in the same echelon).

� Different facilities may hold various commodities in stock.

The mathematical models proposed in Melo et al. (2006) couldbe solved to optimality for small and medium-sized probleminstances using commercial optimization software within a rea-sonable time limit. However, due to the combinatorial nature ofthe problem, it becomes very difficult to solve to optimalityinstances with many locations, customers, and commodities overan extended planning horizon. In particular, when re-optimiza-tion is required for performing ‘‘what-if’’ analyses, a problem issolved repeatedly and the computational burden becomes aneven more important issue. In these cases, heuristic methodsprovide a viable alternative.

The goal of this paper is to present an efficient procedure tofind good feasible solutions to realistically sized multi-periodSCND problems. We propose a tabu search heuristic that exploresthe space of the binary facility location variables. During thecourse of the search, the heuristic presents a strategic oscillationbehavior that permits alternating between feasible and infeasiblesolutions. The latter solutions are achieved by exceeding thebudget available for network reconfiguration over the planninghorizon, and are penalized for their infeasibility.

Our study is – to the best of our knowledge – the first attemptto investigate the suitability of tabu search for tackling a large-scale multi-period supply chain network redesign problem. Com-parisons with a sophisticated rounding procedure developed byMelo et al. (2011), which is based on the linear relaxation of theproblem at hand, indicate that the new heuristic finds feasiblesolutions of superior quality within reasonable time for randomlygenerated test instances of realistic size.

The remainder of the paper is organized as follows. In the nextsection we briefly review the literature dedicated to comprehen-sive location models for SCND and to the application of tabusearch to solve them. In Section 3 we present a mathematicalformulation of the problem that stems from the modeling frame-work developed by Melo et al. (2006). The tabu search heuristic isdetailed in Section 4, followed by an analysis of the computa-tional results in Section 5. We conclude with a few generalremarks and present future research directions.

2. Literature review

Over the last decades, facility location decisions have attracteda great deal of attention from researchers. In recent years,increasing attention has also been paid to the interaction of thesedecisions with key features to strategic supply chain planningsuch as supplier selection, production planning, technologyacquisition, inventory planning, transportation mode selection,and vehicle routing. The importance of integrating locationdecisions with other decisions relevant to SCND has been under-lined by Daskin et al. (2005). Economic globalization has alsoprompted the development of more comprehensive facility loca-tion models as shown by the surveys of Goetschalckx et al. (2002)and Meixell and Gargeya (2005). A recent survey by Melo et al.(2009) highlights the contributions that capture various aspectsof SCND in conjunction with facility location. Among the mostcomprehensive studies are Cordeau et al. (2008), Cordeau et al.

(2006), Thanh et al. (2008), Vila et al. (2006), and Wilhelm et al.(2005). Nevertheless, several research directions still requireintensive research. In particular, models addressing the designof multi-commodity, multi-echelon supply chain networksthrough determining the timing of facility locations, expansionsand relocations over an extended time horizon have receivedconsiderably less attention than their static counterpart (seeBallou, 2001). As shown by Melo et al. (2009) and Melo et al.(2011), within this problem class, focus has been mainly given tosimple networks with at most two facility layers and a singlecommodity that flows between adjacent echelons to satisfycustomer demands.

Multi-period SCND problems place increased demands onmathematical tools used to solve them. In recent years, significantgains have been made in the size and complexity of the problemsthat can be solved with classical mathematical programmingtechniques (e.g. branch-and-bound and decomposition methods),mainly due to increased computing power. However, time andcomputing resources to solve such problems repeatedly in prac-tical applications become prohibitive. This situation has given riseto a multitude of heuristic solution techniques that seek toprovide good approximate solutions within a reasonable amountof time. As the solution methodology that we will describe inSection 4 uses tabu search, we will focus the remainder of thissection on the literature dedicated to the application of thistechnique in the context of facility location and strategic supplychain planning.

Over the past two decades metaheuristics have becomeimportant tools for solving various combinatorial problemsencountered in many practical settings. Among the differentmetaheuristic methodologies that exist, Tabu Search (TS) (seee.g., Glover and Laguna, 1997) has become a very popularapproach as it identifies high-quality solutions to many problems.Although the TS literature is very rich, only a few papers addressthe application of this master strategy to solve facility locationproblems in the context of SCND. Wang et al. (2003) consider avariant of the p-median problem, where the decisions to opennew facilities and to close existing facilities are subject to abudget constraint. Three heuristic approaches are developed: agreedy interchange, a TS and a Lagrangian relaxation heuristic.Numerical experiments indicate that TS generates the mostsatisfactory results in terms of solution quality. For the combina-tion of depot location and vehicle routing decisions in a singlemodel, Tuzun and Burke (1999) and Albareda-Sambola et al.(2005) developed efficient TS algorithms. The latter authorsconsider strategic oscillation in the implementation of their TSheuristic, a technique that is used when intermediate infeasiblesolutions are allowed to be visited during the search process.Caballero et al. (2007) describe a case study in Andalusia (Spain)for the location of incineration plants and the design of vehicleroutes to collect animal waste from slaughterhouses. A multi-objective model is developed, which comprises economic andsocial objectives, and is solved with TS using an adaptive memoryprocedure.

Recently, Lee and Dong (2008) presented a TS approach for thedesign of a two-echelon network. The proposed model includessome practical elements of SCND such as the direct shipment of asingle commodity from plants to customers. Moreover, locationdecisions concern both plants and warehouses. Keskin and Ulster(2007) also addressed a two-echelon SCND problem but withmultiple commodities. Among various heuristics aimed towardssolving this problem, TS was preferred in terms of both solutionquality and computational time. In some cases, TS could evenidentify optimal solutions. In the context of reverse logistics, Araset al. (2008) developed a mixed-integer nonlinear model to findthe optimal locations of collection centers, the number of vehicles

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230220

required to collect used products from customer zones, theamount of products to be transported, and the financial incentivesto be offered to product holders. The latter depend on thecondition of the returned items. TS is at the core of the solutionmethodology introduced by the authors and shows a goodperformance.

Besides the contributions described above, the design of TSalgorithms has been confined to classical location problems suchas the uncapacitated facility location problem (UFLP) (see e.g.,Ghosh, 2003; Michel and Van Hentenryck, 2004; Sun, 2006) andthe capacitated facility location problem (CFLP) along with itsvariants (see e.g., Correia and Captivo, 2006; Cortinhal andCaptivo, 2003; Hansen et al., 1992; Li et al., 2009). Although suchproblems do not incorporate practical features arising in SCND,the experience reported in the literature to solve them with TSprovides an important basis for the design of efficient solutionapproaches to strategic network design problems. In a recentempirical study, Arostegui et al. (2006) compared the perfor-mance of tabu search, simulated annealing and genetic algo-rithms for solving three variants of the UFLP: the CFLP, the multi-period and the multi-commodity location problems. Within thesame computational time limit, the performance of TS proved tobe the best in terms of solution quality for all three problemtypes. Thus, based on the robust results obtained, the authorsrecommend TS as the first metaheuristic to be tried. Furthermore,the attractiveness of using TS lies in its ease of development andimplementation. We note that all the characteristics of the abovethree problems are comprised in the problem that we study,which also captures additional features relevant to SCND.

Finally, we observe that apart from the study of Arostegui et al.(2006), no other work has investigated the usefulness of TS in amulti-period location context. The heuristic procedure that wedeveloped is - to the best of our knowledge - the first attempt toinvestigate the suitability of TS for solving a large-scale dynamicSCND problem.

3. Problem formulation

In this section, we first introduce the notation that will be usedthroughout the paper. As the new tabu search algorithm (that will

Table 1Index sets.

