Hande Yaman

The robust spanning tree problem with interval data

Motivated by telecommunications applications we investigate the minimum spanning tree problem where edge costs are interval numbers. Since minimum spanning trees depend on the realization of the edge costs, we define the robust spanning tree problem to hedge against the worst case contingency, and present a mixed integer programming formulation of the problem. We also define some useful optimality concepts, and present characterizations for these entities leading to polynomial time recognition algorithms. These entities are then used to preprocess a given graph with interval data prior to the solution of the robust spanning tree problem. Computational results show that these preprocessing procedures are quite effective in reducing the time to compute a robust spanning tree.

A branch and cut algorithm for hub location problems with single assignment

Abstract.The hub location problem with single assignment is the problem of locating hubs and assigning the terminal nodes to hubs in order to minimize the cost of hub installation and the cost of routing the traffic in the network. There may also be capacity restrictions on the amount of traffic that can transit by hubs. The aim of this paper is to investigate polyhedral properties of these problems and to develop a branch and cut algorithm based on these results.

Formulations and Valid Inequalities for the Heterogeneous Vehicle Routing Problem

We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift some of the constraints to improve the lower bounds. We generalize and strengthen subtour elimination and generalized large multistar inequalities.

Solving the hub location problem with modular link capacities

This paper deals with a capacitated hub location problem arising in the design of telecommunications networks. The problem is different from the classical hub location problem in two ways: the cost of using an edge is not linear but stepwise and the capacity of a hub restricts the amount of traffic transiting through the hub rather than the incoming traffic. In this paper both an exact and a heuristic method are presented. They are compared and combined in a heuristic concentration approach to investigate whether it is possible to improve the results within limited computational times.

Allocation strategies in hub networks

In this paper, we study allocation strategies and their effects on total routing costs in hub networks. Given a set of nodes with pairwise traffic demands, the p-hub median problem is the problem of choosing p nodes as hub locations and routing traffic through these hubs at minimum cost. This problem has two versions; in single allocation problems, each node can send and receive traffic through a single hub, whereas in multiple allocation problems, there is no such restriction and a node may send and receive its traffic through all p hubs. This results in high fixed costs and complicated networks. In this study, we introduce the r-allocation p-hub median problem, where each node can be connected to at most r hubs. This new problem generalizes the two versions of the p-hub median problem. We derive mixed-integer programming formulations for this problem and perform a computational study using well-known datasets. For these datasets, we conclude that single allocation solutions are considerably more expensive than multiple allocation solutions, but significant savings can be achieved by allowing nodes to be allocated to two or three hubs rather than one. We also present models for variations of this problem with service quality considerations, flow thresholds, and non-stop service.

Projecting the flow variables for hub location problems

We consider two formulations for the uncapacitated hub location problem with single assignment (UHL), which use multicommodity flow variables. We project out the flow variables and determine some extreme rays of the projection cones. Then we investigate whether the corresponding inequalities define facets of the UHL polyhedron. We also present two families of facet defining inequalities that dominate some projection inequalities. Finally, we derive a family of valid inequalities that generalizes the facet defining inequalities and that can be separated in polynomial time.

Star p-hub median problem with modular arc capacities

We consider the hub location problem, where p hubs are chosen from a given set of nodes, each nonhub node is connected to exactly one hub and each hub is connected to a central hub. Links are installed on the arcs of the resulting network to route the traffic. The aim is to find the hub locations and the connections to minimize the link installation cost. We propose two formulations and a heuristic algorithm to solve this problem. The heuristic is based on Lagrangian relaxation and local search. We present computational results where formulations are compared and the quality of the heuristic solutions are tested.

A branch-and-cut algorithm for the hub location and routing problem

Abstract We study the hub location and routing problem where we decide on the location of hubs, the allocation of nodes to hubs, and the routing among the nodes allocated to the same hubs, with the aim of minimizing the total transportation cost. Each hub has one vehicle that visits all the nodes assigned to it on a cycle. We propose a mixed integer programming formulation for this problem and strengthen it with valid inequalities. We devise separation routines for these inequalities and develop a branch-and-cut algorithm which is tested on CAB and AP instances from the literature. The results show that the formulation is strong and the branch-and-cut algorithm is able to solve instances with up to 50 nodes.

Star p-hub center problem and star p-hub median problem with bounded path lengths

We consider two problems that arise in designing two-level star networks taking into account service quality considerations. Given a set of nodes with pairwise traffic demand and a central hub, we select p hubs and connect them to the central hub with direct links and then we connect each nonhub node to a hub. This results in a star/star network. In the first problem, called the Star p-hub Center Problem, we would like to minimize the length of the longest path in the resulting network. In the second problem, Star p-hub Median Problem with Bounded Path Lengths, the aim is to minimize the total routing cost subject to upper bound constraints on the path lengths. We propose formulations for these problems and report the outcomes of a computational study where we compare the performances of our formulations.

The Robust Network Loading Problem Under Hose Demand Uncertainty: Formulation, Polyhedral Analysis, and Computations

We consider the network loading problem (NLP) under a polyhedral uncertainty description of traffic demands. After giving a compact multicommodity flow formulation of the problem, we state a decomposition property obtained from projecting out the flow variables. This property considerably simplifies the resulting polyhedral analysis and computations by doing away with metric inequalities. Then we focus on a specific choice of the uncertainty description, called the “hose model,” which specifies aggregate traffic upper bounds for selected endpoints of the network. We study the polyhedral aspects of the NLP under hose demand uncertainty and use the results as the basis of an efficient branch-and-cut algorithm. The results of extensive computational experiments on well-known network design instances are reported.