Benders decomposition for the uncapacitated multiple allocation hub location problem

Transcription

1 Computers & Operations Research 35 (2008) Benders decomposition for the uncapacitated multiple allocation hub location problem R.S. de Camargo a,,1, G. Miranda Jr. b, H.P. Luna c a Departamento da Ciência da Computação, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Prédio do Icex, sala 4010, Pampulha, zip , Belo Horizonte, Minas Gerais, Brazil b Departamento da Engenharia de Produção, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Pavilhão Central de Aulas, Pampulha, zip , Belo Horizonte, Minas Gerais, Brazil c Instituto de Computação, Universidade Federal de Alagoas, Campus A. C. Simões, BR 104, Km 14, Tabuleiro dos Martins, zip , Maceió, Alagoas, Brazil Available online 7 September 2006 Abstract In telecommunication and transportation systems, the uncapacitated multiple allocation hub location problem (UMAHLP) arises when we must flow commodities or information between several origin destination pairs. Instead of establishing a direct node to node connection from an origin to its destination, the flows are concentrated with others at facilities called hubs. These flows are transported on lins established between hubs, being then splitted and delivered to its final destination. Systems with this sort of topology are named hub-and-spoe (HS) systems or hub-and-spoe networs. They are designed to exploit the scale economies attainable through the shared use of high capacity lins between hubs. Therefore, the problem is to find the least expensive HS networ, selecting hubs and assigning traffic to them, given the demands between each origin destination pair and the respective transportation costs. In the present paper, we present efficient Benders decomposition algorithms based on a well nown formulation to tacle the UMAHLP. We have been able to solve some large instances, considered out of reach of other exact methods in reasonable time Elsevier Ltd. All rights reserved. Keywords: Hub-and-spoe networs; Benders decomposition; Large-scale optimization 1. Introduction Since the emergence of hub-and-spoe (HS) networs, an increasing number of researchers have been studying their application in different areas, such as telecommunication networs [1,2], air transportation [3,4], and parcel delivery [5 8]. The HS systems are suitable when commodities (telecommunication data, passengers or parcel cargo) between a pair of nodes of a networ cannot be shipped by an exclusive direct connection or because this sort of linage is too expensive to be carried out. Corresponding author. addresses: rcamargo@dcc.ufmg.br (R.S. de Camargo), miranda@dep.ufmg.br (G. Miranda), pacca@tci.ufal.br (H.P. Luna). 1 Research partially supported by the Grant CNPq/FAPEAL/PRONEX /$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi: /j.cor

