Abstract
Given a set I = {i|i=1,2, ... ,n} of current n items (for example, n green peppers) with their weights wi and priorities p_i, a lexicographic bi-criteria combinatorial food packing problem asks to find a subset I'(⊆I) so that the total weight Σ_<i∈I'>w_i is no less than a specified target amount t for each package, and it is minimized as the primary objective, and further the total priority Σ_<i∈I'>p_i is maximized as the second objective. The food packing problem has been known to be NP-hard, while it can be solved exactly in O(nt) time if all the input data are assumed to be integral. We have designed a heuristic algorithm such that for a given real ε>0, it delivers a feasible subset I' with the total weight at most (1+ε) times the optimum, and it runs in 0(n^2/ε) time. In this paper, we show a complementary property of an optimal solution in rounded instances to be solved by the heuristic algorithm.
