A Compact Linearisation of Euclidean Single Allocation Hub Location Problems

Transcription

1 A Compact Linearisation of Euclidean Single Allocation Hub Location Problems J. Fabian Meier 1,2, Uwe Clausen 1 Institute of Transport Logistics, TU Dortmund, Germany Borzou Rostami 1, Christoph Buchheim 1 Fakultät für Mathematik, TU Dortmund, Germany Abstract Hub location problems are strategic network planning problems. They formalise the challenge of mutually exchanging shipments between a large set of depots. The aim is to choose a set of hubs (out of a given set of possible hubs) and connect every depot to a hub so that the total transport costs for exchanging shipments between the depots are minimised. In classical hub location problems, the unit cost for transport between hubs is proportional to the distance between the hubs. Often these distances are Euclidean distances: Then is possible to replace the quadratic cost term for hub-hub-transport in the objective function by a linear term and a set of linear inequalities. The resulting model can be solved by a row generation scheme. The strength of the method is shown by solving all AP instances to optimality. Keywords: Hub Location, Euclidean Distance, Row Generation 1 This research has been funded by the German Research Foundation (DFG) within the project Lenkung des Güterflusses in durch Gateways gekoppelten Logistik-Service- Netzwerken mittels quadratischer Optimierung (CL 318/14 and BU 2313/2) 2 meier@itl.tu-dortmund.de

2 1 Introduction O Kelly [5] introduced the Uncapacitated Single Allocation p-hub Median Problem (USApHMP) in 1987: A set of p hubs is chosen from n possible hub locations and every depot is connected to exactly one hub. The depots mutually exchange shipments: A shipment is sent from the source depot to the assigned hub, then to the assigned hub of the sink depot and then finally to the sink depot. The aim is to choose hubs and assignments with minimal overall transport costs. We state the problem in the original quadratic form: Consider a complete graph G = (V, E) with V = n. Each node i V of the graph corresponds to origins, destinations and possible hub locations. Let C ij be the transport cost per unit of flow from node i to node j, and W ij be the amount of flow from node i to node j (the shipment from i to j). The cost per unit of flow for each path i k l j from an origin node i to a destination node j which passes hubs k and l respectively, is β 1 C ik + αc kl + β 2 C lj, where β 1, α, and β 2 are the collection, transfer and distribution costs respectively. We define the binary variable x ik to indicate the allocation of node i to the hub located at node k. If node i is assigned to itself, then node i is a hub. To ease the argumentation in the following sections, we define d kl = αc kl K ik = β 1 C ik W ij + β 2 C ki W ji. The quadratic 0 1 formulation of the USApHMP is stated as follows: min W ij d kl x ik x jl i V k V K ik x ik + i V j V j V k V l V j V s.t. x ik = 1 i V (1) k V x ik x kk i, k V (2) x kk = p (3) k V x ik {0, 1} i, k V. (4) Constraints (1) indicate that node i is allocated to precisely one hub node. The inequalities (2) make sure that a node i can only be allocated to a hub node. Constraints (3) force the number of selected hubs to be p.

3 The main difficulty of the problem lies in the quadratic structure of the objective function. Different attempts have been made to linearise the objective function. The MILP formulations of Skorin-Kapov et al. [6] and Ernst and Krishnamoorthy [2] result in O ( n 3) or O ( n 4) variables. Other authors successfully applied metaheuristic techniques to construct good solutions for large instances, recently even for more than 100 nodes [3,4]. In many problem instances (also in those from the usual benchmark instances CAB and AP [1]) d kl is given by the Euclidean distance of the hubs (or a constant multiple of it). Assuming this kind of structure, we can construct a linearisation with only O ( n 2) variables, but O ( n 4) additional constraints. The resulting linearized problem can be solved very efficiently in practice by a row generation procedure. The next section will explain our new approach to the linearisation of the quadratic term in the objective function, while Sect. 3 gives computational evidence. Section 4 gives an outlook for further applications. 2 The Euclidean Hub Location Problem To reduce the number of linearization variables to O ( n 2), we will make the assumption that all distances d kl are calculated by a vector space norm derived from an inner product,, i.e. every hub k is represented by a vector w k in a Hilbert space W and we have d kl = w k w l. Although it is nearly always possible to realise this in an n 1 dimensional vector space, we are more interested in the situation in which costs are proportional to the Euclidean distance in a 2-dimensional landscape. We call this problem Euclidean Hub Location Problem. The Euclidean distance assumption is only relevant for the distance between possible hubs, i.e. the coefficient of x ik x jl in the objective function of (USApHMP). Let us introduce the variables y ij k V W ij d kl x ik x jl (5) l V which represent the total cost of any hub-hub flow induced by the shipment (i, j). The quadratic term of the objective function of (USApHMP) is then equal to i V j V y ij. As the equation (5) is quadratic itself, it is not sensible to add it directly to the problem. Instead, we want to derive a set of linear inequalities which forces y ij to fulfil the constraint (5). In W, we can define the orthogonal projection P w (u) of a vector u W to another vector w W. We know that P w (u 1 ) P w (u 2 ) u 1 u 2 for all

