Ann Oper ResDOI 10.1007/s10479-011-1050-9

An improved Benders decomposition algorithmfor the logistics facility location problem with capacityexpansions

Lixin Tang · Wei Jiang · Georgios K.D. Saharidis

© Springer Science+Business Media, LLC 2012

Abstract We investigate a logistics facility location problem to determine whether the ex-isting facilities remain open or not, what the expansion size of the open facilities shouldbe and which potential facilities should be selected. The problem is formulated as a mixedinteger linear programming model (MILP) with the objective to minimize the sum of thesavings from closing the existing facilities, the expansion costs, the fixed setup costs, thefacility operating costs and the transportation costs. The structure of the model motivatesus to solve the problem using Benders decomposition algorithm. Three groups of valid in-equalities are derived to improve the lower bounds obtained by the Benders master problem.By separating the primal Benders subproblem, different types of disaggregated cuts of theprimal Benders cut are constructed in each iteration. A high density Pareto cut generationmethod is proposed to accelerate the convergence by lifting Pareto-optimal cuts. Compu-tational experiments show that the combination of all the valid inequalities can improvethe lower bounds significantly. By alternately applying the high density Pareto cut gener-ation method based on the best disaggregated cuts, the improved Benders decompositionalgorithm is advantageous in decreasing the total number of iterations and CPU time whencompared to the standard Benders algorithm and optimization solver CPLEX, especially forlarge-scale instances.

Keywords Facility location · Existing facility expansion · Establishment of new facilities ·Benders decomposition · Valid inequalities · Disaggregated cuts · High density Pareto cuts

L. Tang (�) · W. JiangLiaoning Key Laboratory of Manufacturing System and Logistics, The Logistics Institute, NortheasternUniversity, Shenyang, 110004, Chinae-mail: qhjytlx@mail.neu.edu.cn

G.K.D. SaharidisUniversity of Thessaly, Leoforos Athinon, Pedion Areos, 38834 Volos, Greece

G.K.D. SaharidisKathikas Institute of Research & Technology, 303 Ryefield Ridge, Columbia, MO 65203, USA

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1 Introduction

The facility location problems have received extensive attention in the past few decades(see e.g., Erengüç et al. 1999; Melo et al. 2009). Most of researches in the literature aredevoted to determining which sites should be selected to establish new facilities from a setof potential sites under the constraints that the demands of all customers have to be met,and the capacity limits of the suppliers and facilities must not be violated so as to minimizethe total cost. This total cost is the sum of the fixed setup costs, the operating costs ofproducts at each established facility and the transportation costs. In practice, however, it isfrequently encountered that some existing facilities are available but insufficient to satisfythe increasing storage demands, and thus it is necessary to expand the capacities of theexisting facilities and/or establish additional facilities. If these existing facilities where someinfrastructures are already equipped with are ignored when considering the establishment ofnew facilities, it will result in significant waste of the resources and increase of the fixedsetup costs. Therefore, the problem under our consideration is to determine the locations ofnew facilities based on the existing facilities. However, whether the existing facilities wouldbe expanded, closed or unchanged is dependent on the trade-off between their operatingcosts and the fixed setup costs for establishing new facilities.

To the best of our knowledge, little research has been done on the facility location prob-lem with simultaneous consideration of the existing facilities even though the facility loca-tion problems and capacity expansion problems have been considered separately in most ofthe previous studies. Geoffrion and Graves (1974) propose a two-stage distribution systemdesign problem in which only the establishment of new facilities is considered and eachcustomer is only able to be served by a single facility to minimize the total cost in the sys-tem, and Benders decomposition algorithm is first employed to solve it. Hindi and Basta(1994) study the same problem proposed by Geoffrion and Graves (1974) except that eachcustomer can be served by more than one facility, and a branch and bound algorithm is usedto solve it. Üster et al. (2007) investigate a production/distribution system design problemwith a fixed number of capacitated facilities, and multiple meta-heuristic approaches areproposed to solve it. Klose (2000) studies a two-stage capacitated facility location problemand a Lagrangean relax-and-cut approach is proposed to solve it.

The existing capacity expansion problems in the literature can be divided into two cate-gories. One is the node capacity expansion problem in which the capacities of the existingnodes are expanded while the structure of the network remains unchanged. The other isthe node number expansion problem in which new nodes are added to the existing networkwhile the capacities of the existing nodes keep unchanged. For the node capacity expansionproblems, Fond and Srinivasan (1986) study a production capacity expansion problem fora single stage production-distribution system, and an improved heuristic algorithm is pro-posed. Yilmaz and Catay (2006) investigate a three-stage production-distribution networkplanning problem with node capacity expansions, and the MILP relaxation-based heuris-tics are developed to obtain a good feasible solution. Ko and Evans (2007) consider a nodecapacity expansion problem for a distribution network with reverse logistics where the ware-houses and repair centers are regarded as expandable nodes. The problem is formulated asa mixed integer nonlinear programming model and a genetic algorithm-based heuristic isproposed to solve a realistically sized problem. For the node number expansion problems,Singh et al. (1998) investigate a distribution substation and feeder planning problem for anelectricity transmission network in which new substations and feeders need to be installedwhile the capacities of the existing substations and feeders remain unchanged. A mixed 0–1quadratic program model is developed and the generalized Benders decomposition algo-rithm is applied to solving it. Gendreau et al. (2006) consider a telecommunications network

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Fig. 1 The structure of the system under consideration

plan problem to determine the installation of a concentrator at each node so as to minimizethe installation costs.

As compared with the ordinary facility location problems or the capacity expansion prob-lems mentioned above, the problem under our consideration has some different features sothat it is more realistic and challenging. These features are reflected by making integrateddecisions on whether the existing facilities will be open, expanded or closed and whichpotential sites will be selected to establish new facilities.

The remainder of the paper is organized as follows. Section 2 is devoted to describing theMILP model of our problem. Benders decomposition algorithm is applied to our problemin Sect. 3. In Sect. 4, the improved Benders decomposition algorithms with three groups ofvalid inequalities, different disaggregated cuts and high density Pareto cuts are proposed.Section 5 reports the experimental results to evaluate the performance of the improved Ben-ders decomposition algorithms. Finally Sect. 6 concludes the paper.

2 Mathematical formulation

In this section, the detailed description and the mathematical model of the facility locationproblem with capacity expansions are given.

2.1 Problem description

The structure of the logistics system under consideration is shown in Fig. 1. The productsproduced by suppliers will be shipped to customers via facilities to meet customers’ de-mands. Each supplier can supply a wide variety of products to several facilities, and the

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supply capacity for each product is known. Our problem is to determine whether the exist-ing facilities should be open or not, what the expansion size of each open facility shouldbe, which new facilities should be established, and which customers should be served by anopen facility so that the sum of the expansion costs, the savings from closing the existingfacilities, the fixed setup costs, the facility operating costs and the transportation costs isminimized.

2.2 The model

Our problem is formulated as a mixed integer linear programming model (MILP), and thefollowing parameters are used:

Parameters

I Set of suppliers, i ∈ I

J Set of customers, j ∈ J

K Set of all facilities, k ∈ K . Note that K0 ⊆ K and K1 ⊆ K are the sets of the existingfacilities and potential facilities, respectively

L Set of products, l ∈ L

| · | The total number of elements in set ·Sil Capacity of supplier i for product l

Djl Demand of customer j for product l

fk Fixed setup cost of facility k

vk Operating cost per unit of product at facility k

Rk Maximum allowed expansion (additional) amount at facility k

pk Savings from closing facility k

MLk Minimum required throughput at open facility k

MUk Maximum allowed capacity before being expanded at facility k

cikj l Unit cost of shipping product l from supplier i via facility k to customer j

ek Unit expansion cost of facility k

We define the following binary and continuous decision variables to determine whetherfacility k will be open or not, which customers will be served by facility k, the total through-put of facility k and the expansion size of each open facility. Here quadruply subscriptedcontinuous variable xikj l is used to represent the logistics volume of a product from a sup-plier to a customer via a facility, which can easily track the origin of a product after it hasarrived at a facility or a customer and can also give an advantage when Benders decomposi-tion method is applied.

