THE MULTIPLE-CHOICE KNAPSACK PROBLEM

Abstract

This paper treats the multiple-choice (continuous) knapsack problem P: maximize Σ^^n__<i=1> Σ^^<m_i__<j=1>c_<ij>x_<ij>subject to (1) Σ^^n__<i=1> Σ^^<m_i__<j=1>c_<ij>x_<ij>&les; b, (2) 0&les; x_<ij>&les; 1,i = 1,2 ,...,n,j=1,2,.... m_i and (3) at most one of x _<i1> , x _<i2>,..., x _<im_i> is positive for i = 1, 2,..., n, where n, m_i are positive integers ancl a_<i1>, c_<i1>, b are 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 m_i = 2, which are randomly generated from the uniform distribut[on. Exact optimal solutions of these problems with n = 500 and m_i = 2 are also obtainedin less than 0.2 seconds (on FACOM M190).