An improved Benders decomposition algorithm for the logistics facility location problem with capacity expansions

Transcription

1 DOI /s An improved Benders decomposition algorithm for the logistics facility location problem with capacity expansions 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 existing facilities remain open or not, what the expansion size of the open facilities should be and which potential facilities should be selected. The problem is formulated as a mixed integer linear programming model (MILP) with the objective to minimize the sum of the savings from closing the existing facilities, the expansion costs, the fixed setup costs, the facility operating costs and the transportation costs. The structure of the model motivates us to solve the problem using Benders decomposition algorithm. Three groups of valid inequalities 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 the primal Benders cut are constructed in each iteration. A high density Pareto cut generation method is proposed to accelerate the convergence by lifting Pareto-optimal cuts. Computational experiments show that the combination of all the valid inequalities can improve the lower bounds significantly. By alternately applying the high density Pareto cut generation method based on the best disaggregated cuts, the improved Benders decomposition algorithm is advantageous in decreasing the total number of iterations and CPU time when compared to the standard Benders algorithm and optimization solver CPLEX, especially for large-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. Jiang Liaoning Key Laboratory of Manufacturing System and Logistics, The Logistics Institute, Northeastern University, Shenyang, , China qhjytlx@mail.neu.edu.cn G.K.D. Saharidis University of Thessaly, Leoforos Athinon, Pedion Areos, Volos, Greece G.K.D. Saharidis Kathikas Institute of Research & Technology, 303 Ryefield Ridge, Columbia, MO 65203, USA

2 1 Introduction The facility location problems have received extensive attention in the past few decades (see e.g., Erengüç et al. 1999; Meloetal.2009). Most of researches in the literature are devoted to determining which sites should be selected to establish new facilities from a set of 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 minimize the total cost. This total cost is the sum of the fixed setup costs, the operating costs of products at each established facility and the transportation costs. In practice, however, it is frequently encountered that some existing facilities are available but insufficient to satisfy the increasing storage demands, and thus it is necessary to expand the capacities of the existing facilities and/or establish additional facilities. If these existing facilities where some infrastructures are already equipped with are ignored when considering the establishment of new facilities, it will result in significant waste of the resources and increase of the fixed setup costs. Therefore, the problem under our consideration is to determine the locations of new facilities based on the existing facilities. However, whether the existing facilities would be expanded, closed or unchanged is dependent on the trade-off between their operating costs and the fixed setup costs for establishing new facilities. To the best of our knowledge, little research has been done on the facility location problem with simultaneous consideration of the existing facilities even though the facility location problems and capacity expansion problems have been considered separately in most of the previous studies. Geoffrion and Graves (1974) propose a two-stage distribution system design problem in which only the establishment of new facilities is considered and each customer is only able to be served by a single facility to minimize the total cost in the system, 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 each customer can be served by more than one facility, and a branch and bound algorithm is used to solve it. Üster et al. (2007) investigate a production/distribution system design problem with a fixed number of capacitated facilities, and multiple meta-heuristic approaches are proposed to solve it. Klose (2000) studies a two-stage capacitated facility location problem and a Lagrangean relax-and-cut approach is proposed to solve it. The existing capacity expansion problems in the literature can be divided into two categories. One is the node capacity expansion problem in which the capacities of the existing nodes are expanded while the structure of the network remains unchanged. The other is the node number expansion problem in which new nodes are added to the existing network while the capacities of the existing nodes keep unchanged. For the node capacity expansion problems, Fond and Srinivasan (1986) study a production capacity expansion problem for a single stage production-distribution system, and an improved heuristic algorithm is proposed. Yilmaz and Catay (2006) investigate a three-stage production-distribution network planning problem with node capacity expansions, and the MILP relaxation-based heuristics are developed to obtain a good feasible solution. Ko and Evans (2007) consider a node capacity expansion problem for a distribution network with reverse logistics where the warehouses and repair centers are regarded as expandable nodes. The problem is formulated as a mixed integer nonlinear programming model and a genetic algorithm-based heuristic is proposed 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 an electricity transmission network in which new substations and feeders need to be installed while the capacities of the existing substations and feeders remain unchanged. A mixed 0 1 quadratic program model is developed and the generalized Benders decomposition algorithm is applied to solving it. Gendreau et al. (2006) consider a telecommunications network

3 Fig. 1 The structure of the system under consideration plan problem to determine the installation of a concentrator at each node so as to minimize the installation costs. As compared with the ordinary facility location problems or the capacity expansion problems mentioned above, the problem under our consideration has some different features so that it is more realistic and challenging. These features are reflected by making integrated decisions on whether the existing facilities will be open, expanded or closed and which potential sites will be selected to establish new facilities. The remainder of the paper is organized as follows. Section 2 is devoted to describing the MILP model of our problem. Benders decomposition algorithm is applied to our problem in Sect. 3. In Sect. 4, the improved Benders decomposition algorithms with three groups of valid inequalities, different disaggregated cuts and high density Pareto cuts are proposed. Section 5 reports the experimental results to evaluate the performance of the improved Benders decomposition algorithms. Finally Sect. 6 concludes the paper. 2 Mathematical formulation In this section, the detailed description and the mathematical model of the facility location problem with capacity expansions are given. 2.1 Problem description The structure of the logistics system under consideration is shown in Fig. 1. The products produced by suppliers will be shipped to customers via facilities to meet customers demands. Each supplier can supply a wide variety of products to several facilities, and the

