Redesigning multi-echelon supply chain networks

M.T. Melo, S. Nickel, F. Saldanha-da-Gama

CIO − Working Paper 5/2010

Redesigning multi-echelon supply chain networks

M.T. Meloa,b, S. Nickelc,d, F. Saldanha-da-Gamab,e,∗

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

Abstract

This paper addresses the problem of redesigning a supply chain network with mul-tiple echelons and commodities. Redesign decisions comprise the relocation of existingfacilities to new sites under an available budget over a finite time horizon, the supplyof commodities by upstream facilities, the inventory levels at storage facilities, and theflow of commodities through the network. The problem is modelled as a large-scalemixed-integer linear program. Feasible solutions are obtained by using a tabu searchprocedure that explores the space of the facility location variables. The latter prescribethe time periods in which changes in the network configuration occur. They are triggeredby the setup of new facilities, which operate with capacity transferred from the existingfacilities, and by closing the latter upon their entire relocation. As the problem is highlyconstrained, infeasible solutions with excess budget are allowed during the course of thesearch process. However, such solutions are penalized for their infeasibility. Computa-tional experiments on realistically-sized randomly generated instances indicate that thisstrategic oscillation scheme used in conjunction with tabu search performs very well.

Keywords: supply chain network redesign, facility relocation, metaheuristics, tabu search,strategic oscillation.

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 [4], the

∗Corresponding author. E-mail address : fsgama@fc.ul.pt

1

reconfiguration process can result in a 5 − 15% reduction of the logistics costs. Chandra and

Grabis [7] and Hammami et al. [18] 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, establishing 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, transportation, 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. [25]. 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 modelled as no restrictions on the number of echelons

and on the type of facilities (e.g. plants, warehouses, cross-docks, etc.) are imposed;

• Facility relocation can be planned under fairly general conditions including opening new

facilities and closing existing facilities;

• 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 [25] could be solved to optimality for small and

medium-sized problem instances using commercial optimization software within a reasonable

2

time limit. However, due to the combinatorial nature of the problem, it becomes very difficult

to solve to optimality instances with many locations, customers and commodities over an

extended planning horizon. In particular, when re-optimization is required for performing “what-

if” analyzes, a problem is solved repeatedly and the computational burden becomes an even

more important issue. In these cases, heuristic methods provide a viable alternative.

The goal of this paper is to present an efficient procedure to find good feasible solutions

to realistically-sized multi-period SCND problems. We propose a tabu search heuristic that

explores the space of the binary facility location variables. During the course of the search,

the heuristic presents a strategic oscillation behaviour that permits alternating between feasible

and infeasible solutions. The latter solutions are achieved by exceeding the budget available for

network reconfiguration over the planning horizon, and are penalized for their infeasibility.

Our study is - to the best of our knowledge - the first attempt to investigate the suitability

of tabu search for tackling a large-scale multi-period supply chain network redesign problem.

Comparisons with a sophisticated rounding procedure developed by Melo et al. [27], which

is based on the linear relaxation of the problem at hand, indicate that the new heuristic finds

feasible solutions of superior quality within reasonable time for randomly generated test instances

of realistic size.

The remainder of the paper is organized as follows. In the next section we briefly review

the literature dedicated to comprehensive location models for SCND and to the application of

tabu search to solve them. In Section 3 we present a mathematical formulation of the problem.

The tabu search heuristic is detailed in Section 4, followed by an analysis of the computational

results in Section 5. We conclude with a few general remarks and some directions for future

research.

2 Literature review

Over the last decades, facility location decisions have attracted a great deal of attention from

researchers. In recent years, increasing attention has also been paid to the interaction of these

decisions with key features to strategic supply chain planning such as supplier selection, produc-

tion planning, technology acquisition, inventory planning, transportation mode selection, and

vehicle routing. The importance of integrating location decisions with other decisions relevant

to SCND has been underlined by Daskin et al. [13]. Economic globalization has also prompted

the development of more comprehensive facility location models as shown by the surveys of

3

Goetschalckx et al. [17] and Meixell and Gargeya [24]. A recent survey by Melo et al. [26]

highlights the contributions that capture various aspects of SCND in conjunction with facility

location. Among the most comprehensive studies are Cordeau et al. [9, 10], Thanh et al. [30],

Vila et al. [32], and Wilhelm et al. [34]. Nevertheless, several research directions still require in-

tensive research. In particular, models addressing the design of multi-commodity, multi-echelon

supply chain networks through determining the timing of facility locations, expansions and re-

locations over an extended time horizon have received considerably less attention than their

static counterpart (see Ballou [4]). As shown by Melo et al. [26,27], within this problem class,

focus has been mainly given to simple networks with at most two facility layers and a single

commodity that flows between adjacent echelons to satisfy customer demands.

Multi-period SCND problems place increased demands on mathematical tools used to solve

them. In recent years, significant gains have been made in the size and complexity of the

problems that can be solved with classical mathematical programming techniques (e.g. branch-

and-bound and decomposition methods), mainly due to increased computing power. However,

time and computing resources to solve such problems repeatedly in practical applications become

prohibitive. This situation has given rise to a multitude of heuristic solution techniques that

seek to provide good approximate solutions within a reasonable amount of time. As the solution

methodology that we will describe in Section 4 uses tabu search, we will focus the remainder

of this section on the literature dedicated to the application of this technique in the context of

facility location and strategic supply chain planning.

Over the past two decades metaheuristics have become important tools for solving various

combinatorial problems encountered in many practical settings. Among the different meta-

heuristic methodologies that exist, Tabu Search (TS) (see e.g., Glover and Laguna [16]) has

become a very popular approach as it identifies high-quality solutions to many problems. Al-

though the TS literature is very rich, only a few papers address the application of this master

strategy to solve facility location problems in the context of SCND. Wang et al. [33] consider

a variant of the p-median problem, where the decisions to open new facilities and to close ex-

isting facilities are subject to a budget constraint. Three heuristic approaches are developed: a

greedy interchange, a TS and a lagrangean relaxation heuristic. Numerical experiments indicate

that TS generates the most satisfactory results in terms of solution quality. For the combina-

tion of depot location and vehicle routing decisions in a single model, Tuzun and Burke [31]

and Albareda-Sambola et al. [1] developed efficient TS algorithms. The latter authors con-

sider strategic oscillation in the implementation of their TS heuristic, a technique that is used

4

when intermediate infeasible solutions are allowed to be visited during the search process. Ca-

ballero et al. [6] describe a case study in Andalusia (Spain) for the location of incineration plants

and the design of vehicle routes to collect animal waste from slaughterhouses. A multiobjective

model is developed, which comprises economic and social objectives, and is solved with TS

using an adaptive memory procedure.

Recently, Lee and Dong [21] presented a TS approach for the design of a two-echelon

network. The proposed model includes some practical elements of SCND such as the direct

shipment of a single commodity from plants to customers. Moreover, location decisions concern

both plants and warehouses. Keskin and Ulster [20] also addressed a two-echelon SCND problem

but with multiple commodities. Among various heuristics aimed towards solving this problem,

TS was preferred in terms of both solution quality and computational time. In some cases,

TS could even identify optimal solutions. In the context of reverse logistics, Aras et al. [2]

developed a mixed-integer nonlinear model to find the optimal locations of collection centres,

the number of vehicles required to collect used products from customer zones, the amount

of products to be transported, and the financial incentives to be offered to product holders.

The latter depend on the condition of the returned items. TS is at the core of the solution

methodology introduced by the authors and shows a good performance.

