The mixed general routing polyhedron

 In Arc Routing Problems, ARPs, the aim is to find on a graph a minimum cost traversal satisfying some conditions related to the links of the graph. Due to restrictions to traverse some streets in a specified way, most applications of ARPs must be modeled with a mixed graph. Although several exact algorithms have been proposed, no polyhedral investigations have been done for ARPs on a mixed graph. In this paper we deal with the Mixed General Routing Problem which consists of finding a minimum cost traversal of a given link subset and a given vertex subset of a mixed graph. A formulation is given that uses only one variable for each link (edge or arc) of the graph. Some properties of the associated polyhedron and some large families of facet-inducing inequalities are described. A preliminary cutting-plane algorithm has produced very good lower bounds over a set of 100 randomly generated instances of the Mixed Rural Postman Problem. Finally, applications of this study to other known routing problems are described. Received: June 30, 1999 / Accepted: March 2002 Published online: March 21, 2003 Key Words. polyhedral combinatorics – facets – routing – arc routing – rural postman problem – general routing problem – mixed chinese postman problem     

The mixed capacitated general routing problem under uncertainty

We study the General Routing Problem defined on a mixed graph and with stochastic demands. The problem under investigation is aimed at finding the minimum cost set of routes to satisfy a set of clients whose demand is not deterministically known. Since each vehicle has a limited capacity, the demand uncertainty occurring at some clients affects the satisfaction of the capacity constraints, that, hence, become stochastic. The contribution of this paper is twofold: firstly we present a chance-constrained integer programming formulation of the problem for which a deterministic equivalent is derived. The introduction of uncertainty into the problem poses severe computational challenges addressed by the design of a branch-and-cut algorithm, for the exact solution of limited size instances, and of a heuristic solution approach exploring promising parts of the search space. The effectiveness of the solution approaches is shown on a probabilistically constrained version of the benchmark instances proposed in the literature for the mixed capacitated general routing problem.     

Optimum matching forests III: Facets of matching forest polyhedra

In [3] we presented a linear system which definesP(G), the convex hull of incidence vectors of matching forests of a mixed graphG. However, many of the inequalities of this system may be redundant. Here we describe the dimension of the facets ofP(G) obtained by setting one inequality of the defining system forP(G) to an equation. This leads to a presentation of a minimal defining linear system forP(G), i.e., to a presentation of the facets ofP(G). This generalizes earlier characterizations of facets of 1-matching polyhedra and of branching polyhedra.Research partially supported by a N.R.C. Canada Postdoctorate Fellowship.     

The capacitated general windy routing problem with turn penalties

In this paper we present the capacitated general windy routing problem with turn penalties. This new problem subsumes many important and well-known arc and node routing problems, and it takes into account turn penalties and forbidden turns, which are crucial in many real-life applications, particularly in downtown areas and for large vehicles. We provide a way to solve this problem both optimally and heuristically by transforming it into a generalized vehicle routing problem.     

Modeling and solving the mixed capacitated general routing problem

We tackle the mixed capacitated general routing problem (MCGRP) which generalizes many other routing problems. We propose an integer programming model for the MCGRP and extend some inequalities originally introduced for the capacitated arc routing problem (CARP). Identification procedures for these inequalities and for some relaxed constraints are also discussed. Finally, we describe a branch and cut algorithm including the identification procedures and present computational experiments over instances derived from the CARP.     

The time dependent traveling salesman problem: polyhedra and algorithm

The time dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.     

A transformation for the mixed general routing problem with turn penalties

In this paper, we study a generalization of the Mixed General Routing Problem (MGRP) with turn penalties and forbidden turns. Thus, we present a unified model of this kind of extended versions for both node- and arc-routing problems with a single vehicle. We provide a polynomial transformation of this generalization into an asymmetric travelling salesman problem, which can be considered a particular case of the MGRP. We show computational results on the exact resolution on a set of 128 instances of the new problem using a recently developed code for the MGRP.     

The skorokhod oblique reflection problem in a convex polyhedron

The Skorokhod oblique reflection problem is studied in the case ofn-dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to construct in a direct way the reflected (in a convex polyhedral domain) diffusion processes possessing the nice properties.     

The multi-trip vehicle routing problem

The basic vehicle routing problem is concerned with the design of a set of routes to serve a given number of customers, minimising the total distance travelled. In that problem, each vehicle is assumed to be used only once during a planning period, which is typically a day, and therefore is unrepresentative of many practical situations, where a vehicle makes several journeys during a day. The present authors have previously published an algorithm which outperformed an experienced load planner working on the complex, real-life problems of Burton's Biscuits, where vehicles make more than one trip each day. This present paper uses a simplified version of that general algorithm, in order to compare it with a recently published heuristic specially designed for the theoretical multi-trip vehicle routing problem.     

