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Coordinating a three-level supply chain with effort and price dependent stochastic demand under random yield

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Abstract

One of the major objectives of modern supply chain management is dealing with the negative impact of decentralization among the involved entities and minimizing double marginalization effect within the chain, especially when the end-customers’ demand is not deterministic. This paper investigates coordination issue in a three-level supply chain with one raw-material supplier, one manufacturer, and one retailer. The retailer determines the retail price, sales effort, and order quantity simultaneously before the selling season starts. Both the supplier and the manufacturer face random yield in production. A composite contract having two components—a contingent buyback with target sales rebate and penalty between the retailer and the manufacturer, and a revenue sharing contract between the manufacturer and the supplier is proposed. The proposed composite contract is shown to achieve supply chain coordination and allows arbitrary allocation of total channel profit among all the chain members. The impact of randomness in both demand and production, and the impact of non-existence of emergency resource for the final product on the performance of the entire supply chain are analyzed. A numerical example is provided to illustrate the developed model and draw some important managerial insights.

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Abbreviations

\(c_s\)::

Unit production cost at the raw material supplier

\(c_s^{\prime }\)::

Unit procurement cost of raw material from the secondary market

\(c_m\)::

Unit manufacturing cost at the manufacturer

v::

Unit salvage value of the final product

\(v_s\)::

Unit salvage value of excess product at the raw material supplier

\(g_r\)::

Retailer’s goodwill cost for unit unmet demand

X::

A positive random variable with range [l, u], pdf \(f(\cdot )\), cdf \(F(\cdot )\), mean \(\bar{x}\), and variance \(\sigma ^2_x\) representing the stochastic portion of the customer demand

Y::

A random variable with range [c,d], \(0\le c<d \le 1\), having pdf \(g(\cdot )\) and cdf \(G(\cdot )\), denoting the randomness of the production quantity produced by the manufacturer

Z::

A random variable with range [a, b], \(0\le a<b \le 1\), with pdf \(h(\cdot )\) and cdf \(H(\cdot )\), denoting the randomness of the production quantity of the raw material produced by the supplier

R::

Planned production quantity of the raw-material supplier, a decision variable (in units)

Q::

Ordered quantity of the retailer, a decision variable (in units)

p::

Unit retail price of the final product charged by the retailer, a decision variable

e::

Effort level to summarize the retailer’s activities to influence market demand, a decision variable. We assume J(e) To be the retailer’s cost of exerting an effort level e with \(J(0)=0\), \(J'(e)>0\) and \(J''(e)>0\) when \(e>0\)

\(w_s\)::

Unit wholesale price of the raw material offered by the supplier to the manufacturer, a decision variable

\(w_m\)::

Unit wholesale price of the finished product charged by the manufacturer to the retailer, a decision variable

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    B. C. Giri, J. K. Majhi & K. S. Chaudhuri

  2. Department of Mathematics, Kanchrapara College, Kanchrapara, 743145, India

    S. Bardhan

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  1. B. C. Giri
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  4. K. S. Chaudhuri
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Giri, B.C., Majhi, J.K., Bardhan, S. et al. Coordinating a three-level supply chain with effort and price dependent stochastic demand under random yield. Ann Oper Res 307, 175–206 (2021). https://doi.org/10.1007/s10479-021-04257-z

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