Dynamic Programming Algorithms with Data Rounding for Combinatorial Food Packing Problems

Abstract

The lexicographic bi-criteria combinatorial food packing problem to be discussed in this paper is described as follows. 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 γ i , the problem asks to find a subset I (⊆ I) so that the total weight i∈I w i is no less than a specified target weight T for each package, and it is minimized as the primary objective, and further the total priority i∈I γ i is maximized as the second objective. The 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. In this paper, we design a heuristic algorithm for the problem by applying a data rounding technique to an O(nT ) time dynamic programming procedure. We also conduct numerical experiments to examine the empirical performance such as execution time and solution quality.

Heuristic algorithms with rounded weights for a combinatorial food packing problem

“…In this section, we review a problem formulation of the lexicographic bi-criteria food packing and a known rounded instance which has been introduced to the problem by Karuno et al (2013). We call the food packing problem with respect to the lexicographic bi-criteria (that is, the total weight of chosen items as the primary objective and total priority of the chosen items as the second objective) LEXICO for short.…”

“…That is, there is a relationship of trade-off between the quality of the heuristic total weight of chosen items and the execution time of the heuristic algorithm with rounded weights. Formally, the rounded instance of a given instance of problem LEXICO introduced by Karuno et al (2013) is summarized as follows:…”

“…We expect the items with longer durations in hoppers to be preferably chosen by means of the second objective. In fact, the previous numerical results have indicated the effectiveness of the second objective upon reducing the durations in hoppers of items (e.g., see Imahori et al, 2011;Karuno et al, 2013).…”

“…However, the food packing algorithms employed in their papers basically enumerate O(2 n ) subsets I ′ of the set I from the viewpoint of O notation (e.g., see Ibaraki, 1999 for the definition of O notation). For a given real ε > 0, Karuno et al (2013) have designed an O(n 2 /ε) time heuristic algorithm such that the heuristic total weight is at most (2 + ε) times the optimal total weight. The heuristic algorithm has been obtained by applying a rounding technique to an O(nt) time dynamic programming procedure (e.g., see Imahori et al, 2011).…”

“…Afterward, Karuno and Tateishi (2014) have showed that for a given real ε > 0, the problem admits an O(n 2 /ε) time heuristic algorithm such that the heuristic total weight is at most (1 + ε) times the optimal total weight. However, the latter heuristic algorithm with rounded weights can hardly regard the total priority objective, since it prefers to utilize the original weights of current items rather than their priorities, while the former heuristic algorithm by Karuno et al (2013) does not use the original weights of current items in the computation process, but it regards the priorities as well as the rounded weights.…”