4 supply capacity for each product is known. Our problem is to determine whether the existing facilities should be open or not, what the expansion size of each open facility should be, which new facilities should be established, and which customers should be served by an open facility so that the sum of the expansion costs, the savings from closing the existing facilities, the fixed setup costs, the facility operating costs and the transportation costs is minimized. 2.2 The model Our problem is formulated as a mixed integer linear programming model (MILP), and the following 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 K 0 K and K 1 K are the sets of the existing facilities and potential facilities, respectively L Set of products, l L The total number of elements in set S il Capacity of supplier i for product l D jl Demand of customer j for product l f k Fixed setup cost of facility k v k Operating cost per unit of product at facility k R k Maximum allowed expansion (additional) amount at facility k p k Savings from closing facility k Mk L Minimum required throughput at open facility k Mk U Maximum allowed capacity before being expanded at facility k c ikjl Unit cost of shipping product l from supplier i via facility k to customer j Unit expansion cost of facility k e k We define the following binary and continuous decision variables to determine whether facility k will be open or not, which customers will be served by facility k, the total throughput of facility k and the expansion size of each open facility. Here quadruply subscripted continuous variable x ikjl is used to represent the logistics volume of a product from a supplier to a customer via a facility, which can easily track the origin of a product after it has arrived at a facility or a customer and can also give an advantage when Benders decomposition method is applied. Decision variables { 1 if facility k is open z k = k K 0 otherwise 0 K 1. { 1 ifcustomer j is served by facility k u kj = 0 otherwise k K 0 K 1 ; j J. x ikjl s k Continuous variable which corresponds to the amount of product l shipped from supplier i via facility k to customer j Continuous variable which corresponds to the amount of capacity expansion of facility k

5 According to the above notations, the logistics facility location problem with capacity expansions of the existing facilities can be formulated as follows: Min e k s k + c ikjl x ikjl + f k z k k K 0 K 1 i I k K 1 + D jl u kj p k (1 z k ) (1) k K 0 v k x ikjl S il i I; l L. (2) k K 0 K 1 j J x ikjl = D jl u kj k K 0 K 1 ; j J ; l L. (3) i I Mk L z k D jl u kj Mk U z k + s k j J l L k K 0 K 1. (4) s k R k z k k K 0 K 1. (5) u kj = 1 j J. (6) k K 0 K 1 x ikjl 0 i I; k K 0 K 1 ; j J ; l L. (7) s k 0 k K 0 K 1. (8) z k {0, 1} k K 0 K 1. (9) u kj {0, 1} k K 0 K 1 ; j J. (10) In the above model, objective function (1) is to minimize the sum of the capacity expansion costs, the transportation costs, the fixed setup costs for establishing new facilities, the operating costs for handling products at facilities and the savings from closing the existing facilities. Constraints (2) are the supply capacity constraints of supplier i for product l to guarantee that the amount of product l shipped from supplier i cannot exceed the supply capacity of supplier i for product l. Constraints (3) are the flow conservation constraints at each open facility, and they ensure that the total amount of product l shipped from all suppliers to customer j via facility k should exactly match the demand of customer j for product l. Constraints (4) are the capacity constraints for all the facilities to ensure that the throughput of a facility must be in the range of its minimum required throughput and maximum allowed capacity before being expanded plus its capacity expansion amount. If a facility s throughput is less than its minimum required throughput, closing it will save the operating cost or fixed setup cost. On the other hand, a facility s throughput cannot increase unlimitedly because of the limited available land, and thus it can not exceed its maximum allowed capacity. The expression of Constraints (4) allows the existing and new facilities to be expanded, although in general new facilities should be directly designed to reach to its required size so that the potential expansion is no longer needed (this means s k = 0, k K 1 ). Constraints (5) ensure that the capacity expansion amount of facility k cannot exceed its maximum allowed expansion amount. Constraints (6) ensure that each customer should be served by a single facility to meet the economical requirement on transportation operation cost. 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 is characterized by binary variables z k and u kj, continuous variables x ikjl and s k, and coupling constraints (3), the right-hand side of (4) and(5). Due to the complexity of the model, it