Symbol Description

L Set of all facilities

Sc Set of existing facilities that can be closed

So Set of potential sites for establishing new facilities

S Set of selectable facilities; S¼ Sc[ So , S� L

L\S Set of non-selectable facilities

P Set of product families

T Set of time periods; j T j ¼ n

Table 2Costs.

Symbol Description

PCti,p

Variable cost of supplying one unit of product pAP by facility iAL in pe

TCti,j,p

Variable cost of shipping one unit of product pAP from facility iAL to f

ICti,p

Variable inventory carrying cost per unit on hand of product pAP in fac

MCti,j Variable cost of moving one unit of capacity at the beginning of period

OCit Fixed cost of operating facility iAL in period tAT

FCit

Fixed setup cost charged in period tAT\fng when a new facility establish

SCit

Fixed cost charged in period tAT\f1g for closing the existing facility iAS

be presented in Section 4) is based on a mixed-integer linearprogramming formulation for the problem, we briefly present thisformulation, which is based on the modeling framework devel-oped by Melo et al. (2006). In the text below, we highlight thedifferences between the model in use in this paper and themodels proposed by Melo et al. (2006). Details regarding themotivating assumptions and the underlying supply chain rede-sign context can be found in Melo et al. (2006) and thus weomit them.

3.1. Notation

The starting point for our SCND problem is a network com-prising various types of operating facilities (any number of facilitylayers may be considered as well as any system of transportationchannels). In addition, a finite set of candidate sites for locatingnew facilities is available. Over the planning horizon, facilityrelocation takes place by gradually moving capacity from existingfacilities to new sites. Table 1 introduces the index sets to be used.

Non-selectable facilities refer to facilities that are not subject tocapacity relocation. Such facilities may include suppliers, plantsand warehouses that must operate over the time horizon. Wenote that customer locations always belong to this class.

Table 2 summarizes all costs. Since establishing a new facilityis often a time-consuming process, it is assumed that it takesplace in the period immediately preceding the start-up of opera-tions. On the other hand, when an existing facility ceases operat-ing, the corresponding fixed closing costs are charged in thefollowing period. Relocation costs are incurred to capacity shiftsand depend on the amount moved from an existing location to anew site. They account, for example, for workforce and equipmenttransfers. The assumption of a linear capacity moving cost is agood approximation for many applications in which the size ofthe capacity shifts is large. In contrast, when transfer sizes arerestricted to discrete amounts subject to economies of scale, Meloet al. (2006) propose a simple adaptation of the model to bepresented in Section 3.2.

Capacities moved to new sites cannot be withdrawn in laterperiods. This assumption is also not too restrictive as our modelcan be easily adapted to situations in which capacity removal maybe desirable over the time horizon. In particular, this occursduring periods of economic downturns leading to demand con-traction. This case is captured by extending the set of selectablefacilities with a new fictitious facility that absorbs all excesscapacity. Melo et al. (2006) describe the required model adapta-tion to this case.

Table 3 introduces additional input parameters. Due to therelocation nature of the problem, the capacity of each existingfacility is assumed to be non-increasing over the planninghorizon. Similarly, potential new facilities have non-decreasingcapacities throughout the time horizon.

Table 4 describes the decision variables. Existing facilities mayhave an initial positive inventory level which in that case fixes the

riod tAT

acility jAL (ia j) in period tAT

ility iAL at the end of period tAT

tAT\f1g from the existing facility iASc to a new facility established at site jASo

ed at site iASo starts its operation at the beginning of period tþ1c at the end of period t�1

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230 221

values of the inventory variables y0i,p for every iAL\So and pAP.

Since potential sites do not hold initial stock it follows that y0i,p ¼ 0

for every iASo and pAP.The statuses of the facilities over the time horizon are ruled by

the binary variables g. If an existing facility iASc ceases to operateat the end of period t (as a result of entire relocation) then Zt

i ¼ 1.Similarly, if a new facility starts to operate in site iASo at thebeginning of period t then Zt

i ¼ 1. Observe that a new facility cannever operate in the first period since that would incur a setupcost prior to the beginning of the planning horizon. Analogously,an existing facility cannot be closed at the end of the last periodsince the fixed closing cost would be charged in a period beyondthe time horizon. Hence, z1

i,j ¼ 0 for every iASc and jASo. More-over, Z1

i ¼ 0 for every iASo and Zni ¼ 0 for every iASc .

3.2. A mixed-integer linear programming formulation

The multi-period SCND problem is formulated by the followingmixed-integer linear program, which is based on the modelingframework introduced by Melo et al. (2006).

ðPÞ MINXtAT

XiAL

XpAP

PCti,pbt

i,pþXtAT

XiAL

XjAL\fig

XpAP

TCti,j,pxt

i,j,pþXtAT

XiAL

XpAP

ICti,pyt

i,p

þXtAT

XiASc

OCti 1�

Xt�1

t ¼ 1

Zti

!þXtAT

XiASo

OCti

Xt

t ¼ 1

Zti

!�xn

ð1Þ

s:t:

bti,pþ

XjAL\fig

xtj,i,pþyt�1

i,p ¼Dti,pþ

XjA L\fig

xti,j,pþyt

i,p iAL, pAP, tAT

ð2Þ

K1

i �Xt

t ¼ 1

XjA So

zti,jrKt

i 1�Xt�1

t ¼ 1

Zti

!iASc , tAT ð3Þ

Xt

t ¼ 1

XiA Sc

zti,jrKt

j

Xt

t ¼ 1

Ztj jASo, tAT ð4Þ

Table 3Other input parameters.

Symbol Description

Kt

iMaximum capacity of facility iAL in period tAT

K ti

Lower limit on the total amount shipped by facility iAS in period

tAT

mi,p Amount of capacity required by one unit of product pAP at facility

iAL

Dti,p

Demand of customer/facility iAL for product pAP in period tAT

Bt Available budget in period tAT

at Unit return factor on capital not invested in period tAT\fng

E Sufficiently small positive number

Table 4Decision variables.

