The integrated size and price optimization problem

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2012, 2(4): 669-693. doi: 10.3934/naco.2012.2.669

The integrated size and price optimization problem

Miriam Kiessling ^1, , Sascha Kurz ^2, and Jörg Rambau ^2,

1.	Universität Bayreuth, 95440 Bayreuth, Germany
2.	Universität Bayreuth, 95440 Bayreuth, Germany, Germany

Received December 2011 Revised October 2012 Published November 2012

Abstract
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We present the Integrated Size and Price Optimization Problem (ISPO) for a fashion discounter with many branches. Based on a two-stage stochastic programming model with recourse, we develop an exact algorithm and a production-compliant heuristic that produces small optimality gaps. In a field study we show that a distribution of supply over branches and sizes based on ISPO solutions is significantly better than a one-stage optimization of the distribution ignoring the possibility of optimal pricing.

Keywords: fashion discounter, Supply chain, lot-type, stochastic optimization with recourse, mark-down, field study., pre-pack.

Mathematics Subject Classification: Primary: 90B90, Secondary: 90B0.

Citation: Miriam Kiessling, Sascha Kurz, Jörg Rambau. The integrated size and price optimization problem. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 669-693. doi: 10.3934/naco.2012.2.669

References:

[1]	Elodie Adida and Georgia Perakis, A robust optimization approach to dynamic pricing and inventory control with no backorders, Math. Program., 107 (2006), 97-129. doi: 10.1007/s10107-005-0681-5.
[2]	Elodie Adida and Georgia Perakis, Dynamic pricing and inventory control: Uncertainty and competition, Oper. Res., 58 (2010), 289-302. doi: 10.1287/opre.1090.0718.
[3]	Dimitris Bertsimas and Sanne de Boer, Dynamic pricing and inventory control for multiple products, JRPM, 3 (2005), 303-319. doi: 10.1057/palgrave.rpm.5170117.
[4]	John R. Birge and Francois Louveaux, "Introduction To Stochastic Programming," 2^nd ed, Springer Series in Operations Research and Financial Engineering, New York, 2011. doi: 10.1007/978-1-4614-0237-4.
[5]	Gabriel Bitran and Reneé Caldentey, An overview of pricing models for revenue management, MSOM, 5 (2003), 203-229. doi: 10.1287/msom.5.3.203.16031.
[6]	Lap M. A. Chan, Zuo-Jun M. Shen, David Simchi-Levi and Julie Swann, Coordination of pricing and inventory decisions: A survey and classification, Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, (2004), 335-392.
[7]	Andreas Christmann and Ingo Steinwart, "Support Vector Machines," Information Science and Statistics, Springer, 2008.
[8]	Antonio J. Conejo, Enrique Castillo, Roberto Minguez and Raquel Garcia-Bertrand, "Decomposition Techniques in Mathematical Programming: Engineering and Science Applications," Springer, 2010.
[9]	Awi Federgruen and Aliza Heching, Combined pricing and inventory control under uncertainty, Oper. Res., 47 (1999), 454-475. doi: 10.1287/opre.47.3.454.
[10]	David Freedman, Robert Pisani and Roger Purves, "Statistics," 4^th Edition, W. W. Norton & Company, 2007.
[11]	Constantin Gaul, Sascha Kurz and Jörg Rambau, On the lot-type design problem, Optim. Methods Softw., 25 (2010), 217-227. doi: 10.1080/10556780902965163.
[12]	Guillermo Gallego and Garrett van Ryzin, Optimal dynamic pricing of inventories with stochastic demand over finite horizons, Manage. Sci., 40 (1994), 999-1020. doi: 10.1287/mnsc.40.8.999.
[13]	Guillermo Gallego and Garrett van Ryzin, A multiproduct dynamic pricing problem and its applications to network yield management, Oper. Res., 45 (1997), 24-41. doi: 10.1287/opre.45.1.24.
[14]	Lars Grüne and Jürgen Pannek, "Nonlinear Model Predictive Control," Communications and Control Engineering, Springer, 2011. doi: 10.1007/978-0-85729-501-9.
[15]	Miriam Kießling, Sascha Kurz and Jörg Rambau, An exact column-generation approach for the lot-type design problem, Preprint, Universität Bayreuth, 2012.
[16]	Kaisa M. Miettinen, "Evolutionary Algorithms In Engineering & Computer Science," John Wiley & Sons LTD, 1999.
[17]	Alan L. Montgomery, The implementation challenge of pricing decision support systems for retail managers, Appl. Stoch. Models Bus. Ind., 21 (2005), 367-378. doi: 10.1002/asmb.572.
[18]	Serguei Netessine, Dynamic pricing of inventory/capacity with infrequent price changes, Eur. J. Oper. Res., 174 (2006), 553-580. doi: 10.1016/j.ejor.2004.12.015.
[19]	Frank Wilcoxon, Individual comparisons by ranking methods, Biometrics Bulletin, 1 (1945), 80-83. doi: 10.2307/3001968.
[20]	Rui Yin, Yossi Aviv, Amat Pazgal and Christopher S. Tang, Optimal markdown pricing: implications of inventory display formats in the presence of strategic customers, Manage. Sci., 55 (2009), 1391-1408. doi: 10.1287/mnsc.1090.1029.
[21]	Wen Zhao and Yu-Sheng Zheng, Optimal dynamic pricing for perishable assets with nonhomogeneous demand, Manage. Sci., 46 (2000), 375-388. doi: 10.1287/mnsc.46.3.375.12063.