6 cannot be efficiently solved by an optimization package like CPLEX solver, especially for large 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 decomposition method which is suitable for solving the complicated mixed integer programming problem with integer variables and coupling constraints. It is based on the ideas of partition and delayed constraint generation. In each iteration, by fixing the integer variables, the primary problem with only continuous variables becomes a Benders subproblem that can be solved easily. The Benders master problem with only integer variables and an auxiliary variable is a relaxation of the primary problem. An optimal solution of the Benders master problem which provides a lower bound (in the case of minimization) is transmitted to the subproblem for constructing a new subproblem. By resolving the subproblem, the values of its dual variables and a valid upper bound (in the case of minimization) are obtained, and then a Benders cut is formed and appended to the Benders master problem. By introducing the new Benders cut, the Benders master problem is resolved and Benders decomposition algorithm is iterated continuously until the difference between the lower bound and the best upper bound in some iteration is small enough or zero. In the past few decades, some acceleration techniques for Benders decomposition algorithm have been successfully developed. McDaniel and Devine (1977) propose a method to quickly 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 on the variables of the master problem to guarantee the convergence of the algorithm. Cote and Laughton (1984) use the first integer feasible solution rather than an optimal solution of the master problem to construct the Benders subproblem to accelerate Benders decomposition algorithm. Because sometimes this method can cause Benders decomposition algorithm to fail to converge, a heuristic method for choosing an iteration in which the master problem has to be solved to optimality is developed. Poojari and Beasley (2009) use a genetic algorithm to obtain feasible solutions of the master problem to reduce the CPU time. The above three acceleration strategies are suitable for the problem where the master problem is hard to be solved. The following previous studies focus on how to improve the quality of Benders cuts. Magnanti and Wong (1981) first define a cut as Pareto-optimal if no other cut dominates it. By applying Pareto-optimal cuts to a problem in which the Benders subproblem is degenerate, Benders decomposition algorithm can be significantly improved. Rei et al. (2009) investigate a local branching method to strengthen or replace Benders feasible cuts by local branching constraints, and the lower and upper bounds can be improved simultaneously. Saharidis et al. (2010) present a Covering Cut Bundle strategy to accelerate Benders decomposition algorithm by generating a bundle of cuts in each iteration in order to cover all the decision variables of the master problem. Saharidis et al. (2011) also propose a series of generic valid inequalities which are valid for the general fixed charge network problem to initialize the master problem. For the facility location problems, Benders decomposition algorithm has been successfully applied by the following researchers. Geoffrion and Graves (1974) first employ Benders decomposition algorithm to solve the two-stage distribution system design problem. Dogan and Goetschalckx (1999) study the integrated design problem of a multi-period production-distribution system based on Benders decomposition algorithm in which the disaggregated cuts according to products and seasons and an initial cut set by relaxing the

7 integer restrictions for the integer variables are simultaneously applied to accelerate the Benders decomposition algorithm. Cordeau et al. (2006) propose an integrated model for a logistics network design problem and Benders decomposition algorithm is applied to solving it. Some valid inequalities are proposed to strengthen the linear relaxation of the primary model, and a set of initial cuts obtained by relaxing the integrality constraints of the master problem are used to accelerate Benders decomposition algorithm. Wentges (1996) uses Pareto-optimal cuts and a procedure to strengthen Benders cuts to speed up Benders decomposition algorithm for a discrete capacitated facility location problem. From the examples mentioned above, it can be concluded that Benders decomposition algorithm is very suitable for solving the facility location problems. In this paper, we will use Benders decomposition algorithm to solve our problem. A series of valid inequalities are first added to the Benders master problem to restrict its solution space. According to the structure features of our MILP model, different disaggregated cuts of the primal Benders cut are applied to our problem. For some instances that were solved slowly, a novel strategy called High Density Pareto (HDP) cut generation is proposed to reduce the number of iterations and the CPU time. For all the instances, a hybrid strategy is proposed to improve the average performance of the HDP cut generation method. Without loss of generality, we consider the following mixed integer linear programming problem: Min c1 T x + ct 2 y (11) Ax + By b (12) x 0 (13) y {0, 1}. (14) Where c 1 R n 1, x R n 1 +, c 2 R n 2, y R n 2 +, b R m, A R m n 1, B R m n 2 and 0 is the m-dimensional null vector. Fixing integer variables y =ȳ in the problem given by (11) (14), the general form of the Benders subproblem is as follows: Min c1 T x (15) Ax b Bȳ (16) x 0. (17) The dual problem of the Benders subproblem can be written as: Max u T (b Bȳ) (18) A T u c 1 (19) u 0. (20) The general form of the Benders master problem is as follows: Min c2 T y + z (21) u T (b By) z u P U (22) u T (b By) 0 u R U. (23)