Symbol Description

bti,p

Amount of product pAP supplied by facility iAL in p

xti,j,p

Amount of product pAP shipped from facility iAL to

yti,p Amount of product pAP held in stock in facility iAL

zti,j Amount of capacity shifted at the beginning of period

xt Amount of capital not invested in period tAT

Zti

¼ 1 if the selectable facility iAS changes its status in

Xt

t ¼ 1

XjA So

zti,jþE 1�Xt�1

t ¼ 1

Zti

!rK

1

i iASc , tAT ð5Þ

XpAP

mi,p bti,pþ

XjAL\fig

xtj,i,pþyt�1

i,p

0@ 1ArK1

i �Xt

t ¼ 1

XjA So

zti,j iASc , tAT ð6Þ

XpAP

mi,p bti,pþ

XjAL\fig

xtj,i,pþyt�1

i,p

0@ 1ArXt

t ¼ 1

XjASc

ztj,i iASo, tAT ð7Þ

XpAP

mi,p bti,pþ

XjAL\fig

xtj,i,pþyt�1

i,p

0@ 1ArKt

i iAL\S, tAT ð8Þ

XpAP

mi,p bti,pþ

XjAL\fig

xtj,i,pþyt�1

i,p

0@ 1AZK ti

1�Xt�1

t ¼ 1

Zti

!iASc , tAT ð9Þ

XpAP

mi,p bti,pþ

XjAL\fig

xtj,i,pþyt�1

i,p

0@ 1AZK ti

Xt

t ¼ 1

Zti jASo, tAT ð10Þ

XtAT

Zti r1 iAS ð11Þ

XiA So

FC1i

X2

t ¼ 1

Zti

!þx1¼ B1

ð12Þ

XiA Sc

XjA So

MCti,jz

ti,jþ

XiASc

SCtiZ

t�1i þ

XjA So

FCtjZ

tþ1j þxt

¼ Btþat�1xt�1 tAT\f1,ng ð13ÞX

iA Sc

XjA So

MCni,jz

ni,jþ

XiASc

SCni Z

n�1i þxn

¼ Bnþan�1xn�1

ð14Þ

bti,pZ0,yt

i,pZ0,xti,j,pZ0, iAL, jAL\fig, pAP, tAT ð15Þ

zti,jZ0 iASc , jASo, tAT ð16Þ

xtZ0 tAT ð17Þ

Zti Af0;1g iAS, tAT ð18Þ

The objective function (1) minimizes the total net supply chain

cost. Variable costs comprise supply, transportation and inventoryholding costs, while fixed costs account for operating the facil-ities. The sum of the variable and fixed costs is reduced by thetotal revenue obtained at the end of the time horizon. Revenue isgenerated by the fraction of the available budget that has notbeen invested and has thus gained interest. The above objectivefunction differs from that of the models in Melo et al. (2006) bythe revenue term, which encourages the minimization of expen-ditures on capacity transfer, facility opening and facility closing(see (12)–(14)). Moreover, in contrast to Melo et al. (2006), the

eriod tAT

facility jAL (ia j) in period tAT

at the end of period tAT [ f0g; y0i,p denotes the initial inventory level

tAT from the existing facility iASc to a newly established facility at site jASo

period tAT; 0 otherwise

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230222

fixed operating costs of non-selectable facilities are not consid-ered explicitly as they correspond to a fixed term.

Constraints (2) are the usual flow conservation conditions andalso ensure the satisfaction of customer demands. Inequalities (3)guarantee that only operating existing facilities can have theircapacities transferred to new facilities. Constraints (4) state that anew facility can only start receiving capacity after its setup, whileconstraints (5) ensure that an existing facility is only closed afterentire removal of its capacity. Capacity constraints are imposedby inequalities (6)–(8). Constraints (9)–(10) guarantee that aselectable facility operates with at least a given throughput.Constraints (11) allow the status of each selectable facility tochange at most once over the time horizon. This means that afacility that is removed cannot be re-opened and once open, anew facility cannot be closed. Conditions (12)–(14) guarantee thatthe available budget is invested in capacity transfers, the setup ofnew facilities and the removal of existing facilities upon entirerelocation. The amount of capital not used in a given period earnsinterest and can later be invested. Finally, constraints (15)–(18)represent non-negativity and binary conditions.

4. Tabu search algorithm

The feasible space of our problem is defined by a set ofconstraints involving binary and continuous variables. For anyfeasible set of values for the facility status variables, the optimalvalues for the continuous variables are easily obtained by solvingthe associated linear problem. In this context, an attractive searchspace is the set of feasible facility status variables gAf0;1gn�9S9.Any solution in this space can then be ‘‘completed’’ to yield a(hopefully) feasible solution to problem ðPÞ by computing theassociated optimal values of the continuous variables b, x, y, z andn. As these values correspond to the optimal solution of a linearprogramming problem we use an optimization software packageto determine them. Details regarding the binary choice for facilityoperation will be given later on. Since our problem is highlyconstrained, we will not restrict the search space to feasiblesolutions. As will be described in Section 4.2, letting the searchmove to solutions that violate the domain of the variables rulingthe amount of budget remaining at the end of each time period(i.e. constraints (17)) allows for a meaningful exploration of thesearch space. This technique, known as strategic oscillation, wasintroduced by Glover (1977) and used since in many successfultabu search procedures for solving various combinatorial pro-blems (see e.g., Albareda-Sambola et al., 2005; Cordeau andLaporte, 2004).

4.1. Construction of an initial solution

Since our algorithm allows infeasible intermediate solutions, itcan be initialized with any network configuration with over-budget in a single or even several time periods of the planninghorizon. Starting from the optimal solution of the linear relaxa-tion, denoted ðLPÞ, to ðPÞ we apply a simple rounding procedure tothe facility status variables in an attempt to obtain an initialsolution not necessarily feasible. Infeasibility is associated to theviolation of the available budget. This is a solution to the relaxedproblem ðPBÞ, a problem that coincides with ðPÞ except thatconstraints (17) are replaced by nARn�9S9 and the facility statusvariables g are fixed with a particular realization. The procedure isrepeated by dynamically adjusting a threshold E0 for variablefixing, which is initialized with a value between zero and one,denoted E. Details regarding the adjustment mechanism are givenbelow.

Algorithm 1. Rounding procedure.

1 Solve (LP) and denote by ðg,bn,xn,yn,zn,nnÞ its optimal

solution;2 E0 :¼ E;3 O :¼ fiAS : some Zti is fractional ðtATÞ};

4 X :¼ |;

5 foreach facility iAS do

6

7

8

9

if i=2O then

9bZti :¼ Zt

i for every tAT

else

9bZti :¼ 0 for every tAT

����������10 foreach new facility iASo do

11

12

Find the first time period tAT satisfyingPt�1t ¼ 1 Z

ti oE0 and

Ptt ¼ 1 Z

ti ZE0;

if t exists then bZti :¼ 1;X :¼ X [ fig;

��������13 foreach existing facility iASc do

14

15

Find the first time period tAT satisfyingPt�1t ¼ 1 Z

ti o1�E0 and

Ptt ¼ 1 Z

ti Z1�E0;

if t exists then bZti :¼ 1;X :¼ X [ fig;

��������16 Solve ðPBÞ with the realization bg of variables g;17 if ðPBÞ is infeasible and E040 then

18

19

20

21

Zti :¼ bZt

i for every iAX and tAT;

O :¼ O\X;E0 :¼ E0�gE;go to line 4

����������22 else

23 9return s0 :¼ ðbg,bn,xn,yn,zn,nnÞ

The pseudo-code in Algorithm 1 details the rounding proce-dure. At each iteration, a set O is built with those facilities havingat least one fractional status variable in some time period.Facilities associated with fractional variables that are made binaryin the course of an iteration are gathered in the set X. Lines 10–12,resp. 13–15, prescribe the conditions under which a new facility,resp. an existing facility, is selected for a status change.

A new facility can only be established in a certain time periodif at least a minimum level of ‘‘activity’’ has been observed untilthat period. The ‘‘activity level’’ of site i in period t is measured bythe fractional value of the corresponding status variable Zt

i , whilethe minimum activity imposed is defined by the threshold E0.Hence, the time period for setting up a new facility in a potentialsite iASo is the first period for which the sum of the activity levelsin that site is at least E0. When the status variables have relativelysmall values, the cumulative activity level in site i is notsignificant to justify opening a new facility there. As will bedescribed in Section 5, exploratory testing showed that 0.25 is apromising initial value for the threshold E0, thus imposing arelatively low demand on the minimum cumulative activity.

The choice of closing an existing facility is taken in a similarway. Since at the beginning of the planning horizon an existingfacility operates (and so the corresponding status variable takestypically the value one), the gradual transfer of its capacity to newsites decreases its activity level. Hence, facility iASc is only closedat the end of some period t if a sufficiently large amount ofcapacity has been removed until that time point. This condition issatisfied when the cumulative activity of the facility is at leastequal to the threshold 1�E0.

In each iteration of the rounding procedure the supply chainnetwork is redesigned. In one or several time periods, the network

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230 223

configuration may exceed the available budget. By construction,there is no guarantee that a feasible solution to ðPBÞ will beidentified. In particular, the minimum facility throughput con-straints (9)–(10) may be violated. In this case (see lines 17–21),the threshold E0 is decreased by gE with g denoting a user-definedfactor (0ogo0:5). Observe that the threshold adjustment corre-sponds to lowering the requirement on the minimum activitylevel imposed on a facility in the previous iteration of thealgorithm. If at some iteration the threshold E0 becomes zero ornegative then further reduction is not possible and as a result, aninitial solution is not available. However, such an outcome wasnever observed in all the numerical tests that we performed (cf.Section 5).