Besides the contributions described above, the design of TS algorithms has been confined

to classical location problems such as the uncapacitated facility location problem (UFLP) (see

e.g., Ghosh [14], Michel and Van Hentenryck [28], Sun [29]) and the capacitated facility lo-

cation problem (CFLP) along with its variants (see e.g., Correia and Captivo [11], Cortinhal

and Captivo [12], Hansen et al. [19], Li et al. [22]). Although such problems do not incorpo-

rate practical features arising in SCND, the experience reported in the literature to solve them

with TS provides an important basis for the design of efficient solution approaches to strategic

network design problems. In a recent empirical study, Arostegui et al. [3] compared the per-

formance of tabu search, simulated annealing and genetic algorithms for solving three variants

of the UFLP: the CFLP, the multi-period and the multi-commodity location problems. Within

the same computational time limit, the performance of TS proved to be the best in terms of

solution quality for all three problem types. Thus, based on the robust results obtained, the

authors recommend TS be the first metaheuristic to be tried. Furthermore, the attractiveness

of using TS lies in its ease of development and implementation. We note that all the charac-

teristics of the above three problems are comprised in the problem that we study, which also

captures additional features relevant to SCND.

5

Finally, we observe that apart from the study of Arostegui et al. [3], no other work has

investigated the usefulness of TS in a multi-period location context. The heuristic procedure

that we developed is - to the best of our knowledge - the first attempt to investigate the

suitability of TS for solving a large-scale dynamic SCND problem.

3 Problem formulation

In this section we first introduce the notation that will be used throughout the paper. As the

new tabu search algorithm (that will be presented in Section 4) is based on a mixed integer

linear programming formulation for the problem, we briefly present this formulation, which is

based on the modelling framework developed by Melo et al. [25]. In the text below we highlight

the differences between the model in use in this paper and the models proposed by Melo et

al. [25]. Details regarding the motivating assumptions and the underlying supply chain redesign

context can be found in [25] and thus we omit them.

3.1 Notation

The starting point for our SCND problem is a network comprising various types of operating

facilities (any number of facility layers may be considered as well as any system of transportation

channels). In addition, a finite set of candidate sites for locating new facilities is available. Over

the planning horizon, facility relocation takes place by gradually moving capacity from existing

facilities to new sites. Table 1 introduces the index sets to be used.

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; S = Sc ∪ So, S ⊂ L

L \ S Set of non-selectable facilities

P Set of product families

T Set of time periods; | T |= n

Table 1: Index sets

Non-selectable facilities refer to facilities that are not subject to capacity relocation. Such

facilities may include plants and warehouses that must operate over the time horizon. We note

6

that customer locations always belong to this class.

Table 2 summarizes all costs. Since establishing a new facility is often a time-consuming

process, it is assumed that it takes place in the period immediately preceding the start-up of

operations. On the other hand, when an existing facility ceases operating, the corresponding

fixed closing costs are charged in the following period. Relocation costs are incurred to capacity

shifts and depend on the amount moved from an existing location to a new site. They account,

for example, for workforce and equipment transfers. Capacities moved to new sites cannot be

withdrawn in later periods.

Symbol Description

PCti,p Variable cost of supplying one unit of product p ∈ P by facility i ∈ L in period

t ∈ T

TCti,j,p Variable cost of shipping one unit of product p ∈ P from facility i ∈ L to facility

j ∈ L (i 6= j) in period t ∈ T

ICti,p Variable inventory carrying cost per unit on hand of product p ∈ P in facility

i ∈ L at the end of period t ∈ T

MCti,j Variable cost of moving one unit of capacity at the beginning of period t ∈ T \ 1

from the existing facility i ∈ Sc to a new facility established at site j ∈ So

OCti Fixed cost of operating facility i ∈ L in period t ∈ T

FCti Fixed setup cost charged in period t ∈ T \ n when a new facility established

at site i ∈ So starts its operation at the beginning of period t + 1

SCti Fixed cost charged in period t ∈ T \ 1 for closing the existing facility i ∈ Sc

at the end of period t − 1

Table 2: Costs

Table 3 introduces additional input parameters. Due to the relocation nature of the problem,

the capacity of each existing facility is assumed to be non-increasing over the planning horizon.

Similarly, potential new facilities have non-decreasing capacities throughout the time horizon.

Table 4 describes the decision variables. Existing facilities may have an initial positive

inventory level which in that case fixes the values of the inventory variables y0i,p for every

i ∈ L \ So and p ∈ P . Since potential sites do not hold initial stock it follows that y0i,p = 0 for

every i ∈ So and p ∈ P .

The statuses of the facilities over the time horizon are ruled by the binary variables η. If an

existing facility i ∈ Sc ceases to operate at the end of period t (as a result of entire relocation)

then ηti = 1. Similarly, if a new facility starts to operate in site i ∈ So at the beginning of

period t then ηti = 1. Observe that a new facility can never operate in the first period since

7

Symbol Description

Kt

i Maximum capacity of facility i ∈ L in period t ∈ T

Kti Lower limit on the total amount shipped by facility i ∈ S in period t ∈ T

µi,p Amount of capacity required by one unit of product p ∈ P at facility i ∈ L

Dti,p Demand of customer/facility i ∈ L for product p ∈ P in period t ∈ T

Bt Available budget in period t ∈ T

αt Unit return factor on capital not invested in period t ∈ T \ n

ǫ Sufficiently small positive number

Table 3: Other input parameters

that would incur a setup cost prior to the beginning of the planning horizon. Analogously, an

existing facility cannot be closed at the end of the last period since the fixed closing cost would

be charged in a period beyond the time horizon. Hence, z1i,j = 0 for every i ∈ Sc and j ∈ So.

Moreover, η1i = 0 for every i ∈ So and ηn

i = 0 for every i ∈ Sc.

Symbol Description

bti,p Amount of product p ∈ P supplied by facility i ∈ L in period t ∈ T

xti,j,p Amount of product p ∈ P shipped from facility i ∈ L to facility j ∈ L (i 6= j)

in period t ∈ T

yti,p Amount of product p ∈ P held in stock in facility i ∈ L at the end of period

t ∈ T ∪ 0; y0i,p denotes the initial inventory level

zti,j Amount of capacity shifted at the beginning of period t ∈ T from the existing

facility i ∈ Sc to a newly established facility at site j ∈ So

ξt Amount of capital not invested in period t ∈ T

ηti = 1 if the selectable facility i ∈ S changes its status in period t ∈ T ; 0 otherwise

Table 4: Decision variables

3.2 A mixed-integer linear programming formulation

The multi-period SCND problem is formulated by the following mixed-integer linear program,

which is based on the modelling framework introduced by Melo et al. [25].

(P ) MIN∑

t∈T

∑

i∈L

∑

p∈P

PCti,p bt

i,p +∑

t∈T

∑

i∈L

∑

j∈L\i

∑

p∈P

TCti,j,p xt

i,j,p +∑

t∈T

∑

i∈L

∑

p∈P

ICti,p yt

i,p

+∑

t∈T

∑

i∈Sc

OCti

(1 −

t−1∑

τ=1

ητi

)+∑

t∈T

∑

i∈So

(OCt

i

t∑

τ=1

ητi

)− ξn (1)

8

s.t.

bti,p +

∑

j∈L\i

xtj,i,p + yt−1

i,p = Dti,p +

∑

j∈L\i

xti,j,p + yt

i,p i ∈ L, p ∈ P, t ∈ T (2)

K1i −

t∑

τ=1

∑

j∈So

zτi,j ≤ K

t

i

(1 −

t−1∑

τ=1

ητi

)i ∈ Sc, t ∈ T (3)

t∑

τ=1

∑

i∈Sc

zτi,j ≤ K

t

j

t∑

τ=1

ητj j ∈ So, t ∈ T (4)

t∑

τ=1

∑

j∈So

zτi,j + ǫ

(1 −

t−1∑

τ=1

ητi

)≤ K

1i i ∈ Sc, t ∈ T (5)