2 1048 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) In HS networs, the traffic between two nodes is not shipped directly, but instead it is usually routed through one or two nodes, designated as hubs, before being delivered to its final destination. These hubs or central transshipment facilities centralize the commodity handling and reduce the networ installation and operational costs by replacing direct node to node connections with indirect lins. The design of HS networs consists generally of the selection of which nodes of the networ will be acting as hubs and how the other non-hubs will be allocated to the hubs. The former is a location problem, while the later is an allocation problem, i.e., the determination of the spoes between non-hubs and hubs. In most of the HS networs, hubs are fully connected among hub nodes, however non-hubs can be either single allocated to a hub, that is, a non-hub can be assigned to one hub only; or allocated to multiple hubs, meaning that a non-hub can be connected to more than one hub of the networ. Since a HS networ has fewer lins than a fully meshed networ, it allows the increase of traffic on inter-hub connections. As a consequence of it, scale economies can be exploited resulting in lower transportation costs per unit of flow and potential savings in the overall design and operational costs of a networ. Hence, the design of such networs has to be carefully done, once it involves great amounts of resources and has major impact in the operational costs afterwards. So, spending a few hours to solve huge HS networ problems optimally or near optimally is considered reasonable. Usually, designers would rather spend more time getting an optimal solution or near assured optimal solution instead of using heuristic approaches to get an inferior solution for such important problem. In this paper, we consider an efficient algorithm based on Benders decomposition method for the uncapacitated multiple allocation hub location problem (UMAHLP). Our algorithm is based on the Hamacher et al. [9] formulation of the problem and is able to solve the UMAHLP optimally or within a specified ε of optimality of large scale problems. Although there are alternative models in the literature [5,6,10,11], our choice is driven by the linear programming bound quality of the Hamacher et al. formulation. In order to tae into account scale economies, most of these models apply a constant discount factor on the inter-hub lins of the networ, such that the transportation cost on these lins are lower than that on the lins connecting the non-hub nodes to the hubs. However the assumption of a flow-independent scale economies on inter-hub lins is not suitable for practical problems, once it allows large discounts on lins with low flows. O Kelly and Bryan [12], Horner and O Kelly [13], Klincewicz [14] and Racunica and Wynter [15] have proposed more sophisticated formulations tacling this issue, being our subject of a future paper. Surveys of the various problem types arising in the context can be found in Campbell [10,16] and Campbell et al. [17] and classification schemes can be found in O Kelly and Miller [18]. Most of the wors of the literature has been using efficient heuristics to solve the problem (see [19]), since the pioneering wors of O Kelly [20,21]. This is partially explained due to the large size of the problems. Klincewicz [22] has proposed two different heuristics and a meta-heuristic, tabu-search, to find good solutions. Sorin-Kapov and Sorin-Kapov [23] have proposed a tabu-search algorithm. Ayin [24] has developed a enumeration method for the multiple allocation problem, where flows of a node can be routed through different hubs. Campbell [25] has presented a greedy exchange heuristic for the problem, while Ernst and Krishnamoorthy [5] have used simulating annealing to obtain good upper bounds that are used in a branch-and-bound algorithm. O Kelly et al. [26] were the first ones to publish a methodology for calculating lower bounds on the single allocation hub problem, allowing the quality measurement of a heuristic solution. Klincewicz [27], using Campbelĺs model, develop a dual ascent and dual adjustment method, while Sorin-Kapov et al. [28] reformulate Campbell s models [25] for single and multiple allocation. Pirul and Schilling [29] propose a subgradient optimization based on a Lagrangian relaxation of the model of Sorin-Kapov et al. More recently, Mayer and Wagner [30] develop a branch-and-bound using the Klincewicz [27] method but also based on the model of Sorin-Kapov et al. [28]. Mayer and Wagner [30] use the same methodology used by Klincewicz [27], however with superior performance. In this paper the UMAHLP is decomposed in two smaller problems: at a higher level, named as master problem (MP), the location decisions are made; while at an inferior level, nown as subproblem (SP), the determination of the spoes is done. The MP is a mixed-integer programing problem, while the SP is a transportation problem, efficiently solved by inspection. To enrich the discussion, we also present three variants of the algorithm. The first one is the classical Benders decomposition (BD1) due to Benders [31], the second is the multi-cut version (BD2) proposed by Birge and Louveaux [32], while the last (BD3) is the one devised by Geoffrion and Graves [33]. Further, as most of the early wors in HS

3 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) problems limits the set of hub node candidates to be much smaller than the number of nodes in the networ, we stretch this boundary out by allowing all nodes to be hub candidates. We have tested our algorithm using the standard AP and CAB data sets with the addition of fixed cost as published in Ernst and Krishnamoorthy [8] with minor modifications. This paper is organized as follows: in Section 2 the model formulation and the notation used is introduced. In Section 3, the three variants of the Benders decomposition method are developed to solve the problem and it is also demonstrated how the SP can be solved optimally by inspection. Computational results and conclusion remars are presented in Sections 4 and 5, respectively. 2. Model formulation The model formulation is based on the following definitions: let N be the set of node locations that exchange traffic and let K be the set of hub candidates, K N. For any pair of nodes i and j (i, j N), we have w ij, the flow from node i to node j that must be routed through either one or two installed hubs. Usually, we have w ij = w ji. Further, let a be the fixed cost of installing a hub at node K and let c ij m be the transportation cost per unit of flow from node i to j routed via hubs and m (i, j N and, m K). The transportation cost is the composition of three cost segments: c ij m = c i + αc m + c mj, where c i and c mj are the standard transportation cost per unit from location i (j)tohub(m), and αc m is the discounted transportation cost between hubs and m. The discount factor 0 α 1 represents the scale economies on the inter-hub linage. If only one hub is used, we have = m and no discount factor is applied. We define the following decision variables: { 1 if hub K is installed, y = 0 otherwise. x ij m 0 is the flow from origin i to destination j (i, j N) that is routed through hubs and m (, m K) in that order. In the above notation, i, j are used as node indexes, while, m are used as hub indexes, so the UMAHLP can be formulated as follows (see Hamacher et al. [9]): min a y + c ij m x ij m (1) i j m s.t. x ij m + x ij m w ij y i, j N, K, (2) m m = m x ij m = w ij i, j N, (3) m x ij m 0 i, j N,,m K, (4) y {0, 1} K. (5) In the above model, the objective function (1) minimizes the total cost. The total cost is the sum of installation and transportation costs. Constraints (2) guarantee that the flow which goes through a hub only happens if that hub is installed. Constraints (3) assure that the flow for every pair i j is routed via some hub pair. Once again, if only one hub is used, we have = m. Constraints (4) are the non-negativity of the continuous variables x ij m, while constraints (5) restrict the integer variables y tobe0or1. Hamacher et al. [9] have shown that constraints (2) (3) are facets of the integer polytope, maing the above formulation one of the tightest linear relaxation nown so far for the problem. Examining the model further, one can see that if we use all the nodes as hub candidates, we have n 4 n 2 flow variables x ij m ( i, j N,,m K i = j). For small values of n, we have an incredible number of flow variables, when comparing to the number of integer variables, y. We have a significant difference in magnitude. Moreover, as there are not any capacity constraints, either on the hubs or on the arcs, model (1) (5) has always a feasible solution and, for any given fixed structure of hubs, i.e., a feasible vector y, the model transforms into a

