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April  2011, 7(2): 425-434. doi: 10.3934/jimo.2011.7.425

## A market selection and inventory ordering problem under demand uncertainty

 1 Department of Management Science and Engineering, School of Economics and Management, Southeast University, Nanjing 211189, China, China, China

Received  October 2010 Revised  January 2011 Published  April 2011

We study an integrated market selection and inventory control problem that was initially proposed by Geunes et al. [Naval Research Logistics, 51(1):117-136, 2004]. This problem generalizes the classical EOQ problem by incorporating the market choice decisions. In this note, we further consider the problem with stochastic demand in which we assume the demand mean and variance are known for each market. We show that the problem can be formulated as an unconstrained nonlinear binary IP model. Its special structure leads to efficient solution algorithms and we summarize some interesting observations via numerical experiments.
Citation: Jia Shu, Zhengyi Li, Weijun Zhong. A market selection and inventory ordering problem under demand uncertainty. Journal of Industrial and Management Optimization, 2011, 7 (2) : 425-434. doi: 10.3934/jimo.2011.7.425
##### References:
 [1] I. S. Bakal, J. Geunes and H. E. Romeijn, Market selection decisions for inventory models with price-sensitive demand, Journal of Global Optimization, 4 (2008), 633-657. doi: 10.1007/s10898-007-9269-3. [2] K. Chahar and K. Taaffe, Risk averse demand selection with all-or-nothing orders, OMEGA-International Journal of Management Science, 37 (2009), 996-1006. doi: 10.1016/j.omega.2008.11.004. [3] A. K. Chakravarty, J. B. Orlin and U. G. Rothblum, Consecutive optimizers for a partitioning problem with applications to optimal inventory groupings for joint replenishment, Operations Research, 33 (1985), 820-834. doi: 10.1287/opre.33.4.820. [4] M. S. Daskin, C.R. Coullard and Z. J. Max Shen, An inventory-location model: formulation, solution algorithm and computational results, Recent developments in the theory and applications of location models, Part I. Ann. Oper. Res., 110 (2002), 83-106. doi: 10.1023/A:1020763400324. [5] J. Geunes, Z. J. Max Shen and H. E. Romeijn, Economic ordering decisions with market selection flexibility, Naval Research Logistics, 51 (2004), 117-136. doi: 10.1002/nav.10109. [6] J. Geunes, H. E. Romeijn and K. Taaffe, Requirements planning with dynamic pricing and order selection flexibility, Operations Research, 54 (2006), 394-401. doi: 10.1287/opre.1050.0255. [7] J. Geunes, R. Levi, H. E. Romeijn and D. Shmoys, Approximation algorithms for supply chain planning problems with market choice, Mathematical Programming, forthcoming. [8] S. Nahmias, "Production and Operations Management," Irwin, Chicago, 1997. [9] M. Önal and H. E. Romeijn, Two-echelon requirements planning with pricing decisions, Journal of Industrial and Management Optimization, 5 (2009), 767-781. doi: 10.3934/jimo.2009.5.767. [10] L. Ozsen, C. R. Coullard and M. S. Daskin, Capacitated warehouse location model with risk pooling, Naval Research Logistics, 55 (2008), 295-312. doi: 10.1002/nav.20282. [11] L. Ozsen, M. S. Daskin and C. R. Coullard, Facility location modeling and inventory management with multisourcing, Transportation Science, 43 (2009), 455-472. doi: 10.1287/trsc.1090.0268. [12] Z. J. Max Shen, A multi-commodity supply chain design problem, IIE Transactions, 37 (2005), 753-762. doi: 10.1080/07408170590961120. [13] Z. J. Max Shen, C. R. Coullard and M. S. Daskin, A joint location-inventory model, Transportation Science, 37 (2003), 40-55. doi: 10.1287/trsc.37.1.40.12823. [14] J. Shu, C. P. Teo and Z. J. Max Shen, Stochastic transportation-inventory network design problem, Operations Research, 53 (2005), 48-60. doi: 10.1287/opre.1040.0140. [15] L. V. Snyder, M. S. Daskin and C. P. Teo, The stochastic location model with risk pooling, European Journal of Operational Research, 179 (2007), 1221-1238. doi: 10.1016/j.ejor.2005.03.076. [16] K. Taaffe, J. Geunes and H. E. Romeijn, Target market selection and marketing effort under uncertainty: the selective newsvendor, European Journal of Operational Research, 189 (2008), 987-1003. doi: 10.1016/j.ejor.2006.11.049. [17] K. Taaffe, H. E. Romeijn and D. Tirumalasetty, A selective newsvendor approach to order management, Naval Research Logistics, 55 (2008), 769-784. doi: 10.1002/nav.20320. [18] V. N. Vapnik and A. Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and Its Applications, 16 (1971), 264-280. doi: 10.1137/1116025. [19] L. Zhang and S.-Y. Wu, Robust solutions to euclidean facility location problems with uncertain data, Journal of Industrial and Management Optimization, 6 (2010), 751-760. doi: 10.3934/jimo.2010.6.751.