∑

p∈P

µi,p

bt

i,p +∑

j∈L\i

xtj,i,p + yt−1

i,p

≤ K

1i −

t∑

τ=1

∑

j∈So

zτi,j i ∈ Sc, t ∈ T (6)

∑

p∈P

µi,p

bt

i,p +∑

j∈L\i

xtj,i,p + yt−1

i,p

≤

t∑

τ=1

∑

j∈Sc

zτj,i i ∈ So, t ∈ T (7)

∑

p∈P

µi,p

bt

i,p +∑

j∈L\i

xtj,i,p + yt−1

i,p

≤ K

t

i i ∈ L \ S, t ∈ T (8)

∑

p∈P

µi,p

bt

i,p +∑

j∈L\i

xtj,i,p + yt−1

i,p

≥ Kt

i

(1 −

t−1∑

τ=1

ητi

)i ∈ Sc, t ∈ T (9)

∑

p∈P

µi,p

bt

i,p +∑

j∈L\i

xtj,i,p + yt−1

i,p

≥ Kt

i

t∑

τ=1

ητi j ∈ So, t ∈ T (10)

∑

t∈T

ηti ≤ 1 i ∈ S (11)

∑

i∈So

FC1i

(2∑

τ=1

ητi

)+ ξ1 = B1 (12)

∑

i∈Sc

∑

j∈So

MCti,j zt

i,j +∑

i∈Sc

SCtiη

t−1i +

∑

j∈So

FCtj ηt+1

j + ξt

= Bt + αt−1 ξt−1 t ∈ T \ 1, n (13)∑

i∈Sc

∑

j∈So

MCni,j zn

i,j +∑

i∈Sc

SCni ηn−1

i + ξn = Bn + αn−1 ξn−1 (14)

bti,p ≥ 0, yt

i,p ≥ 0, xti,j,p ≥ 0, i ∈ L, j ∈ L \ i,

p ∈ P, t ∈ T (15)

zti,j ≥ 0 i ∈ Sc, j ∈ So, t ∈ T (16)

ξt ≥ 0 t ∈ T (17)

9

ηti ∈ 0, 1 i ∈ S, t ∈ T (18)

The objective function (1) minimizes the total net supply chain cost. Variable costs comprise

supply, transportation and inventory holding costs, while fixed costs account for operating the

facilities. The sum of the variable and fixed costs is reduced by the total revenue obtained at

the end of the time horizon. Revenue is generated by the fraction of the available budget that

has not been invested and has thus gained interest. The above objective function differs from

that of the models in Melo et al. [25] by the revenue term, which encourages the minimization

of expenditures on capacity transfer, facility opening and facility closing (see (12)–(14)). More-

over, in contrast to [25], the fixed operating costs of non-selectable facilities are not considered

explicitly as they correspond to a fixed term.

Constraints (2) are the usual flow conservation conditions and also ensure the satisfaction

of customer demands. Inequalities (3) guarantee that only operating existing facilities can

have their capacities transferred to new facilities. Constraints (4) state that a new facility

can only start receiving capacity after its setup, while constraints (5) ensure that an existing

facility is only closed after entire removal of its capacity. Capacity constraints are imposed by

inequalities (6)–(8). Constraints (9)–(10) guarantee that a selectable facility operates with at

least a given throughput. Constraints (11) allow the status of each selectable facility to change

at most once over the time horizon. This means that a facility that is removed cannot be

re-opened and once open, a new facility cannot be closed. Conditions (12)–(14) guarantee that

the available budget is invested in capacity transfers, the setup of new facilities and the removal

of existing facilities upon entire relocation. The amount of capital not used in a given period

earns interest 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 of constraints involving binary and contin-

uous variables. For any feasible set of values for the facility status variables, the optimal values

for the continuous variables are easily obtained by solving the associated linear problem. In this

context, an attractive search space is the set of feasible facility status variables η ∈ 0, 1n·|S|.

Any solution in this space can then “completed” to yield a (hopefully) feasible solution to

problem (P ) by computing the associated optimal values of the continuous variables b, x, y,

10

z and ξ. This aspect will be detailed later on. Since our problem is highly constrained, we will

not restrict the search space to feasible solutions. As will be described in Section 4.2, letting

the search move to solutions that violate the domain of the variables ruling the amount of

budget remaining at the end of each time period (i.e. constraints (17)) allows for a meaningful

exploration of the search space. This technique, known as strategic oscillation, was introduced

by Glover [15] and used since in many successful tabu search procedures for solving various

combinatorial problems (see e.g., Albareda-Sambola et al. [1], Cordeau and Laporte [8]).

4.1 Construction of an initial solution

Since our algorithm allows infeasible intermediate solutions, it can be initialized with any network

configuration with overbudget in at least one time period of the planning horizon. Starting from

the optimal solution of the linear relaxation, denoted (LP ), to (P ) we apply a simple rounding

procedure to the facility status variables in an attempt to obtain an initial solution that violates

at most the available budget. This is a solution to the relaxed problem (PB), a problem that

coincides with (P ) except that constraints (17) are replaced by ξ ∈ Rn·|S| and the facility status

variables η are fixed with a particular realization. The procedure is repeated by dynamically

adjusting a threshold ǫ′ for variable fixing, which is initialized with a value between zero and

one, denoted ǫ. Details regarding the adjustment mechanism are given below.

The pseudo-code in Algorithm 1 details the rounding procedure. At each iteration, a set Ω

is built with those facilities having at least one fractional status variable in some time period.

Facilities associated with fractional variables that are made binary in the course of an iteration

are gathered in the set Ξ. 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 period if at least a minimum level

of “activity” has been observed until that period. The “activity level” of site i in period t is

measured by the fractional value of the corresponding status variable ηti , while the minimum

activity imposed is defined by the threshold ǫ′. Hence, the time period for setting up a new

facility in a potential site i ∈ So is the first period for which the sum of the activity levels in that

site is at least ǫ′. When the status variables have relatively small values, the cumulative activity

level in site i is not significant to justify opening a new facility there. As will be described in

Section 5, exploratory testing showed that 0.25 is a promising initial value for the threshold ǫ′,

thus imposing a relatively low demand on the minimum cumulative activity.

11

Algorithm 1: Rounding procedure

1 Solve (LP ) and denote by (η, b∗, x∗, y∗, z∗, ξ∗) its optimal solution;

2 ǫ′ := ǫ;

3 Ω := i ∈ S : some ητi is fractional (τ ∈ T );

4 Ξ := ∅;

5 foreach facility i ∈ S do

6 if i /∈ Ω then

7 ηti := ηt

i for every t ∈ T

8 else

9 ηti := 0 for every t ∈ T

10 foreach new facility i ∈ So do

11 Find the first time period t ∈ T satisfying∑t−1

τ=1 ητi < ǫ′ and

∑t

τ=1 ητi ≥ ǫ′;

12 if t exists then ηti := 1; Ξ := Ξ ∪ i;

13 foreach existing facility i ∈ Sc do

14 Find the first time period t ∈ T satisfying∑t−1

τ=1 ητi < 1− ǫ′ and

∑t

τ=1 ητi ≥ 1− ǫ′;

15 if t exists then ηti := 1; Ξ := Ξ ∪ i;

16 Solve (PB) with the realization η of variables η;

17 if (PB) is infeasible and ǫ′ > 0 then

18 ηti := ηt

i for every i ∈ Ξ and t ∈ T ;

19 Ω := Ω \ Ξ;

20 ǫ′ := ǫ′ − γ ǫ;

21 go to line 4

22 else

23 return s0 := (η, b∗, x∗, y∗, z∗, ξ∗)

12

The choice of closing an existing facility is taken in a similar way. Since at the beginning

of the planning horizon an existing facility operates (and so the corresponding status variable

takes typically the value one), the gradual transfer of its capacity to new sites decreases its

activity level. Hence, facility i ∈ Sc is only closed at the end of some period t if a sufficiently

large amount of capacity has been removed until that time point. This condition is satisfied

when the cumulative activity of the facility is at least equal to the threshold 1 − ǫ′.