4 1050 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) transportation model. This transportation problem can then be solved by an all pairs shortest path algorithm. Gathering all these features, we have been motivated in deploying Benders decomposition method [31] to tacle the problem. 3. Benders decomposition method for the UMAHLP In 1962, Benders [31] proposed a partitioning method for solving mixed linear and nonlinear integer programming problems. Benders approach defined a relaxation algorithm for solving a problem partitioning it into two simpler problems: an integer problem, nown as MP, and a linear problem, nown as SP. The MP is a relaxed version of the original problem with the set of integer variables and its associated constraints, while SP is the original problem with the values of the integer variables temporarily fixed by the MP. The algorithm solves each one of the two simpler problems iteratively, one at a time. At each iteration, a new constraint, nown as Benders cut, is added to the MP. This new constraint is originated by the dual problem of the SP. The algorithm goes on until the objective function value of the optimal solution to the MP is equal to that of the SP, when it stops obtaining the optimal solution of the original mixed integer problem. The computational success of Benders decomposition on solving large scale problems has been confirmed since the pioneering paper of Geoffrion and Graves [33], the uncapacitated networ design problem with undirected arcs of Magnanti, Mirchand and Wong [34], the locomotive and car assignment problem of Cordeau et al. [35,36], the non-convex water resource management problem of Cai et al. [37], the cellular manufacturing system design of Heragu and Chen [38], the multicasting networ design problem of Miranda [39] and Randazzo and Luna [40]. In Sections we present a formal description of the algorithm and different strategies of resolution for the MP and the SP Master problem and the subproblem By fixing the vector of integer variables y = y h at iteration h to yield a feasible solution of model (1) (5), we obtain the following primal linear problem: min c ij m x ij m + s h (6) s.t. i j m x ij m + x ij m w ij y h i, j N, K, (7) m m = m x ij m = w ij i, j N, (8) m x ij m 0 i, j N,,m K, (9) where s h is the hub installation cost associated to the vector y. Deriving the dual problem from model (6) (9) by associating the dual variables u ij to constraints (7) and the dual variables α ij to constraints (8), we have the following dual problem at iteration h: max w ij α ij y h u ij (10) i j s.t. α ij u ij u ij m c ij m i, j N, = m,, m K, (11) α ij u ij c ij i, j N, m K, (12) u ij 0 i, j N, K, (13) α ij R i, j N. (14)

5 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) The dual problem (10) (14) is a linear problem. From the dual objective function (10), for a given iteration h, wecan formulate the following constraint nown as Benders cut: η + i w ij y u h ij j i w ij α h ij, (15) j where u h ij and αh ij are the optimal value of the dual variables obtained by solving the SP at iteration h, and η is a continuous variable and a transportation cost under-estimator. Therefore, we have the following MP: min η + a y (16) s. t. η + w ij y u h ij w ij α h ij, (17) i j i j y 1, (18) η 0, (19) y {0, 1}, K, (20) where h = 1,...,H and H is the maximum number of Benders iterations. In order to avoid any infeasible solution at any Benders iteration, we have added constraint (18) to the MP. This constraint assures the installation of at least one hub and avoids the generation of extreme rays. In Section 3.2, the formalization of the Benders decomposition algorithm is presented The classical Benders decomposition algorithm The Benders decomposition algorithm is formally stated below, where UB is the upper bound, LB is the lower bound, zmp and z SP are the optimal solutions obtained by solving the current MP and SP, respectively: Classical Benders Algorithm: BD1. Step 1: Set UB =+, LB = 0. Step 2: If LB=UB, stop. Terminate, we have obtained the optimal solution of the original problem (1) (5). Step 3: Solve the MP (16) (20), obtaining zmp and the optimal values for the integer variables y. Step 4: Set LB = zmp and update y in a new dual problem (10) (14). Step 5: Solve the dual problem (10) (14). Step 6: Add a new Benders cut to the MP (16) (20) using (15). Step 7: If zsp + a y <UB, set UB = zsp + a y. Go to Step 2. A few remars about the above algorithm BD1 are in order. As we have included in our MP model constraint (18), we can guarantee that the MP obtains only feasible solutions regarding the variables y. So the MPs solution is a lower bound to the objective function value of the original problem (1) (5). This lower bound is then improved by the addition of a stronger cut, such as the Benders cut (15), at each iteration. When the lower bound converges to the upper bound, we have found the optimal solution of the original problem. The computational efficiency of the above algorithm depends mainly on three issues: (i) the number of iterations required to attain global convergence; (ii) the time needed to solve each SP at each iteration; and (iii) the time and computer effort demanded at solving each MP on each iteration. One way of tacling issue (i) is to add more than a stronger cut per iteration. This can be done when: (a) the dual problem (10) (14) has multiple optimal solutions and one can generate cuts that are not dominated by any other constraint of the MP, allowing these multiple cuts to be then added to the MP; (b) or when we reformulate the Benders cut originating a new stronger cut for each pair i j. While the former per se is a difficult tas involving the solution of a linear program (see [34]), the later is addressed in Section 3.3.

