Stochastic comparisons in a price-quantity setting firm with uncertain demand and emergency procurement

Abstract

We consider a newsvendor problem with price-dependent demand, in either additive or multiplicative format. The newsvendor has two modes of purchasing: regular ordering at the beginning of the selling season and emergency ordering (if the realized demand exceeds the initial order quantity) at the end of the selling season. By stochastic comparisons, we systematically investigate the effects of demand magnitude and demand randomness on pricing and ordering quantity decisions as well as expected profit of the newsvendor, under both usual stochastic order (first order stochastic dominance) and convex order (less variable). Our key findings include: (i) in contrary to the case where price is exogenous, a stochastically larger demand shock may even lead to a lower order quantity; (ii) a stochastically larger demand shock leads to a higher price for the additive demand case, but may lead to a lower price for the multiplicative demand case; (iii) a stochastically larger demand shock leads to a higher expected profit for both demand models; (iv) in general, a less variable demand leads to a higher expected profit for both demand models; and furthermore, a less variable demand shock has no effect on price for the additive demand model, but leads to a higher price for the multiplicative demand model. The implications of all these findings for pricing and order quantity are discussed in detail.

Keywords

Price-dependent newsvendor / stochastic comparison / demand uncertainty / pricing / emergency supply option

References

[1] Agrawal V, Seshadri S. Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. Manufacturing & Service Operations Management, 2000, 2(4): 410-423.

[2] Barankin EW. A delivery-lag inventory model with an emergency provision (the single-period case). Naval Research Logistics Quarterly, 1961, 8(3): 285-311.

[3] Chen J, Bell PC. The impact of customer returns on pricing and order decisions. European Journal of Operational Research, 2009, 195(1): 280-295.

[4] Chen Y, Xu M, Zhang ZG. A risk-averse newsvendor model under the CVaR criterion. Operations Research, 2009, 57(4): 1040-1044.

[5] Eeckhoudt L, Gollier C, Schlesinger H. Risk averse (and prudent) newsboy. Management Science, 1995, 41(5): 786-794.

[6] Gallego G, Moon I. The distribution free newsboy problem: review and extensions. Journal of the Operational Research Society, 1993, 44(8): 825-834.

[7] Gerchak Y, He Q. On the relation between the benefits of risk pooling. IIE Transactions, 2003, 35(11): 1027-1031.

[8] Gerchak Y, Mossman D. On the effect of demand randomness on inventories and costs. Operations Research, 1992, 40(4): 804-807.

[9] Granot D, Yin S. On the effectiveness of returns policies in the price-dependent newsvendor model. Naval Research Logistics, 2005, 52(8): 765-779.

[10] Granot D, Yin S. Price and order postponement in a decentralized newsvendor model with multiplicative and price-dependent demand. Operations Research, 2008, 56(1): 121-139.

[11] Gupta D, Cooper WL. Stochastic comparisons in production yield management. Operations Research, 2005, 53(2): 377-384.

[12] He Y, Zhang J. Random yield risk sharing in a two-level supply chain. International Journal of Production Economics, 2008, 112(2): 769-781.

[13] He Y, Zhang J. Random yield supply chain with a yield dependent secondary market. European Journal of Operational Research, 2010, 206(1): 221-230.

[14] He Y, Zhao X. Coordination in multi-echelon supply chain under supply and demand uncertainty. International Journal of Production Economics, 2012, 139(1): 106-115.

[15] Khouja M. A note on the newsboy problem with an emergency supply option. Journal of the Operational Research Society, 1996, 47(12): 1530-1534.

[16] Lee H, Lodree EJ. Modeling customer impatience in a newsboy problem with time-sensitive shortages. European Journal of Operational Research, 2010, 205(3): 595-603.

[17] Li Q, Atkins D. On the effect of demand randomness on a price/quantity setting firm. IIE Transactions, 2005, 37(12): 1143-1153.

[18] Liu S, So KC, Zhang F. Effect of supply reliability in a retail setting with joint marketing and inventory decisions. Manufacturing & Service Operations Management, 2010, 12(1): 19-32.

[19] Lodree E, Kim Y, Jang W. Time and quantity dependent waiting costs in a newsvendor problem with backlogged shortages. Mathematical & Computer Modeling, 2008, 47(1–2): 60-71.

[20] Petruzzi NC, Dada M. Pricing and the newsvendor problem: a review with extensions. Operations Research, 1999, 47(2): 183-194.

[21] Porteus EL. Foundations of Stochastic Inventory Theory, 2002, Stanford, CA: Stanford University Press

[22] Ridder A, Van Der Laan E, Salomon M. How larger demand variability may lead to lower costs in the newsvendor problem. Operations Research, 1998, 46(6): 934-936.

[23] Ross SM. Stochastic Processes (2nd), 1995, New York: Wiley, John & Sons Inc.

[24] Shaked M, Shanthikumar JG. Stochastic Orders, 2006, New York: Springer

[25] Song J-S. The effect of lead time uncertainty in a simple stochastic inventory model. Management Science, 1994, 40(5): 603-613.

[26] Song J-S, Zhang H, Hou Y, Wang M. The effect of lead time and demand uncertainties in (r, q) inventory systems. Operations Research, 2010, 58(1): 68-80.

[27] Xu H. Managing production and procurement through option contracts in supply chains with random yield. International Journal of Production Economics, 2010, 126(2): 306-313.

[28] Xu M, Chen Y, Xu X. The effect of demand uncertainty in a price-setting newsvendor model. European Journal of Operational Research, 2010, 207(2): 946-957.

[29] Xu M, Lu Y. The effect of supply uncertainty in price-setting newsvendor models. European Journal of Operational Research, 2013, 227(3): 423-433.

[30] Yano CA, Gilbert SM. Chakravarty A, Eliashberg J. Coordinated pricing and production/procurement decisions: a review. Managing Business Interfaces: Marketing, Engineering and Manufacturing Perspectives, 2003, Boston, MA: Kluwer Academic Publishers 65-104.

[31] Yao L, Chen Y, Yan H. The newsvendor problem with pricing: extensions. International Journal of Management Science and Engineering Management, 2006, 1(1): 3-16.