8 Where U is the polyhedron defined by (19) (20), and P and R are the sets of the extreme points and the extreme rays of U respectively, and z is an auxiliary variable complementing the objective function of the master problem. Following the Benders decomposition algorithm described above, for our model, by fixing binary variables z k = z k and u kj =ū kj, the Benders subproblem in our problem (SP) can be written as follows: (SP) Min e k s k + c ikjl x ikjl k K 0 i I (24) x ikjl S il i I; l L. (25) k K 0 K 1 j J x ikjl = D jl ū kj k K 0 K 1 ; j J ; l L. (26) i I s k D jl ū kj Mk U z k j J l L k K 0. (27) s k R k z k k K 0. (28) x ikjl 0 i I; k K 0 K 1 ; j J ; l L. (29) s k 0 k K 0. (30) Define dual variables μ il associated with constraints (25), r kj l associated with constraints (26), φ k associated with constraints (27), and α k associated with constraints (28). Therefore, the dual of the Benders subproblem in our problem (DSP) is as follows: (DSP) Max S il μ il + D jl ū kj r kj l i I l L + ( ) D jl ū kj Mk U z k φ k + R k z k α k (31) k K 0 j J l L k K 0 μ il + r kj l c ikjl i I; k K 0 K 1 ; j J ; l L. (32) φ k + α k e k k K 0. (33) μ il 0 i I; l L. (34) φ k 0 k K 0. (35) α k 0 k K 0. (36) The constraints of the DSP given by (32) (36) constitute a polyhedron denoted as. If the SP is feasible for the fixed values z k ( k K 0 )andū kj, the DSP has a bounded solution which is an extreme point of the polyhedron in the light of duality theory, and thus an optimality Benders cut is deduced. On the contrary, if the SP is infeasible, the DSP has an unbounded solution, an extreme ray of the polyhedron can be identified and a feasibility Benders cut will be generated. Let P and Q be the sets of all the extreme points and extreme rays of polyhedron, respectively.

9 Optimality Benders cut: y S il μ il + D jl u kj rkj l i I l L + ( ) D jl u kj Mk U z k φk + R k z k αk. k K 0 j J l L k K 0 Feasibility Benders cut: S il μ il + D jl u kj r kj l i I l L + ( ) D jl u kj Mk U z k φ k + R k z k α k 0. k K 0 j J l L k 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 problem. Saharidis and Ierapetritou (2010) show that producing optimality instead of feasibility cuts would lead to faster convergence of Benders decomposition algorithm and generating more 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 problem in our problem (MP) can be written as: (MP) Min k K 1 f k z k + v k D jl u kj p k (1 z k ) + y (37) k K 0 D jl u kj Mk L z k k K 0. (38) j J l L Mk L z k D jl u kj Mk U z k k K 1. (39) j J l L u kj = 1 j J. (40) k K 0 K 1 y S il μ il + D jl u kj rkj l i I l L + ( ) D jl u kj Mk U z k φk + R k z k αk. (41) k K 0 j J l L k K 0 S il μ il + D jl u kj r kj l i I l L + ( ) D jl u kj Mk U z k φ k + R k z k α k 0. (42) k K 0 j J l L k K 0 z k {0, 1} k K 0 K 1. (43)

10 u kj {0, 1} k K 0 K 1 ; j J. (44) y 0. (45) Due to the special structure of our problem, using Benders decomposition algorithm directly leads to slow convergence. In the following sections, we propose some strategies to accelerate the Benders decomposition algorithm. 4 Improved Benders decomposition algorithms In this section, three groups of valid inequalities, different disaggregated cuts and high density Pareto cuts are proposed and added to the Benders master problem to accelerate convergence of the Benders decomposition algorithm. 4.1 Valid inequalities Aside from the quality of the produced Benders cuts which was the subject under study in Magnanti and Wong (1981) and Saharidis et al. (2010), the other main reason leading to slow convergence of Benders decomposition algorithm is that the LB (in the case of minimization) obtained from the master problem without strong valid inequalities is relatively weak (Saharidis et al. 2011). Introducing a series of valid inequalities to obtain a restricted master problem from the first iteration is an effective way to accelerate convergence, and in this case the infeasible cases of the master problem may be eliminated a priori and the first lower bound derived by the master problem will be significantly improved. As a result, the gap between the lower bound and the upper bound will be narrowed and the algorithm will converge 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 the master problem and obtain improved lower bounds, a series of valid inequalities which combine all binary variables of the master problem, are developed according to the features of our problem. (1) Supply-demand valid inequalities for the existing facilities D jl u kj z k (Mk U + R k) k K 0. (46) j J l L Valid inequalities (46) ensure that the total demands of the customers served by an existing facility cannot be greater than its maximum allowed capacity after being expanded. Because constraints j J l L D jlu kj Mk U z k + s k ( k K 0 ) will be included in the Benders subproblem rather than the Benders master problem after decomposition, these valid inequalities are to recover the function of these constraints in the Benders master problem as much as possible. (2) Supply-demand valid inequalities for all the facilities z k (Mk U + R k) + Mk U z k D jl (47) k K 0 k K 1 j J l L Valid inequalities (47) ensure that the sum of the maximum allowed capacities after being expanded for all the facilities must be greater than or equal to the total demands of the customers. Valid inequalities (47) in conjunction with valid inequalities (46) can further reduce the solution space of the MP and make the SP feasible.

