The tree-of-hubs location problem: A comparison of formulations

Transcription

1 1 The tree-of-hubs location problem: A comparison of formulations Iván Contreras, Elena Fernández Departament d Estadística i Investigació Operativa. Universitat Politècnica de Catalunya, Barcelona, Spain. (ivan.contreras,e.fernandez)@upc.edu Alfredo Marín Departamento de Estadística e Investigación Operativa, Universidad de Murcia, Murcia, Spain. amarin@um.es Key words: Hub location, Integer Programming, Spanning Tree. Abstract The Tree-of-Hubs Location Problem is studied. This problem, which combines several aspects of some location, network design and routing problems, is inspired by those transportation systems in which (i) several locations send and receive a product, and (ii) the performance can be improved by using transshipment points (hubs) where the product is collected and distributed. We want to locate a fixed number of hubs and, instead of assuming that the transshipment points are fully interconnected, we require them to be connected by means of a tree. Potential applications appear when the set-up costs for the links between hubs are very high, so full interconnection is prohibitive. Three Integer Programming formulations are proposed and computationally compared. All of them are carefully implemented, with the natural constraints tightened as much as possible. The problem turns out to be difficult since in this initial stage of the research, only instances of size 20 can be approached and, sometimes, optimally solved.

2 2 1 Introduction Hub location problems have their origin in transportation systems where several sites send and receive some product and where it is convenient to use transshipment points (hubs), where the product is collected and distributed to reduce transportation costs. A common feature of hub location problems is that a discount factor is applied to the usage of the links that connect any pair of hubs. Two basic models appeared in the literature more than a decade ago: the multiple allocation and the single allocation uncapacitated hub location problems, with and without a constraint on the number of hubs. Since then, many papers have studied the development of Integer Programming formulations which can be used by either commercial or specialized solvers to optimize the systems. The first formulation for the multiple allocation case was given by Campbell [5]. Later, Klincewicz [24], Skorin-Kapov et al. [39], O Kelly et al. [35], Ernst and Krishnamoorthy [14, 15] and Mayer and Wagner [33] studied several improvements with the main objective of reducing the number of variables, while Hamacher et al. [19] and Cánovas et al. [10] began the polyhedral study of the problem. Recent advances in the formulations for this problem have been presented in Marín [31] and Cánovas et al. [9]. The single allocation case has been studied by O Kelly [36], O Kelly et al. [35], Klincewicz [22, 23], Skorin-Kapov et al. [39], Aykin [3], and Ernst and Krishnamoorthy [14]. More elaborated hub location problems have also been studied. The capacitated multiple allocation case was studied by Aykin [2], Ebery et al. [13], Campbell [5], Boland et al. [4], and by Marín [30]. The capacitated single allocation problem has also been studied by Ernst and Krishnamoorthy [16], Labbé and Yaman [27], Labbé et al. [28], and by Contreras et al. [12]. We also refer the reader to two recent surveys on the matter, Alumur and Kara [1] and Campbell et al. [8]. Traditionally, hub location models assume that the hubs are fully interconnected, that is to say, that there exists a link connecting any pair of hubs, which can be used by applying the corresponding discount factor. However, some hub location models have been proposed recently where two hubs are not necessarily connected by some link (see, for instance [34]). For example, the so called hub-arc location models [6, 7] impose no condition on the structure of the arcs that connect hubs, and they do not even require these arcs to define one single connected component. We propose a single allocation model for locating a fixed number of hubs on a network. The particularity of our model is that we require the hubs to be connected by means of a (non-

3 3 directed) tree. The model that we propose can therefore be seen as a compromise between hub location models that require full interconnection between hubs and hub-arc location models where the connectivity of the links between hubs is not required. To the best of our knowledge, the location of a tree on a network has never been considered previously in the context of hub location. The problem we address involves location, design, and routing decisions. The location decisions refer to the location of the hubs on the nodes of the network. The design decisions refer to the way the selected hubs are interconnected so as to define a tree, and to the (single) allocation of the nodes to the selected hubs. The routing decisions refer to the way in which the flows between vertices of the network are routed through the tree of hubs. Note that in the problem in question the location and the design decisions determine the routing decisions, since for each pair of nodes of the network there exists only one path connecting them in the tree of hubs. Hence, in addition to hub-location problems, our problem is also related to other types of problems studied in the literature where location, design or routing decisions are considered. To mention just a few: the location of tree-shaped facilities (also called extensive facilities) on a network, which has been studied within location theory [18, 21, 38]; network design problems that study the arcs to be used in the optimal routing of flows within pairs of nodes on a network [20, 25, 40]; rapid transit network design problems that locate lines in rail transit systems [32]; and some location-routing problems, where a route is located and customers are assigned to the closest node in the route, like the ring-star problem [26, 29]. As in the case of tree location, potential applications of our model arise when the cost of the links between hubs is very high, and as a consequence full interconnection is prohibitive. Since all the flows in the network must be sent through the hubs, a path must exist between each pair of hubs, i.e., a connected graph must be built. But due to the high cost of the connections, connectivity must be achieved using the minimum number of links. One example of such an application is the design of the high-speed train network in Spain, which is currently under construction and which is planned to be completed by This high-speed train network has been designed with a tree structure and it is intended that when finished each city with more than 10,000 inhabitants will be within 50 km of some high-speed train station. Another potential application is the design of rapid transit systems for urban areas where there are flows of users (citizens) who must travel between origin/destination pairs. More often than not, such

