The Optimal Diversity Management Problem

Abstract

In some industries, a certain part can be needed in a very large number of different configurations. This is the case, e.g., for the electrical wirings in European car factories. A given configuration can be replaced by a more complete, therefore more expensive, one. The diversity management problem consists of choosing an optimal set of some given number k of configurations that will be produced, any nonproduced configuration being replaced by the cheapest produced one that is compatible with it. We model the problem as an integer linear program. Our aim is to solve those problems to optimality. The large-scale instances we are interested in lead to difficult LP relaxations, which seem to be intractable by the best direct methods currently available. Most of this paper deals with the use of Lagrangean relaxation to reduce the size of the problem in order to be able, subsequently, to solve it to optimality via classical integer optimization. Subject classifications: large-scale integer programming and relaxations: Lagrangean relaxation of a large linear integer program arising from an application; production planning: choice of production. Area of review: Optimization.

Data aggregation for p-median problems

“…The first column shows the library from which the instance is taken, the second column gives the name of the instance, the third column gives the number of plants considered in the problem, and the fourth provides the number of entries in the cost matrix of the instance. The three libraries considered here, i.e., OR, ODM, and TSP instances are available from ORLibrary (http://mscmga.ms.ic.ac.uk/info.html), Briant and Naddef (2004), and TSPlibrary (http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95), respectively. The fifth column in the table gives the number of terms in the reduced Hammer-Beresnev polynomial for the instance.…”

“…Problem reduction is a common technique in integer programming and combinatorial optimization; see, for example, Briant and Naddef (2004) and references within. Classical reduction techniques for PMP instances are based either on good lower bounds (see e.g., Briant and Naddef 2004) or on reduction tests (see e.g., Avella and Sforya 1999).…”

“…Classical reduction techniques for PMP instances are based either on good lower bounds (see e.g., Briant and Naddef 2004) or on reduction tests (see e.g., Avella and Sforya 1999). In this paper, we present two reduction techniques for PMP instances using a pseudo-Boolean formulation of PMP due to Hammer (1968) and Beresnev (1973).…”

The minimum weight t-composition of an integer

“…This study was motivated by a problem arising from the automobile industry called the optimal diversity management problem [1,2,3,4]. For this problem there is a natural decomposition into several problems such that the optimality depends entirely on the way the second issue above is addressed.…”

A decomposition approach for a very large scale optimal diversity management problem

“…1 Introduction Briant and Naddef (2004) introduced the optimal diversity management problem (ODM) in the car industry. Normally, each car can be provided with a different set of options.…”