Dynamic programming algorithms for duplex food packing problems

In this paper, we discuss a lexicographic bi-criteria combinatorial optimization problem arising in automated food packing systems known as so-called automatic combination weighers. A typical food packing system possesses n weighing hoppers. Some amount of foods is thrown into each hopper, and it is called an item. We deal with a duplex packing operation such that the food packing system chooses two disjoint subsets I' and I" from the set I of the current n items to produce two packages of foods. After choosing two subsets I' and I", the resulting empty hoppers are supplied with next new items, and the set I is updated. By repeating the duplex packing operation, a large number of packages are produced two by two. The primary objective of lexicographic bi-criteria duplex food packing problem is to minimize the total weight of chosen items for two packages, making the total weight of each package no less than a specified target weight T. The second objective is to maximize the total priority of chosen items for two packages so that items with longer durations in hoppers are preferably chosen. The priority of an item is given as its duration in hopper. In this paper, we prove that the lexicographic bi-criteria duplex food packing problem can be solved in O(nT²) time by dynamic programming if all input data are integral.