Dynamic programming algorithms for producing food mixture packages by automatic combination weighers

Abstract

The lexicographic bi-criteria combinatorial optimization problem to be discussed in this paper is a mathematical model of the food mixture packing performed by so-called automatic combination weighers, and it is described as follows. We are given a union I = I₁ ∪ I₂ ∪ • • • ∪ I_m of m sets of items, where for each i = 1,2,...,m, I_i = {I_ik | k = 1,2,...,n} denotes a set of n items of the i-th type and I_ik denotes the k-th item of the i-th type. Each item I_ik has an integral weight w_ik and an integral priority γ_ik. The 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 I′ is no less than an integral target weight T, and the sum weight of chosen items of the i-th type for I′_i is no less than an integral indispensable weight b_i. The total weight of chosen items for I′ is minimized as the primary objective, and further 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), we propose an O(nT) time dynamic programming algorithm, applying a linear search technique. We also conduct numerical experiments to demonstrate the empirical performance.