4 u 1, u 2 W. Furthermore, P w (u) = λ u w w w with λ u w = u, w w R (6) and P w (u 1 ) P w (u 2 ) = λ u 1 w λ u 2 w. (7) The value λ u w can be understood as co-ordinate in direction w. We see that it can be directly computed from the knowledge of the involved vectors. This implies, that for any k, l V and w W, w 0, we have: d kl = w k w l P w (w k ) P w (w l ) λ w k w If w = w k w l we have equality everywhere. This implies that y ij = W ij d kl x ik x jl k V l V ( ) W ij λ w k w λ w l w xik x jl k V l V = W ij λ w k w x ik x jl W ij λ w l w x ik x jl k V l V k V l V where we can reduce the sums due to (1): λ w l w (8) = k V W ij λ w k w x ik l V W ij λ w l w x jl (9) Let λ k mh be the co-ordinate associated with the projection of w k to w m w h (see also Fig. 1). Then we know that y ij k V W ij λ k mh x ik l V W ij λ l mh x jl i, j, m, h V (I mh ij ) If x im = x jh = 1, then (Iij mh ) is an equality. This means that the constraints (Iij mh ) completely determine y ij if the values of x ik are binary and chosen according to the constraint (1), so that we can insert y ij into the objective function and get an equivalent MIP to USApHMP which we call EUSApHMP. The LP formulation of EUSApHMP may be weaker than in the original formulation. The number of variables of

5 EUSApHMP is O ( n 2). Unfortunately, the number of inequalities resulting from our construction is O ( n 4). Although we can halve the number of variables and inequalities by exploiting the symmetry in the indices of y ij, it is still not sensible to add all these inequalities at once. Hence we use a row generation scheme and add the inequalities step by step. From a theoretical point of view, an efficient separation algorithm is given trivially, as the number of constraints to consider is polynomial. However, from a practical point of view, checking O ( n 4) inequalities in every call of a separation algorithm is still too expensive. We thus decided to adopt the following heuristic separation method in order to limit both iterations and added inequalities, splitting the solution process into two rounds: In the first round, we iteratively solve the LP relaxation and determine a solution for USApHMP by rounding. We start by considering only the inequalities ( ) I ij ij for all i j V. After every iteration, we add those inequalities that are violated by the rounded solution, i.e. those ( ) Iij kl for ˆx ik = ˆx jl = 1, where {ˆx ik } ik is the solution of the USApHMP. In this way, we restrict the number of new inequalities to at most n 2. We stop the first round when there are no more inequalities to add. Note that this method does not guarantee that the final LP solution solves the LP relaxation of EUSApHMP; but the rounded solutions can be used as initial values for the second round. In the second round, we use a MIP solver. We start by adding all inequalities we used so far in the LP relaxation. Then we proceed as before: In every step we determine a MIP solution {ˆx ik } ik for USApHMP from the given solution and add all inequalities for ˆx ik = ˆx jl = 1. If we have no more inequalities to add, we stop, because this proves the solution is optimal. As shown by the computational results presented in the following section, this approach is able to effectively limit the number of added inequalities while at the same time leading to a very small number of iterations Fig. 1. Illustration of the orthogonal projection of all hub positions onto the vector w 5 w 4. The distance of any two vectors exceeds the difference of the λ values.

6 3 Computational Results We code the procedures in C# using the Gurobi 5.6 solver for the LPs and MIPs. We used a 3.4 GHz computer with 16GB RAM and four threads. The famous AP data served as our test data set [1]; we only give results for instances starting at 50 nodes. We see in Table 1 and 2 that all instances are solved to optimality. After stating the size of V and the number of hubs, we give two specific iterations of the calculation: (i) The first iteration which reaches less than 1% gap between the best found MIP solution and the last lower bound. (ii) The iteration after which optimality is proven. Iterations are written as L/M where L and M are the iterations of the first and second round. For each iteration we also give the total time in seconds and the total number of added inequalities. Table 1 AP results up to 90 nodes. n p Gap 1% Optimality Optimal Value It s (Iij mh ) It s (Iij mh ) / / / / / / / / / / / / / / / / / / / /

7 Table 2 AP results for more than 90 nodes. No published optimal solutions for these instances could be found. n p Gap 1% Optimality Optimal Value It s (Iij mh ) It s (Iij mh ) / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

8 We see that at most 4n 2 of the possible n 4 /2 inequalities of (Iij mh )-type are used. Often, rounding one of the first LP iterations gives the optimal value, so that the optimal value is reached long before optimality is proven. In [3] we find the best known values for 100 and 200 nodes, found by a metaheuristic method: Our optimal results are identical to these best known results, proving them to be optimal. 4 Conclusion Constructing linearisations from the Euclidean structure that is present in many instances of hub location problems leads to compact formulations with few variables. These can be solved even for a large number of nodes. As future work, we plan to apply this method to even larger instances and other variants of single allocation problems, like capacitated or stochastic problems. Furthermore, the method can be extended to the case where the distances are not Euclidean but just fulfil the triangle inequality, i.e., the metric case. Setting λ k mh = d kh leads to the same kind of model, but turned out to be weaker in preliminary numerical experiments. Nevertheless it is worth exploring these techniques for more general cases. References [1] Beasley, J. E., OR library (2012). URL [2] Ernst, A. T. and M. Krishnamoorthy, Efficient algorithms for the uncapacitated single allocation p-hub median problem, Location science 4 (1996), pp [3] Ilić, A., D. Urošević, J. Brimberg and N. Mladenović, A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem, European Journal of Operational Research 206 (2010), pp [4] Kratica, J., Z. Stanimirović, D. Tošić and V. Filipović, Two genetic algorithms for solving the uncapacitated single allocation p-hub median problem, European Journal of Operational Research 182 (2007), pp [5] O Kelly, M. E., A quadratic integer program for the location of interacting hub facilities, European Journal of Operational Research 32 (1987), pp [6] Skorin-Kapov, D., J. Skorin-Kapov and M. O Kelly, Tight linear programming relaxations of uncapacitated p-hub median problems, European Journal of Operational Research 94 (1996), pp