11 (3) The operating state valid inequalities for all the facilities u kj J z k k K 0 K 1. (48) j J Valid inequalities (48) guarantee that facility k must be operated if facility k provides service 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 to an extra expansion cost, or (2) establishing the new facility will lead to an extra fixed setup cost. If a facility is determined not to be operated, two cases would appear: (1) the existing facility will be closed and its operating cost will be saved, and (2) the new facility will not be operated. After adding valid inequalities (48) tothemp,z k = 1 will become true if u kj = 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 productiondistribution system by Dogan and Goetschalckx (1999) and the closed-loop supply chain network design by Üster et al. (2007). The disaggregated cuts can be obtained only from the problems with a specific structure, i.e., the Benders subproblem must be able to be separated into some independent subproblems. In each iteration, the multiple disaggregated cuts formed by the dual optimal solutions corresponding to the independent subproblems will be appended to the master problem simultaneously. These cuts which include exactly the same information as the primal Benders cut will restrict the solution space of the master problem in a more accurate-exact way. The disaggregation of the primal Benders cut is essentially a multi-generation of cuts method. The idea of the multi-generation of cuts is to add multiple cuts to Benders master 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 special case of min cost multi-commodity flow problem. Saharidis et al. (2010) and Saharidis and Ierapetritou (2010) present a general applicable method in which a bundle of cuts is generated in each iteration and an optimality type cut is generated in addition to a feasibility Benders cut. When a multi-generation of cuts method is developed and applied, a good strategy to converge to optimality faster than the standard Benders algorithm is to maintain a balance between the number of iterations and the amount of time spent in each iteration for the generation of the additional cuts. This balance should be based on the idea that for producing an additional cut more time is spent to complete an iteration but the total time spent to solve iteratively the master problem decreases due to smaller number of iterations needed. As we mentioned above, Benders decomposition algorithm could converge to optimality in an iteration if all Benders cuts are added to the master problem in the beginning of the algorithm. This is the reason that makes the multi-generation of cuts method to decrease the total number of iterations and also the CPU time if the cuts generated are of good quality and the time spent to produce them keeps the appropriate balance described before. The only difference between the generation of the disaggregated cuts and the multi-generation of cuts is 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 decomposition algorithm for solving the location problems with capacity expansions. When the disaggregation of the primal Benders cut is applied to our problem, the Benders subproblem given by (24) (30) can be exactly separated into two independent subproblems SP1 and SP2

12 according to two different sets of continuous variables x ikjl and s k (variable separation). SP1 can be further separated into L disconnected single-product subproblems (product separation), and SP2 can be further separated into K 0 disconnected single-facility subproblems (facility separation). Therefore, three types of Benders cuts are generated based on different separations mentioned above. The first type of Benders cuts is obtained using only variable separation. The second type of Benders cuts is obtained using variable separation and facility separation. The third type of Benders cuts is obtained using both facility separation and product separation. The following are the details of how to obtain these different types of cuts 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 auxiliary variables corresponding to the different types of cuts should be included in the objective function of the master problem. The detailed forms of the two independent subproblems (SP1 and SP2) are given respectively, as follows: (SP1) (SP2) Min i I k K 0 K 1 j J c ikjl x ikjl (49) x ikjl S il i I; l L. (50) x ikjl = D jl ū kj k K 0 K 1 ; j J ; l L. (51) i I x ikjl 0 i I; k K 0 K 1 ; j J ; l L. (52) Min k K 0 e k s k (53) s k D jl ū kj Mk U z k k K 0. (54) j J l L s k R k z k k K 0. (55) s k 0 k K 0. (56) Based on the models of SP1 and SP2, define dual variables μ il associated with constraints (50), r kj l associated with constraints (51), φ k associated with constraints (54), and α k associated with constraints (55), and then their dual problems DP1 and DP2 take the following forms: (DP1) Max S il μ il + D jl ū kj r kj l i I l L (57) μ il + r kj l c ikjl i I; k K 0 K 1 ; j J ; l L. (58) μ il 0 i I; l L. (59)

13 (DP2) Max ( ) D jl ū kj Mk U z k φ k + R k z k α k k K 0 j J l L k K 0 (60) φ k + α k e k k K 0. (61) φ k 0 k K 0. (62) α k 0 k K 0. (63) Because valid inequalities (46) (48) presented in Sect. 4.1 have been added to the Benders master problem, SP1 and SP2 are always feasible. Let ( ˆμ il, ˆr kj l ) and ( ˆφ k, ˆα k ) be the optimal solutions obtained by DP1 and DP2, respectively. Therefore, the first type of cuts is given by: Type 1 cuts: y 1 S il ˆμ il + D jlˆr kj l u kj. (64) i I l L y 2 D jl ˆφ k u kj + (R k ˆα k M U ˆφ k k )z k. (65) k K 0 k K 0 j J l L Where y 1 0andy 2 0 are two auxiliary variables. When Type 1 cuts are used in the general procedure of Benders decomposition algorithm, y 1 + y 2 will be included in the objective function of the MP. The second type of cuts obtained by further separating SP2 into K 0 disconnected singlefacility subproblems is given by: Type 2 cuts: y 1 S il ˆμ il + D jlˆr kj l u kj. (66) i I l L yk 2 D jl ˆφ k u kj + (R k ˆα k M U ˆφ k k )z k k K 0. (67) j J l L Where yk 2 ( k K 0) are auxiliary variables. When Type 2 cuts are used in the general procedure of Benders decomposition algorithm, y 1 + k K 0 yk 2 will be included in the objective function of the MP. The third type of cuts obtained by further separating SP1 into L disconnected singleproduct subproblems is given by: Type 3 cuts: yl 1 S il ˆμ il + D jlˆr kj l u kj l L. (68) i I k K 0 K 1 j J yk 2 D jl ˆφ k u kj + (R k ˆα k M U ˆφ k k )z k k K 0. (69) j J l L Where yl 1 ( l L) are auxiliary variables. When Type 3 cuts are used in the general procedure of Benders decomposition algorithm, l L y1 l + k K 0 yk 2 will be included in the objective function of the MP.