In each iteration of the rounding procedure the supply chain network is redesigned. In

one or several time periods, the network configuration may exceed the available budget. By

construction, there is no guarantee that a feasible solution to (PB) will be identified. In

particular, the minimum facility throughput constraints (9)–(10) may be violated. In this case

(see lines 17–21), the threshold ǫ′ is decreased by γ ǫ with γ denoting a user-defined factor

(0 < γ < 0.5). Observe that the threshold adjustment corresponds to lowering the requirement

on the minimum activity level imposed on a facility in the previous iteration of the algorithm.

If at some iteration the threshold ǫ′ becomes zero or negative then further reduction is not

possible and as a result, an initial solution is not available. However, such an outcome was

never observed in all the numerical tests that we performed (cf. Section 5).

4.2 Relaxation mechanism

Previous experience with a similar problem to (P ) (see Melo et al. [27]) has shown that re-

stricting the search process solely to feasible solutions may lead to a fairly small number of

promising solutions or even to none at all. Therefore, we developed a diversification mechanism

based on strategic oscillation to force the search into yet unexplored areas of the search space.

This is accomplished by relaxing constraints (17) regarding the unspent budget. Our choice of

these constraints is derived from the observation that a tight budget (as is usually the case in

relocation projects) strongly limits the investment options in a given period. This leads to a

restricted number of feasible network configurations with respect to opening new facilities and

closing existing facilities.

Although the relaxation of constraints (17) allows a budget consumption beyond the avail-

able limit over the time horizon, the structure of the relaxed problem still encourages network

configurations without excess budget. This goal is enforced by equalities (12)–(14) along with

the last term in the objective function, which maximizes revenue generated by non-invested

capital funds. Thus, problem (PB) corresponds to a “soft” relaxation of the original problem

13

which has a significant impact on the quality of the results obtained as will be seen in Section 5.

Through the relaxation of constraints (17) the search space is enlarged and a simple neigh-

bourhood structure is used to explore it. As constraint violation is reflected in negative values

for variables ξ, the following penalty term is added to the objective function of (P ) that weighs

the violation:

v′(s) = v(s) + α∑t∈T

max (0,−ξt) (19)

where s denotes any solution, v(s) is the objective value of (P ), and α is a self-adjusting

positive penalty factor, which alternates between values that encourage or discourage infeasible

solutions. By dynamically updating the value of α during the local search process, this relax-

ation mechanism facilitates the exploration of the search space and, as mentioned before, is

particularly useful for tightly constrained instances. It also stimulates the use of simple exchange

operators as the complex modification of a feasible solution into another feasible solution can

then be achieved by a series of simpler modifications through intermediate infeasible solutions.

This issue will be detailed below.

Function (19) is called the fitness of solution s. When we obtain a solution s feasible to (P )

the values of v′(s) and v(s) coincide. In this case, the penalty factor is decreased by setting

α := α/(1 + θ) with θ a user-defined parameter. Otherwise, v′(s) includes a penalty term

proportional to the total excess budget. In this case, the penalty factor is increased by taking

α := α (1 + θ).

4.2.1 Neighbourhood structure for solutions that violate the available budget

Since intermediate solutions may consume more budget in one or several time periods than the

available limit, one of the goals of our local search is to reduce infeasibility when it occurs.

In order to recover feasibility it is necessary to modify the status of at least one facility over

the planning horizon. This is achieved by exploring a neighbourhood, that we denote by N1,

when at an iteration of the TS algorithm the current solution s = (η, b, x, y, z, ξ) is feasible

to (PB) but not to (P ). Neighbouring solutions are obtained by performing moves that change

the status of one site in a single period. As will be shown next, these moves make partial use

of information on budget expenditures.

Suppose that we have a solution that is feasible to (PB) but not to (P ). Let τ be the first

period in which the total investment exceeds the available budget that is, τ = arg mint∈T

ξt :

ξt < 0. Expenditures in period τ include capacity moving costs as well as fixed setup costs for

14

new facilities that start operating in period τ + 1 (assuming τ < n) and fixed closing costs for

existing facilities that cease operating in period τ − 1 (assuming τ > 1). Since a large portion

of the budget is absorbed by the fixed costs, we will try to move part of the investment made

in period τ to another period. Let F oτ and F c

τ denote the sets of facilities for which fixed costs

are paid in period τ :

F oτ =

i ∈ So : ητ+1

i = 1

F cτ =

i ∈ Sc : ητ−1

i = 1

Observe that F oτ = ∅ for τ = n and F c

τ = ∅ for τ = 1. The local transformations applied to

solution s, and which define its neighbourhood N1(s), are obtained by the following exchanges.

The decision to open a new facility i ∈ F oτ in period τ+1 is overridden (by setting ητ+1

i = 0)

and one of the following local modifications is selected:

1. Destructive move: facility i is not open over the planning horizon.

2. Postpone opening facility i to some period k with τ + 1 < k ≤ n, i.e. set ηki = 1.

3. Schedule opening facility i for an earlier period k with 1 < k ≤ τ if there is sufficient

unspent budget in period k − 1. In other words, if FCk−1i ≤ ξk−1 then set ηk

i = 1.

For an existing facility i ∈ F cτ , similar changes are performed. The decision to close this

facility in period τ − 1 is overridden (i.e. ητ−1i = 0) and one of the following local moves is

performed:

1. Destructive move: facility i is not closed over the planning horizon.

2. Postpone closing facility i to some period k with τ ≤ k < n, i.e. set ηki = 1.

3. Schedule closing facility i for an earlier period k with 1 ≤ k < τ − 1 if there is sufficient

unspent budget in period k + 1. This means that if SCk+1i ≤ ξk+1 then set ηk

i = 1.

The size of the neighbourhood N1(s) is bounded by (n − 1) · |S|, but generally only a

few selectable facilities are eligible for further examination, which means that the sets F oτ

and F cτ tend to be small. Furthermore, due to the strategic nature of location problems, the

planning horizon is typically not too long which turns the complete exploration of N1(s) fairly

undemanding.

15

4.3 Neighbourhood structure for solutions that satisfy the avail-

able budget

Throughout the search, feasible solutions may also be identified to the original problem. In this

case, the structure of the neighbourhood N1 is inappropriate and local search is performed in a

new neighbourhood that we denote by N2. The transition from a feasible solution s to (P ) to

a solution s′ ∈ N2(s) is expressed by modifying the status of a single facility. This is achieved

by first choosing a facility at random. Next, the time period in which that facility has its status

changed (if any) in the current solution s will be either postponed to the following period or

scheduled for the previous period.

Assume that the randomly selected facility i belongs to the set Sc. Let τ denote the time

period in which this existing facility is closed (i.e. ητi = 1 and ηt

i = 0 for every t ∈ T , t 6= τ

in solution s). In the case that facility i is never closed and thus operates over the whole time

horizon, we set τ = 0. In order to move to a neighbouring solution s′, the current decision is

overruled by choosing another time period, denoted τ ′, as follows:

(i) If τ = 0 then τ ′ = n− 1, meaning that the facility is closed at the latest possible period;

(ii) If τ = 1 then τ ′ = 2;

(iii) If τ ∈ 2, . . . , n−2 then τ ′ = τ+1 with probability p and τ ′ = τ−1 with probability 1−p;

(iv) If τ = n − 1 then τ ′ = 0 with probability p (the facility operates in every period) and

τ ′ = τ − 1 with probability 1 − p.

