Production planning problem with pricing under random yield: CVaR criterion

Abstract

In this paper, we address a basic production planning problem with price dependent demand and stochastic yield of production. We use price and target quantity as decision variables to lower the risk of low yield. The value of risk control becomes more important especially for products with short life cycle. This is because, the profit implications of low yield might be unbearable in the short run. We apply Conditional Value at Risk (CVaR) to model the risk. CVaR measure is a coherent risk measure and thereby having nice conceptual and mathematical underpinnings. It is also widely used in practice. We consider the problem under general demand function and general distribution function of yield and find sufficient conditions under which the problem has a unique local maximum. We also both analytically and numerically analyze the impact of parameter change on the optimal solution. Among our results, we analytically show that with increasing risk aversion, the optimal price increases. This relation is opposite to that of in Newsvendor problem where the uncertainty lies in demand side.

Keywords

Production planning problem / coherent risk measures / conditional value at risk / random yield / pricing

References

[1] Arifglu K, Deo S, Iravani S MR. Consumption externality and yield uncertainty in the influenza vaccine supply chain: interventions in demand and supply sides. Management Science, 2012, 58(6): 1072-1091.

[2] Arrow KJ. Essays in the Theory of Risk-bearing, 1971, Chicago: Markham Pub. Co

[3] Artzner, P. (2007). Coherent multiperiod risk adjusted values and bellmans principle. Annals of Operations Research 152

[4] Artzner P, Delbaen F, Eber J-M, Heath D. Coherent measures of risk. Mathematical Finance, 1999, 9(3): 203-228.

[5] Artzner P, Delbaen F, Eber J-M, Heath D, Ku H. Coherent Multiperiod Risk Measurement, 2002

[6] Bakal IS, Akcali E. Effects of random yield in remanufacturing with price-sensitive supply and demand. Production and operations management, 2006, 15(3): 407-420.

[7] Ben-Tal A, Teboulle M. An old-new concept of convex risk measures: the optimized certainty equivalent. Mathematical Finance, 2007, 17(3): 449-476.

[8] Bertsimas D, Lauprete GJ, Samarov A. Shortfall as a risk measure: properties, optimization and applications. Journal of Economic Dynamics and Control, 2004, 28(7): 1353-1381.

[9] Chen Y, Xu M, Zhang ZG. Technical note—a risk-averse newsvendor model under the cvar criterion. Operations Research, 2009, 57(4): 1040-1044.

[10] Chick SE, Mamani H, Simchi-Levi D. Supply chain coordination and influenza vaccination. Operations Research, 2008, 56(6): 1493-1506.

[11] Chiu C-H, Choi T-M. Optimal pricing and stocking decisions for newsvendor problem with valueat-risk consideration. Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on, 2010, 40(5): 1116-1119.

[12] Chiu C-H, Choi T-M, Li X. Supply chain coordination with risk sensitive retailer under target sales rebate. Automatica, 2011, 47(8): 1617-1625.

[13] Cho S-H, Tang CS. Advance Selling in a Supply Chain under Uncertain Supply and Demand, 2011.

[14] Choi S, Ruszczynski A, Zhao Y. A multiproduct risk-averse newsvendor with law-invariant coherent measures of risk. Operations Research, 2011, 59(2): 346-364.

[15] Choi T-M. Optimal return service charging policy for a fashion mass customization program, 2013

[16] Choi T-M, Chiu C-H. Mean-downside-risk and mean-variance newsvendor models: implications for sustainable fashion retailing. International Journal of Production Economics, 2012, 135(2): 552-560.

[17] Choi T-M, Li D, Yan H. Mean-variance analysis for the newsvendor problem. Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on, 2008, 38(5): 1169-1180.

[18] Dada M, Petruzzi NC, Schwarz LB. A newsvendor’s procurement problem when suppliers are unreliable. Manufacturing and Service Operations Management, 2007, 9(1): 9-32.

[19] Deo S, Corbett CJ. Cournot competition under yield uncertainty: the case of the U.S. influenza vaccine market. Manufacturing and Service Operations Management, 2009, 11(4): 563-576.

[20] Detlefsen K, Scandolo G. Conditional and dynamic convex risk measures. Finance and Stochastics, 2005, 9(4): 539-561.

[21] Follmer H, Schied A. Convex measures of risk and trading constraints. Finance and Stochastics, 2002, 6: 429-447.

[22] Follmer H, Schied A. Stochastic Finance. An Introduction in Discrete Time, 2ed. de gruyter studies in mathematik, 2004, Berlin, New York: Walter de Gruyter.

[23] Frittelli M, Gianin MR. Szego G P. Dynamic convex risk measures. Risk Measures for the 21st Century, 2004, Chichester, Hoboken, NJ: Wiley

[24] Gotoh JY, Takano Y. Newsvendor solutions via conditional value-at-risk minimization. European Journal of Operational Research, 2007, 179(1): 80-96.

[25] Kahneman D, Tversky A. Prospect theory: an analysis of decision under risk. Econometrica, 1979, 47(2): 263-292.

[26] Kazaz B, Webster S. The impact of yield-dependent trading costs on pricing and production planning under supply uncertainty. Manufacturing and Service Operations Management, 2011, 13(3): 404-417.

[27] Keren B, Pliskin JS. A benchmark solution for the risk-averse newsvendor problem. European Journal of Operational Research, 2006, 174(3): 1643-1650.

[28] Li Q, Zheng S. Joint inventory replenishment and pricing control for systems with uncertain yield and demand. Operations Research, 2006, 54(4): 696-705.

[29] Machina MJ. Choice under uncertainty: problems solved and unsolved. The Journal of Economic Perspectives, 1987, 1(1): 121-154.

[30] Rockafellar RT, Uryasev S. Optimization of conditional value-at-risk. Journal of Risk, 2000, 2: 21-41.

[31] Rockafellar RT, Uryasev S. Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 2002, 26(7): 1443-1471.

[32] Rockafellar RT, Uryasev S, Zabarankin M. Generalized deviations in risk analysis. Finance and Stochastics, 2006, 10: 51-74.

[33] Tang CS, Yin R. Responsive pricing under supply uncertainty. European Journal of Operational Research, 2007, 182(1): 239-255.

[34] Tang O, Musa SN, Li J. Dynamic pricing in the newsvendor problem with yield risks. International Journal of Production Economics, 2011

[35] Tomlin B. on the value of mitigation and contingency strategies for managing supply chain disruption risks. Management Science, 2006, 52(5): 639-657.

[36] Tomlin B. Impact of supply learning when suppliers are unreliable. Manufacturing and Service Operations Management, 2009, 11(2): 192-209.

[37] Tomlin B, Wang Y. on the value of mix flexibility and dual sourcing in unreliable newsvendor networks. Manufacturing and Service Operations Management, 2005, 7(1): 37-57.

[38] Wang CX, Webster S, Suresh NC. Would a risk-averse newsvendor order less at a higher selling price?. European Journal of Operational Research, 2009, 196(2): 544-553.

[39] Wei Y, Choi T-M. Mean-variance analysis of supply chains under wholesale pricing and profit sharing schemes. European Journal of Operational Research, 2010, 204(2): 255-262.

[40] Xu M. A price-setting newsvendor model under cvar decision criterion with emergency procurement. Journal of Systems Science and Systems Engineering, 2010, 19: 85-104.

[41] Xu M, Lu Y. The effect of supply uncertainty in price-setting newsvendor models. European Journal of Operational Research, 2013

[42] Yano C A, Lee H L. Lot sizing with random yields: a review. Operations Research, 1995, 43(2): 311-334.