Decision variables

zk ={

1 if facility k is open0 otherwise

∀k ∈ K0 ∪ K1.

ukj ={

1 if customer j is served by facility k

0 otherwise∀k ∈ K0 ∪ K1; ∀j ∈ J.

xikj l Continuous variable which corresponds to the amount of product l shipped from sup-plier i via facility k to customer j

sk Continuous variable which corresponds to the amount of capacity expansion of facil-ity k

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According to the above notations, the logistics facility location problem with capacityexpansions of the existing facilities can be formulated as follows:

Min∑

k∈K0∪K1

eksk +∑i∈I

∑k∈K0∪K1

∑j∈J

∑l∈L

cikj lxikj l +∑k∈K1

fkzk

+∑

k∈K0∪K1

vk

∑j∈J

∑l∈L

Djlukj −∑k∈K0

pk(1 − zk) (1)

s.t. ∑k∈K0∪K1

∑j∈J

xikj l ≤ Sil ∀i ∈ I ; ∀l ∈ L. (2)

∑i∈I

xikj l = Djlukj ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (3)

MLk zk ≤

∑j∈J

∑l∈L

Djlukj ≤ MUk zk + sk ∀k ∈ K0 ∪ K1. (4)

sk ≤ Rkzk ∀k ∈ K0 ∪ K1. (5)∑k∈K0∪K1

ukj = 1 ∀j ∈ J. (6)

xikj l ≥ 0 ∀i ∈ I ; ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (7)

sk ≥ 0 ∀k ∈ K0 ∪ K1. (8)

zk ∈ {0,1} ∀k ∈ K0 ∪ K1. (9)

ukj ∈ {0,1} ∀k ∈ K0 ∪ K1; ∀j ∈ J. (10)

In the above model, objective function (1) is to minimize the sum of the capacity expan-sion costs, the transportation costs, the fixed setup costs for establishing new facilities, theoperating costs for handling products at facilities and the savings from closing the existingfacilities. Constraints (2) are the supply capacity constraints of supplier i for product l toguarantee that the amount of product l shipped from supplier i cannot exceed the supplycapacity of supplier i for product l. Constraints (3) are the flow conservation constraintsat each open facility, and they ensure that the total amount of product l shipped from allsuppliers to customer j via facility k should exactly match the demand of customer j forproduct l. Constraints (4) are the capacity constraints for all the facilities to ensure thatthe throughput of a facility must be in the range of its minimum required throughput andmaximum allowed capacity before being expanded plus its capacity expansion amount. If afacility’s throughput is less than its minimum required throughput, closing it will save theoperating cost or fixed setup cost. On the other hand, a facility’s throughput cannot increaseunlimitedly because of the limited available land, and thus it can not exceed its maximumallowed capacity. The expression of Constraints (4) allows the existing and new facilities tobe expanded, although in general new facilities should be directly designed to reach to its re-quired size so that the potential expansion is no longer needed (this means sk = 0, ∀k ∈ K1).Constraints (5) ensure that the capacity expansion amount of facility k cannot exceed itsmaximum allowed expansion amount. Constraints (6) ensure that each customer should beserved by a single facility to meet the economical requirement on transportation operationcost. Finally, constraints (7–10) define the nature and range of the decision variables.

The model developed in this paper is a mixed integer linear programming model and ischaracterized by binary variables zk and ukj , continuous variables xikj l and sk , and couplingconstraints (3), the right-hand side of (4) and (5). Due to the complexity of the model, it

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cannot be efficiently solved by an optimization package like CPLEX solver, especially forlarge scale instances, which drives us to adopt Benders decomposition algorithm to solve it.

3 Benders decomposition algorithm

Benders decomposition algorithm proposed by Benders (1962) is a typical decompositionmethod which is suitable for solving the complicated mixed integer programming problemwith integer variables and coupling constraints. It is based on the ideas of partition anddelayed constraint generation. In each iteration, by fixing the integer variables, the primaryproblem with only continuous variables becomes a Benders subproblem that can be solvedeasily. The Benders master problem with only integer variables and an auxiliary variableis a relaxation of the primary problem. An optimal solution of the Benders master problemwhich provides a lower bound (in the case of minimization) is transmitted to the subproblemfor constructing a new subproblem. By resolving the subproblem, the values of its dualvariables and a valid upper bound (in the case of minimization) are obtained, and then aBenders cut is formed and appended to the Benders master problem. By introducing the newBenders cut, the Benders master problem is resolved and Benders decomposition algorithmis iterated continuously until the difference between the lower bound and the best upperbound in some iteration is small enough or zero.

In the past few decades, some acceleration techniques for Benders decomposition algo-rithm have been successfully developed. McDaniel and Devine (1977) propose a method toquickly generate an initial Benders cut set by solving the relaxation of the master problem.Some heuristic rules are also developed to determine when integrality should be imposed onthe variables of the master problem to guarantee the convergence of the algorithm. Cote andLaughton (1984) use the first integer feasible solution rather than an optimal solution of themaster problem to construct the Benders subproblem to accelerate Benders decompositionalgorithm. Because sometimes this method can cause Benders decomposition algorithm tofail to converge, a heuristic method for choosing an iteration in which the master problemhas to be solved to optimality is developed. Poojari and Beasley (2009) use a genetic algo-rithm to obtain feasible solutions of the master problem to reduce the CPU time. The abovethree acceleration strategies are suitable for the problem where the master problem is hardto be solved. The following previous studies focus on how to improve the quality of Benderscuts. Magnanti and Wong (1981) first define a cut as Pareto-optimal if no other cut domi-nates it. By applying Pareto-optimal cuts to a problem in which the Benders subproblemis degenerate, Benders decomposition algorithm can be significantly improved. Rei et al.(2009) investigate a local branching method to strengthen or replace Benders feasible cutsby local branching constraints, and the lower and upper bounds can be improved simultane-ously. Saharidis et al. (2010) present a Covering Cut Bundle strategy to accelerate Bendersdecomposition algorithm by generating a bundle of cuts in each iteration in order to coverall the decision variables of the master problem. Saharidis et al. (2011) also propose a seriesof generic valid inequalities which are valid for the general fixed charge network problem toinitialize the master problem.

For the facility location problems, Benders decomposition algorithm has been success-fully applied by the following researchers. Geoffrion and Graves (1974) first employ Ben-ders decomposition algorithm to solve the two-stage distribution system design problem.Dogan and Goetschalckx (1999) study the integrated design problem of a multi-periodproduction-distribution system based on Benders decomposition algorithm in which the dis-aggregated cuts according to products and seasons and an initial cut set by relaxing the

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integer restrictions for the integer variables are simultaneously applied to accelerate theBenders decomposition algorithm. Cordeau et al. (2006) propose an integrated model for alogistics network design problem and Benders decomposition algorithm is applied to solv-ing it. Some valid inequalities are proposed to strengthen the linear relaxation of the primarymodel, and a set of initial cuts obtained by relaxing the integrality constraints of the mas-ter problem are used to accelerate Benders decomposition algorithm. Wentges (1996) usesPareto-optimal cuts and a procedure to strengthen Benders cuts to speed up Benders decom-position algorithm for a discrete capacitated facility location problem. From the examplesmentioned above, it can be concluded that Benders decomposition algorithm is very suitablefor solving the facility location problems.

In this paper, we will use Benders decomposition algorithm to solve our problem. A se-ries of valid inequalities are first added to the Benders master problem to restrict its solutionspace. According to the structure features of our MILP model, different disaggregated cutsof the primal Benders cut are applied to our problem. For some instances that were solvedslowly, a novel strategy called High Density Pareto (HDP) cut generation is proposed toreduce the number of iterations and the CPU time. For all the instances, a hybrid strategy isproposed to improve the average performance of the HDP cut generation method.

Without loss of generality, we consider the following mixed integer linear programmingproblem:

Min cT1 x + cT

2 y (11)

s.t.

Ax + By ≤ b (12)

x ≥ 0 (13)

y ∈ {0,1}. (14)

Where c1 ∈ Rn1 , x ∈ Rn1+ , c2 ∈ Rn2 , y ∈ R

n2+ , b ∈ Rm, A ∈ Rm×n1 , B ∈ Rm×n2 and 0 is them-dimensional null vector. Fixing integer variables y = y in the problem given by (11)–(14),the general form of the Benders subproblem is as follows:

Min cT1 x (15)

s.t.

Ax ≤ b − By (16)

x ≥ 0. (17)

The dual problem of the Benders subproblem can be written as:

Max uT (b − By) (18)

s.t.

AT u ≤ c1 (19)

u ≤ 0. (20)

The general form of the Benders master problem is as follows:

Min cT2 y + z (21)

s.t.

uT (b − By) ≤ z ∀u ∈ P ⊆ U (22)

uT (b − By) ≤ 0 ∀u ∈ R ⊆ U. (23)

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Where U is the polyhedron defined by (19)–(20), and P and R are the sets of the extremepoints and the extreme rays of U respectively, and z is an auxiliary variable complementingthe objective function of the master problem.

