Abstract
In this paper, a lexicographic bi-criteria combinatorial optimization problem arising in actual food packaging equipments and its solutions are considered. In each packaging operation, a set I = {i | i = 1, 2, . . . , n} of current n items (for example, n green peppers) with their weights wi and priorities pi , and a target weight t of a package are given. Then the problem asks to find a subset I ′ ⊆ I so that the total weight ∑ i∈I ′ wi is minimized as the primary objective, meeting the target weight constraint ∑ i∈I ′ wi ≥ t, and the total priority ∑ i∈I ′ pi is maximized as the second objective. Such a problem is repeatedly solved during an operating run of the food packing system, in which a large number of packages are produced one by one. The priority of an item is defined to be the duration of the item in the food packing system, and it is expected to avoid the frequent occurrence of items with longer durations. In this paper, all input data are assumed to be integral, and the effectiveness of the lexicographic bi-criteria solutions in an operating run is empirically observed by means of an O(nt) time exact algorithm for the food packing problem.
Highlights
A directed bipartite graph with a set I of m items and a set J of n players is given
The weighted item collecting problem to be discussed in this paper asks to find an arc reversing strategy of the n players which maximizes the objective function value of the total profit of collected items minus a penalty value with the total cost paid by the players
We considered the weighted item collecting problem (WIC for short) in directed bipartite structure containing a set of items with profits and a set of players with costs
Summary
A directed bipartite graph with a set I of m items and a set J of n players is given. The weighted item collecting problem to be discussed in this paper asks to find an arc reversing strategy of the n players which maximizes the objective function value of the total profit of collected items minus a penalty value with the total cost paid by the players. A modification of the total profit objective is considered, i.e., as mentioned in the above, the objective function consists of the total profit term of collected items and a penalty term with the total cost of the players Such an objective may tend to regard a smaller change from the initial directions of arcs (e.g., from the initial circuit design) as desirable in the final solution of an are reversing strategy. Numerical experiments are conducted to demonstrate the effectiveness of the preprocessing procedure and the performance of the proposed greedy heuristic algorithm
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