A Two-Level Purchase Problem for Food Retailing in Japan


In this paper, we deal with a purchase problem for food retailing, and formulate a two-level linear programming problem with a food retailer and a distributer. The food retailer deals with vegetables and fruits which are purchased from the distributer; the distributer buys vegetables and fruits ordered from the food retailer at the central wholesale markets in several cities, and transports them by truck from each of the central wholesaler markets to the food retailer’s storehouse. We solve the two-level linear programming problem in which the profits of the food retailer and the distributer are maximized.

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M. Sakawa, I. Nishizaki, T. Matsui and T. Hayashida, "A Two-Level Purchase Problem for Food Retailing in Japan," American Journal of Operations Research, Vol. 2 No. 4, 2012, pp. 482-494. doi: 10.4236/ajor.2012.24057.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] M. Simaan and J. B. Cruz, “On the Stackelberg Strategy in Nonzero-Sum Games,” Journal of Optimization Theory and Applications, Vol. 11, No. 5, 1973, pp. 533-555. doi:10.1007/BF00935665
[2] J. F. Bard, “Some properties of the bilevel programming problem,’’ Journal of Optimization Theory and Applications, Vol. 68, No. 2, 1991, pp. 371-378. doi:10.1007/BF00941574
[3] R. G. Jeroslow, “The Polynomial Hierarchy and a Simple Model for Competitive Analysis,” Mathematical Programming, Vol. 32, No. 2, 1985, pp. 146-164. doi:10.1007/BF01586088
[4] J. F. Bard, “An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem,” Operations Research, Vol. 31, No. 4, 1983, pp. 556-560. doi:10.1287/opre.31.4.670
[5] J. F. Bard and J. E. Falk, “An Explicit Solution to the Multi-Level Programming Problem,” Computers and Operations Research, Vol. 9, No. 1, 1982, pp. 77-100. doi:10.1016/0305-0548(82)90007-7
[6] J. F. Bard and J. T. Moore, “A Branch and Bound Algorithm for the Bilevel Programming Problem,” SIAM Journal on Scientific and Statistical Computing, Vol. 11, No. 2, 1990, pp. 281-292. doi:10.1137/0911017
[7] W. F. Bialas and M. H. Karwan, “Two-Level Linear Programming,” Management Science, Vol. 30, No. 8, 1984, pp. 1004-1020. doi:10.1287/mnsc.30.8.1004
[8] P. Hansen, B. Jaumard and G. Savard, “New Branch-and-Bound Rules for Liner Bilevel Programming,” SIAM Journal of Scientific and Statistical Computing, Vol. 13, No. 5, 1992, pp. 1194-1217. doi:10.1137/0913069
[9] J. J. J’udice and A. M. Faustino, “A Sequential LCP Method for Bilevel Linear Programming,” Annals of Operations Research, Vol. 34, No. 1, 1992, pp. 89-106. doi:10.1007/BF02098174
[10] D. J. White and G. Anandalingam, “A Penalty Function Approach for Solving Bilevel Linear Programs,” Journal of Global Optimization, Vol. 3, No. 4, 1993, pp. 397-419. doi:10.1007/BF01096412
[11] J. F. Bard, “Practical Bilevel Optimization: Algorithms and Applications,” Kluwer Academic Publisher, Dordrecht, 1998.
[12] J. F. Bard and J. T. Moore, “Production Planning with Variable Demand,” Omega, Vol. 18, No. 1, 1990, pp. 35- 42. doi:10.1016/0305-0483(90)90016-3
[13] O. Ben-Ayed, D. E. Boyce and C. E. Blair, “A General Bilevel Linear Programming Formulation of the Network Design Problem,” Transportation Research, Vol. 22, No. 4, 1988, pp. 311-318. doi:10.1016/0191-2615(88)90006-9
