## Abstract

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.

This is a preview of subscription content, log in to check access.

## References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Author information

## Appendix

### Appendix

*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:

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

and by definition

By condition of optimality, we have

The function *G*(*w*) would be

We characterize the solution and provide condition for optimality

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

## Rights and permissions

## About this article

### Cite this article

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

Received:

Revised:

Accepted:

Published:

Issue Date:

### Keywords

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