The Multiple Knapsack Problem With Color ConstraintsMilind

LIMITED DISTRIBUTION NOTICE This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and speciic requests. After outside publication, requests should be lled only by reprints or legally obtained copies of the article (e.g., payment of royalties). 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 deening inequalities and four classes of other valid inequalities are discussed. We establish necessary and suucient conditions under which a color constraint is redundant and also provide separate necessary conditions and suucient conditions under which a color constraint is facet deening. In addition, we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet deening inequality associated with an individual knapsack constraint.