6 1052 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) The issue (ii) is addressed in Section 3.5, where we demonstrate how one can solve the SP efficiently in polynomial time. The solution of the SP has to be done with extreme care, since the Benders algorithm is very sensitive to the choice of the dual variable values. If care is not taen, poor behavior of the Benders algorithm may be observed. Finally, the computer effort and time demanded, issue (iii), at solving each MP is a critical factor in the overall performance of the algorithm. Although the MP is a mixed-integer programming, we can reformulate it transforming into a pure integer problem. Even though this conversion is possible, it still taes a considerable time to obtain optimal solutions at each iteration. This transformation is addressed in Section 3.4, where we use the modified Benders decomposition algorithm proposed by Geoffrion and Graves [33] The multi-cut Benders decomposition algorithm Analyzing the SP, one can see that the SP can be decomposed in smaller problems, one for each pair i j. Exploiting this fact, we can reformulate the Benders cut (15), as proposed in [32], originating the following set of Benders cuts: η ij + w ij y u h ij w ij α h ij i, j N, h H. (21) So rather than applying the Benders decomposition directly, we can tae advantage of the special structure of the SP and add more than one constraint to the MP at a time, obtaining the following MP: min η ij + a y (22) i j s.t. η ij + w ij y u h ij w ij α h ij i, j N, (23) y 1, (24) η ij 0 i, j N, (25) y {0, 1} K, (26) where u h ij and αh ij are still the optimal values of the dual variables of iteration h. The continuous variables η ij are a transportation cost under-estimators for pair i j. The multi-cut Benders decomposition algorithm is now stated as: Multi-cut Benders Algorithm: BD2. Step 1: Set UB =+, LB = 0. Step 2: If LB = UB, stop. Terminate, we have obtained the optimal solution of the original problem (1) (5). Step 3: Solve the MP (22) (26), obtaining zmp and the optimal values for the integer variables y. Step 4: Set LB = zmp and update y in a new dual problem (10) (14). Step 5: Solve the dual problem (10) (14). Step 6: Add the set of constraints (21) to the MP (22) (26). Step 7: If zsp + a y <UB, set UB = zsp + a y. Go to Step 2. The multi-cut Benders decomposition algorithm (BD2) has a huge number of constraints when compared to the classical one (BD1). However, its number of major iterations is expected to be very small. A detailed description about reformulating Benders decomposition when facing problems with special structure can be found in Birge and Louveaux [32] and Nemhauser and Wolsey [41] Modified Benders decomposition algorithm In 1974, Geoffrion and Graves [33] presented a variant of the Benders decomposition algorithm for a multicommodity distribution system design problem. Geoffrion and Graves have shown that the MP does not need to be solve to optimality. It can be stopped whenever a first feasible solution better than the best incumbent solution found so far is reached. The MP is now a feasibility seeing problem.

