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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 and Management Optimization, 2016, 12 (3) : 1057-1073. doi: 10.3934/jimo.2016.12.1057
##### References:
 [1] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, San Francisco, 1979. [2] S. Imahori and Y. Karuno, Pseudo-polynomial time algorithms for food mixture packing by automatic combination weighers, in Proceedings of International Symposium on Scheduling 2013 (ISS 2013), 2013, 59-64. [3] S. Imahori, Y. Karuno, H. Nagamochi and X. Wang, Kansei engineering, humans and computers: Efficient dynamic programming algorithms for combinatorial food packing problems, International Journal of Biometrics, 3 (2011), 228-245. doi: 10.1504/IJBM.2011.040817. [4] S. Imahori, Y. Karuno, R. Nishizaki and Y. Yoshimoto, Duplex and quasi-duplex operations in automated food packing systems, in IEEE Xplore of the Fifth IEEE/SICE International Symposium on System Integration (SII 2012), 2012, 810-815. doi: 10.1109/SII.2012.6427267. [5] S. Imahori, Y. Karuno and K. Tateishi, Dynamic programming algorithms for producing food mixture packages by automatic combination weighers, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 8 (2014), 1-11. doi: 10.1299/jamdsm.2014jamdsm0065. [6] Ishida Co., Ltd., Products (Total System Solutions), Weighing and Packaging, 2015. Available from: http://www.ishida.com/products/. [7] K. Kameoka and M. Nakatani, Feed control criterion for a combination weigher and its effects (in Japanese), Transactions of the Society of Instrument and Control Engineers, 37 (2001), 911-915. [8] K. Kameoka, M. Nakatani and N. Inui, Phenomena in probability and statistics found in a combinatorial weigher (in Japanese), Transactions of the Society of Instrument and Control Engineers, 36 (2000), 388-394. [9] Y. Karuno, H. Nagamochi and X. Wang, Bi-criteria food packing by dynamic programming, Journal of the Operations Research Society of Japan, 50 (2007), 376-389. [10] Y. Karuno, H. Nagamochi and X. Wang, Optimization problems and algorithms in double-layered food packing systems, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 4 (2010), 605-615. doi: 10.1299/jamdsm.4.605. [11] Y. Karuno, K. Takahashi and A. Yamada, Dynamic programming algorithms with data rounding for combinatorial food packing problems, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 7 (2013), 233-243. doi: 10.1299/jamdsm.7.233. [12] H. Morinaka, Automatic combination weigher for product foods (in Japanese), Journal of the Japan Society of Mechanical Engineers, 103 (2000), 130-131. [13] H. A. Wurdemann, V. Aminzadeh, J. S. Dai, J. Reed and G. Purnell, Category-based food ordering processes, Trends in Food Science & Technology, 22 (2011), 14-20. doi: 10.1016/j.tifs.2010.10.003. [14] Yamato Scale Co., Ltd., Category Search, Filling and Packaging, 2015. Available from: http://www.yamato-scale.co.jp/en/products/index.

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##### References:
 [1] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, San Francisco, 1979. [2] S. Imahori and Y. Karuno, Pseudo-polynomial time algorithms for food mixture packing by automatic combination weighers, in Proceedings of International Symposium on Scheduling 2013 (ISS 2013), 2013, 59-64. [3] S. Imahori, Y. Karuno, H. Nagamochi and X. Wang, Kansei engineering, humans and computers: Efficient dynamic programming algorithms for combinatorial food packing problems, International Journal of Biometrics, 3 (2011), 228-245. doi: 10.1504/IJBM.2011.040817. [4] S. Imahori, Y. Karuno, R. Nishizaki and Y. Yoshimoto, Duplex and quasi-duplex operations in automated food packing systems, in IEEE Xplore of the Fifth IEEE/SICE International Symposium on System Integration (SII 2012), 2012, 810-815. doi: 10.1109/SII.2012.6427267. [5] S. Imahori, Y. Karuno and K. Tateishi, Dynamic programming algorithms for producing food mixture packages by automatic combination weighers, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 8 (2014), 1-11. doi: 10.1299/jamdsm.2014jamdsm0065. [6] Ishida Co., Ltd., Products (Total System Solutions), Weighing and Packaging, 2015. Available from: http://www.ishida.com/products/. [7] K. Kameoka and M. Nakatani, Feed control criterion for a combination weigher and its effects (in Japanese), Transactions of the Society of Instrument and Control Engineers, 37 (2001), 911-915. [8] K. Kameoka, M. Nakatani and N. Inui, Phenomena in probability and statistics found in a combinatorial weigher (in Japanese), Transactions of the Society of Instrument and Control Engineers, 36 (2000), 388-394. [9] Y. Karuno, H. Nagamochi and X. Wang, Bi-criteria food packing by dynamic programming, Journal of the Operations Research Society of Japan, 50 (2007), 376-389. [10] Y. Karuno, H. Nagamochi and X. Wang, Optimization problems and algorithms in double-layered food packing systems, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 4 (2010), 605-615. doi: 10.1299/jamdsm.4.605. [11] Y. Karuno, K. Takahashi and A. Yamada, Dynamic programming algorithms with data rounding for combinatorial food packing problems, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 7 (2013), 233-243. doi: 10.1299/jamdsm.7.233. [12] H. Morinaka, Automatic combination weigher for product foods (in Japanese), Journal of the Japan Society of Mechanical Engineers, 103 (2000), 130-131. [13] H. A. Wurdemann, V. Aminzadeh, J. S. Dai, J. Reed and G. Purnell, Category-based food ordering processes, Trends in Food Science & Technology, 22 (2011), 14-20. doi: 10.1016/j.tifs.2010.10.003. [14] Yamato Scale Co., Ltd., Category Search, Filling and Packaging, 2015. Available from: http://www.yamato-scale.co.jp/en/products/index.
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