14 It can be seen that the numbers of Type 1 3 cuts in each iteration are two, K 0 +1and K 0 + L, respectively. As compared with the primal Benders cut, each of Type 1 3 cuts can effectively restrict the solution space of the master problem, and thus result in better convergence behavior, which is verified by the computational results in Sect Generation of high density Pareto cuts A cut is called Pareto-optimal if no other cut dominates it, which is first presented by Magnanti and Wong (1981). A Pareto-optimal cut only exists in which the dual of the Benders subproblem has several optimal solutions, and it is the strongest cut among all the alternative Benders cuts in the same iteration. Therefore, Pareto-optimal cuts can improve the performance of Benders decomposition algorithm effectively. Many researchers have employed the method presented by Magnanti and Wong (1981)to generate Pareto-optimal cuts to accelerate Benders decomposition algorithm for their problems. Recently Papadakos (2008) presents a new improved method for the generation of Pareto-optimal cuts, and he proves that it is not necessary to find a core point of the solution space of Benders master problem to produce a Pareto-optimal cut. Although the generation method of Pareto-optimal cuts is improved, there is little research on improving the quality of Pareto-optimal cuts. In fact, a Pareto-optimal cut is lifted from the corresponding optimality Benders cut, but the lifted effect is limited because it is dependent on the solution of the dual problem. It is possible to further lift Pareto-optimal cuts. In this section, we develop a new method referred to as High Density Pareto (HDP) cut generation, and it can produce Pareto cuts with high density. A high density cut is a cut which covers a high number of decision variables of the master problem, and thus it further restricts the solution space of the master problem and results in a better convergence behavior to the algorithm (Saharidis et al. 2010). In the following subsections, we will give how to generate Pareto-optimal cuts and HDP cuts 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 Generation of Pareto-optimal cuts Magnanti and Wong (1981) first give a method to generate Pareto-optimal cuts. Let V = u T (b Bȳ) be the optimal objective value of the dual subproblem and y c be a core point of the solution space of the master problem. A Pareto-optimal cut can be obtained by solving the following problem: Max u T (b By c ) (70) A T u c 1 (71) u T (b Bȳ) = u T (b Bȳ) (72) u 0. (73) Based on the idea that any extreme point or any extreme ray of the dual subproblem gives a valid Benders cut (Benders 1962) as well as that it is not necessary to use a core point 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:

15 (AP) Max u T (Coef ) (74) A T u c 1 (75) u T (b Bȳ) = u T (b Bȳ) (76) u 0. (77) Where Coef is m-vector with constant values which defines the direction of AP s objective function and can take any value High density Pareto cut generation The generation of the HDP cuts is based on the auxiliary problem given by (74) (77), but coefficient matrix Coef needs to be specified in advance and the AP needs to be solved optimally 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 feasible bounded solution, and the following optimality Benders cut can be obtained: u T (b By) z u P U u T b z u T By (78) Decision variable y k corresponding to the kth element of n 2 -column vector y is defined as a covered decision variable if the kth row (u T B) k of n 2 -row vector u T B has a value not equal to zero and at the same order as the other non-zero coefficients corresponding to the other decision variables of the master problem. The value of the coefficient corresponding to y k depends on m-row vector u T and m n 2 matrix B. If the structure of the problem (represented by the matrix B) allows us (e.g., n 2 q=1 B k,q 0) to cover decision variable y k, in order to cover y k, we ask from the auxiliary problem (AP) to find (if any) a non-zero dual solution (u T ) k among the optimal solutions. In order to cover a high number of decision variables in the master problem, we need to obtain the solution of AP with the maximum non-zero dual solution for Coef = ( 1 b 1 ( 1 b k ( 1 b m ) 1 n2 q=1 B 1,q. 1 n2 q=1 B k,q. ) 1 n2 q=1 Bm,q ) each (u T ) k with n 2 q=1 B k,q 0. Therefore, the AP with Coef k = ( 1 k=1 b k (79) 1 n2 ) needs to q=1 B k,q 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: m ( ) 1 1 Max u k b n2 k q=1 B k,q (80) A T u c 1 (81) u T (b Bȳ) = u T (b Bȳ) (82) u 0. (83)