As the shutdown of a facility compels capacity transfers as well as the setup of new facilities

to receive the transferred capacity, and these actions call for considerable budget expenditure,

a careful choice of a new time period τ ′ is imperative. This is particularly the case with the

above rules (i) and (iv) which attempt to reduce the impact on the network configuration by

postponing capital investments. Hence, network reconfiguration is to occur as late as possible.

If the random selection yields a new facility i ∈ So, then a move is defined by overruling

the time period in which this facility is established according to solution s. Should no facility

be located in site i over the time horizon then we set τ = 0. The neighbouring solution s′ is

achieved by fixing a new time period τ ′ as follows:

(v) If τ = 0 then τ ′ = n, so that the facility is opened in the latest possible period;

(vi) If τ = 2 then τ ′ = 3;

16

(vii) If τ ∈ 3, . . . , n−1 then τ ′ = τ+1 with probability p and τ ′ = τ−1 with probability 1−p;

(viii) If τ = n then τ ′ = 0 with probability p (the facility is not open over the time horizon)

and τ ′ = τ − 1 with probability 1 − p.

The motivation for using rules (v) and (viii) is to interfere as late as possible with the

network configuration. Changes caused by establishing new facilities early in the time horizon

put a significant strain on the budget and may even lead to overbudget situations.

We opted for an unbiased selection of a new time period for a status change of a facility

and hence, the parameter p takes the value 0.5 in rules (iii), (iv), (vii) and (viii). The size

of the neighbourhood N2(s) is bounded by 2 · |S|. In order to avoid the computational effort

incurred by exploring the entire neighbourhood of a solution, we only consider a random sample

N ′2(s) of N2(s). A user-defined parameter β determines the fraction of the neighbourhood to

be examined. In addition to reducing the computational burden, the evaluation of N ′2(s) allows

us to use a shorter tabu list than would be necessary if complete examination of N2(s) were to

be performed.

As the choice of the neighbourhood structure is by far the most critical step in the design of

a TS heuristic, we also explored an extension of N2(·) in which both the choice of a facility and

of a time period are taken randomly. This alternative neighbourhood structure introduces some

additional randomization which later proved not to be beneficial both from a computational

viewpoint and in terms of the quality of the solutions identified. Hence, the numerical results

to be presented in Section 5 omit the exploration of this alternative neighbourhood.

4.4 Backtracking scheme

At an iteration of the TS algorithm, the “admissible” subset of the explored neighbourhood

may be empty. This occurs when all the local transformations applied to the current solution s:

(i) refer to tabu moves that cannot be revoked (see Section 4.5 for details on the aspiration

criterion), or (ii) yield infeasible solutions to the relaxed problem (PB). In either case, the

local search is trapped in a local minimum. Usually, s is a mediocre solution or not even

feasible to (P ) if the available budget is exceeded in at least one time period. To overcome this

problem, once a neighbourhood has been explored, its m best neighbours, denoted s1, . . . , sm,

are retained. The user-defined parameter m typically takes a small value (e.g. two or three)

to reduce the computational burden. The search is restarted from the j-th neighbour, when it

has failed for all solutions s1, . . . , sj−1 and 1 < j ≤ m. This backtracking mechanism allows

17

previously visited neighbouring solutions to be considered, even though their fitness is not the

best since v′(sm) ≥ . . . ≥ v′(s1).

In the worst case, the backtracking strategy does not help the search move away from a local

minimum. The algorithm is then restarted with a new initial solution. The latter is obtained

by the rounding procedure (cf. Section 4.1) for a user-defined threshold ǫ such that ǫ < ǫ. The

algorithm is restarted at most once during the whole search process.

4.5 Short-term memory

When the status of a facility changes according to one of the moves described before (cf. Sec-

tions 4.2.1 and 4.3), the reverse move is forbidden (and thus, tabu) for a fixed number of

iterations. To prevent the search from tracing back its steps to where it came from, a tabu list

of fixed size h is kept. The tabu tenure of a move is also set equal to this parameter h.

An aspiration criterion is applied to revoke a tabu move if it results in a solution that

improves the incumbent. The latter is the feasible solution with the best objective value found

so far. When no feasible solution is available (and this typically occurs at the earlier stages of

the search), the incumbent solution is taken to be the least infeasible (the one with smallest

value v′(·) according to (19)) known so far.

4.6 Intensification phase

This phase is basically oriented to identify good quality feasible solutions when a pre-specified

solution quality threshold has not been achieved after a given number of iterations. In this case,

the tabu search algorithm is restarted with the best feasible incumbent solution, denoted s∗.

Since neighbourhood sampling is used while exploring N2(·), new network configurations are

likely to be visited as the result of exchanges that were not considered when the local search

was first performed in N2(s∗). To restrict the overall computational burden, the size of the

tabu list is reduced in this phase to the length h′ < h, with h denoting the initial tabu list size.

Moreover, less iterations are allowed to be performed when the local search is restarted. The

intensification process is applied at most once during the whole search process.

4.7 Algorithm

The pseudo-code in Algorithm 2 summarizes the TS heuristic. It starts from an initial solution s0

identified by the rounding procedure (cf. Algorithm 1) and returns, after execution, the best

18

feasible solution found s∗, if any (line 22, resp. line 3). At each iteration, an appropriate

neighbourhood of the incumbent solution s is selected (line 7). If s violates constraints (17)

then N1(s) is fully explored, otherwise N ′2(s) is examined. At the end of each iteration, the

penalty factor α is updated (lines 14–18). The local search stops when one of the following

conditions is met:

T1. The maximum number of iterations kmax is achieved (this value is decreased to kintens

when the local search is prompted by the intensification phase);

T2. The best identified solution is feasible and satisfies a pre-specified solution quality thresh-

old δ, that is, v(s)−v(LP )v(LP )

≤ δ, with v(LP ) denoting the optimal value of the linear

relaxation.

The intensification phase is triggered by the best feasible solution identified by the loop in

lines 6–18 when the latter does not meet the quality criterion T2. In this case, the local search

is restarted and a smaller number of iterations (kintens) is performed.

Finally, we note that for each particular realization of the facility status variables we solve

the associated linear problem (PB) with commercial optimization software.

5 Computational results

To evaluate the quality of the solutions identified by the proposed TS heuristic we performed

a series of computational experiments with randomly generated test problems, as benchmark

instances are not available for the problem at hand. We first present the instances generated

and then describe the calibration process of parameter values. We conclude this section with

a summary and analysis of the results obtained. The TS heuristic is also compared to a

sophisticated LP-based procedure proposed by Melo et al. [27].

5.1 Test instances

Realistically-sized test instances were randomly generated to depict three-echelon networks.

Upstream facilities either refer to plants or suppliers while the intermediate echelons comprise

distribution centres (DCs). The latter are classified into central and regional facilities and are

subject to relocation decisions. Table 7 in the Appendix lists the main characteristics of the

53 generated instances. All costs follow a non-decreasing pattern over the time horizon since in

our view this reflects real-world situations better, as supply chain networks are often redesigned

19

Algorithm 2: Tabu search procedure

1 Build an initial solution s0 starting with the threshold ǫ;

2 if s0 is not feasible to (PB) then

3 return initial solution not found

4 else

5 s := s0; v′(s) := v′(s0); s∗ := s0; v∗ := v(s0); intens := false

6 repeat

7 Choose a neighbourhood definition;

8 Apply local search to s to obtain the m best non-tabu solutions s′1, . . . , s′m

among the explored solutions;

9 if no new neighbouring solution is found then

10 apply backtracking;

11 if backtracking fails to identify a new solution then go to line 1 with ǫ := ǫ

12 Update the tabu list and the current solution s with s′1; retain the best m − 1

solutions s′2, . . . , s′m;

13 if v′(s) < v∗ then s∗ := s; v∗ := v′(s);

14 if s is feasible then

15 α := α/(1 + θ)

16 else

17 α := α (1 + θ)

18 until at least one stopping criterion is satisfied ;

19 if s∗ is feasible and quality of s∗ > δ and intens = false then

20 intens := true; s := s∗; v′(s) := v∗;

21 go to line 7 and consider a tabu list of size h′

22 return s∗ (if feasible)

20

to cope with rising costs driven by an expanding global economy. A distinctive feature of the test

instances is the magnitude of the facility closing costs compared to facility opening costs. The

former are significantly lower and may even take negative values to account for revenues due

to the termination of leasing contracts or the selling of property. Furthermore, the number of

network arcs is restricted to the values given in Table 5. In addition to inter-echelon transports,

commodities can also be shipped directly to customers as well as distributed among facilities

in the same echelon. Finally, 70 − 80% of the commodities can actually be shipped over each

generated arc. This limits the flows through the network and thus mimics real-world situations.

