Heuristic algorithms with rounded weights for a combinatorial food packing problem

Abstract

In this paper, a lexicographic bi-criteria food packing problem arising in actual packaging equipments is considered. 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 p i , the problem asks to find a subset I ′ (⊆ I) so that the total weight ∑ i∈I ′ w i is no less than a given positive t which denotes a target weight for each package, and it is minimized as the primary objective, and further the total priority ∑ i∈I ′ p 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 integral. In this paper, for a given real ε > 0, an O(n 2 /ε) time heuristic algorithm with rounded weights is proposed such that the heuristic total weight is at most (1 + ε) times the optimal total weight. Numerical experiments are also conducted to compare the proposed and known heuristic algorithms with rounded weights, and the results are reported.

An improved performance of greedy heuristic solutions for a bi-criteria mixture packaging problem of two types of items with bounded weights

“…In this paper, we are treating the problem with m = 2. Karuno and Saito (2017) have considered the problem with m = 1, and showed that a greedy heuristic solution with the total weight at most twice the minimum attains the total priority at least the conditionally maximum. As mentioned in the previous section, Karuno and Nakahama (2017) have started the performance guarantee analysis of greedy heuristic solutions for the problem with m = 2, but they have only illustrated an instance-dependent ratio of the greedy heuristic total priority to the conditionally maximum, which is unfortunately less than one.…”

Optimisation algorithms for improvement of a multihead weighing process

“…This algorithm aimed to minimise the maximum time items spent in the system heuristically, while also keeping the total weight of each package as close to the target weight as possible. Some authors (Imahori et al, 2011(Imahori et al, , 2012Karuno et al, 2013;Karuno and Tateishi, 2014;Karuno and Saito, 2017) have studied the possibility of improving this bi-objective optimisation model. Other authors, such as Imahori et al (2012) and Karuno et al (2010), have investigated different types of packaging operations, developing several algorithms for double-layered and duplex packaging systems.…”

Performance of a Heuristic Total Weight in Combinatorial Mixture Packaging of Two Types of Items

“…Problem R-i is the minimum knapsack problem and also for the problem, there is a pseudo-polynomial time dynamic programming procedure (8,10,11) . By using the dynamic programming procedure with a small modification, as Lemma 1, we also have the following lemma : Lemma 2.…”