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

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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.

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Correspondence to Asif Muzaffar.

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:

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

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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Keywords

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