Journal of Mathematical Sciences, Vol. 161, No. 6, 2009 SOLVING HUGE SIZE INSTANCES OF THE OPTIMAL DIVERSITY MANAGEMENT PROBLEM A. Agra, D. M. Cardoso, J. O. Cerdeira, M. Miranda, and E. Rocha UDC 517.977.5 Abstract. We report a real application project in a car industry optimization problem known as the optimal diversity management problem. We provide an alternative proof of NP-hardness, and we give and discuss the results obtained from a greedy algorithm applied to huge size instances. 1. Introduction This paper reports the result of a project with a car industry company that produces the cables for connecting electrical components. Cars are assembled with the necessary wire connections to activate a set of requested options such as airbag, air conditioner, radio, etc. A configuration is the aggregate of minimum connections allowing one to activate a given group of options. For technical reasons, it is not reasonable (and it would certainly be very expensive) to produce a large variety of different configurations. In practice, a number p of different configurations is agreed to, and customers are supplied with cars with configurations that usually exceed their options demand. This gives rise to production extra costs, making the selection of the p configurations an important issue. This problem is called the optimal diversity management problem (ODMP); it was introduced in [3] and [4], and can be stated as follows. Let G =(V,A) be the transitive acyclic digraph of the ordered set (V, ). The elements of V represent the configurations, and v  u iff every option that configuration u can be also activated by v. (It is convenient to interpret each configuration v as the set of options that v activates.) Each arc a =(u, v) of G has a weight w a , which can be viewed as the cost of using the configuration v to substitute u. In [4], w a = c v d u , where c v is the production cost of the configuration v and d u is the demand for the configuration u. A spanning star forest (SSF) of G is a spanning sub-digraph of G where every connected component is a star. In each star, there is a vertex v to which every other vertex is adjacent. We call v the center of the star. Note that every maximal (with respect to the order ) element of V must be the center of some star. The weight of a SSF F is the sum of the weights of the arcs in F .A p-SSF is a SSF with p stars. The ODMP seeks a minimum weight p-SSF. This paper has two major contributions. (i) An alternative proof of the NP-hardness of ODMP, which settles the NP-hardness of the special case of the ODMP where V is a collection of subsets of a finite set N , v  u iff v ⊇ u, and w a = |v|−|u| for every arc a =(u, v) of G. (ii) A report of the computational experience carried out on huge size real instances (we run in- stances whose dimensions are greater than twenty times the greatest dimensions previously reported). Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applica- tions), Vol. 63, Optimal Control, 2009. 956 1072–3374/09/1616–0956 c  2009 Springer Science+Business Media, Inc.