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July  2016, 12(3): 1057-1073. doi: 10.3934/jimo.2016.12.1057

## Pseudo-polynomial time algorithms for combinatorial food mixture packing problems

 1 Faculty of Science and Engineering, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan 2 Faculty of Mechanical Engineering, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan 3 Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan

Received  November 2014 Revised  April 2015 Published  September 2015

A union $\mathcal{I}=\mathcal{I}_{1}\cup \mathcal{I}_{2}\cup \cdots \cup \mathcal{I}_{m}$ of $m$ sets of items is given, where for each $i=1,2,\ldots,m$, $\mathcal{I}_{i}=\{I_{ik} \mid k=1,2,\ldots,n\}$ denotes a set of $n$ items of the $i$-th type and $I_{ik}$ denotes the $k$-th item of the $i$-th type. Each item $I_{ik}$ has an integral weight $w_{ik}$ and an integral priority $p_{ik}$. The food mixture packing problem to be discussed in this paper asks to find a union $\mathcal{I}'=\mathcal{I}'_{1}\cup \mathcal{I}'_{2}\cup \cdots \cup \mathcal{I}'_{m}$ of $m$ subsets of items so that for each $i=1,2,\ldots,m$, the sum weight of chosen items of the $i$-th type for $\mathcal{I}'_{i} \subseteq \mathcal{I}_{i}$ is no less than an integral indispensable bound $b_{i}$, and the total weight of chosen items for $\mathcal{I}'$ is no less than an integral target weight $t$. The total weight of chosen items for $\mathcal{I'}$ is minimized as the primary objective, and further the total priority of chosen items for $\mathcal{I'}$ is maximized as the second objective. In this paper, the known time complexity $O(mnt+mt^{m})$ is improved to $O(mnt+mt^{2})$ for an arbitrary $m\geq 3$ by a two-stage constitution algorithm with dynamic programming procedures. The improved time complexity figures out the weak NP-hardness of the food mixture packing problem.
Citation: Shinji Imahori, Yoshiyuki Karuno, Kenju Tateishi. Pseudo-polynomial time algorithms for combinatorial food mixture packing problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1057-1073. doi: 10.3934/jimo.2016.12.1057
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