Distribution of Solutions of Multi-objective Assignment Problem and Links with the Efficiency of Solving Methods.
Résumé
In a preceeding work, a two phase method using a ranking algorithm as main routine in Phase 2 has been proposed by Przybylski, Gandibleux and Ehrgott for the exact solution of Assignment Problem with p objectives (pAP). To validate their method, they have used instances the coefficients of which were generated randomly and independantly following an uniform distribution. This implies in particular that the objective functions are independantly generated. Experimentaly, they have observed that the distribution of solutions of pAP, measured by their objective value, follows a p-dimensionnal normal distribution. This assumption has justified the few solutions enumerated in practice by this method and therefore its efficiency. Generally, the instances used to test a solution method for a multi-objective optimization problem are constructed from mono-objective instances. In the case of pAP, the mono-objective problem belongs to the class P, thus no classification of instances is necessary. This explains why most of the authors working on this problem have only considered purely random instances. However, it is possible to construct in a different way very difficult pAP instances for the two phase method. Our goal is to be able to recognize a difficult instance for a solution method. For this, we consider several classes of instances of pAP: the usual purely random instances (class A), instances the coefficients of which are not generated independantly (class B), instances the cost matrices of which are correlated (class C). We will close the opened conjecture about the instances of class A, and we will study the other classes of instances using probabilistic methods, that allows to highlight properties explaining precisely the difficulties of each class of instance.