Observe that with this feature problem (P ) becomes more tightly constrained as the number

of transportation channels available for product distribution is reduced. As a result, finding

feasible solutions is an even more difficult task. Further details about the random generation

of the test instances can be found in Melo et al. [27].

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

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

5.2 Parameter calibration

Exploratory testing to find good ranges of parameters was performed by running the TS algo-

rithm with a variety of parameter settings. Table 6 displays the values that not only led to

good results but also appear to be robust as they do not seem to have a significant impact on

the performance of the heuristic with respect to two measures: (i) the deviation to the lower

bound provided by the linear relaxation, and (ii) the computational time.

At first sight, the tabu list size may seem small but experiments performed with larger

lists (with sizes 5 and 7) did not yield better solutions. Furthermore, the total number of

iterations kmax is also small. This results from the observation that the rate of improvement

in solution quality, after a few iterations, becomes negligible and therefore, it is not worth to

invest in additional computational resources. Nevertheless, a sufficient number of neighbours

21

Parameter Description Value

ǫ Initial value of the threshold ǫ′ for variable fixing in the rounding

procedure (Algorithm 1)

0.25

γ Factor used to update ǫ′ 1/6

ǫ Threshold for variable fixing when the tabu search fails to find a

feasible solution

0.05

α Penalty factor for overbudget 1000

θ Parameter used to update α θ ∈ (0, 1]

h Size of the tabu list and tabu tenure 3

h′ 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 neighbourhood

3

β Percentage of solutions that are examined in the neighbourhood

N2(·)

15%

p Probability factor used in conjunction with neighbourhood N2(·) 0.5

kmax Total number of iterations 15

kintens Total number of iterations in the intensification phase 5

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

Table 6: Parameter values

is explored during the local search (in total 45 neighbours given the values of m and kmax).

The fact that good quality solutions are identified early in the search process demonstrates that

the neighbourhood structures N1(·) and N2(·) are particularly adequate and thus, strategic

oscillation proves to be a very effective mechanism for the problem at hand.

The value of the threshold δ, which is used in the termination criterion T2, may seem at

first glance rather tight. Our choice is motivated by the results obtained by Melo et al. [25] for

a similar class of network redesign problems for which the optimal solution proved to be within

1% of the LP bound.

Finally, we note that the parameter θ used for dynamically adjusting the penalty factor α in

the fitness function (19) is selected randomly in the interval (0, 1]. Preliminary testing showed

that its value does not impact on the results of the TS heuristic.

5.3 Performance analysis

We categorize the 53 test instances into three sets based on the number of customers. The

performance of the new TS heuristic is evaluated with respect to the quality of the solutions

obtained compared to the linear relaxation bound (denoted “LP-gap” and defined in the ter-

mination criterion T2) and to the total computational time. Set 1 comprises 23 instances with

22

50 customers, while set 2, resp. 3, includes 17 instances with 100 customers, resp. 13 instances

with 200 customers. The relaxed problems (LP ) and (PB) were modeled using ILOG Concert

Technology 2.0 [23] and solved with CPLEX 10.2. The TS algorithm was implemented in C++

following the framework proposed by Blesa and Xhafa [5]. All experiments were conducted on

a Pentium III with a 2.6 GHz processor and 2 GB RAM.

Figure 1 displays the results obtained with the parameter settings given in Table 6. Detailed

results are reported in Tables 8–10 in the Appendix. The effectiveness of our solution method

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25

Set 1: P1-P23

Set 2: P24-P40

Set 3: P41-P53

LP-gap (%)

CPU (seconds)

Figure 1: Results obtained using the new TS procedure.

is demonstrated by the high quality of the best solutions identified, which have an LP-gap

below 1% in 38 out of 53 instances (71.7%). Moreover, in more than half of the instances

(28 out of 53) the best solution deviates less than 0.02% from the LP bound. In particular,

instances with 100 and 200 customers (sets 2 and 3) exhibit this feature, with the exception of

two, which have an LP-gap of 2.34%, resp. 21.43%. The latter refer to three-echelon networks

with 100 customers and 50 suppliers (see P39 and P40 in Table 9). Large LP-gaps were also

obtained for some of the instances with 50 customers (set 1). Regarding the computational

performance of the TS heuristic, half of the instances were solved in less than 2.7 minutes

23

which can be seen as a reasonable CPU time for large-scale instances.

Four outliers were excluded to produce Figure 1 due to their poor solution quality or their

high computational demands. They refer to instances P1 and P3 (with LP-gaps above 100%)

along with P8 and P31 (with CPU times of 17 resp. 27.6 minutes).

To better understand the behaviour of the TS heuristic, we also collected information on

how often the intensification phase and the backtracking scheme were used. Furthermore,

we kept track of the instances in which the heuristic had to be restarted. Recall that this

occurs when at an iteration the exploration of the neighbourhood of all of the m non-tabu

solutions fails to identify a feasible solution to (PB). Figure 2 summarizes the frequencies

of these three cases (Tables 8–10 in the Appendix report the detailed results). The solution

Intensification Backtracking

Restart

12

2

21

0 5

0

Figure 2: Number of instances in which the intensification phase, the backtracking scheme

and the restart strategy were used.

quality threshold δ is not satisfied by 28.3% of the instances (in total 15, 13 of which belong

to networks with 50 customers) and as a result, the local search is intensified in promising

regions for a few iterations. Although further solution improvement is obtained, the procedure

ends with a solution that still does not meet the quality criterion. Numerical tests with an

increased number of iterations during the intensification phase did not produce better results,

thus indicating that the initial local search has a powerful effect on the quality of the solutions

identified. Observe that the maximum number of iterations (kmax + kintens) is performed in

each one of the above 15 instances which impacts the overall CPU time. The backtracking

scheme is only used in 10 out of 53 instances (18.8%) and in more than half of these the local

search is restarted with a new initial solution. This strategy turns out to be a suitable way to

escape from local minima as good quality solutions are then identified in 7 out of 10 instances.

24

5.4 Comparison with an LP-based heuristic

In order to evaluate the benefit of allowing infeasible intermediate solutions during the local

search, the results of the TS heuristic are compared to a two-phase heuristic approach proposed

by Melo et al. [27], in which solution feasibility is maintained throughout the whole procedure.

In the first phase of this heuristic, a sophisticated linear programming rounding strategy is

applied to find initial values for the binary facility status variables. The second phase uses

local search in an attempt to improve the initial solution when its quality does not meet given

criteria. The second phase is also used when a feasible solution was not identified before. In

this case, the initial variable choices are corrected by means of local search. In what follows we

will denote this approach as the LP-based procedure.