4.2. Relaxation mechanism

Previous experience with a similar problem to ðPÞ (see Meloet al., 2011) has shown that restricting the search process solelyto feasible solutions may lead to a fairly small number ofpromising solutions or even to none at all. Therefore, we devel-oped a diversification mechanism based on strategic oscillation toforce the search into yet unexplored areas of the search space.This is accomplished by relaxing constraints (17) regarding theunspent budget. Our choice of these constraints is derived fromthe observation that a tight budget (as is usually the case inrelocation projects) strongly limits the investment options in agiven period. This leads to a restricted number of feasible networkconfigurations with respect to opening new facilities and closingexisting facilities.

Although the relaxation of constraints (17) allows a budgetconsumption beyond the available limit over the time horizon,the structure of the relaxed problem still encourages networkconfigurations without excess budget. This goal is enforced byequalities (12)–(14) along with the last term in the objectivefunction, which maximizes revenue generated by non-investedcapital funds. Thus, problem ðPBÞ corresponds to a ‘‘soft’’ relaxa-tion of the original problem which has a significant impact on thequality of the results obtained as will be seen in Section 5.

Through the relaxation of constraints (17) the search space isenlarged and a simple neighborhood structure is used to exploreit. As constraint violation is reflected in negative values forvariables n, the following penalty term is added to the objectivefunction of ðPÞ that weighs the violation:

v0ðsÞ ¼ vðsÞþaXtAT

maxð0,�xtÞ ð19Þ

where s denotes any solution, v(s) is the objective value of ðPÞ, anda is a self-adjusting positive penalty factor, which alternatesbetween values that encourage or discourage infeasible solutions.By dynamically updating the value of a during the local searchprocess, this relaxation mechanism facilitates the exploration ofthe search space and, as mentioned before, is particularly usefulfor tightly constrained instances. It also stimulates the use ofsimple exchange operators as the complex modification of afeasible solution into another feasible solution can then beachieved by a series of simpler modifications through intermedi-ate infeasible solutions. This issue will be detailed below.

Function (19) is called the fitness of solution s. When we obtaina solution s feasible to ðPÞ the values of v0ðsÞ and v(s) coincide. Inthis case, the penalty factor is decreased by setting a :¼ a=ð1þyÞwith y a user-defined parameter. Otherwise, v0ðsÞ includes apenalty term proportional to the total excess budget. In this case,the penalty factor is increased by taking a :¼ að1þyÞ.

4.3. Neighborhood structure for solutions that violate the available

budget

Since intermediate solutions may consume more budget inone or several time periods than the available limit, one of thegoals of our local search is to reduce infeasibility when it occurs.In order to recover feasibility it is necessary to modify the statusof at least one facility over the planning horizon. This is achievedby exploring a neighborhood, that we denote by N1, when at aniteration of the TS algorithm the current solution s¼ ðg,b,x,y,z,nÞis feasible to ðPBÞ but not to ðPÞ. Neighboring solutions areobtained by performing moves that change the status of one sitein a single period. As will be shown next, these moves makepartial use of information on budget expenditures.

Suppose that we have a solution that is feasible to ðPBÞ but notto ðPÞ. Let t be the first period in which the total investmentexceeds the available budget that is, t¼ argmintATfx

t : xt o0g.Expenditures in period t include capacity moving costs as well asfixed setup costs for new facilities that start operating in periodtþ1 (assuming ton) and fixed closing costs for existing facilitiesthat cease operating in period t�1 (assuming t41). Since a largeportion of the budget is absorbed by the fixed costs, we will try tomove part of the investment made in period t to another period.Let Fo

t and Fct denote the sets of facilities for which fixed costs are

paid in period t

Fot ¼ fiASo : Ztþ1

i ¼ 1g

Fct ¼ fiASc : Zt�1

i ¼ 1g

Observe that Fot ¼ | for t¼ n and Fc

t ¼ | for t¼ 1. The localtransformations applied to solution s, and which define itsneighborhood N1ðsÞ, are obtained by the following exchanges.

The decision to open a new facility iAFot in period tþ1 is

overridden (by setting Ztþ1i ¼ 0) and one of the following local

modifications is selected:

1.

Destructive move: facility i is not open over the planninghorizon.

2.

Postpone opening facility i to some period k with tþ1okrn,i.e. set Zk

i ¼ 1.

3. Schedule opening facility i for an earlier period k with 1okrt

if there is sufficient unspent budget in period k�1. In otherwords, if FCk�1

i rxk�1 then set Zki ¼ 1.

For an existing facility iAFct, similar changes are performed.

The decision to close this facility in period t�1 is overridden (i.e.Zt�1

i ¼ 0) and one of the following local moves is performed:

1.

Destructive move: facility i is not closed over the planninghorizon.

2.

Postpone closing facility i to some period k with trkon, i.e.set Zk

i ¼ 1.

3. Schedule closing facility i for an earlier period k with

1rkot�1 if there is sufficient unspent budget in periodkþ1. This means that if SCkþ1

i rxkþ1 then set Zki ¼ 1.

The size of the neighborhood N1ðsÞ is bounded by ðn�1Þ � 9S9,but generally only a few selectable facilities are eligible for furtherexamination, which means that the sets Fo

t and Fct tend to be

small. Furthermore, due to the strategic nature of locationproblems, the planning horizon is typically not too long whichturns the complete exploration of N1ðsÞ fairly undemanding.

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230224

4.4. Neighborhood structure for solutions that satisfy the available

budget

Throughout the search, feasible solutions may also be identi-fied to the original problem. In this case, the structure of theneighborhood N1 is inappropriate and local search is performed ina new neighborhood that we denote by N2. The transition from afeasible solution s to ðPÞ to a solution s0AN2ðsÞ is expressed bymodifying the status of a single facility. This is achieved by firstchoosing a facility at random. Next, the time period in which thatfacility has its status changed (if any) in the current solution s willbe either postponed to the following period or scheduled for theprevious period.

Assume that the randomly selected facility i belongs to the setSc. Let t denote the time period in which this existing facility isclosed (i.e. Zti ¼ 1 and Zt

i ¼ 0 for every tAT , tat in solution s). Inthe case that facility i is never closed and thus operates over thewhole time horizon, we set t¼ 0. In order to move to aneighboring solution s0, the current decision is overruled bychoosing another time period, denoted t0, as follows:

(i)

If t¼ 0 then t0 ¼ n�1, meaning that the facility is closed atthe latest possible period.

(ii)

If t¼ 1 then t0 ¼ 2. (iii) If tAf2, . . . ,n�2g then t0 ¼ tþ1 with probability p and

t0 ¼ t�1 with probability 1�p.

(iv) If t¼ n�1 then t0 ¼ 0 with probability p (the facility operates

in every period) and t0 ¼ t�1 with probability 1�p.

As the shutdown of a facility compels capacity transfers aswell as the setup of new facilities to receive the transferredcapacity, and these actions call for considerable budget expendi-ture, a careful choice of a new time period t0 is imperative. This isparticularly the case with the above rules (i) and (iv) whichattempt to reduce the impact on the network configuration bypostponing capital investments. Hence, network reconfigurationis to occur as late as possible.

If the random selection yields a new facility iASo, then a moveis defined by overruling the time period in which this facility isestablished according to solution s. Should no facility be locatedin site i over the time horizon then we set t¼ 0. The neighboringsolution s0 is achieved by fixing a new time period t0 as follows:

(v)

If t¼ 0 then t0 ¼ n, so that the facility is opened in the latestpossible period.

