The Multiple-choice Knapsack Problem

This paper treats the multiple-choice (continuous) knapsack problem P: n mi n mi maximize L .L cijxijsubjectto(l) I I aij x ij";b,(2)0,,;xij";1,i=I,2, i=l J=l i=l J=1 ... , n, j = 1,2, .... mi and (3) at most one of x il, x i2' ... , x im. is positive for i = 1,2, ., ., n, , where n, mi are positive integers and aij' Cij' bare nonnegative real numbers. Two approximate algorithms and an exact branch-and-bound algorithm are proposed, by making use of the property that the LP relaxation of P provides considerably accurate upper and lower bounds of the optimal value of P. Although the multiple-choice knapsack problem is known to be NP-complete, computation results are quite encouraging. For example, approximate solutions withing 0.001% of the optimal values are obtained in less than one second (on FACOM 230/60) for problems with n = 1000 and mi = 2, which are randomly generated from the uniform distribution. Exact optimal solutions of these problems with n = 500 and mi = 2 are also obtained in less th-an 0.2 seconds (on FACOM M190). 59 © 1978 The Operations Research Society of Japan 60 T. Ibaraki, T. Hasegawa, K. Teranaka and J. Iwase