Solving the Orienteering Problem through Branch-and-Cut

Abstract

In the Orienteering Problem (OP), we are given an undirected graph with edge weights and node prizes. The problem calls for a simple cycle whose total edge weight does not exceed a given threshold, while visiting a subset of nodes with maximum total prize. This NP-hard problem arises in routing and scheduling applications. We describe a branch-and-cut algorithm for finding an optimal OP solution. The algorithm is based on several families of valid inequalities. We also introduce a family of cuts, called conditional cuts, which can cut off the optimal OP solution, and propose an effective way to use them within the overall branch-and-cut framework. Exact and heuristic separation algorithms are described, as well as heuristic procedures to produce near-optimal OP solutions. An extensive computational analysis on several classes of both real-world and random instances is reported. The algorithm proved to be able to solve to optimality large-scale instances involving up to 500 nodes, within acceptable computing time. This compares favorably with previous published methods.

A branch and cut approach to the cardinality constrained circuit problem

“…This is a somewhat naive approach to solving vehicle routing CCCP instances. As pointed out by Bixby et al [9] and Fischetti et al [14], many classes of inequalities can be improved by using the knowledge that the depot must be included in any circuit. Also, specialized primal heuristics can be developed.…”

“…The objective is to find a route R starting and ending at v 0 that maximizes the total collected reward ( v∈V(R) p v ) subject to the constraint that the total travel cost ( e∈R t e ) is less than Q. Heuristics for solving the OP are given by Golden et al [17] and Ramesh et al [34]; and polyhedral approaches for OP are given by Fischetti et al [14] and Leifer and Rosenwein [26].…”

Orientazio problema eta algoritmo genetikoak

“…Hurrengo (3a)-(3e) modelizazioa Fischetti, Salazar-González eta Toth egileek 1998an argitaratutako OPren modelizazioa da; ikusi [13]. Ohar gaitezen modelizazioa bakarrik dela baliogarria norantza gabeko grafoentzat.…”

“…[10] artikuluan agertzen direnak baztertu ditugu, tamainaren aldetik txikiegiak baitira. [13]-ko datu multzoa baztertu egin dugu, herrien sariak ausaz kalkulatuta daudelako eta, beraz, ez errepikagarriak direlako.…”

An Ant Colony Approach to the Orienteering Problem

“…After solving the LP relaxation, a cutting plane algorithm is added and the LP is solved again. Fischetti, et al [24] propose a branch-and-cut algorithm by adding inequalities such as the matching inequality, the cover inequality, and the path inequality. Then, the overall branch-and-cut algorithm is used to find the optimal solution.…”