(vi)

If t¼ 2 then t0 ¼ 3. (vii) If tAf3, . . . ,n�1g then t0 ¼ tþ1 with probability p and

t0 ¼ t�1 with probability 1�p.

(viii) If t¼ n then t0 ¼ 0 with probability p (the facility is not open

over the time horizon) and t0 ¼ t�1 with probability 1�p.

The motivation for using rules (v) and (viii) is to interfere aslate as possible with the network configuration. Changes causedby establishing new facilities early in the time horizon put asignificant strain on the budget and may even lead to overbudgetsituations.

We opted for an unbiased selection of a new time period for astatus change of a facility and hence, the parameter p takes thevalue 0.5 in rules (iii), (iv), (vii) and (viii). The size of theneighborhood N2ðsÞ is bounded by 2 � 9S9. In order to avoid thecomputational effort incurred by exploring the entire neighbor-hood of a solution, we only consider a random sample N02ðsÞ ofN2ðsÞ. A user-defined parameter b determines the fraction of theneighborhood to be examined. In addition to reducing thecomputational burden, the evaluation of N02ðsÞ allows us to use a

shorter tabu list than would be necessary if complete examina-tion of N2ðsÞ were to be performed.

As the choice of the neighborhood structure is by far the mostcritical step in the design of a TS heuristic, we also explored anextension of N2ð�Þ in which both the choice of a facility and of atime period are taken randomly. This alternative neighborhoodstructure introduces some additional randomization which laterproved not to be beneficial both from a computational viewpointand in terms of the quality of the solutions identified. Hence, thenumerical results to be presented in Section 5 omit the explora-tion of this alternative neighborhood.

4.5. Backtracking scheme

At an iteration of the TS algorithm, the ‘‘admissible’’ subset ofthe explored neighborhood may be empty. This occurs when allthe local transformations applied to the current solution s:(i) refer to tabu moves that cannot be revoked (see Section 4.6for details on the aspiration criterion), or (ii) yield infeasiblesolutions to the relaxed problem ðPBÞ. In either case, the localsearch is trapped in a local minimum. Usually, s is a mediocresolution or not even feasible to ðPÞ if the available budget isexceeded in at least one time period. To overcome this problem,once a neighborhood has been explored, its m best neighbors,denoted s1, . . . ,sm, are retained. The user-defined parameter m

typically takes a small value (e.g. two or three) to reduce thecomputational burden. The search is restarted from the jthneighbor, when it has failed for all solutions s1, . . . ,sj�1 and1o jrm. This backtracking mechanism allows previously visitedneighboring solutions to be considered, even though their fitnessis not the best since v0ðsmÞZ � � �Zv0ðs1Þ.

In the worst case, the backtracking strategy does not help thesearch move away from a local minimum. The algorithm is thenrestarted with a new initial solution. The latter is obtained by therounding procedure (cf. Section 4.1) for a user-defined threshold ~Esuch that ~EoE. The algorithm is restarted at most once during thewhole search process.

4.6. Short-term memory

When the status of a facility changes according to one of themoves described before (cf. Sections 4.3 and 4.4), the reversemove is forbidden (and thus, tabu) for a fixed number ofiterations. To prevent the search from tracing back its steps towhere it came from, a tabu list of fixed size h is kept. The tabutenure of a move is also set equal to this parameter h.

An aspiration criterion is applied to revoke a tabu move if itresults in a solution that improves the incumbent. The latter is thefeasible solution with the best objective value found so far. Whenno feasible solution is available (and this typically occurs at theearlier stages of the search), the incumbent solution is taken to bethe least infeasible (the one with smallest value v0ð�Þ according to(19)) known so far.

4.7. Intensification phase

This phase is basically oriented to identify good qualityfeasible solutions when a pre-specified solution quality thresholdhas not been achieved after a given number of iterations. In thiscase, the tabu search algorithm is restarted with the best feasible

incumbent solution, denoted sn. Since neighborhood sampling isused while exploring N2ð�Þ, new network configurations are likelyto be visited as the result of exchanges that were not consideredwhen the local search was first performed in N2ðs

nÞ. To restrict theoverall computational burden, the size of the tabu list is reducedin this phase to the length h0oh, with h denoting the initial tabu

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230 225

list size. Moreover, less iterations are allowed to be performedwhen the local search is restarted. The intensification process isapplied at most once during the whole search process.

4.8. Algorithm

The pseudo-code in Algorithm 2 summarizes the TS heuristic.It starts from an initial solution s0 identified by the roundingprocedure (cf. Algorithm 1) and returns, after execution, the bestfeasible solution found sn, if any (line 21, resp. line 3). At eachiteration, an appropriate neighborhood of the incumbent solutions is selected (line 7). If s violates constraints (17) then N1ðsÞ is fullyexplored, otherwise N02ðsÞ is examined. At the end of eachiteration, the penalty factor a is updated (lines 14–17). The localsearch stops when one of the following conditions is met:

T1.

The maximum number of iterations kmax is achieved (thisvalue is decreased to kintens when the local search is promptedby the intensification phase).

T2.

The best identified solution is feasible and satisfies a pre-specified solution quality threshold d, that is, ðvðsÞ�vðLPÞÞ=

vðLPÞrd, with v(LP) denoting the optimal value of the linearrelaxation.

Table 5Arc density used to generate three-echelon networks.

Source Destination Arc density (%)

Plants/suppliers Central DCs 70

Central DCs Regional DCs 40

Central DCs Central DCs 100

Central DCs Customers 5

Regional DCs Customers 50

Regional DCs Regional DCs 40

Algorithm 2. Tabu search procedure.

1 Build an initial solution s0 starting with the threshold E;2 if s0 is not feasible to ðPBÞ then

3 9return initial solution not found

4 else

5 9s :¼ s0;v0ðsÞ :¼ v0ðs0Þ; s

n :¼ s0;vn :¼ vðs0Þ; intens :¼ false

6 while no stopping criterion is satisfied do

7

8

9

10

11

12

13

14

15

16

17

Choose a neighborhood definition

Apply local search to s to obtain the m best non-tabu

solutions s01, . . . ,s0m among the explored solutions;

if no new neighboring solution is found then

apply backtracking;

if backtracking fails to identify a new solution then

go to line 1 with E :¼ ~E

�������Update the tabu list and the current solution s

with s01; retain the best m�1 solutions s02, . . . ,s0m;

if v0ðsÞovn then sn :¼ s;vn :¼ v0ðsÞ;

if s is feasible then

9a :¼ a=ð1þyÞelse

9a :¼ að1þyÞ

�����������������������������������18 if sn is feasible and quality of sn4d and intens ¼ false then

19

20

intens :¼ true; s :¼ sn;v0ðsÞ :¼ vn;

go to line 7 and consider a tabu list of size h0

�����21 return sn (if feasible)

The intensification phase is triggered by the best feasiblesolution identified by the loop in lines 6–17 when the latter doesnot meet the quality criterion T2. In this case, the local search isrestarted and a smaller number of iterations (kintens) is performed.

Finally, we note that for each particular realization of thefacility status variables we solve the associated linear problemðPBÞ with commercial optimization software.

5. Computational results

To evaluate the quality of the solutions identified by theproposed TS heuristic we performed a series of computationalexperiments with randomly generated test problems, as bench-mark instances are not available for the problem at hand. We firstpresent the instances generated and then describe the calibrationprocess of parameter values. We conclude this section with asummary and analysis of the results obtained. The TS heuristic isalso compared to a sophisticated LP-based procedure proposed byMelo et al. (2011).

