Single-period inventory model with discrete stochastic demand based on prospect theory

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July 2012, 8(3): 577-590. doi: 10.3934/jimo.2012.8.577

Single-period inventory model with discrete stochastic demand based on prospect theory

Wei Liu ^1, , Shiji Song ^1, and Cheng Wu ^1,

1.	Department of Automation, Tsinghua University, Beijing 100084, China, China, China

Received March 2011 Revised November 2011 Published June 2012

Abstract
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This paper studies a single-period inventory (newsvendor) problem with discrete stochastic demand. In general, most of the previous works are based on the expected profit/cost criterion or expected utility criterion. We consider the effect of irrational factor under uncertainty and therefore incorporate prospect theory into inventory model. Our objective is to maximize the overall value of the prospect, which can be calculated by using the value function and the weighting function. For any given initial inventory level, it can be shown that a state-dependent order-up-to policy is optimal. Further, the optimal policy has a simple structure, and the retailer can easily decide whether to place an order or not. Moreover, the impacts of parameters on the optimal policy are illustrated through numerical experiments.

Keywords: Prospect theory, inventory, the overall value of the prospect, order-up-to policy..

Mathematics Subject Classification: Primary: 90B05; Secondary: 91B4.

Citation: Wei Liu, Shiji Song, Cheng Wu. Single-period inventory model with discrete stochastic demand based on prospect theory. Journal of Industrial and Management Optimization, 2012, 8 (3) : 577-590. doi: 10.3934/jimo.2012.8.577

References:

[1]	V. Agrawal and S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management, 2 (2000), 410-423.
[2]	M. Allais, Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'ecole americaine, Econometrica, 21 (1953), 503-546. doi: 10.2307/1907921.
[3]	N. Barberis and M. Huang, Mental accounting, loss aversion, and individual stock returns, Journal of Finance, 56 (2001), 1247-1292. doi: 10.1111/0022-1082.00367.
[4]	A. O. Brown and C. S. Tang, The impact of alternative performance measures on single-period inventory policy, Journal of Industrial and Management Optimization, 2 (2006), 297-318.
[5]	L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 41 (1995), 786-794. doi: 10.1287/mnsc.41.5.786.
[6]	M. Fisher and A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99. doi: 10.1287/opre.44.1.87.
[7]	W. Geng, X. Zhao and D. Gao, A single-period inventory system with a general s-shaped utility and exponential demand, Journal of Systems Science and Systems Engineering, 19 (2010), 227-236. doi: 10.1007/s11518-010-5128-8.
[8]	J. A. Kahn, Why is production more volatile than sales? Theory and evidence on the stockout-avoidance motive for inventory holding, Quarterly Journal of Economics, 107 (1992), 481-510. doi: 10.2307/2118479.
[9]	D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291. doi: 10.2307/1914185.
[10]	B. Keren and J. S. Pliskin, A benchmark solution for the risk-averse newsvendor problem, European Journal of Operational Research, 174 (2006), 1643-1650. doi: 10.1016/j.ejor.2005.03.047.
[11]	M. Khouja, The single-period (news-vendor) problem: Literature review and suggestions for future research, Omega-International Journal of Management Science, 27 (1999), 537-553. doi: 10.1016/S0305-0483(99)00017-1.
[12]	H. Lau and A. H. Lau, The newsstand problem: A capacitated multiple-product single-period inventory problem, European Journal of Operational Research, 94 (1996), 29-42. doi: 10.1016/0377-2217(95)00192-1.
[13]	S. A. Lippman and K. F. McCardle, The competitive newsboy, Operations Research, 45 (1997), 54-65. doi: 10.1287/opre.45.1.54.
[14]	Y. Liu, M. Peng and L. Li, A prospect theory-based order model under stochastic demand, Chinese Control and Decision Conference, (2009), 2539-2544.
[15]	N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194. doi: 10.1287/opre.47.2.183.
[16]	M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 404-420. doi: 10.1287/mnsc.46.3.404.12070.
[17]	E. Stavrulaki, Inventory decisions for substitutable products with stock-dependent demand, International Journal of Production Economics, 129 (2011), 65-78. doi: 10.1016/j.ijpe.2010.09.002.
[18]	A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.
[19]	C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452. doi: 10.1016/j.ijpe.2009.12.009.
[20]	C. X. Wang and S. Webster, The loss-averse newsvendor problem, Omega-International Journal of Management Science, 37 (2009), 93-105. doi: 10.1016/j.omega.2006.10.003.
[21]	C. X. Wang, S. Webster and N. C. Suresh, Would a risk-averse newsvendor order less at a higher selling price?, European Journal of Operational Research, 196 (2009), 544-553. doi: 10.1016/j.ejor.2008.04.002.
[22]	X. Zhao, W. Gen and X. Chao and D. Gao, A periodic review inventory system with s-shaped utility function, Working paper, 2009. Available from: http://ioe.engin.umich.edu/techrprt/pdf/TR09-04.pdf.