7 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) As pointed out by Geoffrion and Graves [33], it taes several Benders cuts in order of the MP to obtain accurate information about the dual problems and the transportation costs. Hence the first MPs iterations are not worth optimizing until optimality is proved. This strategy of solving the MP implies that the MP no longer provides a lower bound for the optimal value of the original problem (1) (5), forcing the alteration of the termination criteria of the algorithm. Introducing an allowed error margin ε > 0, the algorithm terminate whenever the MP does not find any feasible solution better than the value UB ε. So, the algorithm quits and the best incumbent solution is an ε-optimal solution of the original problem (1) (5). The MP minimizes the sum η+ a y, where η is the greatest under-estimator of the transportation costs. Replacing η at the objective function of (16) we get a y + w ij α h ij w ij y u h ij. (27) i j i j We can now use this modified objective function as a ind of surrogate constraint, allowing the MP to obtain a solution that is better than the current upper bound: a y + i UB + i w ij α h ij j i w ij w ij u h ij a y j i j y u h ij (UB ε), (28) w ij α h ij + ε. (29) j Constraint (29) is the new Benders cut. We can now introduce a new more appealing objective function, for example, function (30). The traditional MP is transformed into (30) (33): min z = a i w ij u h ij y + j i w ij α h ij (30) j s.t. w ij u h ij a y i j i w ij α h ij (UB ε), (31) j y 1, (32) y {0, 1} K, (33) Geoffrion and Graves [33] have demonstrated that when ε > 0 the Benders cut, here represented by constraint (31), does not regenerate the same integer solution. So, the MP is now a feasibility-seeing problem and the Benders algorithm is given by: Modified Benders Algorithm: BD3. Step 1: Step 2: Set UB =+. Solve the MP (30) (33). If solution is infeasible, terminate! The best incumbent solution is a ε-optimal solution. Else get the optimal values for the integer variables y. Step 3: Update y in a new dual problem (10) (14). Step 4: Solve the dual problem (10) (14). Step 5: Add a new Benders cut to the MP (30) (33) using (29). Step 6: Update the UB if necessary, eeping the best incumbent solution. Go to Step 2.

8 1054 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) hub client j client i hub m Fig. 1. Possible paths to route commodity ij Subproblems Instead of solving the SP (Eqs. (10) (14)) using simplex or interior point methods, we can efficiently solve it by means of a specialized algorithm which relies on the complementary slacness condition and on the following proposition: Proposition 1. The primal problem (6) (9) is always feasible and bounded for any fixed feasible y satisfying constraints (2) (5). Proof. The original problem (1) (5) is modeled to find the minimum cost related to the transportation among i j pairs and to the hub installation. One can easily see that model (1) (5) is always feasible for any given hub structure installed y satisfying constraints (2) (5). One can easily use an all pairs shortest path algorithm to compute the cost of transportation and implicitly assign the nodes to the hubs. Since the transportation costs are finite and non-negative, any feasible solution to the primal problem must be bounded. Therefore, the primal problem is feasible and bounded for any given y satisfying the set of constraints (2) (5). Corollary. The dual problem (10) (14) is feasible and bounded, having at least one optimal solution. Proposition 2. The optimal solution of the dual problem (10) (14) can be found by inspection. Proof. Recalling from duality theory and the complementary slacness property, we have for any feasible y h iteration h: min c ij m x h ij m (2) (5) holds i j m [ = max w ij α h ij ] u h ij yh (11) (14) holds i j. given at For a given pair i j, considering the installed hubs, we have a simple routing problem to solve, that can be seen in Fig. 1: The optimal route is: w ij min {c ij m y h =,m K yh m = 1}. (34) It is possible to write the optimal values of α h ij as α h ij = min {c ij m y h =,m K yh m = 1}. (35) Resulting zero duality gap. We must now determine the values of the other dual variables u h ij. The dual variables computation with economical interpretation (meaningful values) is a clue for an efficient implementation. Here, α h ij is the highest possible price difference between i and j considering the installed hub structure. Therefore, u h ij is an

9 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) additional tax paid to route the flow i j through alternative uninstalled hub. To produce the tightest values of these dual variables, we recall that α h ij has already been computed by equation (35). The remaining problem, for a given i j pair, is min w ij y h uh ij (36) s.t. =m u h ij + uh ij m αh ij c ij m = m,, m K, (37) u h ij αh ij c ij K, (38) u h ij 0 K. (39) This program objective function (36) is associated to the y variable coefficients on the MP. According to Magnanti and Wong [42] the solution of program (36) (39) produces the tightest possible Benders cut. A way to solve efficiently this program is to consider constraints (38) and (39). They enable us to determine the starting values of variables u h ij as u h ij = max{0, αh ij c ij }, K, y h = 0. For the sae of the complementary slacness condition, u h ij = 0 whenever yh = 1. Considering now the paths that contain two different hubs, we must observe constraints (37). There are two distinct cases: one can go through two hubs, one of them already installed, or go through two uninstalled hubs. In the first case, we remar that the introduction of a second hub = m on the route of pair i j implies the inspection of two different routes: i m j, i m j (see Fig. 1). This apparently innocuous but fundamental detail leads us to the following calculation: } u h ij {u = max h ij, max {α h m K m =,ym h =1 ij c ij m, α h ij c ij m}, K, y h = 0. The second case, where the two hubs in the path are not installed, yields: } u h ij {u = max h ij, max {α h m K m =, ym h =0 ij c ij m u h ij m, αh ij c ij m u h ij m }, K, y h = 0. The subproblem algorithm is presented below: Subproblem s Algorithm: SP. For each pair i j Step 1: Compute α h ij = min {c ij m y h,m K = yh m = 1}. Step 2: Compute u h ij = max{0, αh ij c ij }, K, y h = 0. Set u h ij = 0 If yh = 1 Step 3: Compute u h ij = max{uh ij, max m K m =,ym h =1{αh ij c ij m, α h ij c ij m}}, K,y h = 0. Step 4: Compute u h ij = max{uh ij, max m K m = ym h =0{αh ij c ij m u h ij m, αh ij c ij m u h ij m }}, K, yh = 0. The computational effort of solving the SPs at each iteration h is considerable. The proposed solution algorithm has a complexity of o(n 4 ). Although the theory expresses the hardness of integer programs, lie the MP, many of the Benders MP iterations are easier to solve than the respective SPs. 4. Computational results In order to test the variants of the Benders decomposition method, we have used the CAB data set of the US Civil Aeronautics Board and the AP data set introduced by Ernst and Krishnamoorthy [5,8] with minor modifications. The