5.1. Test instances

Realistically sized test instances were randomly generated todepict three-echelon networks. Upstream facilities either refer toplants or suppliers while the intermediate echelons comprisedistribution centers (DCs). The latter are classified into centraland regional facilities and are subject to relocation decisions.Table A1 in the Appendix lists the main characteristics of the 53generated instances. All costs follow a non-decreasing patternover the time horizon since in our view this reflects real-worldsituations better, as supply chain networks are often redesignedto cope with rising costs driven by an expanding global economy.A distinctive feature of the test instances is the magnitude of thefacility closing costs compared to facility opening costs. Theformer are significantly lower and may even take negative valuesto account for revenues due to the termination of leasingcontracts or the selling of property. Furthermore, the number ofnetwork arcs is restricted to the values given in Table 5. Inaddition to inter-echelon transports, commodities can also beshipped directly to customers as well as distributed amongfacilities in the same echelon. Finally, 70–80% of the commoditiescan actually be shipped over each generated arc. This limits theflows through the network and thus mimics real-world situations.Observe that with this feature problem (P) becomes more tightlyconstrained as the number of transportation channels availablefor product distribution is reduced. As a result, finding feasiblesolutions is an even more difficult task. Further details about therandom generation of the test instances can be found in Meloet al. (2009).

5.2. Parameter calibration

Exploratory testing to find good ranges of parameters wasperformed by running the TS algorithm with a variety of para-meter settings. Table 6 displays the values that not only led togood results but also appear to be robust as they do not seem tohave a significant impact on the performance of the heuristic withrespect to two measures: (i) the deviation to the lower boundprovided by the linear relaxation, and (ii) the computing time.

At first sight, the tabu list size may seem small but experi-ments performed with larger lists (with sizes 5 and 7) did notyield better solutions. Furthermore, the total number of iterationskmax is also small. This results from the observation that the rate

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Table 6Parameter values.

Parameter Description Value

E Initial value of the threshold E0 for variable fixing in the rounding

procedure (Algorithm 1)

0.25

g Factor used to update E0 1/6~E Threshold for variable fixing when the tabu search fails to find a

feasible solution

0.05

a Penalty factor for overbudget 1000

y Parameter used to update a yAð0;1�h Size of the tabu list and tabu tenure 3

h0 Size of the tabu list and tabu tenure in the intensification phase 2

m Number of best solutions that are retained during the exploration

of a neighborhood

3

b Percentage of solutions that are examined in the neighborhood

N2ð�Þ

15%

p Probability factor used in conjunction with neighborhood N2ð�Þ 0.5

kmax Total number of iterations 15

kintens Total number of iterations in the intensification phase 5

d Solution quality threshold w.r.t. the linear relaxation bound 0.01

Table 7Results for the instances in set 1 (50 customers).

Instance Tabu search heuristic LP-based heuristic CPLEX

LP-gap (%) CPU (s) Intens. Restart Backtr. # Iter. LP-gap (%) CPU (s) CPU (s)

P1 135.43 8 n 20 7.90 20 580

P2 0.11 18 1 0.10 16 209

P3 239.01 157 n 20 10.07 2606 –

P4 0.18 959 5 2.97 802 234

P5 6.78 21 n n 20 4.24 27 375

P6 9.03 190 n 20 – – 11,375

P7 10.02 588 n 20 7.70 780 –

P8 11.06 1655 n 20 8.41 9530 –

P9 0.01 39 1 0.02 34 94

P10 2.40 119 n 20 4.26 188 1286

P11 0.01 37 0 0.01 41 64

P12 0.01 124 2 0.01 100 825

P13 o 0.01 249 2 o 0.01 143 312

P14 o 0.01 182 1 o 0.01 248 816

P15 o 0.01 92 0 o 0.01 105 140

P16 o 0.01 330 0 – – 489

P17 o 0.01 329 2 o 0.01 310 570

P18 4.51 37 n 20 – – 6484

P19 3.44 36 n 20 – – 316

P20 6.38 409 n 20 8.20 684 –

P21 13.52 117 n n n 20 20.11 43 318

P22 15.8 100 n 20 21.32 185 1057

P23 3.89 146 n 20 3.87 204 –

M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230226

of improvement in solution quality, after a few iterations,becomes negligible and therefore, it is not worth to invest inadditional computational resources. Nevertheless, a sufficientnumber of neighbors is explored during the local search (in total45 neighbors given the values of m and kmax). The fact that goodquality solutions are identified early in the search processdemonstrates that the neighborhood structures N1ð�Þ and N2ð�Þ

are particularly adequate and thus, strategic oscillation proves tobe a very effective mechanism for the problem at hand.

The value of the threshold d, which is used in the terminationcriterion T2, may seem at first glance rather tight. Our choice ismotivated by the results obtained by Melo et al. (2006) for asimilar class of network redesign problems for which the optimalsolution proved to be within 1% of the LP bound.

Finally, we note that the parameter y used for dynamicallyadjusting the penalty factor a in the fitness function (19) is selectedrandomly in the interval ð0;1�. Preliminary testing showed that itsvalue does not impact on the results of the TS heuristic.

5.3. Performance analysis

We categorize the 53 test instances into three sets based onthe number of customers. The performance of the new TSheuristic is evaluated with respect to the quality of the solutionsobtained compared to the linear relaxation bound (denoted ‘‘LP-gap’’ and defined in the termination criterion T2) and to the totalcomputing time. Set 1 comprises 23 instances with 50 customers,while set 2, resp. 3, includes 17 instances with 100 customers,resp. 13 instances with 200 customers. The relaxed problems ðLPÞ

and ðPBÞ were modeled using ILOG Concert Technology 2.0 (ILOGConcert Technology 2.0 User’s Manual, 2003) and solved withCPLEX 10.2. The TS algorithm was implemented in Cþþ followingthe framework proposed by Blesa and Xhafa (2001). All experi-ments were conducted on a Pentium III with a 2.6 GHz processorand 2 GB RAM.

Tables 7–9 display detailed results obtained with the para-meter settings given in Table 6. Fig. 1 provides a global overview

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Table 8Results for the instances in set 2 (100 customers).

Instance

Tabu Search heuristic LP-based

heuristic

CPLEX

LP-gap

(%)

CPU

(s)

Intens. Restart Backtr. #

Iter.

LP-gap

(%)

CPU

(s)

CPU

(s)

P24 0.34 21 1 0.02 21 31

P25 0.30 54 2 0.12 58 232

P26 0.03 247 2 0.03 167 895

P27 0.04 330 1 0.04 436 2948

P28 0.02 51 3 0.01 65 176

P29 0.02 66 0 – – 116

P30 0.05 209 2 0.03 351 1414

P31 0.04 1017 4 – – 3305

P32 0.69 180 n n 9 o 0.01 55 300

P33 0.01 322 n n 10 o 0.01 108 920

P34 o 0.01 41 0 o 0.01 51 69

P35 o 0.01 105 1 – – 155

P36 o 0.01 264 1 o 0.01 347 336

P37 o 0.01 348 n n 5 – – 167

P38 o 0.01 431 2 – – 403

P39 21.43 42 n n n 20 21.35 35 131

P40 2.34 278 n 20 2.65 86 329

Table 9Results for the instances in set 3 (200 customers).

Instance

Tabu search heuristic LP-based

heuristic

CPLEX

LP-gap

(%)

CPU

(s)

Intens. Restart Backtr. #

Iter.