Following the Benders decomposition algorithm described above, for our model, by fix-ing binary variables zk = zk and ukj = ukj , the Benders subproblem in our problem (SP) canbe written as follows:

(SP)

Min∑k∈K0

eksk +∑i∈I

∑k∈K0∪K1

∑j∈J

∑l∈L

cikj lxikj l (24)

s.t. ∑k∈K0∪K1

∑j∈J

xikj l ≤ Sil ∀i ∈ I ; ∀l ∈ L. (25)

∑i∈I

xikj l = Djlukj ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (26)

sk ≥∑j∈J

∑l∈L

Djlukj − MUk zk ∀k ∈ K0. (27)

sk ≤ Rkzk ∀k ∈ K0. (28)

xikj l ≥ 0 ∀i ∈ I ; ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (29)

sk ≥ 0 ∀k ∈ K0. (30)

Define dual variables μil associated with constraints (25), rkjl associated with constraints(26), φk associated with constraints (27), and αk associated with constraints (28). Therefore,the dual of the Benders subproblem in our problem (DSP) is as follows:

(DSP)

Max∑i∈I

∑l∈L

Silμil +∑

k∈K0∪K1

∑j∈J

∑l∈L

Djlukj rkj l

+∑k∈K0

(∑j∈J

∑l∈L

Djlukj − MUk zk

)φk +

∑k∈K0

Rkzkαk (31)

s.t.

μil + rkjl ≤ cikj l ∀i ∈ I ; ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (32)

φk + αk ≤ ek ∀k ∈ K0. (33)

μil ≤ 0 ∀i ∈ I ; ∀l ∈ L. (34)

φk ≥ 0 ∀k ∈ K0. (35)

αk ≤ 0 ∀k ∈ K0. (36)

The constraints of the DSP given by (32)–(36) constitute a polyhedron denoted as �.If the SP is feasible for the fixed values zk (∀k ∈ K0) and ukj , the DSP has a boundedsolution which is an extreme point of the polyhedron in the light of duality theory, and thusan optimality Benders cut is deduced. On the contrary, if the SP is infeasible, the DSP hasan unbounded solution, an extreme ray of the polyhedron can be identified and a feasibilityBenders cut will be generated. Let P� and Q� be the sets of all the extreme points andextreme rays of polyhedron �, respectively.

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Optimality Benders cut:

y ≥∑i∈I

∑l∈L

Silμ∗il +

∑k∈K0∪K1

∑j∈J

∑l∈L

Djlukj r∗kj l

+∑k∈K0

(∑j∈J

∑l∈L

Djlukj − MUk zk

)φ∗

k +∑k∈K0

Rkzkα∗k .

Feasibility Benders cut:∑i∈I

∑l∈L

Silμ′il +

∑k∈K0∪K1

∑j∈J

∑l∈L

Djlukj r′kj l

+∑k∈K0

(∑j∈J

∑l∈L

Djlukj − MUk zk

)φ′

k +∑k∈K0

Rkzkα′k ≤ 0.

Where vector (μ∗il , r

∗kj l, φ

∗k , α

∗k ) ∈ P� corresponds to an extreme point of polyhedron �,

vector (μ′il , r

′kj l, φ

′k, α

′k) ∈ Q� corresponds to an extreme ray of polyhedron � and y is an

auxiliary variable complementing the objective function of the master problem.The optimality Benders cuts can strengthen the lower bound obtained from the master

problem while the feasibility Benders cuts make the lower bound valid for the primary prob-lem. Saharidis and Ierapetritou (2010) show that producing optimality instead of feasibilitycuts would lead to faster convergence of Benders decomposition algorithm and generatingmore optimality than feasibility cuts is a way to improve Benders decomposition algorithm.Based on the two types of the Benders cuts mentioned above, the Benders master problemin our problem (MP) can be written as:

(MP)

Min∑k∈K1

fkzk +∑

k∈K0∪K1

vk

∑j∈J

∑l∈L

Djlukj −∑k∈K0

pk(1 − zk) + y (37)

s.t.∑j∈J

∑l∈L

Djlukj ≥ MLk zk ∀k ∈ K0. (38)

MLk zk ≤

∑j∈J

∑l∈L

Djlukj ≤ MUk zk ∀k ∈ K1. (39)

∑k∈K0∪K1

ukj = 1 ∀j ∈ J. (40)

y ≥∑i∈I

∑l∈L

Silμ∗il +

∑k∈K0∪K1

∑j∈J

∑l∈L

Djlukj r∗kj l

+∑k∈K0

(∑j∈J

∑l∈L

Djlukj − MUk zk

)φ∗

k +∑k∈K0

Rkzkα∗k . (41)

∑i∈I

∑l∈L

Silμ′il +

∑k∈K0∪K1

∑j∈J

∑l∈L

Djlukj r′kj l

+∑k∈K0

(∑j∈J

∑l∈L

Djlukj − MUk zk

)φ′

k +∑k∈K0

Rkzkα′k ≤ 0. (42)

zk ∈ {0,1} ∀k ∈ K0 ∪ K1. (43)

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ukj ∈ {0,1} ∀k ∈ K0 ∪ K1; ∀j ∈ J. (44)

y ≥ 0. (45)

Due to the special structure of our problem, using Benders decomposition algorithmdirectly leads to slow convergence. In the following sections, we propose some strategies toaccelerate the Benders decomposition algorithm.

4 Improved Benders decomposition algorithms

In this section, three groups of valid inequalities, different disaggregated cuts and high den-sity Pareto cuts are proposed and added to the Benders master problem to accelerate conver-gence of the Benders decomposition algorithm.

4.1 Valid inequalities

Aside from the quality of the produced Benders cuts which was the subject under study inMagnanti and Wong (1981) and Saharidis et al. (2010), the other main reason leading toslow convergence of Benders decomposition algorithm is that the LB (in the case of min-imization) obtained from the master problem without strong valid inequalities is relativelyweak (Saharidis et al. 2011). Introducing a series of valid inequalities to obtain a restrictedmaster problem from the first iteration is an effective way to accelerate convergence, and inthis case the infeasible cases of the master problem may be eliminated a priori and the firstlower bound derived by the master problem will be significantly improved. As a result, thegap between the lower bound and the upper bound will be narrowed and the algorithm willconverge to an optimal solution faster.

For our problem, the initial master problem (MP of the first iteration), has only constraints(38)–(40) to restrict its solution space. In order to further narrow the solution space of themaster problem and obtain improved lower bounds, a series of valid inequalities whichcombine all binary variables of the master problem, are developed according to the featuresof our problem.

(1) Supply-demand valid inequalities for the existing facilities∑j∈J

∑l∈L

Djlukj ≤ zk(MUk + Rk) ∀k ∈ K0. (46)

Valid inequalities (46) ensure that the total demands of the customers served byan existing facility cannot be greater than its maximum allowed capacity after beingexpanded. Because constraints

∑j∈J

∑l∈L Djlukj ≤ MU

k zk + sk (∀k ∈ K0) will be in-cluded in the Benders subproblem rather than the Benders master problem after decom-position, these valid inequalities are to recover the function of these constraints in theBenders master problem as much as possible.

(2) Supply-demand valid inequalities for all the facilities∑k∈K0

zk(MUk + Rk) +

∑k∈K1

MUk zk ≥

∑j∈J

∑l∈L

Djl (47)

Valid inequalities (47) ensure that the sum of the maximum allowed capacities afterbeing expanded for all the facilities must be greater than or equal to the total demandsof the customers. Valid inequalities (47) in conjunction with valid inequalities (46) canfurther reduce the solution space of the MP and make the SP feasible.

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(3) The operating state valid inequalities for all the facilities∑j∈J

ukj ≤ |J | · zk ∀k ∈ K0 ∪ K1. (48)

Valid inequalities (48) guarantee that facility k must be operated if facility k providesservice for some customer j . In our problem, if a facility is determined to be operated,two cases would occur: (1) remaining the existing facility being operated may lead toan extra expansion cost, or (2) establishing the new facility will lead to an extra fixedsetup cost. If a facility is determined not to be operated, two cases would appear: (1) theexisting facility will be closed and its operating cost will be saved, and (2) the newfacility will not be operated. After adding valid inequalities (48) to the MP, zk = 1 willbecome true if ukj = 1 under the same k.

