Improved Heuristics with Data Rounding for Combinatorial Food Packing Problems

Given a set I = {i | i = 1, 2, . . . , n} of current n items (for example, n green peppers) with their weights w_i and priorities r_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 bound b for each package, and it is minimized as the primary objective, and further the total priority Σ_i∈I' r_i is maximized as the second objective. The problem has been known to be NP-hard, while it can be solved exactly in O(nb) time if all the input data are assumed to be integral. For a given real ε > 0, an O(n²/ε) time heuristic algorithm with a data rounding technique has been designed and the heuristic total weight has been shown to be at most (2+ε) times the optimal total weight. In this paper, a modification of the data rounding heuristic is proposed, and it is shown that the proposed modification delivers a heuristic solution such that the total weight is at most (1 + ε) times the optimum.