[14] P. Marccote, “Network Design Problem with Congestion Effects: A Case of Bilevel Programming,” Mathematical Programming, Vol. 34, No. 2, 1986, pp. 142-162. doi:10.1007/BF01580580
[15] Ministry of Agriculture, Forestry and Fisheries of Japan, “Summary of Report on Price Formation in Each Stage of Food Distribution 2008,” Ministry of Agriculture, Forestry and Fisheries of Japan, Tokyo, 2008.
[16] M. Kidachi, “Evolution and Development of Retailing- Oriented Distribution Systems,” In: Kidate and Tatsuma, Eds., Theory, History and Analysis on Distributive Trades, Chuo University Press, Tokyo, 2006, pp. 133-174. (In Japanese)
[17] M. Geuens, M. Brengman and R. S’Jegers, “Food Retailing, Now and in the Future: A Consumer Perspective,” Journal of Retailing and Consumer Services, Vol. 10, No. 4, 2003, pp. 241-251. doi:10.1016/S0969-6989(02)00017-6
[18] L. C. Harris and E. Ogbonna, “Competitive Advantage in the UK Food Retailing Sector: Past, Present and Future,” Journal of Retailing and Consumer Services, Vol. 8, No. 3, 2001, pp. 157-173. doi:10.1016/S0969-6989(00)00009-6
[19] T. A. Park and R. P. King, “Evaluating Food Retailing Efficiency: The Role of Information Technology,” Journal of Productivity Analysis, Vol. 27, No. 2, 2007, pp. 101-113. doi:10.1007/s11123-006-0030-6
[20] N. Hibara, “Food Retailing: Ito-Yokado Group; Gaining and Sustaining Long-Term Advantage through Information Technology,” Columbia University, New York, 2000.
[21] M. Conlon, “Japan Retail Food Sector: Japanese Retail Food Sector Report 2006,” Data Resource International Inc., Pompano Beach, 2006.
[22] O. Ahumada andJ. R. Villalobos, “Application of Planning Models in the Agri-Food Supply Chain: A Review,” European Journal of Operational Research, Vol. 195, No. 1, 2009, pp. 1-20. doi:10.1016/j.ejor.2008.02.014
[23] M. Erkoc, E. T. Iakovou and A. E. Spaulding, “Multi- Stage Onboard Inventory Management Policies for Food and Beverage Items in Cruise Liner Operations,” Journal of Food Engineering, Vol. 70, No. 3, 2005, pp. 269-279. doi:10.1016/j.jfoodeng.2004.04.044
[24] P. Georgiadis, D. Vlachos and E. Iakovou, “A System Dynamics Modeling Framework for the Strategic Supply Chain Management of Food Chains,” Journal of Food Engineering, Vol. 70, No. 3, 2005, pp. 351-364. doi:10.1016/j.jfoodeng.2004.06.030
[25] C.-T. Lin and Y. M. Chen, “Hedging Strategic Flexibility in the Distribution Optimization Problem,” Omega, Vol. 37, No. 4, 2009, pp. 826-837. doi:10.1016/j.omega.2008.07.008
[26] E. Davis, M. Freedman, J. Lane, B. McCall, N. Nestoriak and T. Park, “Product Market Competition and Human Resource Practices in the Retail Food Sector,” Industrial Relations, Vol. 48, No. 2, 2009, pp. 350-371. doi:10.1111/j.1468-232X.2009.00561.x
[27] I. Erol, N. Cakar, D. Erel and R. Sari, “Sustainability in the Turkish Retailing Industry,” Sustainable Development, Vol. 17, No. 1, 2009, pp. 49-67. doi:10.1002/sd.369
[28] K. Tamura, “Economic Analysis on Innovation of Distribution Systems in Japan: Sustainable and Selective Innovative Changes of Japanese Style of Distribution,” Kyushu University Press, Kyushu, 1998.
[29] J. F. Bard, “An Investigation of the Linear Three-Level Programming Problem,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-14, No. 5, 1984, pp. 711-717. doi:10.1109/TSMC.1984.6313291

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