Continuous Knapsack Problems

TIle multiple-choice continuous knapsack problem is dermed as follows: maximize z = ~n ~~i CjJ'XiJ' 1=1 J= 1 subject to (1) ~n ~mi aijXij .;;; b, (2) 0 .;;; Xij .;;; I, i = 1,2, ... , n,j = 1. 2, ... , mi, (3) at most one of Xij G = 1,2, ... , mi) 1=1 J=1 is positive for each i = I, 2, ... , n, where n, mi are positive integers and aij are nonnegative integers. In this paper, it is proved that this problem is NP-complete (which strongly suggests the computational intractability to obtain exact optimal solutions). Then, starting with an approximate solution obtained from the LP optimal solution by rounding, two approximate algorithms are proposed and analyzed. TIle first one (called the breadth-l search method) obtains an approximate solution with the (worst case) relative error less than 25%. The required computation time is 0 (mN 10gN), where N = ~~= 1 mi and m = max mi. TIle second one (called the breadth-K search method) obtains an approximate solution within an arbitrarily specified (worst case) error bound € x 100% in 0 (m rl/4(e€2 n N rl/4(e-€2 n ) time.