4 4 systems are designed with a tree structure, where the users are allocated to the closest station (hub). In this paper, we present the tree-of-hubs location problem. The problem is defined on a directed graph where it is assumed that for each pair of nodes there exists a known amount of flow that must be sent through the network. To this end p hubs must be located on the network and connected by means of a tree. Each node must be allocated to one single hub and all the flow from/to the node must leave/enter it through its allocated hub. Excepting the arcs that connect each node with its allocated hub, the only arcs that can be used for routing the flows must be links connecting hubs. There is a unit transportation cost associated with each arc. As usual, if hubs are located at both end-nodes of the arc, a discount factor is applied to the unit cost. We assume that the set-up cost for the hubs is the same at each potential location. Given that there is a fixed number p of hubs to be located, this cost is not taken into account, since it is constant and, thus, it does not affect the optimization of the system. We also assume that there is a set-up cost for establishing a link between hubs, which is the same for each arc of the network. Since the hubs are connected by means of a tree, there is also a fixed number p 1 of such links, so the overall link set-up cost is constant and, thus, it is not taken into account. Therefore, we aim to minimize the operation costs of the system, which depend on the amount of flow that circulates through each arc, and on the links to which the discount factor is applied. We propose and analyze three alternative formulations for the problem. These formulations are compared in terms of the lower bound that they produce and of the computational effort required by a standard optimization solver to solve them exactly. As mentioned, our formulations are provided for the p-hub location case. A similar family of problems in the literature is the uncapacitated hub location family, in which the number of hubs is not predetermined but each node has an associated set-up cost, which must be paid if it is chosen as a hub. Different set-up costs associated with the hubs as well as different set-up costs associated with the links of the tree connecting the hubs could be easily incorporated to our formulations. The paper is organized as follows. First, we formally define the problem. In Section 3 we introduce the common framework used to develop the three different formulations, which are given in Sections 4 to 6. The computational results obtained with the implementation of the formulations together with a comparison between them are given in Section 7. The paper finishes with a section of conclusions and some possible future lines of research.

5 5 (A) (B) (C) (D) Figure 1: Details of the model. Filled nodes are hubs 2 Description of the problem Consider a complete digraph without loops G = (N, A) whose set of nodes, N = {1,..., n}, represents the set of origins and destinations of a certain product that is routed through G via some transshipment nodes that are called hubs. Distributing the product through all pairs of origin/destination nodes induces a cost that depends on the nodes selected as hubs and on the amount of product that traverses each arc of the graph. In particular, when some of the end-nodes of the arc (r, s) A is not a hub, each unit of product that traverses (r, s) incurs a cost c rs 0, whereas when both r and s are hubs a discount factor 0 α 1 is applied, and the unit cost associated with arc (r, s) is αc rs. Any of the nodes of N can be chosen to become a hub, and there is a fixed number p of nodes of N, 1 p n 1, that must be chosen to be hubs. Like in all hub location problems, we require the flow from origin i to destination j, denoted W ij 0, to go through one or more intermediate hubs. No node which is not a hub can be used to transship the product. We also require single allocation and, thus, every node i which is not a hub must be assigned (allocated) to one single hub, so that all the flow from/to i to/from any other node must pass through this hub. The particularity of our problem is that we require the hubs to be connected by means of a (non-directed) tree, see Figure 1(A). Thus, the path between each pair i and j will be unique. Note that in this path all the transshipment nodes will be hubs so all the arcs, excepting (possibly) the first and last ones, link hubs. The cost per unit of flow of the path between a pair i and j is the sum of the costs of the arcs in the path, where the discount factor is applied to all the inter-hubs links. Therefore, if i and j are both hubs, W ij could be sent from i to j (directly, if there is such an arc in the tree, or through additional intermediate hubs, otherwise) and the discount factor will be applied to all the arcs in the route. If either i or j are hubs, the product could be sent directly (if such