show all references

References:

[1]	V. Agrawal and S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management, 2 (2000), 410-423.
[2]	M. Allais, Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'ecole americaine, Econometrica, 21 (1953), 503-546. doi: 10.2307/1907921.
[3]	N. Barberis and M. Huang, Mental accounting, loss aversion, and individual stock returns, Journal of Finance, 56 (2001), 1247-1292. doi: 10.1111/0022-1082.00367.
[4]	A. O. Brown and C. S. Tang, The impact of alternative performance measures on single-period inventory policy, Journal of Industrial and Management Optimization, 2 (2006), 297-318.
[5]	L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 41 (1995), 786-794. doi: 10.1287/mnsc.41.5.786.
[6]	M. Fisher and A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99. doi: 10.1287/opre.44.1.87.
[7]	W. Geng, X. Zhao and D. Gao, A single-period inventory system with a general s-shaped utility and exponential demand, Journal of Systems Science and Systems Engineering, 19 (2010), 227-236. doi: 10.1007/s11518-010-5128-8.
[8]	J. A. Kahn, Why is production more volatile than sales? Theory and evidence on the stockout-avoidance motive for inventory holding, Quarterly Journal of Economics, 107 (1992), 481-510. doi: 10.2307/2118479.
[9]	D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-291. doi: 10.2307/1914185.
[10]	B. Keren and J. S. Pliskin, A benchmark solution for the risk-averse newsvendor problem, European Journal of Operational Research, 174 (2006), 1643-1650. doi: 10.1016/j.ejor.2005.03.047.
[11]	M. Khouja, The single-period (news-vendor) problem: Literature review and suggestions for future research, Omega-International Journal of Management Science, 27 (1999), 537-553. doi: 10.1016/S0305-0483(99)00017-1.
[12]	H. Lau and A. H. Lau, The newsstand problem: A capacitated multiple-product single-period inventory problem, European Journal of Operational Research, 94 (1996), 29-42. doi: 10.1016/0377-2217(95)00192-1.
[13]	S. A. Lippman and K. F. McCardle, The competitive newsboy, Operations Research, 45 (1997), 54-65. doi: 10.1287/opre.45.1.54.
[14]	Y. Liu, M. Peng and L. Li, A prospect theory-based order model under stochastic demand, Chinese Control and Decision Conference, (2009), 2539-2544.
[15]	N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 47 (1999), 183-194. doi: 10.1287/opre.47.2.183.
[16]	M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 404-420. doi: 10.1287/mnsc.46.3.404.12070.
[17]	E. Stavrulaki, Inventory decisions for substitutable products with stock-dependent demand, International Journal of Production Economics, 129 (2011), 65-78. doi: 10.1016/j.ijpe.2010.09.002.
[18]	A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.
[19]	C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452. doi: 10.1016/j.ijpe.2009.12.009.
[20]	C. X. Wang and S. Webster, The loss-averse newsvendor problem, Omega-International Journal of Management Science, 37 (2009), 93-105. doi: 10.1016/j.omega.2006.10.003.
[21]	C. X. Wang, S. Webster and N. C. Suresh, Would a risk-averse newsvendor order less at a higher selling price?, European Journal of Operational Research, 196 (2009), 544-553. doi: 10.1016/j.ejor.2008.04.002.
[22]	X. Zhao, W. Gen and X. Chao and D. Gao, A periodic review inventory system with s-shaped utility function, Working paper, 2009. Available from: http://ioe.engin.umich.edu/techrprt/pdf/TR09-04.pdf.

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