Nicolas Jozefowiez

Multi-objective vehicle routing problems

Routing problems, such as the traveling salesman problem and the vehicle routing problem, are widely studied both because of their classic academic appeal and their numerous real-life applications. Similarly, the field of multi-objective optimization is attracting more and more attention, notably because it offers new opportunities for defining problems. This article surveys the existing research related to multi-objective optimization in routing problems. It examines routing problems in terms of their definitions, their objectives, and the multi-objective algorithms proposed for solving them.

Parallel and Hybrid Models for Multi-objective Optimization: Application to the Vehicle Routing Problem

Solving a multi-objective problem means to find a set of solutions called the Pareto frontier. Since evolutionary algorithms work on a population of solutions, they are well-adapted to multi-objective problems. When they are designed, two purposes are taken into account: they have to reach the Pareto frontier but they also have to find solutions all along the frontier. It is the intensification task and the diversification task. Mechanisms dealing with these goals exist. But with very hard problems or benchmarks of great size, they may not be effective enough. In this paper, we investigate the utilization of parallel and hybrid models to improve the intensification task and the diversification task. First, a new technique inspired by the elitism is used to improve the diversification task. This new method must be implemented by a parallel model to be useful. Second, in order to amplify the diversification task and the intensification task, the parallel model is extended to a more general island model. To help the intensification task, a hybrid model is also used. In this model, a specially defined parallel tabu search is applied to the Pareto frontier reached by an evolutionary algorithm. Finally, those models are implemented and tested on a bi-objective vehicle routing problem.

An evolutionary algorithm for the vehicle routing problem with route balancing

In this paper, we address a bi-objective vehicle routing problem in which the total length of routes is minimized as well as the balance of routes, i.e. the difference between the maximal route length and the minimal route length. We propose a meta-heuristic method based on an evolutionary algorithm involving classical multi-objective operators. To improve its efficiency, two mechanisms, which favor the diversification of the search, have been added. First, an elitist diversification mechanism is used in cooperation with classical diversification methodologies. Second, a parallel model designed to take into account the elitist diversification is proposed. Our method is tested on standard benchmarks for the vehicle routing problem. The contribution of the introduced mechanisms is evaluated by different performance metrics. All the experimentations indicate a strict improvement of the generated Pareto set.

The bi-objective covering tour problem

The paper discusses the definition and solution of a bi-objective routing problem, namely the bi-objective covering tour problem. The bi-objective CTP is a generalization of the covering tour problem, which means that the covering distance and the associated constraints have been replaced by a new objective. We propose a two-phase cooperative strategy that combines a multi-objective evolutionary algorithm with a branch-and-cut algorithm initially designed to solve a single-objective covering tour problem. Experiments were conducted using both randomly generated instances and real data. Optimal Pareto sets were determined using a bi-objective exact method based on an @e-constraint approach with a branch-and-cut algorithm.

Multi-objective Meta-heuristics for the Traveling Salesman Problem with Profits

We introduce and test a new approach for the bi-objective routing problem known as the traveling salesman problem with profits. This problem deals with the optimization of two conflicting objectives: the minimization of the tour length and the maximization of the collected profits. This problem has been studied in the form of a single objective problem, where either the two objectives have been combined or one of the objectives has been treated as a constraint. The purpose of our study is to find solutions to this problem using the notion of Pareto optimality, i.e. by searching for efficient solutions and constructing an efficient frontier. We have developed an ejection chain local search and combined it with a multi-objective evolutionary algorithm which is used to generate diversified starting solutions in the objective space. We apply our hybrid meta-heuristic to synthetic data sets and demonstrate its effectiveness by comparing our results with a procedure that employs one of the best single-objective approaches.

Target aiming Pareto search and its application to the vehicle routing problem with route balancing

Abstract In this paper, we present a solution method for a bi-objective vehicle routing problem, called the vehicle routing problem with route balancing (VRPRB), in which the total length and balance of the route lengths are the objectives under consideration. The method, called Target Aiming Pareto Search, is defined to hybridize a multi-objective genetic algorithm for the VRPRB using local searches. The method is based on repeated local searches with their own appropriate goals. We also propose an implementation of the Target Aiming Pareto Search using tabu searches, which are efficient meta-heuristics for the vehicle routing problem. Assessments with standard metrics on classical benchmarks demonstrate the importance of hybridization as well as the efficiency of the Target Aiming Pareto Search.

Enhancements of NSGA II and its application to the vehicle routing problem with route balancing

In this paper, we address a bi-objective vehicle routing problem in which the total length of routes is minimized as well as the balance of routes, i.e. the difference between the maximal route length and the minimal route length. For this problem, we propose an implementation of the standard multi-objective evolutionary algorithm NSGA II. To improve its efficiency, two mechanisms have been added. First, a parallelization of NSGA II by means of an island model is proposed. Second, an elitist diversification mechanism is adapted to be used with NSGA II. Our method is tested on standard benchmarks for the vehicle routing problem. The contribution of the introduced mechanisms is evaluated by different performance metrics. All the experimentations indicate a strict improvement of the generated Pareto set.

A Generic Branch-and-Cut Algorithm for Multiobjective Optimization Problems: Application to the Multilabel Traveling Salesman Problem

This paper describes a generic branch-and-cut algorithm applicable to the solution of multiobjective optimization problems for which a lower bound can be defined as a polynomially solvable multiobjective problem. The algorithm closely follows standard branch and cut except for the definition of the lower and upper bounds and some optional speed-up mechanisms. It is applied to a routing problem called the multilabel traveling salesman problem, a variant of the traveling salesman problem in which labels are attributed to the edges. The goal is to find a Hamiltonian cycle that minimizes the tour length and the number of labels in the tour. Implementations of the generic multiobjective branch-and-cut algorithm and speed-up mechanisms are described. Computational experiments are conducted, and the method is compared to the classical e-constraint method.

From Single-Objective to Multi-Objective Vehicle Routing Problems: Motivations, Case Studies, and Methods

Multi-objective optimization knows a fast growing interest for both academic researches and real-life problems. An important domain is the one of vehicle routing problems. In this chapter, we present the possible motivations for applying multi-objective optimization on vehicle routing problems and the potential uses and benefits of doing so. To illustrate this fact, we also describe two problems, namely the vehicle routing problem with route balancing and the bi-objective covering tour problem. We also propose a two-phased approach based on the combination of a multi-objective evolutionary algorithm and single-objective techniques that respectively provide diversification and intensification for the search in the objective space. Examples of implementation of this method are provided on the two problems.

A branch-and-cut algorithm for the minimum labeling Hamiltonian cycle problem and two variants

This paper proposes a mathematical model, valid inequalities and polyhedral results for the minimum labeling Hamiltonian cycle problem. This problem is defined on an unweighted graph in which each edge has a label. The aim is to determine a Hamiltonian cycle with the least number of labels. We also define two variants of this problem by assigning weights to the edges and by considering the tour length either as an objective or as a constraint. A branch-and-cut algorithm for the three problems is developed, and computational results are reported on randomly generated instances and on modified instances from TSPLIB.