10 1056 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) Computer Effort x Changes on Installation Costs Computational Time [log(s)] Increments of the Installation Costs (%) Fig. 2. Computational effort. former includes problems of size n = 10, 15, 20 and 25 nodes, while from the later one can obtain size problems from 10 to 200 nodes, in multiples of 10. Although both are standard data sets that have been used in almost all previous studies (see [12,43 45]), the CAB data set does not have fixed costs for the nodes, while the AP data set only includes fixed costs for the first 50 nodes. We have generated the fixed costs of all instances using a Gaussian distribution with average equal to f o and the coefficient of variance set to 40% to assess how the different fixed costs vary in real problems. Where f o has been introduced by Ebery et al. [44] and represents the scaled difference in objective value between a scenario in which there is a virtual hub located in the center of mass and a scenario in which all nodes are hubs. We also, following what have been done by Ebery et al. [44], set the nodes with higher flows to higher fixed costs. Doing so maes, in general, harder to select which nodes should be hubs. Further, as most of the demand matrices of the instances of the AP data set are fractional, we have normalized them to be coherent to the CAB data set. We have generated 96 test problems, 80 tests are base on the AP data set and 16 are from the CAB data set. The name of the test problems is denoted by CABn.α or APn.α, where n is the number of nodes on the networ and α is the discount used (2 means 20%, 4 means 40%,...). All computational tests have been carried out on a Sun Blade 100 with a 500 MHz Ultra-SPARC processor and 1 Gbyte of RAM memory, running the operational system Solaris 5.8. Also, all the variants of the Benders decomposition have been implemented in C++using CPLEX 7.0 to solve the MPs with a time limit of 10 h of CPU. Further, the ε parameter of the algorithm based on the Geoffrion and Graves was set as 1%. The computational experience stresses the fundamental importance of installation costs when solving large instances. Fig. 2 depicts the CPLEX computational effort when the installation costs, ranging from five to five hundred percent of the original values of instance CAB15.2, are increased in steps of 5%. Fig. 3 presents the number of installed hubs for each increment. Fig. 4 demonstrates how the CPLEX computational effort is affected, when the coefficient of variation (CV) is increased from 0% to 150%. Again, we have used the instance CAB15.2 and the value of f o as average. The CV represents how the installation costs differ from each other. It can be pointed out how the computational effort decreases when the CV increases. Our point is: when dealing with large scale real world applications, one must conceive good metrics for evaluating the installation costs if they are not available; instead of prescribing them ad hoc. The heterogeneity of the installation costs induces an hierarchy of probable hub locations. If the installation costs differ only slightly, the number of natural hub candidates increases dramatically, maing the instance hard to solve. We believe that it is possible to develop a raning metric for the hub candidates as the one proposed by Mateus and Thizy [46] to networ location problems. A similar analysis can be done considering the CV of the demand matrix. Demand matrices with small heterogeneity may lead to harder instances, once an hierarchy of probable hub locations

