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A Column-Generation Approach to the Multiple Knapsack Problem with Color Constraints

IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA
IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA
IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA
jjforre{at}us.ibm.com
jayant{at}us.ibm.com
ladanyi{at}us.ibm.com

In this paper, we study a new problem that we refer to as the multiple knapsack with color constraints (MKCP). Motivated by a real application from the steel industry, the MKCP can be formulated by generalizing the multiple knapsack problem. A real-life instance (called mkc) of this problem class is available through MIPLIB (Bixby 2004) and a larger instance (mkc7) is downloadable from the COIN site (IBM 2004). The focus of this paper is to present improved computational results for the two mentioned instances of this problem using a column-generation approach. We solve mkc to optimality and use Dantzig-Wolfe decomposition for upper bounding the other instance. Solving mkc to optimality took less time than it takes to solve the LP relaxation of the original formulation. The larger instance is solved to near optimality (within 0.5% of optimality) in a fraction of the time required to solve the original relaxed LP.

Key words: integer programming; column generation; inventory production applications; multiple knapsack problem
History: received January 2002; revised August 2002; revised October 2002; accepted March 2004.