6 6 an arc is part of the tree) without discount, or through some other hubs, otherwise; if i (resp. j) was the hub, the discount will be applied to all the arcs except the last (resp. first) one. If neither i nor j are hubs, W ij will again be sent from i to j through at least one intermediate hub. Now, W ij must go from i to some hub, then possibly to some other hub(s), and finally to j, and the discount factor will be applied to all the intermediate arcs, but will not be applied either to the first or to the last arcs. Note that there can be a flow of product W ii with origin and destination at the same node i. We assume free self service, i.e., c ii = 0 i N. Thus, if i is a hub, W ii can stay in i at null cost, whereas if i is not a hub, W ii must go to some hub before returning to i. Figures 1(B) and 1(C) illustrate the situation when neither i nor j are hubs, and, respectively, when only one of them is a hub. As already mentioned, when i is not a hub, the flow with origin and destination in i, will go to/come from the (only) hub to which i is assigned (Figure 1(D)). Some observations follow: Triangle inequality of costs c ij is not assumed. Some formulations in the literature (see [9]) are only valid if this property holds, but our formulations are also valid without this assumption. We do not assume symmetry either, i.e., costs c ij and c ji can be different. We define the graph of flows, G F, as the undirected graph with vertex set N and an edge associated with each pair (i, j) N N that is connected by some flow, i.e. such that W ij + W ji > 0. Without loss of generality we assume that the graph of flows has one single connected component. If G F had more than one component it would be difficult to justify the design decision of connecting all the nodes of the network via one single hub tree. Instead, it would seem more natural to define as many hub trees as connected components in G F, thus decomposing the original problem into as many subproblems as connected components. Each of these trees would be used to route the flows between the nodes of the component. However, if for some strategic decision we were required to define just one tree of hubs then, we would consider an arbitrarily small value ɛ > 0, and then, for each pair of unconnected components in G F, we would select any pair of vertices, one in each of the components, and define an artificial flow of value ɛ between them in the original graph. At the end of this process our assumption will hold. We will assume throughout that p 2. Our formulations can be easily adapted to the case p = 1, but this means taking additional details into account. Since we are trying

7 7 to help the reader to gain insight into the formulations, we avoid this case for ease of presentation. For the sake of simplicity, hereafter the total flow with origin and destination in node i is defined as O i = j W ij i and respectively. We also define D j = i W ij j, m ikm = min{w ik, W im } i, k, m. We want: (i) to locate p hubs (at null or equal costs); (ii) to link them (at null or equal costs) so as to define a tree the small tree; and, (iii) to allocate every non-hub node to one single hub obtaining the so-called large tree in such a way that the overall cost be minimized. The cost is the sum of the routing costs associated with the flows between all the elements in N N. Given (i, j) N N, the routing cost of the flow W ij is given by that of the unique path from i to j in the large tree which traverses at least one hub times the amount of flow W ij. Our aim is to obtain good Integer Programming formulations for the Tree-of-Hubs Location Problem, hereafter referred to as THLP. Proposition 2.1 The THLP is NP-hard. Proof: When the set of hubs to be located is given, the subproblem of connecting them so as to define a tree, and of routing the flows between the nodes at minimum total cost is the so called Minimum Spanning Tree Communication Cost [20], which is known to be NP hard [11], even if there is an unit flow between each pair of nodes (W ij = 1, (i, j) N N), and hence the result. Q.E.D. 3 The rationale of the formulations The aim of the problem we are dealing with is the minimization of the total cost of sending the flow between each pair of nodes. The product sent from an origin to a destination flows through a route with an undetermined number of arcs, whose costs depend on whether or not the end-nodes of the arcs are both hubs. Therefore, in order to evaluate the objective function, the formulations must (i) represent (or be able to reproduce) the complete routes note here

8 8 that the routes have an orientation and (ii) know if the discount must be applied to each arc of a route or not. Moreover, on putting together all the arcs used in all the routes, the resulting graph must be a tree. To be more precise, there are two trees involved in the systems considered: that given by the hubs plus the links between hubs, which must be a tree as described in the problem definition, and a larger one which is given by all the nodes and links. We have called the first one small tree and the second one large tree. Any formulation must cope with the fact that all the nodes which are not hubs must be leaves of the large tree, since this is the effect of assigning them to one single hub (see again Figure 1). In order to represent the routes between origins and destinations, we will not consider four times indexed variables, which have been commonly used in hub location (see e.g. [31]), but three times indexed variables, also used in the hub location literature (see for instance Ernst and Krishnamoorthy [16]). The three indices will denote, respectively, the origin of the product and the end-nodes of the arc considered. Three index variables can be defined both as continuous variables, representing the amount of flow with the origin given which traversex the arc given, and as binary variables indicating whether or not the arc is used in some route with the origin given. From our connectivity assumption on the graph of flows, we know a priori that the graph of routes for the flows between nodes will also be connected. Hence, to represent the trees that must be determined, we will simply need to constrain the number of edges of the graph of links between nodes. Integer Programming formulations for several kinds of spanning tree problems have been studied, for example, in Gouveia [17], Pop et al. [37] and references therein. In our work we have followed two different strategies for building the graph of routes. One of them consists of building a tree and imposing all the routes to use it. The other one is to build an arborescence, i.e., a directed tree with a given root in some node R. The idea is quite similar, but the implementation can be quite different. With all these ingredients we have checked several formulations and we present three proposals.

