On step fixed-charge hub location problem

On step fixed-charge hub location problem Marcos Roberto Silva DEOP - Departamento de Engenharia Operacional Patrus Transportes Urgentes Ltda. 07934-000, Guarulhos, SP E-mail: marcos.roberto.silva@uol.com.br Keywords: mathematical modeling, hub location, fixed-charge, step function, tabu search Abstract: In this paper we introduce the on step fixed-charge hub location problem. We believe that this problem was never studied before, where the hub location problem has a step objective function, and fixed cost is imposed for every arc of the hub-and-spoke network. A mathematical model, and a solution approach based on tabu search heuristic is proposed, showing promising results, considering the potential of practical application in different areas, such as transportation and telecommunication network design. 1 Introduction In this paper we study the problem of configuring a hub-and-spoke network where fixed cost is incurred for every arc that is used in the network. The step fixed-charge hub location problem is a variation of the hub location problem, where the fixed cost is in the form of a step function dependent on the load in a given route. Hubs are special facilities that serve as switching, transshipment and sorting points in manyto-many distribution systems. Instead of serving each origin-destination pair directly, hub facilities concentrate flows in order to take advantage of economies of scale. Flows from the same origin with different destinations are consolidated on their route to the hub and are combined with flows that have different origins but the same destination. The consolidation is on the route from the origin to the hub and from the hub to the destination as well as between hubs [1]. A recent survey of hub location problems is presented in [6]. The Uncapacitated Single Allocation p-hub Median Problem (USApHMP), first introduced by [13], was chosen to be used as a case study in this paper, with the inclusion of fixed costs in the arcs of the hub-and-spoke network. The USApHMP was largely studied in the literature with more than 20 papers published until 2008 [1], with exact and heuristic solution methods proposed, including branch-and-bound [4], tabu search [16], simulated annealing [3], Lagrangean relaxation heuristic [14], local search [10], to name a few. We believe that this is the first work that studies the fixed-charge hub location problem where the objetive function is a step function. This problem has various application areas in transportation (air passenger, cargo) and telecommunication network design. In less-than-truckload (LTL) trucking transportation, when a route between terminals is opened, it is necessary to define the kind of vehicle to use, depending on the flow, incurring on different fixed costs for different models of vehicles. In the telecommunications industry, it represents the design of private networks that use digital transmission facilities (called T1 circuits) to carry voice and data traffic between locations. Given an organization s forecast for data and voice traffic between its various locations, the 660

problem consists of defining the configuration of transmission facilities between the locations (nodes) providing the necessary link capacities to carry this traffic at minimum cost [12]. In Section 2 we present the mathematical formulation of the problem. In Section 3 the computational results are presented, and in Section 4 the conclusions and some further directions are outlined. 2 Mathematical formulation The on step fixed-charge hub location problem was formulated based on the approach used by [5] for the capacitated single allocation hub location problem (CSAHLP) and in [11] for the on step fixed-charge transportation problem. Let N be the set of nodes, C ij be the transportation cost per unit flow between nodes i and j, and W ij be the amount of flow from origin i to destination j. The path from an origin spoke node i to a destination spoke node j includes three components: collection from spoke node i to its designated hub k, transfer between hubs k and l, and distribution from hub l to destination spoke j. The cost per unit flow along this route (i k l j) is given by χc ik + αc kl + δc lj, where χ, α and δ denote cost multipliers on the collection, transfer and distribution, respectively. Usually α is much smaller than χ and δ due to volume discount on inter-hub links. A factor α < 1 was originally proposed by [13] to represent an economy of scale on the transportation cost between hubs; the two other factors, χ and δ, were later introduced by [3] to properly represent the reality of postal services costs, especially different modes that can be used in mail collection and distribution. We define the variable Ykl i as the total amount of flow of commodity i (i.e. traffic emanating from node i) that is routed between hubs k and l. The variable Z ik equals one if node i is assigned to hub k and zero otherwise. The amount of flow from origin i to destination j is given by W ij, while O i, D i represents the total amount of flow originated and destined at node i, respectively. Thus, the on step fixed charge hub location problem can be formulated as the following mathematical programming problem: min Z = ( ) C ik Z ik χo i + δd i i N k N + αc kl Ykl i i N k N l N + Z ik U ik + N kl V kl i N k N k N l N (1) Subject to: where l N Z ik = 1, i N (2) k N Z kk = p, (3) k N Z ik Z kk, i N, k N (4) Ykl i Ylk i l N = O i Z ik W ij Z jk, j N i N, k N (5) Z ik {0, 1}, i N, k N (6) Ykl i 0, i N, k N, l N (7) 661