Both heuristics are compared in terms of the overall CPU times and the best solutions found

with respect to their LP-gaps. The corresponding ratios are determined by

LP-gap ratio =LP-gap of best TS solution

LP-gap of best LP-based solution

CPU ratio =CPU time of TS heuristic

CPU time of LP-based procedure

and shown in Figure 3. Note that a ratio lower than one indicates that the TS heuristic

outperforms the LP-based procedure. Ratios higher than 3 correspond to instances considered

to be outliers and therefore are not depicted in Figure 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 diamond at the bottom left side of

Figure 3. The LP-based procedure by Melo et al. [27] could not identify a single feasible solution

for any of these instances. As this feature is observed across all three sets, it concerns any

network size independently of the number of customers. In contrast, the TS heuristic is able

to solve all 53 instances, thus clearly demonstrating that it is beneficial to drive the search

towards and away from boundaries of the feasible space. It is also worth pointing out that

the TS algorithm yields better solutions with less computational effort for further 12 instances

(22.6%, bottom left side of the diagram). Hence, the TS heuristic outperforms the LP-based

procedure in almost half of the instances. The upper left and bottom right sides of Figure 3

refer to instances for which the TS heuristic is superior either in terms of solution quality or

computational time. Finally, the LP-based procedure outperforms the TS algorithm in only

7 instances (13.2%). In conclusion, we can say that the proposed TS heuristic exhibits a good

and robust performance.

25

0

1

2

3

0 1 2 3

14 instances

LP-gap ratio

CPU ratio

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

6 Conclusions

This study described a tabu search heuristic for solving a large-scale multi-echelon network

redesign problem. To the best of our knowledge, this is the first attempt to develop such

a metaheuristic for a comprehensive problem capturing several features relevant to strategic

supply chain planning under a facility relocation scenario over a multi-period horizon. The

numerical results show that the new procedure performs well and can reach solutions within

1% of the linear relaxation bound in reasonable computational times. The strength of the new

approach lies in the fact that during the search process the boundary of feasibility can be crossed

in order to explore the infeasible region for a certain number of moves. This is accomplished

by allowing financial investments on network reconfiguration to exceed the available budget in

one or several time periods. This strategic oscillation feature of the new algorithm has shown

to be particularly suitable to tackle a problem with fragmented feasible regions. In our case

- as in many other hard optimization problems - some areas of the feasible space containing

26

attractive solutions seem to be only separated by a narrow infeasible region. By crossing such

a region feasible solutions of superior quality are achieved.

We believe that our solution methodology is flexible and can be successfully adapted to

handle other multi-period optimization problems involving similar phase-in/phase-out decisions

as in our problem. For example, technology acquisition and equipment installation decisions

often arise in production planning and in-house logistics contexts. Such problems are typically

highly constrained and even finding a feasible solution is a difficult task. It would be interesting

to investigate the application of a relaxation mechanism similar to the one proposed in this

paper. Observe that the robustness of our method results from the fact that the relaxed

problem keeps to a greater extent the structure of the original problem. In our view this feature

is critical for obtaining high-quality solutions.

Acknowledgements

This research was partly supported by the German Academic Exchange Service (DAAD) under the

program PPP-Accoes Integradas Luso-Alemas/DAAD-GRICES. This support is gratefully acknowl-

edged. The authors also thank Martin Ducrozet for his valuable help in generating the test instances

and implementing the tabu search algorithm.

References

[1] M. Albareda-Sambola, J.A. Dıaz, and E. Fernandez. A compact model and tight bounds

for a combined location-routing problem. Computers & Operations Research, 32:407–428,

2005.

[2] N. Aras, D. Aksen, and A.G. Tanugur. Locating collection centers for incentive-dependent

returns under a pick-up policy with capacitated vehicles. European Journal of Operational

Research, 191:1223–1240, 2008.

[3] M.A. Arostegui Jr., S.N. Kadipasaoglu, and B.M. Khumawala. An empirical comparison

of tabu search, simulated annealing and genetic algorithms for facilities location problems.

International Journal of Production Economics, 103:742–754, 2006.

[4] R.H. Ballou. Unresolved issues in supply chain network design. Information Systems

Frontiers, 3:417–426, 2001.

27

[5] M. Blesa and F. Xhafa. A skeleton for the tabu search metaheuristic with applications to

problems in software engineering. In Proceedings of the 23rd International Conference on

Software Engineering, pages 62–66, 2001.

[6] R. Caballero, M. Gonzalez, F.M. Guerrero, J. Molina, and C. Paralera. Solving a multi-

objective location routing problem with a metaheuristic based on tabu search. Application

to a real case in Andalusia. European Journal of Operational Research, 177:1751–1763,

2007.

[7] C. Chandra and J. Grabis. Supply Chain Configuration: Concepts, Solutions and Applica-

tions. Springer Science+Business Media, 2007.

[8] J.-F. Cordeau and G. Laporte. Tabu search heuristics for the vehicle routing problem. In

C. Rego and B. Alidaee, editors, Metaheuristic Optimization via Memory and Evolution:

Tabu Search and Scatter Search, chapter 2, pages 145–163. Kluwer, Boston, 2004.

[9] J.-F. Cordeau, G. Laporte, and F. Pasin. An iterated local search heuristic for the logis-

tics network design problem with single assignment. International Journal of Production

Economics, 113:626–640, 2008.

[10] J.-F. Cordeau, F. Pasin, and M.M. Solomon. An integrated model for logistics network

design. Annals of Operations Research, 144:59–82, 2006.

[11] I. Correia and M.E. Captivo. Bounds for the single source modular capacitated plant

location problem. Computers & Operations Research, 33:2991–3003, 2006.

[12] M.J. Cortinhal and M.E. Captivo. Upper and lower bounds for the single source capacitated

location problem. European Journal of Operational Research, 151:333–351, 2003.

[13] M. Daskin, L. Snyder, and R. Berger. Facility location in supply chain design. In A. Langevin

and D. Riopel, editors, Logistics systems: Design and optimization, chapter 2, pages 39–

65. Springer, 2005.

[14] D. Ghosh. Neighborhood search heuristics for the uncapacitated facility location problem.

European Journal of Operational Research, 150:150–162, 2003.

[15] F. Glover. Heuristics for integer programming using surrogate constraints. Decision Sci-

ences, 8:156–166, 1977.

28

[16] F. Glover and M. Laguna. Tabu Search. Kluwer, Boston, 1997.

[17] M. Goetschalckx, C.J. Vidal, and K. Dogan. Modeling and design of global logistics

systems: A review of integrated strategic and tactical models and design algorithms.

European Journal of Operational Research, 143:1–18, 2002.

[18] R. Hammami, Y. Frein, and A.B. Hadj-Alouane. Supply chain design in the delocalization

context: Relevant features and modeling tendencies. International Journal of Production

Economics, 113:641–656, 2008.

[19] P. Hansen, E. Pedrosa Filho, and C. Ribeiro. Location and sizing of offshore platforms for

oil exploration. European Journal of Operational Research, 58:202–214, 1992.

[20] B.B. Keskin and H. Ulster. Meta-heuristic approaches with memory and evolution for a

multi-product production/distribution system design problem. European Journal of Oper-

ational Research, 182:663–682, 2007.

[21] D.-H. Lee and M. Dong. A heuristic approach to logistics network design for end-of-lease

computer products recovery. Transportation Research Part E: Logistics and Transportation

Review, 44:455–474, 2008.

[22] J. Li, F. Chu, and C. Prins. Upper and lower bounds for a capacitated plant location

problem with multicommodity flow. Computers & Operations Research, 36:3019–3030,

2009.

[23] ILOG Concert Technology 2.0 User’s Manual. ILOG, Inc., Incline Village, Nevada: www.

ilog.com/products/optimization/tech/concert.cfm, 2003.

[24] M.J. Meixell and V.B. Gargeya. Global supply chain design: A literature review and critique.

Transportation Research Part E: Logistics and Transportation Review, 41:531–550, 2005.

[25] M.T. Melo, S. Nickel, and F. Saldanha da Gama. Dynamic multi-commodity capacitated

facility location: A mathematical modeling framework for strategic supply chain planning.