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##### References:
 [1] I. S. Bakal, J. Geunes and H. E. Romeijn, Market selection decisions for inventory models with price-sensitive demand, Journal of Global Optimization, 4 (2008), 633-657. doi: 10.1007/s10898-007-9269-3. [2] K. Chahar and K. Taaffe, Risk averse demand selection with all-or-nothing orders, OMEGA-International Journal of Management Science, 37 (2009), 996-1006. doi: 10.1016/j.omega.2008.11.004. [3] A. K. Chakravarty, J. B. Orlin and U. G. Rothblum, Consecutive optimizers for a partitioning problem with applications to optimal inventory groupings for joint replenishment, Operations Research, 33 (1985), 820-834. doi: 10.1287/opre.33.4.820. [4] M. S. Daskin, C.R. Coullard and Z. J. Max Shen, An inventory-location model: formulation, solution algorithm and computational results, Recent developments in the theory and applications of location models, Part I. Ann. Oper. Res., 110 (2002), 83-106. doi: 10.1023/A:1020763400324. [5] J. Geunes, Z. J. Max Shen and H. E. Romeijn, Economic ordering decisions with market selection flexibility, Naval Research Logistics, 51 (2004), 117-136. doi: 10.1002/nav.10109. [6] J. Geunes, H. E. Romeijn and K. Taaffe, Requirements planning with dynamic pricing and order selection flexibility, Operations Research, 54 (2006), 394-401. doi: 10.1287/opre.1050.0255. [7] J. Geunes, R. Levi, H. E. Romeijn and D. Shmoys, Approximation algorithms for supply chain planning problems with market choice, Mathematical Programming, forthcoming. [8] S. Nahmias, "Production and Operations Management," Irwin, Chicago, 1997. [9] M. Önal and H. E. Romeijn, Two-echelon requirements planning with pricing decisions, Journal of Industrial and Management Optimization, 5 (2009), 767-781. doi: 10.3934/jimo.2009.5.767. [10] L. Ozsen, C. R. Coullard and M. S. Daskin, Capacitated warehouse location model with risk pooling, Naval Research Logistics, 55 (2008), 295-312. doi: 10.1002/nav.20282. [11] L. Ozsen, M. S. Daskin and C. R. Coullard, Facility location modeling and inventory management with multisourcing, Transportation Science, 43 (2009), 455-472. doi: 10.1287/trsc.1090.0268. [12] Z. J. Max Shen, A multi-commodity supply chain design problem, IIE Transactions, 37 (2005), 753-762. doi: 10.1080/07408170590961120. [13] Z. J. Max Shen, C. R. Coullard and M. S. Daskin, A joint location-inventory model, Transportation Science, 37 (2003), 40-55. doi: 10.1287/trsc.37.1.40.12823. [14] J. Shu, C. P. Teo and Z. J. Max Shen, Stochastic transportation-inventory network design problem, Operations Research, 53 (2005), 48-60. doi: 10.1287/opre.1040.0140. [15] L. V. Snyder, M. S. Daskin and C. P. Teo, The stochastic location model with risk pooling, European Journal of Operational Research, 179 (2007), 1221-1238. doi: 10.1016/j.ejor.2005.03.076. [16] K. Taaffe, J. Geunes and H. E. Romeijn, Target market selection and marketing effort under uncertainty: the selective newsvendor, European Journal of Operational Research, 189 (2008), 987-1003. doi: 10.1016/j.ejor.2006.11.049. [17] K. Taaffe, H. E. Romeijn and D. Tirumalasetty, A selective newsvendor approach to order management, Naval Research Logistics, 55 (2008), 769-784. doi: 10.1002/nav.20320. [18] V. N. Vapnik and A. Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and Its Applications, 16 (1971), 264-280. doi: 10.1137/1116025. [19] L. Zhang and S.-Y. Wu, Robust solutions to euclidean facility location problems with uncertain data, Journal of Industrial and Management Optimization, 6 (2010), 751-760. doi: 10.3934/jimo.2010.6.751.
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