Abstract
Let I = I_1 ∪ I_2 ∪…∪ Im denote a union of m sets of items, where for each i = 1,2,…,m,I_i = {I_i_k|k=1,2,…,n} denotes a set of n items of the i-th type and Iik denotes the k-th item of the i-th type with an integral weight Wik and an integral priority γik.The food mixture packing problem asks to find a union I' = I'_1 ∪ I'_2 ∪…∪I'm of m subsets of items where I'_i ⊆I'_i so that the total weight of chosen items for r is no less than an integral target weight T, and the sum weight of chosen items of the i-th type for I's is no less than an integral indispensable weight bi. The total weight of chosen items for I' is minimized as the primal objective, and the total priority of chosen items for I' is maximized as the second objective. For the case in which there are two types of items (i.e., m = 2), an O(nT) time dynamic programming algorithm has been designed. A new upper bound on the sum weight of chosen items of each type in an optimal solution is presented, and it empirically improves the execution time of the O(nT) time dynamic programming algorithm.