9 9 i R x=1 x=1 x=1 x=1 (A) (B) (C) (D) y=1 z=1 z=1 y=1 y=1 z=1 z=1 R z=1 y=1 z=1 y=1 y=1 Figure 2: Meaning of several variables in the first formulation z=1 4 First formulation In the first formulation considered here, an arbitrary node R is chosen, and three-indexed x-variables are used to represent directed paths from all the nodes to R, see Figure 2(A): 1 if the path from i to R x ikm = goes through arc (k, m), 0 otherwise, i R, k, m. These paths will guarantee the connection of the resulting graph. Every x-path will be embedded in the large tree given by means of the twice-indexed X-variables (Figure 2(B)): X km = { 1 if the edge {k, m} is in the large tree, 0 otherwise, k, m > k. The z-variables, used to identify nodes which are hubs, and the y-variables, representing the links between hubs, will completely determine the small tree (Figure 2(C)): y km = { 1 if the edge {k, m} is in the small tree, 0 otherwise, k, m > k, { 1 if node k is a hub, z k = k. 0 otherwise, With the above variables, the (undirected) graph of the large tree can be defined in terms of the X variables as T L := (N, {{k, m} : X km = 1}). Similarly, the (undirected) graph of the small tree can be defined in terms of the y and z variables as T S := ({k : z k = 1}, {{k, m} : y km = 1}).

10 10 Finally, in order to build the objective function, a-variables will give the links between hubs and nodes which are not hubs, { 1 if the non hub k is allocated to the hub m, a km = k, m k, 0 otherwise, and the amount of flow with origin in node i traversing arc (k, m) will be given by the continuous variable b ikm. In Figure 2(D), the values of the b-variables with a common first index i have been represented, where the demand from each node with respect to origin i is the amount written inside the node. Our first formulation is the following: (F1) min αc km b ikm + i k m k (1 α)(c ij O i + c ji D i )a ij (1) i j i s.t. x iik = 1 i R (2) k j,r k i,j k i x ikj = x ijk i R, j i, R (3) x ikm + x imk X km i R, k, m > k (4) X km = n 1 (5) k m>k a km + a mk + y km = X km k, m > k (6) a mk + y km z k k, m > k (7) a km + y km z m k, m > k (8) z k = p (9) k y km = p 1 (10) k m>k b ikm O i X km i, k, m > k (11) b imk O i X km i, k, m > k (12) b iim + W ii z i = O i i (13) m i b ikj = W ij + b ijk k j k j i, j i (14) z k {0, 1} k x ikm {0, 1} b ikm 0 a km {0, 1} i, k, m k i, k, m k k, m k

11 11 y km, X km {0, 1} k, m > k. In the objective function (1), the total cost is evaluated using variables a and b. All the amounts of product flowing through the large tree are accounted for in variables b. By multiplying b times αc, the cost of the flows between hubs is obtained, but the cost of the product between leaves which are not hubs and hubs is undervalued. This under-evaluation is corrected by using the a-variables multiplied by 1 α, and by taking into account that, if i is not a hub and j is the hub to which i is assigned, then all the flow with origin in i, O i, leaves i towards j at unitary cost c ij, and all the product demanded by i, D i, arrives at i from j at unitary cost c ji. Constraints (2) and (3) guarantee the existence of a directed path from every node i R to R and, together with inequalities (4), also guarantee the connectivity of the graph T L. Constraint (5) forces T L to be a tree. Constraints (6), (7) and (8) ensure that the graph T S is a subgraph of T L and, consequently, without cycles. These constraints also express the incompatibility between the edges of the small tree and those that link a non-hub with a hub. Note that if the edge {m, k} is part of T S then both m and k must be hubs and, thus, a mk = a km = 0, since the a variables can only take the value 1 when either m or k is not a hub. By the same token, when non-hub m is allocated to hub k, then the edge {m, k} cannot be part of T S (where all the edges link two hubs) but it must form part of T L. Since, due to (9) and (10), the number of nodes in T S is p and the number of edges is p 1, T S will also be a tree. Constraints (11) and (12) cover the relationship between the b-variables, representing the flows through the arcs of the original digraph G and the X-variables associated with the edges of the undirected large tree T L. In particular, in any solution each arc (k, m) of G used to route flow between some pair of nodes, is related to the (only) edge of T L connecting the end-nodes k and m. The two indices of the X-variable associated with this edge will be either km or mk depending on whether k < m or m < k. Since b are continuous variables, an upper bound is needed to bound their values in constraints (11) and (12). Constraints (13) distinguish whether the origin i of a flow O i is a hub (z i = 1), in which case W ii does not flow. The flow conservation constraints (14) guarantee that all the demands are satisfied, by discounting at each node its demanded flow from the total flow entering the node. Note that the structures defined by both the y and X variables at value one in feasible solutions to formulation (F1) are connected, even if the connectivity assumption on the graph of flows does not hold. This is mostly due to the role of the x variables to the way that constraints relate them to the y and the X variables. The formulations defined in the next