N kl = { 1 : if Ylk i > 0, i N, k N, l N 0 : otherwise Formulation of U ik : Assume that a fixed cost to open a route from i to k is u ik,1 if the flow is less than or equal to A ik, and u ik,2 when the flow is higher than A ik. Thus, U ik which is the total fixed charge associated with the route from spoke node i, i N, to the hub k, k N, is: where U ik = b ik,1 u ik,1 + b ik,2 u ik,2 (8) b ik,1 = 1 if Z ik = 1, = 0 otherwise; b ik,2 = 1 if max {O i, D i }Z ik > A ik, = 0 otherwise. Similarly, V kl is the total fixed charge associated with the route between hubs k and l, k N, l N, having two levels of fixed costs: v kl,1 and v kl,2. where V kl = e kl,1 v kl,1 + e kl,2 v kl,2 (9) e kl,1 = 1 if Ykl i i N = 0 otherwise; e kl,2 = 1 if Ykl i kl, i N = 0 otherwise; In the above formulation, constraints (2) impose that each node i is assigned to exactly one hub, while constraints (3) ensure that exactly p hubs needs to be opened. Constrains (4) ensure that no node is assigned to a location unless a hub is opened at that site. Constraints (5) represent the divergence equations for each commodity i at node k in a complete graph, where the demand and supply at the nodes is determined by the allocation variables Z ik. Note that U ik and V kl are step functions, which in this special case has two steps. It could have multiple steps, depending on the problem structure. 3 Preliminary computational experiments Given the nature and complexity of the problem, we opted to develop a tabu search heuristic based on [15] and [16] to solve the on step fixed-charge hub location problem. In both the works, the tabu search heuristic was shown to be effective in solving the hub location problems, being capable of finding the optimal solution in all problems tested. The problem solved in [16] was the USApHMP, and in [15] the uncapacitated single allocation hub location problem (USAHLP). Tabu search (TS) is a local search procedure that uses memory structures to guide the movements from one feasible solution to another, aiming to explore regions of the search space that would be otherwise left unexplored in order to escape local optima. After a move is made, it is classified as tabu (i.e., forbidden) for a certain number of iterations in order to prevent cycling. The fundamental principles underlying TS are fully explained in [7], [8] and in [9]. 662

The data used for the computational experiments was the CAB (Civil Aeronautics Board) data set, first introduced by [13], and this set has been extensively used in the literature as a benchmark for evaluating algorithms for different hub location problems. This data set is based on the airline passenger interactions between 25 US cities in 1970 evaluated by the Civil Aeronautics Board (CAB). The CAB data set is available in the OR Library [2]. We solved problems of four different sizes: the first 10 nodes, the first 15, the first 20 and the full CAB data set with 25 nodes. The number of hubs to be located was tested with p = {2, 3, 4}, and with the discount factor α = {0.2, 0.4, 0.6, 0.8, 1.0}, resulting in 60 problems solved. Since there is no fixed-charge costs provided in the CAB data set, we arbitrarily defined values for U ik and V kl : u ik,1 = 1.0, i N, k N; u ik,2 = 2.0, i N, k N; v kl,1 = 1.0, k N, l N; v kl,2 = 2.0, k N, l N. The values of A ik and F kl were also defined arbitrarily, based on the flow values of the problem at hand. These values are presented in Table 1. The values of A ik were defined as the same for all i N and for all k N, and named only as A for short. The same for F kl, with only one value for all k N and for all l N, also named as F for short. The F value was also defined in function of the number of hubs to be located (p). F n A p = 2 p = 3 p = 4 10 100000 100000 100000 600000 15 150000 400000 200000 130000 20 280000 1000000 500000 300000 25 340000 1000000 700000 450000 Table 1: Values used for A ik and F kl for the experiments with the CAB data set All the experiments were performed on a laptop equipped with 2.4 GHz Intel Core i5 processor (only one processor was used for the experiments), with 4GB of RAM, running under Mac OS X 10.7.5, and the tabu search heuristic was coded using Fortran programming language. We implemented the tabu search method only with short-term memory. The tabu tenure and the maximum number of iterations was set to five and ten respectively, for both the locational and allocational part of the problem, regardless of their sizes. These values are the same used in [15]. In Table 2 we present the results obtained with tabu search heuristic on solving the set of problems with n = 25 nodes. As can be seen in this table, the CPU time increases as the number of hubs increases (column CPUt(s) ), and the objective function value (column Obj.func. ) decreases because of the lower variable costs. 4 Conclusions and further research In this paper we introduced the on step fixed-charge hub location problem and we proposed a mathematical model for the special case when the objective function has two steps, and the hub location problem chosen as a case study was the uncapacitated single allocation p-hub median problem (USApHMP). We believe that this is the first work that has studied the fixed-charge hub location problem where the objetive function is a step function, imposing fixed costs at every arc in the hub-and-spoke network. We developed a tabu search heuristic to solve a set 663