4.2 Disaggregation of the primal Benders cut

The disaggregation of the primal Benders cut has been used for the multi-period production-distribution system by Dogan and Goetschalckx (1999) and the closed-loop supply chainnetwork design by Üster et al. (2007). The disaggregated cuts can be obtained only fromthe problems with a specific structure, i.e., the Benders subproblem must be able to beseparated into some independent subproblems. In each iteration, the multiple disaggregatedcuts formed by the dual optimal solutions corresponding to the independent subproblemswill be appended to the master problem simultaneously. These cuts which include exactlythe same information as the primal Benders cut will restrict the solution space of the masterproblem in a more accurate-exact way.

The disaggregation of the primal Benders cut is essentially a multi-generation of cutsmethod. The idea of the multi-generation of cuts is to add multiple cuts to Benders mas-ter problem simultaneously by solving one or more extra auxiliary problems. Gabrel et al.(1999) and Minoux (2001) have proposed a multi-generation of cuts method for the specialcase of min cost multi-commodity flow problem. Saharidis et al. (2010) and Saharidis andIerapetritou (2010) present a general applicable method in which a bundle of cuts is gen-erated in each iteration and an optimality type cut is generated in addition to a feasibilityBenders cut.

When a multi-generation of cuts method is developed and applied, a good strategy toconverge to optimality faster than the standard Benders algorithm is to maintain a balancebetween the number of iterations and the amount of time spent in each iteration for thegeneration of the additional cuts. This balance should be based on the idea that for producingan additional cut more time is spent to complete an iteration but the total time spent tosolve iteratively the master problem decreases due to smaller number of iterations needed.As we mentioned above, Benders decomposition algorithm could converge to optimalityin an iteration if all Benders cuts are added to the master problem in the beginning of thealgorithm. This is the reason that makes the multi-generation of cuts method to decrease thetotal number of iterations and also the CPU time if the cuts generated are of good qualityand the time spent to produce them keeps the appropriate balance described before. The onlydifference between the generation of the disaggregated cuts and the multi-generation of cutsis that the former doesn’t need to solve any auxiliary problem.

As far as we known, the disaggregated cuts have not been applied to Benders decom-position algorithm for solving the location problems with capacity expansions. When thedisaggregation of the primal Benders cut is applied to our problem, the Benders subproblemgiven by (24)–(30) can be exactly separated into two independent subproblems SP1 and SP2

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according to two different sets of continuous variables xikj l and sk (variable separation). SP1can be further separated into |L| disconnected single-product subproblems (product separa-tion), and SP2 can be further separated into |K0| disconnected single-facility subproblems(facility separation). Therefore, three types of Benders cuts are generated based on differentseparations mentioned above. The first type of Benders cuts is obtained using only variableseparation. The second type of Benders cuts is obtained using variable separation and facil-ity separation. The third type of Benders cuts is obtained using both facility separation andproduct separation. The following are the details of how to obtain these different types ofcuts using the general procedure of Benders decomposition algorithm described in Sect. 3.Note that in the general procedure, the expression concerning the sum of all the auxiliaryvariables corresponding to the different types of cuts should be included in the objectivefunction of the master problem.

The detailed forms of the two independent subproblems (SP1 and SP2) are given respec-tively, as follows:

(SP1)

Min∑i∈I

∑k∈K0∪K1

∑j∈J

∑l∈L

cikj lxikj l (49)

s.t. ∑k∈K0∪K1

∑j∈J

xikj l ≤ Sil ∀i ∈ I ; ∀l ∈ L. (50)

∑i∈I

xikj l = Djlukj ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (51)

xikj l ≥ 0 ∀i ∈ I ; ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (52)

(SP2)

Min∑k∈K0

eksk (53)

s.t.

sk ≥∑j∈J

∑l∈L

Djlukj − MUk zk ∀k ∈ K0. (54)

sk ≤ Rkzk ∀k ∈ K0. (55)

sk ≥ 0 ∀k ∈ K0. (56)

Based on the models of SP1 and SP2, define dual variables μil associated with constraints(50), rkjl associated with constraints (51), φk associated with constraints (54), and αk asso-ciated with constraints (55), and then their dual problems DP1 and DP2 take the followingforms:

(DP1)

Max∑i∈I

∑l∈L

Silμil +∑

k∈K0∪K1

∑j∈J

∑l∈L

Djlukj rkj l (57)

s.t.

μil + rkjl ≤ cikj l ∀i ∈ I ; ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (58)

μil ≤ 0 ∀i ∈ I ; ∀l ∈ L. (59)

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(DP2)

Max∑k∈K0

(∑j∈J

∑l∈L

Djlukj − MUk zk

)φk +

∑k∈K0

Rkzkαk (60)

s.t.

φk + αk ≤ ek ∀k ∈ K0. (61)

φk ≥ 0 ∀k ∈ K0. (62)

αk ≤ 0 ∀k ∈ K0. (63)

Because valid inequalities (46)–(48) presented in Sect. 4.1 have been added to the Ben-ders master problem, SP1 and SP2 are always feasible. Let (μil, rkj l) and (φk, αk) be theoptimal solutions obtained by DP1 and DP2, respectively. Therefore, the first type of cuts isgiven by:

Type 1 cuts:

y1 ≥∑i∈I

∑l∈L

Silμil +∑

k∈K0∪K1

∑j∈J

∑l∈L

Djl rkj lukj . (64)

y2 ≥∑k∈K0

∑j∈J

∑l∈L

Djlφkukj +∑k∈K0

(Rkαk − MUk φk)zk. (65)

Where y1 ≥ 0 and y2 ≥ 0 are two auxiliary variables. When Type 1 cuts are used inthe general procedure of Benders decomposition algorithm, y1 + y2 will be included in theobjective function of the MP.

The second type of cuts obtained by further separating SP2 into |K0| disconnected single-facility subproblems is given by:

Type 2 cuts:

y1 ≥∑i∈I

∑l∈L

Silμil +∑

k∈K0∪K1

∑j∈J

∑l∈L

Djl rkj lukj . (66)

y2k ≥

∑j∈J

∑l∈L

Djlφkukj + (Rkαk − MUk φk)zk ∀k ∈ K0. (67)

Where y2k (∀k ∈ K0) are auxiliary variables. When Type 2 cuts are used in the general

procedure of Benders decomposition algorithm, y1 + ∑k∈K0

y2k will be included in the ob-

jective function of the MP.The third type of cuts obtained by further separating SP1 into |L| disconnected single-

product subproblems is given by:

Type 3 cuts:

y1l ≥

∑i∈I

Silμil +∑

k∈K0∪K1

∑j∈J

Djl rkj lukj ∀l ∈ L. (68)

y2k ≥

∑j∈J

∑l∈L

Djlφkukj + (Rkαk − MUk φk)zk ∀k ∈ K0. (69)

Where y1l (∀l ∈ L) are auxiliary variables. When Type 3 cuts are used in the general

procedure of Benders decomposition algorithm,∑

l∈L y1l +∑

k∈K0y2

k will be included in theobjective function of the MP.

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It can be seen that the numbers of Type 1–3 cuts in each iteration are two, |K0| + 1 and|K0| + |L|, respectively. As compared with the primal Benders cut, each of Type 1–3 cutscan effectively restrict the solution space of the master problem, and thus result in betterconvergence behavior, which is verified by the computational results in Sect. 5.

4.3 Generation of high density Pareto cuts

A cut is called Pareto-optimal if no other cut dominates it, which is first presented by Mag-nanti and Wong (1981). A Pareto-optimal cut only exists in which the dual of the Benderssubproblem has several optimal solutions, and it is the strongest cut among all the alter-native Benders cuts in the same iteration. Therefore, Pareto-optimal cuts can improve theperformance of Benders decomposition algorithm effectively.

Many researchers have employed the method presented by Magnanti and Wong (1981) togenerate Pareto-optimal cuts to accelerate Benders decomposition algorithm for their prob-lems. Recently Papadakos (2008) presents a new improved method for the generation ofPareto-optimal cuts, and he proves that it is not necessary to find a core point of the solutionspace of Benders master problem to produce a Pareto-optimal cut. Although the generationmethod of Pareto-optimal cuts is improved, there is little research on improving the qualityof Pareto-optimal cuts. In fact, a Pareto-optimal cut is lifted from the corresponding opti-mality Benders cut, but the lifted effect is limited because it is dependent on the solution ofthe dual problem. It is possible to further lift Pareto-optimal cuts. In this section, we developa new method referred to as High Density Pareto (HDP) cut generation, and it can producePareto cuts with high density. A high density cut is a cut which covers a high number ofdecision variables of the master problem, and thus it further restricts the solution space ofthe master problem and results in a better convergence behavior to the algorithm (Saharidiset al. 2010).

In the following subsections, we will give how to generate Pareto-optimal cuts and HDPcuts based on the general form of a mixed integer linear programming problem given by(11)–(14), as well as the detailed forms of the HDP cuts in our problem.

