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Contracting mechanism with imperfect information in a two-level supply chain

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This article investigates a two-level supply chain consisting of a single manufacturer and a retailer. The retailer’s ordering patterns are highly influenced by his risk preferences. We discuss ordering policies when the manufacturer has limited demand information and propose a production-commitment contract, which mitigates double marginalization under imperfect information. Demand distribution is private information of the retailer and the manufacturer only assumes an educated guess about the mean and variance. Production-commitment contract is an attractive option for make-to-stock scenarios where quantity is confirmed after the demand is realized. We show that lack of information may not have an adverse effect. We also prove analytically that informational advantage may not necessarily be a supply chain advantage and also provide numerical insights for a win-win situation.

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  1. Arkes HR (1996) The psychology of waste. J Behav Decis Mak 9(3):213–224

  2. Anupindi R, Akella R (1993) An inventory model with commitments. Working paper

  3. Buzacott J, Yan H, Zhang H (2011) Risk analysis of commitment-option contracts with forecast updates. IIE Trans 43(6):415–431

  4. Cachon GP (2003) Handbooks in operations research and management science, Volume 11: supply chain management design, coordination and operation. In: De Kok AG, Graves SC (eds) Supply chain coordination with contracts. Elsevier, The Netherlands, pp 229–339

  5. Cachon GP, Lariviere MA (1999) Capacity choice and allocation: strategic behavior and supply chain performance. Manag Sci 45(8):1091–1108

  6. Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue sharing contracts: strength and limitations. Manag Sci 51(1):30–44

  7. Chen J (2011) Return with whole-sale price discount contract in a newsvendor problem. Int J Prod Econ 130(1):104–111

  8. Cohen M, Ho T, Ren J, Terwiesch C (2003) Measuring inputed costs in the semiconductor equipment supply chain. Working paper, The Wharton School, University of Pennsylvania, Philadelphia, PA

  9. Erkoc M, Wu SD (2005) Managing high-tech capacity expansion via capacity reservation. Prod Oper Manag 14(2):232–251

  10. Fisher M, Raman A (1996) Reducing the cost of demand uncertainty through accurate response to early sales. Oper Res 44(1):87–99

  11. Gallego G, Moon IK (1993) The distribution free newsboy problem: review and extensions. J Oper Res Soc 44(8):825–834

  12. Giri BC, Sarker BR (2016) Coordinatiing a two-echelon supply chain under production disruption when retailes compete with price and service level. Oper Res Int J 16:71–88

  13. Hazra J, Mahadevan B (2009) A procurement model using capacity reservation. Eur J Oper Res 193(1):303–316

  14. Alfares HK, Hassan HE (2005) The distribution-free newsboy problem: extensions to the shortage penalty case. Int J Prod Econ 93–94:465–477

  15. Hou J, Zeng AZ, Zhao L (2010) Coordination with backup supplier through buyback contract under supply disruption. Transp Res Part E 46(6):881–895

  16. Hu W, Li Y, Govindan K (2014) The impact of consumer return polices on consignments contracts with inventory control. Eur J Oper Res 233(2):398–407

  17. Jin M, Wu SD (2007) Capacity reservation contracts for high-tech manufacturing. Eur J Oper Res 176(3):1659–1677

  18. Jinlou Z, Fugen S (2008) Modeling for rebate and penalty contract with retailer’s combined decision bias. International conference on logistics engineering and supply chain, P.R.China

  19. Kalkanci B, Chen K, Erhun F (2011) Contract complexity and performance under asymmetric demand information: an experimental evaluation. Manag Sci 57(4):689–704

  20. Kalkanci B, Erhun F (2012) Pricing games and impact of private demand information in decantralized assembly systems. Oper Res 60(5):1142–1156

  21. Katz, Sadrian A, Tendick P (1994) Telephone companies analyze price quotations with Bellcore PDSS software. Interfaces 24(1):50–63

  22. Khouja M (1999) The single-period (newsvendor) problem: literature review and suggestions for future research. Omega Int J Manag Sci 27:537–553

  23. Lariviere MA (1999) Quantitative models of supply chain management. In: Tayur S, Magazine M, Ganeshan R (eds) Supply chain contracting and coordination with stochastic demand. Kluwer Academic Press, Dordrecht, pp 233–268

  24. Lariviere MA, Porteus EL (2001) Selling to the newsvendor: an analysis of price-only contracts. Manuf Serv Oper Manag 3(4):293–305

  25. Lau H, Lau A (1999) Manufacturer’s pricing strategy and return policy for a single-period commodity. Eur J Oper Res 116(2):291–304

  26. Lau H, Lau A (1996) The newsstand problem: a capacitated multiple-product single period inventory problem. Eur J Oper Res 94(11):29–42

  27. Lee H, Padmanabhan V, Whang S (1997) Information distortion in a supply chain: the bullwhip effect. Manag Sci 43(4):546–558

  28. Li S, Kabadi SN, Nair KPK (2002) Fuzzy models for single-period inventory problem. Fuzzy Sets Syst 132(3):273–289

  29. Murray GR, Silver EA (1966) A Bayesian analysis of the style goods inventory problem. Manag Sci 12(11):785–797

  30. Muzaffar A, Deng S (2012) Tradeoff ordering policy and decision biased newsvendor with uncertain demand. J Supply Chain Oper Manag 10(2):1–13