LP-gap

(%)

CPU

(s)

CPU

(s)

P41 0.01 41 0 0.01 44 114

P42 o 0.01 122 2 0.49 114 561

P43 0.01 437 n 12 0.01 263 1777

P44 0.01 469 1 0.01 855 3152

P45 o 0.01 85 n n 10 – – 278

P46 o 0.01 69 1 – – 380

P47 o 0.01 225 2 o 0.01 204 715

P48 0.01 810 n 12 – – 2169

P49 o 0.01 67 2 o 0.01 81 267

P50 0.81 643 n n 11 o 0.01 167 558

P51 o 0.01 60 0 o 0.01 74 106

P52 o 0.01 158 1 – – 408

P53 o 0.01 464 1 – – 1324

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25

Fig. 1. Results obtained using the new TS procedure.

Fig. 2. Number of instances in which the intensification phase, the backtracking

scheme and the restart strategy were used.

M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230 227

of the behavior of the TS heuristic based on two crucial evaluationcriteria for every instance: the required CPU time and the LP-gapachieved. It can be seen that a good trade-off between solutionquality and computing time is obtained as most instances weresolved in less than 400 seconds and exhibit a low LP-gap.

Four outliers were excluded to produce Fig. 1 due to their poorsolution quality or their high computational demands. They referto instances P1 and P3 (with LP-gaps above 100%, see Table 7)along with P8 and P31 (with CPU times of 17 resp. 27.6 min, seeTables 7 and 8). These cases will be analyzed in Section 5.5.

To better understand the behavior of the TS heuristic, wealso collected information on how often the intensificationphase and the backtracking scheme were used. Furthermore, wekept track of the instances in which the heuristic had to berestarted. Recall that this occurs when at an iteration theexploration of the neighborhood of all of the m non-tabu solutionsfails to identify a feasible solution to ðPBÞ. Fig. 2 summarizes thefrequencies of these three cases (Tables 7–9 report the detailedresults).

The solution quality threshold d is not satisfied by 28.3% of theinstances (in total 15, 13 of which belong to networks with 50customers) and as a result, the local search is intensified inpromising regions for a few iterations. Although further solutionimprovement is obtained, the procedure ends with a solution thatstill does not meet the quality criterion. Numerical tests with anincreased number of iterations during the intensification phasedid not produce better results, thus indicating that the initial localsearch has a powerful effect on the quality of the solutionsidentified. Observe that the maximum number of iterations(kmaxþkintens) is performed in each one of the above 15 instanceswhich impacts the overall CPU time. The backtracking scheme isonly used in 10 out of 53 instances (18.8%) and in more than halfof these the local search is restarted with a new initial solution.This strategy turns out to be a suitable way to escape from localminima as good quality solutions are then identified in 7 out of 10instances (see Tables 7–9).

5.4. Comparison with an LP-based heuristic

In order to evaluate the benefit of allowing infeasible inter-mediate solutions during the local search, the results of the TSheuristic are compared to a two-phase heuristic approach pro-posed by Melo et al. (2011), in which solution feasibility is

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230228

maintained throughout the whole procedure. In the first phase ofthis heuristic, a sophisticated linear programming roundingstrategy is applied to find initial values for the binary facilitystatus variables. The second phase uses local search in an attemptto improve the initial solution when its quality does not meetgiven criteria. The second phase is also used when a feasiblesolution was not identified before. In this case, the initial variablechoices are corrected by means of local search. In what follows wewill denote this approach as the LP-based procedure.

Both heuristics are compared in terms of the overall CPU timesand the best solutions found with respect to their LP-gaps (seeTables 7–9). The corresponding ratios are determined by

LP�gap ratio¼LP� gap of best TS solution

LP� gap of best LP� based solution

CPU ratio¼CPU time of TS heuristic

CPU time of LP� based procedure

and shown in Fig. 3. Note that a ratio lower than one indicatesthat the TS heuristic outperforms the LP-based procedure. Ratioshigher than 3 correspond to instances considered to be outliersand therefore are not depicted in Fig. 3. In total five instances (P1,P3, P24, P32 and P50) exhibit large ratios regarding the LP-gap.

Fourteen instances (26.4%) are highlighted by a large diamondat the bottom left side of Fig. 3. The LP-based procedure by Meloet al. (2011) could not identify a single feasible solution for any ofthese instances. As this feature is observed across all three sets, itconcerns any network size independently of the number ofcustomers. In contrast, the TS heuristic is able to solve all 53instances, thus clearly demonstrating that it is beneficial to drivethe search towards and away from boundaries of the feasiblespace. It is also worth pointing out that the TS algorithm yieldsbetter solutions with less computational effort for further 12instances (22.6%, bottom left side of the diagram). Hence, the TSheuristic outperforms the LP-based procedure in almost half ofthe instances. The upper left and bottom right sides of Fig. 3 referto instances for which the TS heuristic is superior either in termsof solution quality or computing time. Finally, the LP-basedprocedure outperforms the TS algorithm in only seven instances(13.2%). In conclusion, we can say that the proposed TS heuristicexhibits a good and robust performance.

0

1

2

3

3210

Fig. 3. Comparison of the new TS heuristic with the LP-based procedure.

5.5. Detailed results

Tables 7–9 include the detailed results obtained both with thenew TS heuristic and the LP-based procedure developed by Meloet al. (2011). Furthermore, we also report on the computing timerequired by CPLEX to solve each instance. CPLEX was run with thefollowing stopping criteria: a CPU time limit of 5 h and a targetintegrality gap. For the latter, the deviation between the bestsolution and the best lower bound delivered by CPLEX was set toat most 1%. Our choice is prompted by the fact that the dataestimates of a multi-period network redesign problem are oftennot accurate and therefore, it may not be meaningful to solve theproblem to optimality. Since the error margin tends to be largerthan 1%, it is adequate to run an optimization solver such asCPLEX until a feasible solution within 1% optimality has beenidentified.

The first column of Tables 7–9 indicates the test instance.Information regarding the tabu search procedure is given in thecolumns under the heading ‘‘Tabu Search heuristic’’. The secondcolumn presents the LP-gap which is defined as ðzTS�zLPÞ=zLP �

100% with zTS denoting the objective value of the feasible solutionidentified by the TS heuristic and zLP the optimal value of thelinear relaxation to ðPÞ. The computing time is given in seconds inthe third column. Instances for which the intensification phase,the restart mechanism and the backtracking scheme wereexecuted are highlighted with the symbol ‘‘n’’ in columns 4(‘‘Intens.’’), 5 (‘‘Restart’’) and 6 (‘‘Backtr.’’), respectively. Thetotal number of iterations performed by the TS heuristic isshown in column 7. When this number is equal to zero thisindicates that the initial solution is feasible and already satisfiesthe pre-specified quality threshold. This occurred with seveninstances.

The columns under the heading ‘‘LP-based heuristic’’ describethe results obtained with the alternative procedure described inMelo et al. (2011). Column 8 gives the relative difference betweenthe objective value of the best solution identified by thisLP-rounding strategy (denoted zR) and the value of the linearrelaxation, i.e. ðzR�zLPÞ=zLP � 100%. Column 9 presents the CPUtime in seconds. The symbol ‘‘-’’ appears in columns 7 and 8 whenthe LP-based approach fails to identify a feasible solution.

Finally, column 10 presents the CPU time (in seconds) requiredby CPLEX to obtain a feasible solution with an integrality gap of atmost 1%. Whenever CPLEX reaches the time limit of five hourswithout achieving this 1% target gap, the symbol ‘‘-’’ is used.

The effectiveness of the TS heuristic is demonstrated by the highquality of the best solutions identified, which have an LP-gap below1% in 38 out of 53 instances (71.7%). Moreover, in more than half ofthe instances (28 out of 53) the best solution deviates less than0.02% from the LP bound. In particular, instances with 100 and 200customers (sets 2 and 3) exhibit this feature, with the exception oftwo, which have an LP-gap of 2.34%, resp. 21.43%. The latter refer tothree-echelon networks with 100 customers and 50 suppliers (seeP39 and P40 in Table 8). At first sight it may be surprising that 18instances (34%) display an LP-gap smaller than 0.01%. As observedby Melo et al. (2006), the lower bound of the linear relaxation isusually very good for the class of network redesign problems thatalso includes our problem. Recall that this feature motivated ourchoice of the value 0.01 (i.e. 1%) for the threshold d used in thetermination criterion T2 (see Table 6).