n p α Obj.func. CPUt(s) n p α Obj.func. CPUt(s) 10 2 0.20 647.9904 0.0152 20 2 0.20 1025.0869 0.1891 10 3 0.20 535.9343 0.0412 20 3 0.20 782.5380 0.5436 10 4 0.20 449.1304 0.0792 20 4 0.20 666.0948 1.1471 10 2 0.40 706.3079 0.0139 20 2 0.40 1088.5656 0.2084 10 3 0.40 611.9128 0.0390 20 3 0.40 905.7668 0.5949 10 4 0.40 547.7937 0.0770 20 4 0.40 812.7488 1.2075 10 2 0.60 764.6253 0.0139 20 2 0.60 1152.0442 0.1848 10 3 0.60 687.8912 0.0374 20 3 0.60 1028.9958 0.6069 10 4 0.60 631.8312 0.0781 20 4 0.60 958.5887 1.1769 10 2 0.80 822.9427 0.0140 20 2 0.80 1215.5229 0.1866 10 3 0.80 760.9827 0.0417 20 3 0.80 1149.0502 0.6450 10 4 0.80 715.4154 0.0782 20 4 0.80 1092.4916 1.2054 10 2 1.00 867.8129 0.0139 20 2 1.00 1256.0763 0.2399 10 3 1.00 820.6840 0.0392 20 3 1.00 1222.0720 0.6118 10 4 1.00 790.2593 0.0761 20 4 1.00 1195.0154 1.2276 15 2 0.20 1014.2807 0.0675 25 2 0.20 1055.9067 0.5174 15 3 0.20 852.9712 0.1967 25 3 0.20 834.3494 1.3333 15 4 0.20 726.7754 0.3684 25 4 0.20 722.6340 2.4769 15 2 0.40 1095.6266 0.0664 25 2 0.40 1155.3262 0.4901 15 3 0.40 961.3095 0.1861 25 3 0.40 968.6989 1.3834 15 4 0.40 866.7118 0.3621 25 4 0.40 880.5150 2.4994 15 2 0.60 1176.9723 0.0659 25 2 0.60 1248.2053 0.4955 15 3 0.60 1069.6478 0.1855 25 3 0.60 1100.5647 1.3526 15 4 0.60 997.2070 0.3872 25 4 0.60 1032.2056 2.4991 15 2 0.80 1231.7690 0.0662 25 2 0.80 1341.0846 0.5495 15 3 0.80 1152.5073 0.1928 25 3 0.80 1225.8312 1.3926 15 4 0.80 1105.5217 0.4598 25 4 0.80 1180.6616 2.4989 15 2 1.00 1262.9214 0.0712 25 2 1.00 1414.1899 0.5111 15 3 1.00 1221.6777 0.2024 25 3 1.00 1323.6302 1.3673 15 4 1.00 1197.2257 0.4030 25 4 1.00 1304.2318 2.5015 Table 2: Results for the CAB data set of problems created using the CAB (Civil Aeronautics Board) data set. The results obtained showed that this approach can be useful in solving step fixed-charge hub location problems. Real world hub location problems are much larger than those presented in this paper. So, further research must concentrate on solving larger problems, and also on modeling and developing new solution methods to solve different variations of hub location problems. References [1] Alumur, S., and Kara, B. Y. Network hub location problems: The state of the art. European Journal of Operational Research 190, 1 (2008), 1 21. [2] Beasley, J. E. OR-Library: Distributing test problems by electronic mail. Journal of the Operational Research Society 41 (1990), 1069 1072. [3] Ernst, A. T., and Krishnamoorthy, M. Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science 4, 3 (1996), 139 154. Hub Location. [4] Ernst, A. T., and Krishnamoorthy, M. Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem. European Journal of Operational Research 104, 1 (1998), 100 112. 664

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