Bi-criteria Food Packing by Dynamic Programming

In this paper, we deal with a certain type of automated food packing system with n hoppers. The system repeats throwing some amount of foods into empty hoppers to make all hoppers full of foods, and collecting the foods from several hoppers to pack them into a single package. We treat foods thrown into each hopper as an item i with an integer weight wi. Given a set I of n items in the hoppers, the packing system chooses a subset I ′ (⊆ I) of items, and puts them into a package so that the total weight of items is at least a specified target weight B. The packing system then throws new items into the empty hoppers, and the set I is updated to be the union of the remaining items in I − I ′ and the new items. Repeating such packing operations, it produces a large number of packages one by one. In the packing system, an item may stay for a long time in some hopper until it is chosen for packing. To avoid such a situation, we introduce a priority γi for each item i, and formulate the problem of choosing a subset I ′ as a bi-criteria discrete optimization problem in which one objective is to minimize ∑ i∈I′ wi such that ∑ i∈I′ wi ≥ B must be satisfied, and the other is to maximize ∑ i∈I′ γi. In this paper, we propose an O(n wmax) time algorithm based on dynamic programming to obtain a nondominated solution, where wmax is an upper bound on the weight of an item. We also report the results on computational experiments conducted to examine the performance of the proposed approach.