12 12 subsections no longer use these x variables. 5 Second formulation In this section a new formulation, in which less variables are needed, is developed. Some variables of (F1) are used again: z k takes value 1 if k is a hub, X km takes value 1 if edge {k, m} (where k < m) is in the large tree, and b ikm is the amount of flow with origin in i which traverses arc (k, m). A new set of binary variables is defined as follows: 1 if the edge {k, m} is in the large t km = tree and k is a hub, 0 otherwise. k, m k. That is to say, t km := X km z k if k < m, whereas t km := X mk z k if k > m. added: In order to linearize these products in the formulation, the following constraints should be t km X km k, m > k, t km z k k, m > k, t km X km + z k 1 k, m > k, (15) t km X mk k, m < k, t km z k k, m < k, t km X mk + z k 1 k, m < k. (16) Observe that the new t variables are related to the a variables used in (F1). Indeed, if k < m, a km = X km (1 z k ) = X km t km. Similarly, if k > m, a km = X mk (1 z k ) = X mk t km. Therefore, the second term of the objective function of (F1) can be expressed in terms of the X variables and the new t variables as follows: (1 α)(o k c km + D k c mk )a km = k m k (1 α)(o k c km + D k c mk )X km (1 z k ) k m>k + (1 α)(o k c km + D k c mk )X mk (1 z k ) = k m<k α)(o k c km + D k c mk )X km + k m>k(1 (1 α)(o k c km + D k c mk )X mk k m<k

13 13 (1 α)(o k c km + D k c mk )t km = k m k (1 α)((o k + D m )c km + (O m + D k )c mk )X km k m>k (1 α)(o k c km + D k c mk )t km. k m k Due to the negative coefficients of the t-variables in the objective function of the forthcoming formulation (F2), inequalities (15) and (16) will not be needed. The second formulation proposed to solve THLP is: (F2) min αc km b ikm + i k m k ((O k + D m )c km + (O m + D k )c mk )(1 α)x km k m>k (O k c km + D k c mk )(1 α)t km k m k s.t. X km = n 1 (17) k m>k z k = p (18) k X km + X mk (n 2)z k 1 m>k m<k k (19) t km X km k, m > k t km z k t km X mk t km z k k, m k k, m < k k, m < k b ikm O i X km i, k, m > k (20) b ikm O i X mk i, k, m < k (21) b iim + W ii z i = O i i (22) m i b ikj = W ij + b ijk k j k j i, j i (23) z k {0, 1} k t km {0, 1} b ikm 0 k, m k i, k, m k X km {0, 1} k, m > k. Constraints (17-18) and (20-23) were already included in (F1) and their meaning is the same. Constraints (17) and (18) set the number of edges of the large tree and the number

14 14 (A) (B) (C) Figure 3: Meaning of several variables in the third formulation of hubs, respectively, whereas constraints (22) and (23) guarantee that the flow will reach its destination. Along with constraints (20) and (21), they also guarantee the connection of the large graph, which is a tree due to (17). In this formulation (19) is the family of constraints that enforce that a node which is not a hub must be a leaf of the tree, as desired. Indeed, if z k = 0, (19) establishes that k is the end-node of at most one edge (a leaf) and, if z k = 1, the bound on the number of neighbors of k is n 1. 6 Third formulation In the third and last formulation studied, we again consider the variables y km = { 1 if the edge {k, m} links two hubs 0 otherwise, k, m > k, see Figure 3. An extended family of a-variables, similar to that of (F1), will give the allocations of non hubs to hubs, but also the location of hubs, see Figure 3(B): a km = { 1 if the non hub k is allocated to the hub m, 0 otherwise, k, m k, { 1 if node k is a hub, a kk = k. 0 otherwise, Finally, variables b ikm will be used to measure the flow with origin in i going from hub k to hub m. In Figure 3(C), the values of the b-variables with a common first index i have been represented, assuming that the demand of each node with respect to origin i is the amount written inside the node.