4.3.1 Generation of Pareto-optimal cuts

Magnanti and Wong (1981) first give a method to generate Pareto-optimal cuts. Let V ∗ =u∗T (b−By) be the optimal objective value of the dual subproblem and yc be a core point ofthe solution space of the master problem. A Pareto-optimal cut can be obtained by solvingthe following problem:

Max uT (b − Byc) (70)

s.t.

AT u ≤ c1 (71)

uT (b − By) = u∗T (b − By) (72)

u ≤ 0. (73)

Based on the idea that any extreme point or any extreme ray of the dual subproblemgives a valid Benders cut (Benders 1962) as well as that it is not necessary to use a corepoint of the solution space of the master problem to produce a Pareto cut (Papadakos 2008),the following auxiliary problem (AP) gives a valid Pareto cut:

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(AP)

Max uT (Coef ) (74)

s.t.

AT u ≤ c1 (75)

uT (b − By) = u∗T (b − By) (76)

u ≤ 0. (77)

Where Coef is m-vector with constant values which defines the direction of AP’s objec-tive function and can take any value.

4.3.2 High density Pareto cut generation

The generation of the HDP cuts is based on the auxiliary problem given by (74)–(77), butcoefficient matrix Coef needs to be specified in advance and the AP needs to be solvedoptimally in order to cover a high number of decision variables in the master problem.

Without loss of generality, we consider a case where the dual subproblem gives a feasiblebounded solution, and the following optimality Benders cut can be obtained:

uT (b − By) ≤ z ∀u ∈ P ⊆ U ⇒ uT b − z ≤ uT By (78)

Decision variable yk corresponding to the kth element of n2-column vector y is definedas a covered decision variable if the kth row (uT B)k of n2-row vector uT B has a value notequal to zero and at the same order as the other non-zero coefficients corresponding to theother decision variables of the master problem. The value of the coefficient correspondingto yk depends on m-row vector uT and m × n2 matrix B . If the structure of the problem(represented by the matrix B) allows us (e.g.,

∑n2q=1 Bk,q �= 0) to cover decision variable yk ,

in order to cover yk , we ask from the auxiliary problem (AP) to find (if any) a non-zero dualsolution (uT )k among the optimal solutions. In order to cover a high number of decisionvariables in the master problem, we need to obtain the solution of AP with the maximumnon-zero dual solution for

Coef =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(1b1

− 1∑n2q=1 B1,q

)...(

1bk

− 1∑n2q=1 Bk,q

)...(

1bm

− 1∑n2q=1 Bm,q

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(79)

each (uT )k with∑n2

q=1 Bk,q �= 0. Therefore, the AP with Coef k = ( 1bk

− 1∑n2q=1 Bk,q

) needs to

be solved optimally to get the high density Pareto cut. Matrix Coef is defined as (79).The general form for generating the HDP cuts as follows:

Maxm∑

k=1

uk

(1

bk

− 1∑n2q=1 Bk,q

)(80)

s.t.

AT u ≤ c1 (81)

uT (b − By) = u∗T (b − By) (82)

u ≤ 0. (83)

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We have to notice that we prefer to use the values of vector Coef defined by (79) to equi-librate the coefficients that will be involved in the resulting cut. If we don’t use these values,we risk to obtain a coefficient for decision variable yk which will be significantly biggerthan another coefficient corresponding to decision variable y ′

k resulting in a cut where onlyyk is covered in practice even if the coefficient of y ′

k is not equal to zero (due to its smallvalue compared to the coefficient of yk). Notice also that for any k with

∑n2q=1 Bk,q = 0, the

corresponding part of the objective function is deleted because the corresponding decisionvariable yk will not be covered independently on the value of (uT )k . After solving (80)–(83),a HDP cut covering a high number of decision variables is generated. By concluding the pre-sentation of the HDP cut generation method, it has to be noted that the proposed method issuitable for the cases where there are multiple optimal solutions for the Benders subprob-lem, and only these cases make sense to apply it. These cases are encountered frequently inthe problems where the Benders subproblem involves network optimization (Magnanti andWong 1981) while the facility location problem presented here is a typical example amongthem.

For our problem, the HDP cut generation method based on Type 1–3 cuts is implementedas follows. For the convenience of description, let Type 4–6 cuts represent the HDP cuts forType 1–3 cuts, respectively. In order to obtain Type 4–6 cuts, the following two auxiliaryproblems AP1 and AP2 should be solved optimally. Let f ∗(SP 1) and f ∗(SP 2) be theoptimal objective values of SP1 and SP2 respectively, and N be an infinite number. AP1 andAP2 take the following forms:

(AP1)

Max∑

k∈K0∪K1

∑j∈J

∑l∈L

(N − 1

Djl

)rkjl (84)

s.t.

μil + rkjl ≤ cikj l ∀i ∈ I ; ∀k ∈ K0 ∪ K1; ∀j ∈ J ; ∀l ∈ L. (85)∑i∈I

∑l∈L

Silμil +∑

k∈K0∪K1

∑j∈J

∑l∈L

Djlukj rkj l = f ∗(SP 1) (86)

μil ≤ 0 ∀i ∈ I ; ∀l ∈ L. (87)

Note that term∑

i∈I

∑l∈L( 1

Sil−N)μil in the objective function of AP1 is deleted because

the corresponding decision variables will not be covered independently on the values of μil .

(AP2)

Max∑k∈K0

(N − 1

MUk − ∑

j∈J

∑l∈L Djl

)φk +

∑k∈K0

(N + 1

Rk

)αk (88)

s.t.

φk + αk ≤ ek ∀k ∈ K0. (89)∑k∈K0

(∑j∈J

∑l∈L

Djlukj − MUk zk

)φk +

∑k∈K0

Rkzkαk = f ∗(SP 2) (90)

φk ≥ 0 ∀k ∈ K0. (91)

αk ≤ 0 ∀k ∈ K0. (92)

Let (μil, rkj l) and (φk, αk) are the values obtained from AP1 and AP2 respectively. Using(μil , rkj l) and (φk, αk) to replace (μil , rkj l) and (φk, αk) in Type 1–3 cuts, we can obtain thefollowing HDP cuts, respectively.

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Type 4 cuts:

y1 ≥∑i∈I

∑l∈L

Silμil +∑

k∈K0∪K1

∑j∈J

∑l∈L

Djl rkj lukj . (93)

y2 ≥∑k∈K0

∑j∈J

∑l∈L

Djlφkukj +∑k∈K0

(Rkαk − MUk φk)zk. (94)

Type 5 cuts:

y1 ≥∑i∈I

∑l∈L

Silμil +∑

k∈K0∪K1

∑j∈J

∑l∈L

Djl rkj lukj . (95)

y2k ≥

∑j∈J

∑l∈L

Djlφkukj + (Rkαk − MUk φk)zk ∀k ∈ K0. (96)

Type 6 cuts:

y1l ≥

∑i∈I

Silμil +∑

k∈K0∪K1

∑j∈J

Djl rkj lukj ∀l ∈ L. (97)

y2k ≥

∑j∈J

∑l∈L

Djlφkukj + (Rkαk − MUk φk)zk ∀k ∈ K0. (98)

5 Performance evaluation of the improved Benders decomposition algorithms

In this section, in order to verify the performance of the valid inequalities and Type 1–6cuts mentioned above, we carry out the computational experiment on intensive instancesgenerated randomly. The Benders decomposition algorithms are coded in C++ languageand tested on the computer with CPU Intel Core 2, 2.83 GHz and 3.25 GB RAM. Duringimplementing them and solving the model given by (1)–(10), CPLEX 11.0 (with defaultsettings) is used as an optimization solver. For comparing solution quality and runtime,the Benders decomposition algorithms with the different valid inequalities and alternativetypes of cuts are required to obtain an optimal solution. The effects of the different validinequalities and alternative types of cuts on the performance of the Benders decompositionalgorithm are compared and the computational results are reported.

5.1 Description of the tested data

Based on the model given by (1)–(10), the size of a problem is determined by the numberof suppliers (|I |), the number of existing facilities (|K0|), the number of potential facilities(|K1|), the number of customers (|J |) and the number of products (|L|). To evaluate theeffectiveness and efficiency of the Benders decomposition algorithms with the different validinequalities and alternative types of cuts, ten different problem classes ranging from smallto large size are generated randomly according to the uniform distributions in Table 1. Foreach problem classes, 20 random instances are generated.

