J. Fabian Meier

A Compact Linearisation of Euclidean Single Allocation Hub Location Problems

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 it 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.

Minimization and maximization versions of the quadratic travelling salesman problem

The travelling salesman problem (TSP) asks for a shortest tour through all vertices of a graph with respect to the weights of the edges. The symmetric quadratic travelling salesman problem (SQTSP) associates a weight with every three vertices traversed in succession. If these weights correspond to the turning angles of the tour, we speak of the angular-metric travelling salesman problem (Angle TSP). In this paper, we first consider the SQTSP from a computational point of view. In particular, we apply a rather basic algorithmic idea and perform the separation of the classical subtour elimination constraints on integral solutions only. Surprisingly, it turns out that this approach is faster than the standard fractional separation procedure known from the literature. We also test the combination with strengthened subtour elimination constraints for both variants, but these turn out to slow down the computation. Secondly, we provide a completely different, mathematically interesting MILP linearization for the Angle TSP that needs only a linear number of additional variables while the standard linearization requires a cubic one. For medium-sized instances of a variant of the Angle TSP, this formulation yields reduced running times. However, for larger instances or pure Angle TSP instances, the new formulation takes more time to solve than the known standard model. Finally, we introduce MaxSQTSP, the maximization version of the quadratic travelling salesman problem. Here, it turns out that using some of the stronger subtour elimination constraints helps. For the special case of the MaxAngle TSP, we can observe an interesting geometric property if the number of vertices is odd. We show that the sum of inner turning angles in an optimal solution always equals . This implies that the problem can be solved by the standard ILP model without producing any integral subtours. Moreover, we give a simple constructive polynomial time algorithm to find such an optimal solution. If the number of vertices is even, the optimal value lies between 0 and and these two bounds are tight, which can be shown by an analytic solution for a regular n-gon.

Lower Bounding Procedures for the Single Allocation Hub Location Problem

Abstract This paper proposes a new lower bounding procedure for the Uncapacitated Single Allocation p-Hub Median Problem based on Lagrangean relaxation. For solving the resulting Lagrangean subproblem, the given problem structure is exploited: it can be decomposed into smaller subproblems that can be solved efficiently by combinatorial algorithms. Our computational experiments for some benchmark instances demonstrate the strength of the new approach.

Strategic planning in LTL logistics – increasing the capacity utilization of trucks

Abstract A “less than truckload” (LTL) network organises the transport of small shipping volumes by truck between given depots. To be cost-efficient it is necessary to bundle and unbundle goods on their way, using depots as so-called hubs . Our aim is to develop a strategic plan which is cost-optimal for given average shipping volumes. We consider transshipment and transport costs; to give a realistic estimate of the economies of scale, we charge each truck on a specific route equally, whether it is full or (nearly) empty. Real-sized problems become too hard for standard solvers so that we develop a combination of heuristic strategies (which can, in the end, be combined with solvers like CPLEX). We consider the problem in two flavours: MAPIT requires to transport unsplit goods from one depot to another, using at most two intermediate depots as hubs. IO-MAPIT furthermore considers the circulation of trucks.

Solving Single Allocation Hub Location Problems on Euclidean Data

If shipments have to be transported between many sources and sinks, direct connections from each source to each sink are often too expensive. Instead, a small number of nodes are upgraded to hubs that serve as transshipment points. All sources and sinks are connected to these hubs, so that only a few, strongly consolidated transport relations exist. While hubs and detours lead to additional costs, the savings from bundling shipments—i.e., economies of scale—usually outweigh these costs. Typical applications for hub networks are in cargo, air freight, and postal and parcel transport services. In this paper, we consider three classical and two recent formulations of single allocation hub location problems—i.e., hub location problems in which every source and sink is connected to exactly one hub. Solving larger instances of these problems to optimality is difficult because the inherent quadratic structure of the problem has to be linearized: This leads to a sharp rise in the number of variables. Our new appr...

An improved mixed integer program for single allocation hub location problems with stepwise cost function

Recently, a new model for the uncapacitated single allocation p-hub median problem was defined, which uses a more realistic cost structure. Instead of measuring the transport costs as a linear function of the volume, integer variables for the number of used vehicles are introduced. This leads to a more precise model if the number of vehicles is low and capacity utilization plays a major role. We will introduce a new mixed integer program formulation of the problem that uses fewer variables but more constraints. This study shows its numerical advantages.

Linear Models and Computational Experiments for the Quadratic TSP

Abstract We consider the Symmetric Quadratic Traveling Salesman Problem (SQTSP), which is a generalization of the classical TSP where each sequence of two consecutive edges in the tour gives rise to a certain cost value. For the standard linearization we apply a purely integral subtour elimination strategy which outperforms the usual fractional separation routine in computational experiments, even if strengthened inequalities are added. The maximization version of the problem is introduced and turns out to benefit from this strengthening. Finally, a new geometry-based linearization with only a linear number of additional variables is presented for the Angular Metric TSP and variants thereof. It is faster than the other approaches for medium-sized instances of one of the variants.

Heuristic Strategies for a Multi-Allocation Problem in LTL Logistics

We consider a multi-allocation problem where the transport is handled by complete (integer-valued) trucks. It consists of two parts: A number of hubs are chosen out of a given set of depots; then the given transport relations are individually assigned to two hubs, one hub or direct transport. Having four-index variables for the routing and integer variables for the trucks, this MIP becomes difficult. Our heuristic approach gives much better results than a Cplex implementation and can be used to generate a restricted problem which can again be given to Cplex. The idea is as follows: Considering the whole network as collection of n trees sending goods to a chosen depot, we improve the total costs step by step: For that we always take the edge with most expensive transport and try to find a new route for it. In our paper we will explain the theoretic ideas, point out different possibilities and connect them to computational results.

Some Numerical Studies for a Complicated Hub Location Problem

We consider a complicated hub location problem which includes multi-allocation, different hub sizes and different transport volumes on different week days. Furthermore, we consider transport costs per vehicle and not per volume which transforms the cost function into a step function and makes the problem numerically very hard. In our previous work we developed a heuristic approach which we now want to compare to CPLEX results for general and simplified models.