11 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) Number of Installed Hubs x Changes on Installation Costs 14 Number of Installed Hubs Increments of the Installation Costs (%) Fig. 3. Number of installed hubs. 500 Computer Effort x Changes on the Coefficient of Variation Computational Time [s] Increments of the coefficient of variation (%) Fig. 4. Computational effort. may not be so distinguishable. For shortness, this study is not shown here, since the standard data sets already present real heterogeneous demand matrices. Tables 1 5 show the computational results for the carried out tests. The CAB data set appears to be harder to solve than the AP data set. This is true for both the monolithic and the decomposed approach. These small differences of the computational effort are due to numerical issues since the CAB instances have very large demand entries. CPLEX 7.0 emerges as a good tool to solve UMAHLP instances of sizes less than n = 40. For larger instances, it was not even possible to load the problem on the main memory, but this limitation is machine dependent. The Benders decomposition algorithm has been able to solve UMAHLP instances from sizes 40 to 200 under the time constraints proposed. It is important to remar that instances of these sizes have not yet been solved to optimality on the literature. It is not a surprise the suitability of the Benders decomposition method to this class of problems, since its relation to the traditional facility location problems as already pointed out by Klincewicz [27] and Hamacher et al. [9].Onthe other hand, it appears that there is not a common sense on the hub and spoe literature about this matter. As further as we now, there are very few wors on the UMAHLP concerning the use of Benders decomposition [15].

12 1058 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) Table 1 Computational results for the CAB data set Instance CPLEX 7.0 BD1 BD2 BD3 Opt. sol. Time CPU (s) # Iter. Time CPU (s) # Iter. Time CPU (s) # Iter. Time CPU (s) CAB , CAB , CAB , CAB , CAB15.2 2,718, CAB15.4 2,645, CAB15.6 2,352, CAB15.8 1,908, CAB20.2 5,567, CAB20.4 5,618, CAB20.6 5,240, CAB20.8 4,567, CAB25.2 8,570, CAB25.4 8,956, CAB25.6 8,516, CAB25.8 7,214, Table 2 Computational results for the AP instances Instance CPLEX 7.0 BD 1 BD 2 BD 3 Opt. sol. Time CPU (s) # Iter. Time CPU (s) # Iter. Time CPU (s) # Iter. Time CPU (s) AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , The paradigm of using a heuristic approach for solving large instances of this problem is now broen. The ey finding of the present wor is to derive a class of very strong Benders Cuts, enabling an exact solution procedure to overcome this particular problem (Figs. 5 7). A performance comparison of the three algorithms can be seen on Figs for the AP instances. When comparing the three implemented variants of the Benders algorithm, one can see that the multi-cut version (BD2) exhibits the greatest computational times. Although it presents a inferior number of Benders cycles, the computational overhead becomes prohibitive, as n 2 cuts are added to the master program at each iteration. These n 2 cuts bring enhanced dual information to the MP strengthening the lower bound, but they also turn each MP almost as hard as the original formulation, as shown on Fig. 7. A good scheme may be the development of a combined version of algorithms BD1 and BD2. This issue is left for future research. The Benders cuts are strong enough to mae the use of Geoffrion and Graves variant (BD3) uninteresting. This can be explained by the composition of two effects: first, the lower bound convergence rate justifies the complete solution

13 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) Table 3 Computational results for the AP data set Instance Opt. solution BD 1 BD 2 BD 3 # Iter. Time CPU (s) # Iter. Time CPU (s) # Iter. Time CPU (s) AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , AP , Table 4 Computational results for the AP data set Instance Opt. solution BD 1 BD 2 BD 3 # Iter. Time CPU (s) # Iter. Time CPU (s) # Iter. Time CPU (s) AP , AP , AP , AP , AP , AP , AP , AP , AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a , AP , a a AP , a a AP , a a AP , a a 96 13, AP , a a AP , a a AP , a a AP , a a a 10 h time limit exceeded.

14 1060 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) Table 5 Computational results for the AP data set Instance Opt. solution BD 1 BD 2 BD 3 # Iter. Time CPU (s) # Iter. Time CPU (s) # Iter. Time CPU (s) AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a AP , a a 29 11, AP , a a AP , a a AP , a a 39 20, AP , a a AP , a a AP ,084, a a AP , a a 17 10, AP , a a AP , a a 36 22, a 10 h time limit exceeded % sub master 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Fig. 5. Average master versus subproblem time (%) of algorithm BD1 for instances AP20.4, AP30.4, AP40.4, AP50.4 and AP60.4. of the master program as can be seen on Fig. 6; second, the master program of the classical version (BD1) is easier to solve than the SPs, as seen on Fig. 5. Though the approach of just finding feasible solutions in order to generate cuts, the mainstream of Geoffrion and Graves variant (BD3), is not quite attractive. Since the SP solution taes much more time than the master step, and considering that BD3 spends some iterations without improving bounds, it does not mae an optimal use of the computational resources. In spite of this fact, Geoffrion and Graves variant (BD3) cannot be left aside. When facing hard master programs and small lower bound convergence rates, BD3 emerges as one of the most aggressive decision tools. It also has the advantage of allowing the operational research analyst to determine controlled near optimal solutions very fast.