To facilitate describing and understanding, let Dl be the total demands for product l,and D be the total demands for all products. All parameters generated randomly obey thefollowing uniform distributions. Transportation cost (cikj l) is selected from set {1, . . . ,10}.Demand (Djl) is generated from interval [20,100], and facility operating cost of unit prod-uct (vk) is chosen from set {5, . . . ,10}. In order to guarantee that a feasible solution can be

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Table 1 Problem classesgenerated from the followinguniform distributions

Class |I | |K0| |K1| |J | |L|

C1 [2,5] [2,5] [2,5] [5,10] [3,5]C2 [2,5] [2,5] [2,5] [10,30] [3,5]C3 [2,5] [2,5] [5,10] [30,50] [3,5]C4 [5,10] [2,5] [5,10] [30,50] [3,5]C5 [2,5] [5,10] [5,10] [10,30] [5,10]C6 [5,10] [2,5] [5,10] [30,50] [5,10]C7 [5,10] [5,10] [5,10] [30,50] [5,10]C8 [10,15] [5,10] [5,10] [30,50] [10,20]C9 [10,15] [5,10] [15,20] [50,80] [10,20]C10 [15,20] [5,10] [5,10] [50,100] [30,50]

obtained from the randomly generated data, the supply amount of each product (Sil) is gen-erated from [S1l , S2l], where S1l = Dl/|I |, S2l = 2Dl/|I |. Based on the rule that the total ca-pacities of all the existing facilities are less than the total demands, the sum of the maximumallowed capacity before being expanded for all the existing facilities (defined as WT ) is gen-erated randomly from [2/5D,2/3D], the maximum allowed capacity before being expandedfor each existing facility (MU

k (∀k ∈ K0)) is generated from [4WT /5|K0|,6WT /5|K0|], theminimum required throughput for each existing facility (ML

k (∀k ∈ K0)) is generated from[MU

k /3,2MUk /5]. Let NT = D − WT , and the maximum allowed capacity before being

expanded for each potential facility (MUk (∀k ∈ K1)) and the minimum required through-

put for each potential facility (MLk (∀k ∈ K1)) are generated from [NT /|K1|,NT /2] and

[MUk /3,2MU

k /5], respectively. The fixed setup cost for each potential facility (fk (∀k ∈ K1))is generated from [2MU

k ,5MUk ]. Unit expansion cost (ek) and the maximum allowed expan-

sion amount (Rk) are generated from [2fk/MUk ,4fk/M

Uk ] and [2MU

k /5,MUk /2], respec-

tively. The savings from closing each existing facility (pk (∀k ∈ K0)) is generated from[4fk/5, fk].

Table 2 summarizes the characteristics and size of the model given by (1)–(10) for theten different problem classes. As shown in Table 2, the scale of the ten problem classesincreases gradually. Taking the largest scale problem class C10 for example, it includes 610binary variables, 360005 continuous variables and 18685 constraints.

5.2 Effectiveness of valid inequalities

Because different combinations of the valid inequalities may have different effects on re-stricting the solution space of the Benders master problem, Table 3 gives the comparisonresults when various combinations of the valid inequalities are added to the Benders masterproblem. Let B represent the standard Benders decomposition algorithm without any validinequality, and BVI1, BVI2 and BVI3 represent that valid inequalities (46)–(48) are addedto the Benders master problem, respectively. Similarly, BVI123 represents that all the validinequalities are simultaneously added to the Benders master problem.

As shown in Table 3, the instances solved optimally by B were the least. BVI1, BVI12,BVI13 and BVI123 show better performance than B, BVI2, BVI3 and BVI23 because nofeasibility Benders cuts were generated. For the largest scale problem class C10, only 5% ofthe instances were solved optimally by B while the corresponding figures for BVI1, BVI12,BVI13 and BVI123 were 35%, 25%, 30% and 35%. The computational results further in-dicate that producing more optimality Benders cuts than feasibility Benders cuts can ac-celerate the convergence of Benders decomposition algorithm. The Benders decomposition

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Table 2 Characteristics and size of the generated problem classes

Class |I | |K0| |K1| |J | |L| Number of variables Number ofconstraintsBinary Continuous

zk ukj xikj l sk

C1 5 2 3 10 3 5 50 750 2 187

C2 5 3 5 30 5 8 240 6000 3 1274

C3 5 4 6 50 5 10 500 12500 4 2599

C4 10 4 6 50 5 10 500 25000 4 2624

C5 5 8 10 30 8 18 540 21600 8 4434

C6 6 4 5 40 8 9 360 17280 4 2990

C7 5 8 10 40 10 18 720 36000 8 7334

C8 10 6 9 40 10 15 600 60000 6 6176

C9 12 10 15 50 15 25 1250 225000 10 19040

C10 20 5 5 60 30 10 600 360000 5 18685

Table 3 Effects of different combinations of the valid inequalities

Class B BVI1 BVI2 BVI3 BVI12 BVI13 BVI23 BVI123

C1 CPU 0.16 0.12 0.16 0.17 0.11 0.12 0.14 0.10

Iter. 6.65 4.55 6.25 6.4 4.5 4.5 6.0 4.45

P 100% 100% 100% 100% 100% 100% 100% 100%

F\O 2.15\4.5 0\4.55 1.75\4.5 1.9\4.5 0\4.5 0\4.5 1.5\4.5 0\4.45

C2 CPU – 8.07 – 37.40 8.05 9.41 32.22 7.46

Iter. – 6.25 – 8.7 6.1 6.15 9.15 6.0

P 85% 100% 95% 100% 100% 100% 100% 100%

F\O – 0\6.25 – 2.6\6.1 0\6.1 0\6.15 2.35\6.85 0\6.0

C3 CPU – 265.79 – – 390.00 387.35 – 162.40

Iter. – 7.2 – – 7.3 7.45 – 7.2

P 90% 100% 90% 95% 100% 100% 95% 100%

F\O – 0\7.2 – – 0\7.3 0\7.45 – 0\7.2

C4 CPU – 278.03 – – 274.91 282.88 550.41 261.23

Iter. – 9.4 – – 9.8 9.8 13.3 9.55

P 85% 100% 95% 85% 100% 100% 100% 100%

F\O – 0\9.4 – – 0\9.8 0\9.8 3.65\9.65 0\9.55

C5 CPU – – – – 259.83 175.33 – 162.72

Iter. – – – – 9.05 9.35 – 9.15

P 30% 95% 55% 75% 100% 100% 65% 100%

F\O – – – – 0\9.05 0\9.35 – 0/9.15

C6 CPU – 165.03 – 600.03 226.33 169.84 – 160.53

Iter. – 7.1 – 10.65 6.75 7.1 – 7.2

P 95% 100% 95% 100% 100% 100% 95% 100%

F\O – 0\7.1 – 3.6\7.05 0\6.75 0\7.1 – 0\7.2

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Table 3 (Continued)

Class B BVI1 BVI2 BVI3 BVI12 BVI13 BVI23 BVI123

C7 CPU – – – – – – – –

Iter. – – – – – – – –

P 25% 90% 30% 70% 95% 90% 60% 95%

F\O – – – – – – – –

C8 CPU – – – – 996.96 778.97 – 724.55

Iter. – – – – 10.5 10.4 – 10.55

P 55% 95% 75% 85% 100% 100% 85% 100%

F\O – – – – 0\10.5 0\10.4 – 0\10.55

C9 CPU – – – – – – – –

Iter. – – – – – – – –

P 5% 55% 10% 5% 50% 55% 5% 55%

F\O – – – – – – – –

C10 CPU – – – – – – – –

Iter. – – – – – – – –

P 5% 35% 20% 15% 25% 30% 15% 35%

F\O – – – – – – – –

“CPU” = the average CPU time (in seconds)

“Iter.” = the average number of iterations

“P” = the percentage of instances for which an optimal solution was obtained

“–” = some instances were not solved optimally

“F\O ” = the number of feasibility cuts \the number of optimality cuts

Table 4 The relative differencebetween the first lower boundsobtained by B and BVI123

“LB1B” = the lower boundobtained by B in the first iteration

“LB1BVI123” = the lower boundobtained by BVI123 in the firstiteration

“Relative difference” =(LB1BVI123 − LB1B) / LB1B ×100%

“C5-#” = the instance # ofproblem class C5

Example LB1B LB1BVI123 Relative difference (%)

1 (C5-1) 49852 105253 111.13

2 (C5-2) 50872 94694 86.14

3 (C5-3) 50362 109495 117.42

4 (C5-4) 40475 102100 152.26

5 (C5-5) 44109 76544 73.53

6 (C5-6) 34365 106169 208.95

7 (C5-7) 69028 83792 21.39

8 (C5-8) 45811 110659 141.56

9 (C5-9) 46831 80398 71.68

10 (C5-10) 34652 73751 112.83

algorithm where all the valid inequalities were added to the MP (e.g. BVI123) makes mostof the instances solved optimally and requires the shortest CPU time.