  31. Ozer O, Wei W (2006) Strategic commitments for an optimal capacity decision under asymmetric forecast information. Manag Sci 52(8):1239–1258

  32. Pasternack BA (1985) Optimal pricing and return policies for perishable commodities. Mark Sci 4(2):166–176

  33. Petrovic D, Petrovic R, Vujosevic M (1996) Fuzzy models for the newsboy problem. Int J Prod Econ 45(1–3):435–441

  34. Schweitzer ME, Cachon GP (2000) Decision bias in the newsvendor problem with a known demand distribution: experimental evidence. Manag Sci 46(3):404–420

  35. Sethi S, Sorger G (1991) A theory of rolling horizon decision making. Ann Oper Res 29:387–416

  36. Scarf H (1958) A Min–Max solution of an inventory problem. Ch 12 in studies in the mathematical theory of inventory and production. Stanford University Press, Stanford

  37. Tsay A (2001) Managing retail channel overstock: markdown money and return policies. J Retail 77(4):457–492

  38. Wagner RM (2015) Robust purchasing and information asymmetry in supply chains with a price-only contract. IIE Trans 47(8):819–840

  39. Zhao Y, Wang S, Cheng TCE, Yang X, Huang Z (2010) Coordination of supply chains by option contracts: a cooperative game theory approach. Eur J Oper Res 207(2):668–675

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Correspondence to Asif Muzaffar.



Proof for Proposition  1

For continuous probability distribution, self dual exists for probability measure. Let the retailer’s profit function under risk preferences is given as follows:

$$\begin{aligned} {\Pi _R}= \,& {} (P - \upsilon )S(q) - (w - \upsilon )q - \alpha \int \limits _0^q {(q - x)f(x)dx} - \beta \int \limits _q^x {(x - q)f(x)dx}\\=\, & {} (P - \upsilon )(q - \int \limits _0^q {(q - x)f(x)dx} ) - (w - \upsilon )q - \alpha \int \limits _0^q {(q - x)f(x)dx} - \beta \int \limits _q^x {(x - q)(1 - F(x)} )\\ \frac{{\partial {\Pi _R}}}{{\partial q}}=\, & {} (P - \upsilon ) - (P - \upsilon )F(x) - (w - \upsilon ) - \alpha F(x) + \beta - \beta F(x)\\ F(x)=\, & {} \frac{{P - w + \beta }}{{P - \upsilon + \alpha + \beta }}\\ {q^*}=\, & {} {F^{ - 1}}\left( {\frac{{P - w + \beta }}{{P - \upsilon + \alpha + \beta }}} \right) \end{aligned}$$

This completes the proof.

Proof for Lemma 1

Let us define \(G(w) = \left[ {\left( {\frac{{p - w + k}}{{w - \upsilon }}} \right) ^{{ {1 \over 2}}} - \left( {\frac{{w - \upsilon }}{{p - w + k}}} \right) ^{{ {1 \over 2}}} } \right]\) such that

$$\begin{aligned} \Pi _{G(w)}= & {} f.g \\ \Pi ^/ _{G(w)}= & {} f^/ .g + f.g^/ \\ \Pi ^{//} _{G(w)}= & {} f^{//} .g + f.g^{//} + 2f^/ .g^/ \\ \end{aligned}$$

and by definition

$$\begin{aligned} f= & {} (w - c) \\ g= & {} \mu + \frac{\sigma }{2}G(w) \end{aligned}$$

By condition of optimality, we have

$$\begin{aligned}&f^/ = 1,f^{//} = 0 \\&g^/ = \frac{\sigma }{2}G^/ (w) \\&g^{//} = \frac{\sigma }{2}G^{//} (w) \\&\frac{{\partial \Pi ^2 _{G(w)} }}{{\partial ^2 w}} = (w - c)\frac{\sigma }{2}G^{//} (w) + \sigma G^/ (w) \end{aligned}$$

The function G(w) would be

$$\begin{aligned} G^/ (w)= & {} - \frac{1}{2}\frac{{P - \upsilon + k}}{{(P - w + k)(w - \upsilon )^{{ {3 \over 2}}} }} \\ G^{//} (w)= & {} \frac{3}{4}\frac{{(P - \upsilon + k)^2 }}{{(P - w + k)^{{ {3 \over 2}}} (w - \upsilon )^{{ {5 \over 2}}} }} \end{aligned}$$

We characterize the solution and provide condition for optimality

$$\begin{aligned} \frac{{\partial \Pi ^2 _{G(w)} }}{{\partial ^2 w}} = \frac{\sigma }{2}(P - \upsilon + k)\left[ {\frac{{\frac{3}{4}(P - \upsilon + k) - (P - w + k)(w - \upsilon )}}{{(P - w + k)(w - \upsilon )^{{ {5 \over 2}}} }}} \right] \end{aligned}$$

So the optimal wholesale price satisfies the following condition for the manufacturer’s profit function (concave).

$$\begin{aligned} \frac{3}{4}(w - c)(P - \upsilon + k) < (P - w + k)(w - \upsilon ) \end{aligned}.$$

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Muzaffar, A., Deng, S. & Malik, M.N. Contracting mechanism with imperfect information in a two-level supply chain. Oper Res Int J 20, 349–368 (2020). https://doi.org/10.1007/s12351-017-0327-4

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  • Risk preferences
  • Newsboy problem
  • Production-commitment contract
  • Asymmetric information