15 15 The third formulation is (F 3) min k (c km O k + c mk D k )a km m + αc km b ikm i k m k s.t. a kk = p (24) k a km = 1 k (25) m a km + y km a mm k, m > k (26) a mk + y km a kk k, m > k (27) b ikm + b imk (O i W ii m ikm )y km i, k i, m > k, m i (28) b iim (O i W ii )y im i, m > i (29) b iim (O i W ii )y mi i, m < i (30) b iki = 0 i, k i (31) O i a ij + b imj = b ijm + W im a mj i, j i (32) m j m j m y km = p 1 (33) k m b ikm 0 i, k, m a km, y km {0, 1} k, m. Constraints (24-27) imply the constraints of the usual formulation of the p-median problem: there are p hubs and, if k is not a hub, it must be allocated to some hub. In the context of THLP, the usual constraints a km a mm, can be reinforced to (26) when m > k, since when y km = 1 then a hub is located at node k so that a kk = 1 and, thus by constraint (25) a km = 0. Similarly, when m < k the usual constraints can be reinforced to (27). Constraints (28-30) determine that the flow between hubs will only move through the small tree (y-variables). The bounds in the right hand sides of these constraints have been tightened. When i k, m, the maximum amount of flow with origin in i traversing edge {k, m} will be O i minus the flow W ii which traverses only one hub minus the flow with destination in the first hub of the edge. Since we do not know whether such a hub is k or m, we use the value m ikm. The cases where i = k of i = m are written separately, since the bound is different. And, in some very special cases, b-variables are fixed to zero (constraints (31)). Each constraint in family (32) guarantees the conservation of the flow with origin i in node j. On the left hand side we add up the flow coming to j directly from node i (if i is not a hub

16 16 and it is allocated to j), plus the flow with origin in i coming to j from another hub m. On the right hand side, we add up the flow with origin in i going from hub j to another hub (if j is a hub), plus the flow with origin in i and destination in any non-hub node allocated to hub i (again if j is a hub). Finally, through constraints (33) the variables y define a tree, since p 1 edges are chosen which are connected due to constraints (28) and to the flow conservation constraints (32).

17 17 n=10 OPT (F1) (F2) (F3) p α LP gap nodes seconds LP gap nodes seconds LP nodes seconds Average 22.9% 45.8% 10.9% Table 1: Results obtained with formulations (F1), (F2) and (F3), n = 10.

18 18 n p α OPT LP nodes seconds [ ] >10000 > [ ] >12000 > [ ] >10000 > [ ] >20000 > [ ] >22000 > [ ] >20000 > [ ] >24000 > [ ] >26000 >3600 Table 2: Results obtained with formulation (F1), n=20. 7 Computational study A computational study was carried out in order to test the relative performance of the formulations. A set of instances commonly used in the literature, called AP (Australian Post), was downloaded from the web page mscmga.ms.ic.ac.uk/jeb/orlib/phubinfo.html. These instances consist of the Euclidean distances between 200 cities in Australia, a code to reduce the size of the set by grouping cities, and the values of W ij (postal flow between cities). The solver used in the study was Xpress-Mosel v2.0.0, run under Windows, and the processor was a Intel(R) Core(TM)2 T7600 with 2.33GHz and 3 GB of RAM memory. In all the cases we forbide Xpress to use the heuristic search, the preprocessing and the cut generation. In the tables of computational results, n is the number of nodes, p is the number of hubs, α is the discount factor, OPT is the optimal value of the instance (if it could not be reached after some time, the best feasible solution obtained is shown between brackets), LP is the optimal value of the linear relaxation, gap is the duality gap 100((OPT-LP)/OPT), nodes is the number of nodes of the branching tree and seconds is the CPU time, in seconds, needed to solve the instance. The time needed for solving the LP relaxation of the problem was always insignificant. First we checked the instance of the AP data set with 10 nodes combining three values of p, namely 3, 5 and 8 and three discount factors: 0.2, 0.5 and 0.8. Since formulation (F1) depends on which node is chosen to be R, we combined the ten possibilities with each value of p and α. Then, we solved 9 instances of size 10 in 12 different ways. In Table 1, under (F1), we show the minimum and maximum values of nodes and seconds for the ten possible values of R (the lower bound given by the LP relaxation was always the same). The first conclusion we can draw from the results of Table 1 is that THLP is difficult to

19 19 n p α OPT LP nodes seconds [ ] >10500 > [ ] >25000 > [ ] >30000 > [ ] >50000 > [ ] >50000 > [ ] > > [ ] > >3600 Table 3: Results obtained with formulation (F2), n=20. n p α OPT LP nodes seconds [ ] >15000 > [ ] >10000 > [ ] >10000 > [ ] >10000 >3600 Table 4: Results obtained with formulation (F3), n=20. solve. This is not strange, since in general hub location problems, due to the requirement of sending the flow between each pair of nodes and that of calculating the cost of these flows along all the corresponding routes, are difficult to formulate even without the tree shape restriction. Moreover, the instance becomes more difficult when the values of p and α are larger (the results with n = 20 will also confirm this fact). All 9 instances of size 10 were optimally solved with all the 3 formulations and 9 values of R in one hour of CPU. Nevertheless, the times spent on the solution process were large in the cases of (F1) and (F2), and relatively small in the case of (F3). The average of the duality gaps of (F1) was 22.9%, leading to a large number of nodes of the branching tree. In the case of (F2), although the duality gap was even worse, and the number of nodes even larger (probably due to the reduced size of the formulation), the computational times were better. Formulation (F3) achieved a much smaller duality gap (10.9%), leading to a lower number of nodes and much better computational times. It should also be noted that, in (F1), the choice of R drastically affects the computational time, which can be halved if a lucky choice of R is made.