Table 4 shows the comparison of the lower bounds obtained by B and BVI123 in the firstiteration for the first ten instances of the fifth problem class.

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Table 5 Effects of Type 1–3 cuts

Class CPLEX BVI123 + Type1 BVI123 + Type2 BVI123 + Type3

CPU CPU Iter. CPU Iter. CPU Iter.

C1 0.13 0.09 3.4 0.08 3.2 0.07 3.1

C2 7.92 3.68 4.8 3.20 4.35 2.72 3.8

C3 68.36 57.54 4.6 43.97 4.25 43.93 3.6

C4 468.99 146.75 6.4 58.72 5.3 71.24 4.6

C5 49.46 96.26 6.15 18.39 4.95 29.46 4.4

“CPU” = the average CPU time (in seconds)

“Iter.” = the average number of iterations

Experimental results in Table 4 demonstrate that BVI123 can improve the lower boundsubstantially as compared with the standard Benders decomposition algorithm, and thelargest relative difference between them is up to 208.95%. In the following subsection, weevaluate the effects of the three types of disaggregated cuts on the performance of the Ben-ders decomposition algorithm.

5.3 Effectiveness of disaggregated cuts of the primal Benders cut

First of all, problem classes C1 to C5 were used to test the effectiveness of Type 1–3 cuts pro-posed in Sect. 4.2. In the following tables, CPLEX represents optimization solver CPLEX isused, and BVI123 + Type1, BVI123 + Type2 and BVI123 + Type3 represent that Type 1–3cuts are used in the Benders decomposition algorithm with valid inequalities (46)–(48), re-spectively.

Table 5 shows the effects of Type 1–3 cuts on the convergence of the Benders decom-position algorithm. Although all instances were solved optimally as shown in Table 5,BVI123 + Type2 and BVI123 + Type3 took much shorter time than BVI123 + Type1.Therefore, BVI123 + Type2 and BVI123 + Type3 were used to solve the last five problemclasses.

Table 6 gives the comparison results among CPLEX, BVI123 + Type2 and BVI123 +Type3 for problem classes C6 to C10.

From Table 6, it can be seen that the average CPU times of BVI123 + Type2 andBVI123 + Type3 are 140.18 seconds and 152.52 seconds, respectively, while the averageCPU time of CPLEX is up to 607.98 seconds for all the instances which were solved opti-mally by these three methods. BVI123 + Type2 shows better performance than BVI123 +Type3 in most cases. Taking the largest problem class C10 for example, BVI123 + Type2and BVI123 + Type3 can obtain an optimal solution for 20% more instances than CPLEX,and the average CPU time of BVI123 + Type2 for these 20% instances is 699.64 seconds,while that of BVI123 + Type3 is 768.23 seconds. For the instances that an optimal solutionwas obtained by all the three methods, the average CPU time of BVI123 + Type2 is only25.30 seconds while the average time of CPLEX runs up to 1031.18 seconds. Furthermore,BVI123 + Type2 can obtain an optimal solution for 5% more instances than CPLEX andBVI123 + Type3, and the average CPU time of BVI123 + Type2 for these 5% instances is6852.52 seconds. Therefore, Type2 is the most effective, and we evaluate the effectivenessof the HDP cuts for Type 2 cuts.

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Table 6 Comparison results among CPLEX, BVI123 + Type2 and BVI123 + Type3

Class CPLEX BVI123 + Type2 BVI123 + Type3

C6 P 100% 100% 100%

CPU 62.02 26.01 32.06

Iter. \ 3.95 3.7

C7 P 100% 100% 100%

CPU 759.97 146.69 147.32

Iter. \ 4.45 3.95

C8 P 100% 100% 100%

CPU 623.73 179.93 213.31

Iter. \ 6.55 5.25

C9 P 55% 55% 55%

CPU 1141.76 305.45 315.18

Iter. \ 5.27 4.64

Q 10% 10% 10%

CPU* – 1115.86 1081.19

Iter.* \ 8.5 6

C10 P 20% 20% 20%

CPU 1031.18 25.30 29.51

Iter. \ 5.75 4.5

Q 20% 20% 20%

CPU* – 699.64 768.23

Iter.* \ 5.75 4.25

R 5% 5% 5%

CPU# – 6852.52 –

Average 607.98 140.18 152.52

“P” = the percentage of instances which were solved optimally by all the three methods

“CPU” = the average CPU time of instances which were solved optimally by all the three methods (inseconds)

“Iter.” = the average number of iterations for instances that were solved optimally by all the three methods

“–” = the instances were not solved

“\” = no iterations

“Q” = the percentage of instances that were solved optimally by BVI123 + Type2 and BVI123 + Type3

“CPU*” = the average CPU time of instances that were solved optimally by BVI123 + Type2 and BVI123 +Type3 (in seconds)

“Iter.*” = the average number of iterations for instances that were solved optimally by BVI123 + Type2 andBVI123 + Type3

“R” = the percentage of instances that were solved optimally only by BVI123 + Type2

“CPU#” = the average CPU time of instances that were solved optimally only by BVI123 + Type2

“Average” = the average CPU time of all the instances that were solved optimally by all the three methods

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Table 7 Comparison results between BVI123 + Type 2 and BVI123 + Type5

Example CPLEX BVI123 + Type2 BVI123 + Type5 Relative difference

CPU CPU Iter. CPU Iter. CPU (%) Iter. (%)

1(C7-3) 7425.66 1028.73 5 708.08 3 31.2 40

2(C8-2) 560.95 329.22 8 176.16 7 46.5 12.5

3(C8-4) 1954.95 170.83 11 159.94 10 6.4 9.1

4(C8-9) 275.70 200.16 8 86.17 5 56.9 37.5

5(C8-16) 4503.53 1798.58 11 524.22 9 70.9 18.2

6(C9-9) – 438 7 394.22 5 10.0 28.6

7(C9-16) – 1793.72 10 203.13 8 88.7 20

8(C10-1) – 1671.98 6 1543.31 4 7.7 33.3

9(C10-2) – 6852.52 6 4232.33 5 38.2 16.7

10(C10-20) – 804.09 9 320.27 6 60.2 33.3

Average – 1508.78 8.1 834.78 6.2 41.7 24.9

“CPU” = the CPU time

“Iter.” = the number of iterations

“CPU (%)” = the relative difference of CPU time between BVI123 + Type2 and BVI123 + Type5

“Iter. (%)” = the relative difference of the number of iterations between BVI123 + Type2 and BVI123 +Type5

“Average” = the average value of each column

“–” = the instance were not solved optimally

“C*-#” = the instance # of problem class *

5.4 Effectiveness of high density Pareto cuts

Although the average performance of BVI123 + Type2 obviously better than CPLEX asshown in Table 6, there still exist some instances which need relatively intensive number ofiterations and more CPU time to converge to an optimal solution. For these instances, thereason why the convergence is slow is that the density of Type 2 cuts is still lower. In orderto reduce the number of iterations and shorten the CPU time of these instances, we try toapply the HDP cuts generation method to the ten representative instances among them forlifting Type 2 cuts. Type 2 cuts will become Type 5 cuts after being lifted and the Bendersdecomposition algorithm with Type5 is represented by BVI123 + Type5. Table 7 presentsthe comparison results of BVI123 + Type2 and BVI123 + Type5 for the ten representativeinstances.

From the comparison results shown in Table 7, it can be seen that both the average CPUtime and the number of iterations are decreased by using Type 5 cuts. The average rela-tive difference of iterations between BVI123 + Type2 and BVI123 + Type5 is 24.9%, andthe average CPU time is decreased by 41.7%. When BVI123 + Type2 needs an enormousamount of time and intensive number of iterations to find an optimal solution, it is observedthat BVI123 + Type5 has a significant decrease in CPU time. The most obvious examplesare 5 and 7 where BVI123 + Type2 needs 1798.58 seconds, 11 iterations and 1793.72 sec-onds, 10 iterations to find an optimal solution respectively, and the reductions of the CPUtime are up to 70.9% and 88.7% after using Type 5 cuts.

