A new formulation for the Traveling Deliveryman Problem

Abstract

a b s t r a c tThe Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.

The time dependent traveling salesman problem: polyhedra and algorithm

“…Méndez-Díaz et al [18], Bigras et al [4], Miranda Bront et al [20], Godinho et al [9] are recent works providing good computational results on TDP instances. This last work presents interesting extended formulations with Θ(n 4 ) variables and Θ(n 3 ) constraints that obtained very small gaps (usually zero) on several TDP instances with up to 50 vertices.…”

“…Formulations for the TDTSP have been proposed or studied in [23,20,7,10,29,9]. Exact algorithms for the TDTSP are presented in [20,4,29] and, for the special case of the TDP, in [6,17,18]. This paper is organized as follows.…”

Efficient GRASP+VND and GRASP+VNS metaheuristics for the traveling repairman problem

“…The TRP is very similar to the traveling salesman problem (TSP) and several different mathematical programming formulations have been proposed [see, e.g., Fischetti et al 1993;Sarubbi and Luna 2007;Méndez-Díaz et al 2008]. On a metric space, the TRP is NP-hard (Blum et al 1994) like the TSP but surprisingly, the TRP is much harder to solve or approximate.…”

“…In contrast, the best-known approximation ratio for the metric TSP is 1.5, due to Christofides (1976). As stated by Méndez-Díaz et al (2008), the TRP is a very hard problem to solve to proven optimality.…”

Where's waldo at time t ? using spatio-temporal models for mobile robot search

“…Salehipour et al [6] present a meta-heuristic combining General Randomized Adaptive Search (GRASP) with Variable Neighborhood Descent (VNS). A meta-heuristic called GVNS (General Variable Neighborhood Search) is introduced in [7], while integer linear programming is used in [8].…”