The maximin HAZMAT routing problem

The hazardous material routing problem from an origin to a destination in an urban area is addressed. We maximise the distance between the route and its closest vulnerable centre, weighted by the centre’s population. A vulnerable centre is a school, hospital, senior citizens’ residence or the like, concentrating a high population or one that is particularly vulnerable or difficult to evacuate in a short time. The potential consequences on the most exposed centre are thus minimized. Though previously studied in a continuous space, the problem is formulated here over a transport (road) network. We present an exact model for the problem, in which we manage to significantly reduce the required variables, as well as an optimal polynomial time heuristic. The integer programming formulation and the heuristic are tested in a real-world case study set in the transport network in the city of Santiago, Chile.     

The laser-plotter beam routing problem

Laser-plotters are used in the manufacturing industry to draw trademarks and decorations on the metallic surfaces of some products and devices. Given a pattern, the laser-plotter beam routing problem amounts to routing a laser-plotter beam in such a way that the total drawing time is minimised and some additional requirements are met. The aim of this paper is to model and solve the laser-plotter beam routing problem as a constrained Arc Routing Problem. Computational results on test problems with up to 225 vertices are reported.     

The museum visitor routing problem

In the museum visitor routing problem (MVRP), each visitor group has some exhibit rooms of interest. The visiting route of a certain visitor group requires going through all the exhibit rooms that the group wants to visit. Routes need to be scheduled based on certain criteria to avoid congestion and/or prolonged touring time. In this study, the MVRP is formulated as a mixed integer program which is an extension of the open shop scheduling (OSS) problem in which visitor groups and exhibit rooms are treated as jobs and machines, respectively. The time each visitor group spends in an exhibit room is analogous to the processing time required for each job on a particular machine. The travel time required from one exhibit room to another is modeled as the sequence-dependent setup time on a machine, which is not considered in the OSS problem. Due to the intrinsic complexity of the MVRP, a simulated annealing (SA) approach is proposed to solve the problem. Computational results indicate that the proposed SA approach is capable of obtaining high quality MVRP solutions within a reasonable amount of computational time and enables the approach to be used in practice.     

The school bus routing problem: A review

This paper aims to provide a comprehensive review of the school bus routing problem (SBRP). SBRP seeks to plan an efficient schedule for a fleet of school buses where each bus picks up students from various bus stops and delivers them to their designated schools while satisfying various constraints such as the maximum capacity of a bus, the maximum riding time of a student in a bus, and the time window of a school. This class of problem consists of different sub-problems involving data preparation, bus stop selection, bus route generation, school bell time adjustment, and bus scheduling. In this paper, the various assumptions, constraints, and solution methods used in the literature on SBRP are summarized. A list of issues requiring further research is also presented.     

The vehicle routing problem: A book review

We review the recent book, edited by Paolo Toth and Daniele Vigo, The Vehicle Routing Problem, SIAM Monographs on Discrete Mathematics and Applications 2002, ISBN: 0-89871-498-2, price: 95 USD.     

The pyramidal capacitated vehicle routing problem

This paper introduces the pyramidal capacitated vehicle routing problem (PCVRP) as a restricted version of the capacitated vehicle routing problem (CVRP). In the PCVRP each route is required to be pyramidal in a sense generalized from the pyramidal traveling salesman problem (PTSP). A pyramidal route is defined as a route on which the vehicle first visits customers in increasing order of customer index, and on the remaining part of the route visits customers in decreasing order of customer index.     

Facets and valid inequalities for the time-dependent travelling salesman problem

The Time-Dependent Travelling Salesman Problem (TDTSP) is a generalization of the traditional TSP where the travel cost between two cities depends on the moment of the day the arc is travelled. In this paper, we focus on the case where the travel time between two cities depends not only on the distance between them, but also on the position of the arc in the tour. We consider two formulations proposed in the literature, we analyze the relationship between them and derive several families of valid inequalities and facets. In addition to their theoretical properties, they prove to be very effective in the context of a Branch and Cut algorithm.     

The savings algorithm for the vehicle routing problem

Survey is given concerning the savings method for the vehicle routing problem. Results for several methods and data sets are compared. Furthermore, modifications of the savings method are presented which show less CPU time and reduced storage requirements. Therefore, the savings method can be implemented on microcomputers.