It is natural to test the average performance of the HDP cut generation method on allthe instances of the ten problem classes. However, the HDP cut generation method does not

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Table 8 Comparison results between BVI123 + Type2 and the hybrid strategy

Class BVI123 + Type2 Hybrid strategy Relative difference

CPU Iter. CPU Iter. CPU (%) Iter. (%)

C1 0.08 3.2 0.08 3.2 0 0

C2 3.20 4.35 3.06 4.2 4.38 3.45

C3 43.97 4.25 41.46 4.2 5.71 1.18

C4 58.72 5.3 58.00 5.1 1.23 3.78

C5 18.39 4.95 16.04 4.65 12.78 6.06

C6 26.01 3.95 23.16 3.85 10.96 2.53

C7 146.69 4.5 127.89 4.3 12.82 4.44

C8 179.93 6.55 164.20 6.1 8.74 6.87

C9 430.13 5.77 358.15 5.15 16.73 10.75

C10 1083.58 5.78 819.8 5.67 24.34 1.90

“CPU” = the average CPU time

“Iter.” = the average number of iterations

“CPU (%)” = the average relative difference of CPU time between BVI123 + Type2 and the hybrid strategy

“Iter. (%)” = the average relative difference of the number of iterations between BVI123 + Type2 and thehybrid strategy

lead to a consistent performance for all the instances. As compared with BVI123 + Type2,BVI123 + Type5 has great improvement for some instances that were solved slowly, whileit may take much longer CPU time for the instances where BVI123 + Type2 itself can obtainan optimal solution in a shorter time.

In general, Type 2 cuts are often low density cuts where only a small number of decisionvariables of the master problem are involved. Therefore, its contribution to strengthening thesolution space of the master problem is limited. However, for our problem, BVI123 + Type2has produced |K0| + 1 low density cuts in each iteration and sometimes the combination ofthese cuts can probably strengthen the solution space of the master problem obviously, thusleading to converge rapidly. In cases where BVI123 + Type2 performs well, the effect of thelifted cuts generated by the HDP cut generation method is not obvious because of consumingaddition time, and thus the total CPU time may be longer.

In order to maintain the good performance of BVI123 + Type2 and take advantage of theHDP cuts, we propose a hybrid strategy in which Type 2 cuts and Type 5 cuts are appliedalternately. Note that Type 5 cuts are used in even-numbered iterations. Finally, the averageperformances between BVI123 + Type2 and the hybrid strategy for all the instances of theten problem classes are compared, and Table 8 gives their comparison results.

From the results in Table 8, it can be seen that the average performances of the hybridstrategy can exceed BVI123 + Type2 concerning the CPU time and the number of iterations.With the increase of the CPU time and the number of iterations, the improvement of thehybrid strategy is more obvious, which is consistent with the results in Table 7. The reasonwhy the average improvement of the hybrid strategy is less obvious than some instancesin the corresponding problem class as shown in Table 7 is that BVI123 + Type5 performswell for some instances while it has no advantages for some other instances. The hybridstrategy can be able to make better use of their advantages of Type 2 cuts and Type 5 cuts,and perform more stably for all the instances.

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6 Conclusions

In this paper, a mixed integer linear programming model for the facility location problemwith capacity expansions of the existing facilities is formulated, and the improved Ben-ders decomposition algorithms are applied to solving the problem. The three groups of validinequalities and the different types of disaggregated cuts by separating the Benders subprob-lem are added to the Benders master problem to restrict its solution space and improve thelower bounds efficiently. The high density Pareto cut generation method and a hybrid strat-egy which alternately generates the high density Pareto cuts are proposed and applied to ourproblem respectively to further accelerate convergence. Experimental results demonstratethat the combination of all the valid inequalities is the most effective to improve the lowerbounds. The disaggregated cuts obtained using both variable separation and facility sepa-ration show the best performance among all the types of the disaggregated cuts. The HDPcut generation method has significant improvement for some instances where the number ofiterations is relatively intensive and the CPU time is longer, and the hybrid strategy showsthe best average performance for all the instances. The future research may be performed tostudy the facility location problem with stochastic demands apart from expanding capacityand develop efficient metaheuristic algorithms.

Acknowledgements This research is partly supported by State Key Program of National Natural ScienceFoundation of China (71032004), the 111 Project under Grant B08015, Federal Project and (9901011) re-search project of Kathikas Institute of Research and Technology (KIRT), Cyprus.

References

Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Nu-merische Mathematik, 4(1), 238–252.

Cordeau, J. F., Pasin, F., & Solomon, M. M. (2006). An integrated model for logistics network design. Annalsof Operations Research, 144(1), 59–82.

Cote, G., & Laughton, M. (1984). Large-scale mixed integer programming: Benders-type heuristics. Euro-pean Journal of Operational Research, 16(3), 327–333.

Dogan, K., & Goetschalckx, M. (1999). A primal decomposition method for the integrated design of multi-period production-distribution systems. IIE Transactions, 31(11), 1027–1036.

Erengüç, S. S., Simpson, N. C., & Vakharia, A. J. (1999). Integrated production-distribution planning insupply chains: an invited review. European Journal of Operational Research, 115(2), 219–236.

Fond, C. O., & Srinivasan, V. (1986). The multiregion dynamic capacity expansion problem: an improvedheuristic. Management Science, 32(9), 1140–1152.

Gabrel, V., Knippel, A., & Minoux, M. (1999). Exact solution of multicommodity network optimizationproblems with general step cost functions. Operations Research Letters, 25(1), 15–23.

Gendreau, M., Potvin, J. Y., Smires, A., & Soriano, P. (2006). Multi-period capacity expansion for a localaccess telecommunications network. European Journal of Operational Research, 172(3), 1051–1066.

Geoffrion, A. M., & Graves, G. W. (1974). Multicommodity distribution system design by Benders decom-position. Management Science, 20(5), 822–844.

Hindi, K. S., & Basta, T. (1994). Computationally efficient solution of a multiproduct, two-stage distribution-location problem. Journal of the Operational Research Society, 45(11), 1316–1323.

Klose, A. (2000). A Lagrangean relax-and-cut approach for the two-stage capacitated facility location prob-lem. European Journal of Operational Research, 126(2), 408–421.

Ko, H. J., & Evans, G. W. (2007). A genetic algorithm-based heuristic for the dynamic integrated for-ward/reverse logistics network for 3PLs. Computers & Operations Research, 34(2), 346–366.

Magnanti, T. L., & Wong, R. T. (1981). Accelerating Benders decomposition: algorithmic enhancement andmodel selection criteria. Operations Research, 29(3), 464–484.

McDaniel, D., & Devine, M. (1977). A modified Benders’ partitioning algorithm for mixed integer program-ming. Management Science, 24(3), 312–319.

Melo, M. T., Nickel, S., & Saldanha-da-Gama, F. (2009). Facility location and supply chain management—a review. European Journal of Operational Research, 196(2), 401–412.

Ann Oper Res

Minoux, M. (2001). Discrete cost multicommodity network optimization problems and exact solution meth-ods. Annals of Operations Research, 106, 19–46.

Papadakos, N. (2008). Practical enhancements to the Magnanti-Wong method. Operations Research Letters,36(4), 444–449.

Poojari, C. A., & Beasley, J. E. (2009). Improving Benders decomposition using a genetic algorithm. Euro-pean Journal of Operational Research, 199(1), 89–97.

Rei, W., Cordeau, J. F., Gendreau, M., & Soriano, P. (2009). Accelerating Benders decomposition by localbranching. INFORMS Journal on Computing, 21(2), 333–345.

Saharidis, G. K. D., Boile, M., & Theofanis, S. (2011). Initialization of the Benders master problem usingvalid inequalities applied to fixed-charge network problems. Expert Systems with Applications, 38(6),6627–6636.

Saharidis, G. K. D., & Ierapetritou, M. G. (2010). Improving Benders decomposition using maximum feasiblesub-system (MFS) cut generation strategy. Computers & Chemical Engineering, 34(8), 1237–1245.

Saharidis, G. K. D., Minoux, M., & Ierapetritou, M. G. (2010). Accelerating Benders method using coveringcut bundle generation. International Transactions in Operational Research, 17(2), 221–237.

Singh, P., Makram, E. B., & Adams, W. P. (1998). A new technique for optimal time-dynamic distributionsubstation and feeder planning. Electric Power Systems Research, 47(3), 197–204.

Üster, H., Easwaran, G., Akçali, E., & Çetinkaya, S. (2007). Benders decomposition with alternative multiplecuts for a multi-product closed-loop supply chain network design model. Naval Research Logistics,54(8), 890–907.

Wentges, P. (1996). Accelerating Benders’ decomposition for the capacitated facility location problem. Math-ematical Methods of Operations Research, 44(2), 267–290.

Yilmaz, P., & Catay, B. (2006). Strategic level three-stage production distribution planning with capacityexpansion. Computers & Industrial Engineering, 51(4), 609–620.