show all references

References:

[1]	Elodie Adida and Georgia Perakis, A robust optimization approach to dynamic pricing and inventory control with no backorders, Math. Program., 107 (2006), 97-129. doi: 10.1007/s10107-005-0681-5.
[2]	Elodie Adida and Georgia Perakis, Dynamic pricing and inventory control: Uncertainty and competition, Oper. Res., 58 (2010), 289-302. doi: 10.1287/opre.1090.0718.
[3]	Dimitris Bertsimas and Sanne de Boer, Dynamic pricing and inventory control for multiple products, JRPM, 3 (2005), 303-319. doi: 10.1057/palgrave.rpm.5170117.
[4]	John R. Birge and Francois Louveaux, "Introduction To Stochastic Programming," 2^nd ed, Springer Series in Operations Research and Financial Engineering, New York, 2011. doi: 10.1007/978-1-4614-0237-4.
[5]	Gabriel Bitran and Reneé Caldentey, An overview of pricing models for revenue management, MSOM, 5 (2003), 203-229. doi: 10.1287/msom.5.3.203.16031.
[6]	Lap M. A. Chan, Zuo-Jun M. Shen, David Simchi-Levi and Julie Swann, Coordination of pricing and inventory decisions: A survey and classification, Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, (2004), 335-392.
[7]	Andreas Christmann and Ingo Steinwart, "Support Vector Machines," Information Science and Statistics, Springer, 2008.
[8]	Antonio J. Conejo, Enrique Castillo, Roberto Minguez and Raquel Garcia-Bertrand, "Decomposition Techniques in Mathematical Programming: Engineering and Science Applications," Springer, 2010.
[9]	Awi Federgruen and Aliza Heching, Combined pricing and inventory control under uncertainty, Oper. Res., 47 (1999), 454-475. doi: 10.1287/opre.47.3.454.
[10]	David Freedman, Robert Pisani and Roger Purves, "Statistics," 4^th Edition, W. W. Norton & Company, 2007.
[11]	Constantin Gaul, Sascha Kurz and Jörg Rambau, On the lot-type design problem, Optim. Methods Softw., 25 (2010), 217-227. doi: 10.1080/10556780902965163.
[12]	Guillermo Gallego and Garrett van Ryzin, Optimal dynamic pricing of inventories with stochastic demand over finite horizons, Manage. Sci., 40 (1994), 999-1020. doi: 10.1287/mnsc.40.8.999.
[13]	Guillermo Gallego and Garrett van Ryzin, A multiproduct dynamic pricing problem and its applications to network yield management, Oper. Res., 45 (1997), 24-41. doi: 10.1287/opre.45.1.24.
[14]	Lars Grüne and Jürgen Pannek, "Nonlinear Model Predictive Control," Communications and Control Engineering, Springer, 2011. doi: 10.1007/978-0-85729-501-9.
[15]	Miriam Kießling, Sascha Kurz and Jörg Rambau, An exact column-generation approach for the lot-type design problem, Preprint, Universität Bayreuth, 2012.
[16]	Kaisa M. Miettinen, "Evolutionary Algorithms In Engineering & Computer Science," John Wiley & Sons LTD, 1999.
[17]	Alan L. Montgomery, The implementation challenge of pricing decision support systems for retail managers, Appl. Stoch. Models Bus. Ind., 21 (2005), 367-378. doi: 10.1002/asmb.572.
[18]	Serguei Netessine, Dynamic pricing of inventory/capacity with infrequent price changes, Eur. J. Oper. Res., 174 (2006), 553-580. doi: 10.1016/j.ejor.2004.12.015.
[19]	Frank Wilcoxon, Individual comparisons by ranking methods, Biometrics Bulletin, 1 (1945), 80-83. doi: 10.2307/3001968.
[20]	Rui Yin, Yossi Aviv, Amat Pazgal and Christopher S. Tang, Optimal markdown pricing: implications of inventory display formats in the presence of strategic customers, Manage. Sci., 55 (2009), 1391-1408. doi: 10.1287/mnsc.1090.1029.
[21]	Wen Zhao and Yu-Sheng Zheng, Optimal dynamic pricing for perishable assets with nonhomogeneous demand, Manage. Sci., 46 (2000), 375-388. doi: 10.1287/mnsc.46.3.375.12063.

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