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

Abstract

This paper focuses on the solution of the optimal diversity management problem formulated as a p-Median problem. The problem is solved for very large scale real instances arising in the car industry and defined on a graph with several tens of thousands of nodes and with several millions of arcs. The particularity is that the graph can consist of several non connected components. This property is used to decompose the problem into a series of p-Median subproblems of a smaller dimension. We use a greedy heuristic and a Lagrangian heuristic for each subproblem. The solution of the whole problem is obtained by solving a suitable assignment problem using a Branch-and-Bound algorithm.

Using decomposition to improve greedy solutions of the optimal diversity management problem

“…Avella et al. () introduced the decomposition problem, and proposed a branch‐and‐bound algorithm for it.…”

“…The greedy algorithm is a common option to solve large size instances of the ODMP, and greedy solutions are considered to be of quite good quality (Briant, 2000;Avella et al, 2005;. The greedy algorithm starts with the set S of the m maximal elements of V, and while |S| < p chooses a vertex v from V\S for which the cost of using the configurations in S ∪ {v} is minimum and adds it to S. We used heuristic strategies combined with solving a "restrict" decomposition problem to improve the greedy solutions.…”

“…Briant () pointed out that the large size of instances of real problems is a huge barrier to the efficiency of the algorithms. Dealing with the large size instances that appear in real problems is the main concern of the studies on the ODMP (Briant and Naddef, ; Avella et al., , ; Agra et al., ). The greedy algorithm has been used to solve large size instances, and these studies report that the resulting solutions are normally quite good (Briant, ; Avella et al., ; Agra et al., ).…”

The minimum weight t-composition of an integer

“…These problems are N P -hard and several heuristic methods have been developed, some of them exploiting the close relationship with another N P -hard problem designated the dominating set problem [6,9,10]. Since practical location problems have in general a huge dimension, it is a common procedure to decompose the problem into t smaller location problems [2], and produce a solution from the outcomes of each sub-problem. Each subproblem introduces m new problems, each consists of selecting s = 1, 2, .…”

“…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.…”