16 We have to notice that we prefer to use the values of vector Coef defined by (79) to equilibrate 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 y k which will be significantly bigger than another coefficient corresponding to decision variable y k resulting in a cut where only y k is covered in practice even if the coefficient of y k is not equal to zero (due to its small value compared to the coefficient of y k ). Notice also that for any k with n 2 q=1 B k,q = 0, the corresponding part of the objective function is deleted because the corresponding decision variable y k will not be covered independently on the value of (u T ) k. After solving (80) (83), a HDP cut covering a high number of decision variables is generated. By concluding the presentation of the HDP cut generation method, it has to be noted that the proposed method is suitable for the cases where there are multiple optimal solutions for the Benders subproblem, and only these cases make sense to apply it. These cases are encountered frequently in the problems where the Benders subproblem involves network optimization (Magnanti and Wong 1981) while the facility location problem presented here is a typical example among them. For our problem, the HDP cut generation method based on Type 1 3 cuts is implemented as follows. For the convenience of description, let Type 4 6 cuts represent the HDP cuts for Type 1 3 cuts, respectively. In order to obtain Type 4 6 cuts, the following two auxiliary problems AP1 and AP2 should be solved optimally. Let f (SP 1) and f (SP 2) be the optimal objective values of SP1 and SP2 respectively, and N be an infinite number. AP1 and AP2 take the following forms: (AP1) Max ( N 1 D jl ) r kj l (84) μ il + r kj l c ikjl i I; k K 0 K 1 ; j J ; l L. (85) S il μ il + D jl ū kj r kj l = f (SP 1) (86) i I l L μ il 0 i I; l L. (87) Note that term i I l L ( 1 S il N)μ il in the objective function of AP1 is deleted because the corresponding decision variables will not be covered independently on the values of μ il. (AP2) Max ) 1 (N M U k K 0 k )φ j J l L D k + (N k K0 + 1Rk α k (88) jl φ k + α k e k k K 0. (89) ( ) D jl ū kj Mk U z k φ k + R k z k α k = f (SP 2) (90) k K 0 k K 0 j J l L φ k 0 k K 0. (91) α k 0 k K 0. (92) Let ( μ il, r kj l )and( φ k, α k ) are the values obtained from AP1 and AP2 respectively. Using ( μ il, r kj l )and( φ k, α k ) to replace ( ˆμ il, ˆr kj l )and(ˆφ k, ˆα k ) in Type 1 3 cuts, we can obtain the following HDP cuts, respectively.

17 Type 4 cuts: Type 5 cuts: Type 6 cuts: y 1 S il μ il + D jl r kj l u kj. (93) i I l L y 2 D jl φ k u kj + (R k α k M U φ k k )z k. (94) k K 0 k K 0 j J l L y 1 S il μ il + D jl r kj l u kj. (95) i I l L yk 2 D jl φ k u kj + (R k α k M U φ k k )z k k K 0. (96) j J l L yl 1 S il μ il + D jl r kj l u kj l L. (97) i I k K 0 K 1 j J yk 2 D jl φ k u kj + (R k α k M U φ k k )z k k K 0. (98) j J l L 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 6 cuts mentioned above, we carry out the computational experiment on intensive instances generated randomly. The Benders decomposition algorithms are coded in C++ language and tested on the computer with CPU Intel Core 2, 2.83 GHz and 3.25 GB RAM. During implementing them and solving the model given by (1) (10), CPLEX 11.0 (with default settings) is used as an optimization solver. For comparing solution quality and runtime, the Benders decomposition algorithms with the different valid inequalities and alternative types of cuts are required to obtain an optimal solution. The effects of the different valid inequalities and alternative types of cuts on the performance of the Benders decomposition algorithm 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 number of suppliers ( I ), the number of existing facilities ( K 0 ), the number of potential facilities ( K 1 ), the number of customers ( J ) and the number of products ( L ). To evaluate the effectiveness and efficiency of the Benders decomposition algorithms with the different valid inequalities and alternative types of cuts, ten different problem classes ranging from small to large size are generated randomly according to the uniform distributions in Table 1. For each problem classes, 20 random instances are generated. To facilitate describing and understanding, let D l be the total demands for product l, and D be the total demands for all products. All parameters generated randomly obey the following uniform distributions. Transportation cost (c ikjl ) is selected from set {1,...,10}. Demand (D jl ) is generated from interval [20, 100], and facility operating cost of unit product (v k )ischosenfromset{5,...,10}. In order to guarantee that a feasible solution can be