15 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) Fig. 6. Comparison of lower bound convergence rates of algorithms BD1 and BD2 for instances AP % sub master 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Fig. 7. Average master versus subproblem time (%) of algorithm BD2 for instances AP20.4, AP30.4, AP40.4, AP50.4 and AP60.4. Fig. 8. Computational effort (α = 20%).

16 1062 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) Fig. 9. Computational effort (α = 40%). Fig. 10. Computational effort (α = 60%). Fig. 11. Computational effort (α = 80%).

17 R.S. de Camargo et al. / Computers & Operations Research 35 (2008) In most of the carried out tests, the classical version (BD1) has presented a solution time one order of magnitude less than the other ones. As a disadvantage, it has shown some flattening for optimality gaps below 5%. The multi-cut version (BD2) has demonstrated to be less susceptible to this drawbac, recalling the promising direction of developing a hybrid algorithm combining BD1 and BD2. 5. Conclusion In this paper, we have observed the important role of the installation costs during the solution of large scale instances of the UMAHLP. This role indicates that these values should not be defined ad hoc, but, instead, be determined as close to reality as possible. We have also implemented three variants of the Benders decomposition. All the three algorithms have shown to be great engineering tools, being able to solve large scale instances under 10 h of machine. Some of the instances solved here were considered out of reach of exact methods. This fact maes pointless the use of heuristic approaches to get just a good upper-bound for such important class of problems. Another accomplishment of this wor is the derivation of a strong class of Benders cuts for UMAHLP. Our results also suggest that the development of an hybrid version of classical (BD1) and multi-cut (BD2) Benders decomposition is a promising research direction. Further research should also attempt to incorporate congestion costs or flow dependent economies of scale, hence searching for more realistic design capabilities. References [1] Haimi SL. Optimum distribution of switching centres in a communication networ and some related graph theoretic problems. Operations Research 1965;13: [2] Klincewicz JG. Hub location in bacbone/tributary networ design: a review. Location Science 1998;6: [3] Toh RS, Higgins RR. The impact of hub-and-spoe networ centralization and route monopoly on domestic airline profitability. Transportation Journal 1985;24: [4] Ayin T. Networ policies for hub-and-spoe systems with applications to the air transportation system. Transportation Science 1995;29: [5] Ernst AT, Krishnamoorthy M. Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science 1996;4: [6] Ernst AT, Krishnamoorthy M. Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem. European Journal of Operational Research 1998;104: [7] Ernst AT, Krishnamoorthy M. An exact solution approach based on shortest-paths for p-hub median problems. Journal on Computing 1998;10: [8] Ernst AT, Krishnamoorthy M. Solution algorithms for the capacitated single allocation hub location problem. Annals of Operations Research 1999;86: [9] Hamacher HW, Labbé M, Nicel S, Sonneborn T. Polyhedral properties of the uncapacitated multiple allocation hub location problem. Technical report 20, Institut für Techno und Wirtschaftsmathemati (ITWM), [10] Campbell JF. Integer programming formulations of discrete hub location problems. European Journal of Operational Research 1994;72: [11] O Kelly ME, Bryan DL, Sorin-Kapov D, Sorin-Kapov J. Hub networ design with single and multiple allocation: a computational study. Location Science 1996;4(3): [12] O Kelly ME, Bryan DL. Hub location with flow economies of scale. Transportation Research Part B 1998;32(8): [13] Horner MW, O Kelly ME. Embedding economies of scale concepts for hub networ design. Journal of Transport Geography 2001;9: [14] Klincewicz JG. Enumeration and search procedures for a hub location problem with economies of scale. Annals of Operations Research 2002;110: [15] Racunica I, Wynter L. Optimal location of intermodal freight hubs. Transportation Research Part B 2005;39(5): [16] Campbell JF. A survey of networ hub location. Studies in Locational Analysis 1994;6: [17] Campbell JF, Ernst AT, Krishnamoorthy M. Hub location problems, 1st ed. Berlin: Springer; p [chapter 12]. [18] O Kelly ME, Miller HJ. The hub networ design problem. Journal of Transport Geography 1994;2: [19] Campbell JF, Ernst AT, Krishnamoorthy M. Hub location problems. In: Drezner Z, Hamacher HW, editors. Facility location. Berlin: Springer; p [20] O Kelly ME. The location of interacting hub facilities. Transportation Science 1986;20: [21] O Kelly ME. A quadratic integer program for the location of interacting hub facilities. European Journal of Operational Research 1987;32: [22] Klincewicz JG. Heuristics for the p-hub location problem. European Journal of Operational Research 1991;53: [23] Sorin-Kapov D, Sorin-Kapov J. On tabu search for the location of interacting hub facilities. European Journal of Operational Research 1994;73:501 8.

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