Large LP-gaps were also obtained for some of the instanceswith 50 customers (set 1). One possible explanation for this facthas to do with the impact that the number of binary variables hason the problem. In fact, the number of binary variables does notdepend on the number of customers. Accordingly, for a smallernumber of customers, the ratio # binary variables / # continuousvariables is larger so the impact of the binary variables on the

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230 229

structure of the problem becomes stronger. Thus, the results arenot surprising. What is relevant is that in practice we shouldexpect problems with a large number of customers. Therefore, ourresults show that the closer we are to what we expect in practice,the better the quality of the solutions identified by our TSheuristic. This is a characteristic that every well designed heur-istic procedure should exhibit.

Regarding the computational performance of the TS heuristic,half of the instances were solved in less than 2.7 min which canbe seen as a reasonable CPU time for large-scale instances. Incontrast, CPLEX requires a considerably larger computing time todeliver a solution having an integrality gap of at most 1%.

Table A1Characteristics and size of test instances associated with three-echelon networks.

Set Instance Periods Products Plants/suppliers Existing

Central

Set 1 50 cust. P1 3 5 5 4

P2 4 5 5 4

P3 6 5 5 4

P4 8 5 5 4

P5 3 5 5 8

P6 4 5 5 8

P7 6 5 5 8

P8 8 5 5 8

P9 3 10 5 4

P10 4 10 5 4

P11 3 10 5 8

P12 4 10 5 8

P13 3 20 5 4

P14 4 20 5 4

P15 3 20 5 8

P16 4 20 5 8

P17 3 50 5 8

P18 4 5 50 4

P19 3 5 50 8

P20 8 5 50 8

P21 3 10 50 4

P22 4 10 50 4

P23 4 10 50 8

Set 2 100 cust. P24 3 5 5 4

P25 4 5 5 4

P26 6 5 5 4

P27 8 5 5 4

P28 3 5 5 8

P29 4 5 5 8

P30 6 5 5 8

P31 8 5 5 8

P32 3 10 5 4

P33 4 10 5 4

P34 3 10 5 8

P35 4 10 5 8

P36 4 20 5 4

P37 3 20 5 8

P38 4 20 5 8

P39 3 5 50 4

P40 3 10 50 4

Set 3 200 cust. P41 3 5 5 4

P42 4 5 5 4

P43 6 5 5 4

P44 8 5 5 4

P45 3 5 5 8

P46 4 5 5 8

P47 6 5 5 8

P48 8 5 5 8

P49 3 10 5 4

P50 4 10 5 4

P51 3 10 5 8

P52 4 10 5 8

P53 4 20 5 8

Moreover, the pre-defined time limit is reached in five instanceswithout finding a solution with the desired quality.

6. Conclusions

This study described a tabu search heuristic for solving a large-scale multi-echelon network redesign problem. To the best of ourknowledge, this is the first attempt to develop such a metaheur-istic for a comprehensive problem capturing several featuresrelevant to strategic supply chain planning under a facilityrelocation scenario over a multi-period horizon.

DCs Potential DCs Instance size

Regional Central Regional # Cont. var. # Int. var. # Const.

10 8 20 18,888 126 2075

10 8 20 25,314 168 2757

10 8 20 38,166 252 4121

10 8 20 51,018 336 5485

20 12 30 38,978 210 2803

20 12 30 52,362 280 3723

20 12 30 79,130 420 5563

20 12 30 105,898 560 7403

10 8 20 36,981 126 3530

10 8 20 49,438 168 4697

20 12 30 75,593 210 4678

20 12 30 101,182 280 6223

10 8 20 73,167 126 6440

10 8 20 97,986 168 8577

20 12 30 148,823 210 8428

20 12 30 198,822 280 11,223

20 12 30 368,513 210 19,678

10 8 20 32,406 168 3837

20 12 30 46,943 210 3613

20 12 30 127,138 560 9563

10 8 20 47,619 126 5015

10 8 20 63,622 168 6677

20 12 30 122,422 280 8203

10 8 20 28,575 126 2975

10 8 20 38,230 168 3957

10 8 20 57,540 252 5921

10 8 20 76,850 336 7885

20 12 30 54,125 210 3703

20 12 30 72,558 280 4923

20 12 30 109,424 420 7363

20 12 30 146,290 560 9803

10 8 20 56,358 126 5180

10 8 20 75,272 168 6897

20 12 30 105,890 210 6328

20 12 30 141,578 280 8423

10 8 20 149,362 168 12,777

20 12 30 209,420 210 11,578

20 12 30 279,618 280 15,423

10 8 20 33,897 126 3785

10 8 20 66,999 126 6665

10 8 20 47,958 126 4775

10 8 20 64,074 168 6357

10 8 20 96,306 252 9521

10 8 20 128,538 336 12,685

20 12 30 84,428 210 5503

20 12 30 112,962 280 7323

20 12 30 170,030 420 10,963

20 12 30 227,098 560 14,603

10 8 20 95,121 126 8480

10 8 20 126,958 168 11,297

20 12 30 166,493 210 9628

20 12 30 222,382 280 12,823

20 12 30 441,222 280 23,823

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M.T. Melo et al. / Int. J. Production Economics 136 (2012) 218–230230

The numerical results show that the new procedure performswell and can reach solutions within 1% of the linear relaxationbound in reasonable computational times. The economic impactof these results is very important. In a strategic problem such asthe one we address, when millions of euros or dollars may beinvolved, the identification of a good solution leads to significantsavings. Furthermore, the ability to find a good solution within areasonable amount of time is crucial to perform ‘‘what-if’’analyses. Such analyzes are often required to achieve deepermanagerial insights and thus, better help decision-making,namely in terms of the economic benefits that the company incharge of the supply chain earns from network redesign.

The strength of the new solution approach lies in the fact thatduring the search process the boundary of feasibility can becrossed in order to explore the infeasible region for a certainnumber of moves. This is accomplished by allowing financialinvestments on network reconfiguration to exceed the availablebudget in one or several time periods. This strategic oscillationfeature of the new algorithm has shown to be particularly suitableto tackle a problem with fragmented feasible regions. In our case– as in many other hard optimization problems – some areas ofthe feasible space containing attractive solutions seem to be onlyseparated by a narrow infeasible region. By crossing such a regionfeasible solutions of superior quality are achieved.

We believe that our solution methodology is flexible and canbe successfully adapted to handle other multi-period optimiza-tion problems involving similar phase-in/phase-out decisions asin our problem. For example, technology acquisition and equip-ment installation decisions often arise in production planning andin-house logistics contexts. Such problems are typically highlyconstrained and even finding a feasible solution is a difficult task.It would be interesting to investigate the application of a relaxa-tion mechanism similar to the one proposed in this paper.Observe that the robustness of our method results from the factthat the relaxed problem keeps to a greater extent the structure ofthe original problem. In our view this feature is critical forobtaining high-quality solutions.

Acknowledgments

This research was partly supported by the German AcademicExchange Service (DAAD) under the program PPP-Acc- ~oes Integra-

das Luso-Alem ~as/DAAD-GRICES ‘‘New Quantitative Approaches forLogistics Network Design Problems’’. This support is gratefullyacknowledged. The authors also thank Martin Ducrozet for hisvaluable help in generating the test instances and implementingthe tabu search algorithm.

Appendix: Test instances

See Table A1.

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