Computers & Operations Research, 33:181–208, 2006.

[26] M.T. Melo, S. Nickel, and F. Saldanha da Gama. Facility location and supply chain

management - A review. European Journal of Operational Research, 196:401–412, 2009.

29

[27] M.T. Melo, S. Nickel, and F. Saldanha da Gama. An LP-rounding heuristic to solve a

multi-period facility relocation problem. Technical Report 168, Fraunhofer Institute for

Industrial Mathematics, Kaiserslautern, Germany, 2009. Available at www.itwm.fhg.de/

zentral/download/berichte/bericht_168.pdf. Submitted for publication.

[28] L. Michel and P. Van Hentenryck. A simple tabu search for warehouse location. European

Journal of Operational Research, 157:576–591, 2004.

[29] M. Sun. Solving the uncapacitated facility location problem using tabu search. Computers

& Operations Research, 33:2563–2589, 2006.

[30] P.N. Thanh, N. Bostel, and O. Peton. A dynamic model for facility location in the design

of complex supply chains. International Journal of Production Economics, 113:678–693,

2008.

[31] D. Tuzun and L.I. Burke. A two-phase tabu search approach to the location routing

problem. European Journal of Operational Research, 116:87–99, 1999.

[32] D. Vila, A. Martel, and R. Beauregard. Designing logistics networks in divergent process

industries: A methodology and its application to the lumber industry. International Journal

of Production Economics, 102:358–378, 2006.

[33] Q. Wang, R. Batta, J. Bhadury, and C.M. Rump. Budget constrained location problem

with opening and closing of facilities. Computers & Operations Research, 30:2047–2069,

2003.

[34] W. Wilhelm, D. Liang, B. Rao, D. Warrier, X. Zhu, and S. Bulusu. Design of international

assembly systems and their supply chains under NAFTA. Transportation Research Part E:

Logistics and Transportation Review, 41:467–493, 2005.

30

Appendix: Test instances and detailed results

31

Plants / Existing DCs Potential DCs Instance size

Set Instance Periods Products Suppliers Central Regional Central Regional # Var. # Const.

P1 3 5 5 4 10 8 20 19014 2075P2 4 5 5 4 10 8 20 25482 2757P3 6 5 5 4 10 8 20 38418 4121P4 8 5 5 4 10 8 20 51354 5485P5 3 5 5 8 20 12 30 39188 2803P6 4 5 5 8 20 12 30 52642 3723P7 6 5 5 8 20 12 30 79550 5563P8 8 5 5 8 20 12 30 106458 7403

Set 1 P9 3 10 5 4 10 8 20 37107 353050 cust. P10 4 10 5 4 10 8 20 49606 4697

P11 3 10 5 8 20 12 30 75803 4678P12 4 10 5 8 20 12 30 101462 6223P13 3 20 5 4 10 8 20 73293 6440P14 4 20 5 4 10 8 20 98154 8577P15 3 20 5 8 20 12 30 149033 8428P16 4 20 5 8 20 12 30 199102 11223P17 3 50 5 8 20 12 30 368723 19678P18 4 5 50 4 10 8 20 32574 3837P19 3 5 50 8 20 12 30 47153 3613P20 8 5 50 8 20 12 30 127698 9563P21 3 10 50 4 10 8 20 47745 5015P22 4 10 50 4 10 8 20 63790 6677P23 4 10 50 8 20 12 30 122702 8203P24 3 5 5 4 10 8 20 28701 2975P25 4 5 5 4 10 8 20 38398 3957P26 6 5 5 4 10 8 20 57792 5921P27 8 5 5 4 10 8 20 77186 7885P28 3 5 5 8 20 12 30 54335 3703P29 4 5 5 8 20 12 30 72838 4923P30 6 5 5 8 20 12 30 109844 7363

Set 2 P31 8 5 5 8 20 12 30 146850 9803100 cust. P32 3 10 5 4 10 8 20 56484 5180

P33 4 10 5 4 10 8 20 75442 6897P34 3 10 5 8 20 12 30 106100 6328P35 4 10 5 8 20 12 30 141858 8423P36 4 20 5 4 10 8 20 149530 12777P37 3 20 5 8 20 12 30 209630 11578P38 4 20 5 8 20 12 30 279898 15423P39 3 5 50 4 10 8 20 34023 3785P40 3 10 50 4 10 8 20 67125 6665P41 3 5 5 4 10 8 20 48084 4775P42 4 5 5 4 10 8 20 64242 6357P43 6 5 5 4 10 8 20 96558 9521P44 8 5 5 4 10 8 20 128874 12685P45 3 5 5 8 20 12 30 84638 5503

Set 3 P46 4 5 5 8 20 12 30 113242 7323200 cust. P47 6 5 5 8 20 12 30 170450 10963

P48 8 5 5 8 20 12 30 227658 14603P49 3 10 5 4 10 8 20 95247 8480P50 4 10 5 4 10 8 20 127126 11297P51 3 10 5 8 20 12 30 166703 9628P52 4 10 5 8 20 12 30 222662 12823P53 4 20 5 8 20 12 30 441502 23823

Table 7: Characteristics and size of test instances associated with three-echelon networks

32

Tabu Search heuristic LP-based heuristic

Instance LP-Gap (%) CPU (sec) Intensification Restart Backtracking LP-Gap (%) CPU (sec)

P1 135.43 8 * 7.90 20

P2 0.11 18 0.10 16

P3 239.01 157 * 10.07 2606

P4 0.18 959 2.97 802

P5 6.78 21 * * 4.24 27

P6 9.03 190 * – –

P7 10.02 588 * 7.70 780

P8 11.06 1655 * 8.41 9530

P9 0.01 39 0.02 34

P10 2.40 119 * 4.26 188

P11 0.01 37 0.01 41

P12 0.01 124 0.01 100

P13 0.00 249 0.00 143

P14 0.00 182 0.00 248

P15 0.00 92 0.00 105

P16 0.00 330 – –

P17 0.00 329 0.00 310

P18 4.51 37 * – –

P19 3.44 36 * – –

P20 6.38 409 * 8.20 684

P21 13.52 117 * * * 20.11 43

P22 15.8 100 * 21.32 185

P23 3.89 146 * 3.87 204

Table 8: Results for the instances in set 1 (50 customers)

Tabu Search heuristic LP-based heuristic

Instance LP-Gap (%) CPU (sec) Intensification Restart Backtracking LP-Gap (%) CPU (sec)

P24 0.34 21 0.02 21

P25 0.30 54 0.12 58

P26 0.03 247 0.03 167

P27 0.04 330 0.04 436

P28 0.02 51 0.01 65

P29 0.02 66 – –

P30 0.05 209 0.03 351

P31 0.04 1017 – –

P32 0.69 180 * * 0.00 55

P33 0.01 322 * * 0.00 108

P34 0.00 41 0.00 51

P35 0.00 105 – –

P36 0.00 264 0.00 347

P37 0.00 348 * * – –

P38 0.00 431 – –

P39 21.43 42 * * * 21.35 35

P40 2.34 278 * 2.65 86

Table 9: Results for the instances in set 2 (100 customers)

33

Tabu Search heuristic LP-based heuristic

Instance LP-Gap (%) CPU (sec) Intensification Restart Backtracking LP-Gap (%) CPU (sec)

P41 0.01 41 0.01 44

P42 0.00 122 0.49 114

P43 0.01 437 * 0.01 263

P44 0.01 469 0.01 855

P45 0.00 85 * * – –

P46 0.00 69 – –

P47 0.00 225 0.00 204

P48 0.01 810 * – –

P49 0.00 67 0.00 81

P50 0.81 643 * * 0.00 167

P51 0.00 60 0.00 74

P52 0.00 158 – –

P53 0.00 464 – –

Table 10: Results for the instances in set 3 (200 customers)

34