Abstract

In this paper, we study a new problem which we call the multiple knapsack problem with color constraints (MKCP). Motivated by a real application from the steel industry [6], we generalize the multiple knapsack problem by adding a new attribute (called "color" in the paper) to an item and then adding the associated "color" constraints which restrict the number of distinct colors which can be assigned to a knapsack. The color constraints provide a representation for a general class of processing constraints. We consider the MKCP from a polyhedral point of view. Three classes of facet defining inequalities and four classes of other valid inequalities are discussed. We establish necessary and sufficient conditions under which a color constraint is redundant and also provide separate necessary conditions and sufficient conditions under which a color constraint is facet defining. In addition, we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet defining inequality associated with an individual knapsack constraint.

Citations