18 Table 1 Problem classes generated from the following uniform distributions Class I K 0 K 1 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 (S il )isgenerated from [S 1l,S 2l ],wheres 1l = D l / I, S 2l = 2D l / I. Based on the rule that the total capacities of all the existing facilities are less than the total demands, the sum of the maximum allowed capacity before being expanded for all the existing facilities (defined as W T )isgenerated randomly from [2/5D,2/3D], the maximum allowed capacity before being expanded for each existing facility (Mk U ( k K 0 )) is generated from [4W T /5 K 0, 6W T /5 K 0 ], the minimum required throughput for each existing facility (Mk L ( k K 0)) is generated from [Mk U /3, 2MU k /5]. LetN T = D W T, and the maximum allowed capacity before being expanded for each potential facility (Mk U ( k K 1 )) and the minimum required throughput for each potential facility (Mk L ( k K 1 )) are generated from [N T / K 1,N T /2] and [Mk U /3, 2MU k /5], respectively. The fixed setup cost for each potential facility (f k ( k K 1 )) is generated from [2Mk U, 5MU k ]. Unit expansion cost (e k) and the maximum allowed expansion amount (R k ) are generated from [2f k /Mk U, 4f k/mk U ] and [2MU k /5,MU k /2], respectively. The savings from closing each existing facility (p k ( k K 0 )) is generated from [4f k /5,f k ]. Table 2 summarizes the characteristics and size of the model given by (1) (10) forthe ten different problem classes. As shown in Table 2, the scale of the ten problem classes increases gradually. Taking the largest scale problem class C10 for example, it includes 610 binary variables, continuous variables and constraints. 5.2 Effectiveness of valid inequalities Because different combinations of the valid inequalities may have different effects on restricting the solution space of the Benders master problem, Table 3 gives the comparison results when various combinations of the valid inequalities are added to the Benders master problem. Let B represent the standard Benders decomposition algorithm without any valid inequality, and BVI1, BVI2 and BVI3 represent that valid inequalities (46) (48) are added to the Benders master problem, respectively. Similarly, BVI123 represents that all the valid inequalities 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 no feasibility Benders cuts were generated. For the largest scale problem class C10, only 5% of the 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 indicate that producing more optimality Benders cuts than feasibility Benders cuts can accelerate the convergence of Benders decomposition algorithm. The Benders decomposition

19 Table 2 Characteristics and size of the generated problem classes Class I K 0 K 1 J L Number of variables Number of Binary Continuous constraints z k u kj x ikjl s k C C C C C C C C C C Table 3 Effects of different combinations of the valid inequalities Class B BVI1 BVI2 BVI3 BVI12 BVI13 BVI23 BVI123 C1 CPU Iter P 100% 100% 100% 100% 100% 100% 100% 100% F\O 2.15\4.5 0\ \ \4.5 0\4.5 0\ \4.5 0\4.45 C2 CPU Iter P 85% 100% 95% 100% 100% 100% 100% 100% F\O 0\ \6.1 0\6.1 0\ \6.85 0\6.0 C3 CPU Iter 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 Iter P 85% 100% 95% 85% 100% 100% 100% 100% F\O 0\9.4 0\9.8 0\ \9.65 0\9.55 C5 CPU Iter P 30% 95% 55% 75% 100% 100% 65% 100% F\O 0\9.05 0\9.35 0/9.15 C6 CPU Iter P 95% 100% 95% 100% 100% 100% 95% 100% F\O 0\ \7.05 0\6.75 0\7.1 0\7.2

20 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 Iter 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 difference between the first lower bounds obtained by B and BVI123 LB1 B = the lower bound obtained by B in the first iteration LB1 BVI123 = the lower bound obtained by BVI123 in the first iteration Relative difference = (LB1 BVI123 LB1 B ) / LB1 B 100% C5-# = the instance # of problem class C5 Example LB1 B LB1 BVI123 Relative difference (%) 1 (C5-1) (C5-2) (C5-3) (C5-4) (C5-5) (C5-6) (C5-7) (C5-8) (C5-9) (C5-10) algorithm where all the valid inequalities were added to the MP (e.g. BVI123) makes most of 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 first iteration for the first ten instances of the fifth problem class.

21 Table 5 Effects of Type 1 3 cuts Class CPLEX BVI123 + Type1 BVI123 + Type2 BVI123 + Type3 CPU CPU Iter. CPU Iter. CPU Iter. C C C C C 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 bound substantially as compared with the standard Benders decomposition algorithm, and the largest relative difference between them is up to %. In the following subsection, we evaluate the effects of the three types of disaggregated cuts on the performance of the Benders 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 proposed in Sect In the following tables, CPLEX represents optimization solver CPLEX is used, and BVI123 + Type1, BVI123 + Type2 and BVI123 + Type3 represent that Type 1 3 cuts are used in the Benders decomposition algorithm with valid inequalities (46) (48), respectively. Table 5 shows the effects of Type 1 3 cuts on the convergence of the Benders decomposition 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 problem classes. 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 and BVI123 + Type3 are seconds and seconds, respectively, while the average CPU time of CPLEX is up to seconds for all the instances which were solved optimally by these three methods. BVI123 + Type2 shows better performance than BVI123 + Type3 in most cases. Taking the largest problem class C10 for example, BVI123 + Type2 and 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 seconds, while that of BVI123 + Type3 is seconds. For the instances that an optimal solution was obtained by all the three methods, the average CPU time of BVI123 + Type2 is only seconds while the average time of CPLEX runs up to seconds. Furthermore, BVI123 + Type2 can obtain an optimal solution for 5% more instances than CPLEX and BVI123 + Type3, and the average CPU time of BVI123 + Type2 for these 5% instances is seconds. Therefore, Type2 is the most effective, and we evaluate the effectiveness of the HDP cuts for Type 2 cuts.