20 20 When we tried to solve the instances with n = 20 and the same combinations of values of p and α, we established a time limit of one hour. The value of R in (F1) was fixed at 1. We observe that only 1 of the 9 instances could be solved in this time using formulation (F1) (Table 2). Moreover, the values of the best integer solutions obtained in the rest of the instances were a long way from the values of the best lower bounds obtained at that moment (not shown in the tables). We see in Table 3 that (F2) was able to solve two instances inside the time limit. In general, the gaps after one CPU hour were similar (ranging from 1% to 40%) to the gaps obtained with formulation (F1), despite the fact that the initial lower bound obtained with (F1) is greater than that given by (F2). The reason is that the number of nodes (F2) can explore in the same time is larger. Finally, as shown in Table 4, (F3) could solve 5 instances of size 20 and in all the cases better integer solutions were found (since we do not know the optimal value of these instances, we cannot say Moreover, after one CPU hour the gaps where much better (ranging from 1% to 14%) than the gaps obtained with formulations (F1) and (F2). We can conclude, with the natural caution, that (F3) is a more valuable formulation to be developed further and used in more tailored solution approaches than (F1) and (F2). Nevertheless, (F2) still has the advantage of its small size, and in some sense is more adequate for improvement by means of tighter constraints. 8 Conclusions and further research Devising Integer Programming formulations for a given model is one of the main steps during its optimization. This can be done in several ways, and the success of many resolution techniques depends heavily on the goodness of the chosen formulation. There have been important improvements in the formulation of several hub location problems that clearly show this fact, with impressive reductions in the computational times. The Tree-of-Hubs Location Problem was not, to the best of our knowledge, previously studied. This paper is a first attempt to find a good formulation, by means of the computational comparison of several possibilities. Further advances should include the improvement of the best of the proposed formulations, including a polyhedral study, and the design of ad-hoc algorithms which can perform better than the branch-and-bound algorithms based on the LP relaxation used in commercial solvers. In particular, we think that good Lagrangian relaxations based on (F2) or (F3) will be able to solve quite larger instances. Also, as a subject for future research, heuristic methods able to deal with larger and more

21 21 realistic instances whose optimal solution will be hard to obtain should be developed. Finding feasible solutions is not a problem in this model and, again, Lagrangian relaxation methods can be tailored to give good feasible solutions, but several other successful metaheuristics can also be tried with the THLP. More realistic models could incorporate capacities onto the hubs and/or the arcs of the small tree, as well as additional constraints on the performance of the system. These extensions could use our formulations as a basis for further development. Acknowledgements The authors thank Fundación Séneca project 02911/PI/05, Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I+D+I) projects TIC C05-(01,04) and HA and ERDF funds for the support given to the research which led to this article. The research of the first author has been partially supported through grant / from CONACYT, Mexico. References [1] S. Alumur, B.Y. Kara, Network hub location problems: The state of the art, European Journal of Operational Research (2007), doi: /j.ejor [2] T. Aykin, Lagrangian relaxation based approaches to capacitated hub-and-spoke network design problem, European Journal of Operational Research 79 (1994) [3] T. Aykin, Networking policies for hub-and-spoke systems with application to the air transportation system, Transportation Science 29 (1995) [4] N. Boland, M. Krishnamoorthy, A.T. Ernst, J. Ebery, Preprocessing and cutting for multiple allocation hub location problems, European Journal of Operational Research 155 (2004) [5] J.F. Campbell, Integer programming formulations of discrete hub location problems, European Journal of Operational Research 72 (1994) [6] J.F. Campbell, A.T. Ernst, M. Krishnamoorthy, Hub arc location problems: Part I - Introduction and results, Management Science 51(10) (2005) [7] J.F. Campbell, A.T. Ernst, M. Krishnamoorthy, Hub arc location problems: Part II - Formulations and optimal algorithms, Management Science 51(10) (2005)

