Models for multi-depot routing problems

In this dissertation we study two problems. In the first part of the dissertation we study the multi-depot routing problem. In the multi-depot routing problem we are given a set of depots and a set of clients and the objective is to find a set of routes with minimum total cost, one for each depot, such that each route starts and ends at the same depot and all clients are visited in one and only one route. The requirement that routes must start and end at the same depot is modeled by so-called path elimination constraints. We present a formulation which includes a newly developed set of multi-cut path elimination constraints and a branch-and-cut algorithm based on the new formulation that it is able to solve both asymmetric and symmetric instances with up to 300 clients and 60 depots. Additionally, we present other approaches to model path elimination constraints, including a formulation which provides linear programming relaxation values which are close to the optimal value in the instances tested. In the second part of the dissertation we study the Hamiltonian p-median problem. In the Hamiltonian p-median we are given a set of nodes and the objective is to find p circuits with minimum total cost such that each node is in one and only circuit. We propose a formulation based on the concept of acting depot which attributes the role of artificial depot to p of the nodes. This formulation is a non-straightforward adaptation of the new model proposed for the multidepot routing problem and it is based on a novel idea in which the standard arc variables are split into three cases depending on whether none or exactly one of its endpoints is an acting depot. We present a branch-and-cut algorithm based on the new formulation which is able to solve asymmetric instances with up to 171 nodes and symmetric instances with up to 100 nodes.Programa de Bolsas de Doutoramento da Universidade de Lisbo

Revisiting the Hamiltonian p-median problem: a new formulation on directed graphs and a branch-and-cut algorithm

This paper studies the asymmetric Hamiltonian p-median problem, which consists of finding p mutually disjoint circuits of minimum total cost in a directed graph, such that each node of the graph is included in one of the circuits. Earlier formulations view the problem as the intersection of two subproblems, one requiring at most p, and the other requiring at least p circuits, in a feasible solution. This paper makes an explicit connection between the first subproblem and subtour elimination constraints of the traveling salesman problem, and between the second subproblem and the so-called path elimination constraints that arise in multi-depot/location-routing problems. A new formulation is described that builds on this connection, that uses the concept of an acting depot, resulting in a new set of constraints for the first subproblem, and a strong set of (path elimination) constraints for the second subproblem. The variables of the new model also allow for effective symmetry-breaking constraints to deal with two types of symmetries inherent in the problem. The paper describes a branch-and-cut algorithm that uses the new constraints, for which separation procedures are proposed. Theoretical and computational comparisons between the new formulation and an adaptation of an existing formulation originally proposed for the symmetric Hamiltonian p-median problem are presented. Computational results indicate that the algorithm is able to solve asymmetric instances with up to 171 nodes and symmetric instances with up to 100 nodes

Exact algorithms for the matrix bid auction.

In a combinatorial auction, multiple items are for sale simultaneously to a set of buyers. These buyers are allowed to place bids on subsets of the available items. A special kind of combinatorial auction is the so-called matrix bid auction, which was developed by Day (2004). The matrix bid auction imposes restrictions on what a bidder can bid for a subsets of the items. This paper focusses on the winner determination problem, i.e. deciding which bidders should get what items. The winner determination problem of a general combinatorial auction is NP-hard and inapproximable. We discuss the computational complexity of the winner determination problem for a special case of the matrix bid auction. We present two mathematical programming formulations for the general matrix bid auction winner determination problem. Based on one of these formulations, we develop two branch-and-price algorithms to solve the winner determination problem. Finally, we present computational results for these algorithms and compare them with results from a branch-and-cut approach based on Day & Raghavan (2006).Algorithms; Bids; Branch-and-price; Combinatorial auction; Complexity; Computational complexity; Exact algorithm; Mathematical programming; Matrix; Matrix bids; Research; Winner determination;