22 22 [8] J.F. Campbell, A.T. Ernst, M. Krishnamoorthy, Hub location problems, in: Z. Drezner, H.W. Hamacher (Eds.), Facility Location. Applications and Theory, Springer, Heidelberg, 2002, pp [9] L. Cánovas, S. García, A. Marín, Solving the uncapacitated multiple allocation hub location problem by means of a dual ascent technique, European Journal of Operational Research 179 (2007) [10] L. Cánovas, M. Landete, A. Marín, New formulations for the uncapacitated multiple allocation hub location problem, European Journal of Operational Research 172 (2006) [11] H.S. Connamacher, A. Proskurowski, The complexity of minimizing certain cost metrics for k-source spanning trees, Discrete Applied Mathematics 131 (2003) [12] I. Contreras, J.A. Díaz, E. Fernández, A Lagrangean Relaxation Approach for the Capacitated Single Allocation Hub Location Problem, Proceedings of the EURO Winter Institute on Location and Logistics, Estoril, [13] J. Ebery, M. Krishnamoorthy, A. Ernst, N. Boland, The capacitated multiple allocation hub location problem: formulations and algorithms, European Journal of Operational Research 120 (2000) [14] A. Ernst, M. Krishnamoorthy, An exact solution approach based on shortest-paths for p-hub median problems, INFORMS Journal on Computing 10 (1998) [15] A. Ernst, M. Krishnamoorthy, Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem, European Journal of Operational Research 104 (1998) [16] A.T. Ernst, M. Krishnamoorthy, Solution algorithms for the capacitated single allocation hub location problem, Annals of Operations Research, 86 (1999) [17] L. Gouveia, Using the Miller-Tucker-Zemlin constraints to formulate a minimal spanning tree problem with hop constraints, Computers and Operations Research 22 (1995) [18] S.L. Hakimi, E.F. Schmeichel, M. Labbé, On locating path or tree shaped facilities on networks, Networks 23 (1993) [19] H.W. Hamacher, M. Labbé, S. Nickel, T. Sonneborn, Adapting polyhedral properties from facility to hub location problems Discrete Applied Mathematics 145(1) (2004)

23 23 [20] T.C. Hu, Optimum Communication Spanning Trees, SIAM Journal on Computing 3(3) (1974) [21] T.U. Kim, T.J. Lowe, A. Tamir, J.E. Ward, On the location of a tree-shaped facility, Networks 28 (1996) [22] J.G. Klincewicz, Heuristics for the p-hub location problem, European Journal of Operational Research 53 (1991) [23] J.G. Klincewicz, Avoiding local optima in the p-hub location problem using tabu search and grasp, Annals of Operations Research 40 (1992) [24] J.G. Klincewicz, A dual algorithm for the uncapacitated hub location problem, Location Science 4 (1996) [25] J.G. Klincewicz, Hub location in backbone/tributary network design: A review. Location Science 6 (1998) [26] M. Labbé, G. Laporte, I. Rodríguez-Martín, J.J. Salazar-González, The Ring Star Problem: Polyhedral analysis and exact algorithm. Networks 43(3) (2004) [27] M. Labbé, H. Yaman, Polyhedral Analysis for Concentrator Location Problems, Computational Optimization and Applications 34 (2005) [28] M. Labbé, H. Yaman, E. Gourdin, A branch and cut algorithm for hub location problems with single assignment, Mathematical Programming 102 (2005) [29] G. Laporte, I. Rodríguez-Martín, Locating a cycle in a transportation or a telecommunications network, Networks 50(1) (2007) [30] A. Marín, Formulating and solving the splittable capacitated multiple allocation hub location problem. Computers and Operations Research 32 (2005) [31] A. Marín, Uncapacitated Euclidean hub location: Strengthened formulation, new facets and a relax-and-cut algorithm. Journal of Global Optimization 33 (2005) [32] Á. Marín, An extension to rapid transit network design problems, Top (2007), doi: /s [33] G. Mayer, B. Wagner, HubLocator: an exact solution method for the multiple allocation hub location problem, Computers and Operations Research 29 (2002)

24 24 [34] S. Nickel, A. Schöbel, T. Sonneborn, Hub location problems in urban traffic networks. In: J. Niittymaki, M. Pursula (Eds.), Mathematics Methods and Optimization in Transportation Systems. Kluwer Academic Publishers, [35] M.E. O Kelly, D. Bryan, D. Skorin-Kapov, J. Skorin-Kapov, Hub network design with single and multiple allocation: A computational study, Location Science 4 (1996) [36] M.E. O Kelly, A quadratic integer program for the location of interacting hub facilities, European Journal of Operational Research 32 (1987) [37] P.C. Pop, W. Kern, G. Still, A new relaxation method for the generalized minimum spanning tree problem, European Journal of Operational Research 170 (2006) [38] J. Puerto, A. Tamir, Locating tree-shaped facilities using the ordered median objective, Mathematical Programming 102 (2005) [39] D. Skorin-Kapov, J. Skorin-Kapov, M. O Kelly, Tight linear programming relaxations of uncapacitated p-hub median problems, European Journal of Operational Research 94 (1996) [40] B.Y. Wu, Approximation algorithms for the optimal p-source communication spanning tree, Discrete Applied Mathematics 143 (2004)