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A closed-loop supply chain model with uncertain return and learning-forgetting effect in production under consignment stock policy

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Abstract

This paper considers a manufacturer–retailer closed-loop supply chain where the manufacturer’s production process is subject to both learning and forgetting, and inventories of the manufacturer and the retailer are managed by consignment stock policy. The demand of the finished product is linearly dependent on the retail price. The return rate of used product is random and returned used items are inspected to select those which qualify for remanufacturing. The proposed model is demonstrated with a numerical example. A sensitivity analysis is performed to identify the parameters which have significant impacts on the optimal decisions of the supply chain system. Some managerial insights of the model leading to improved strategies beneficial for firms and environment are also discussed.

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Acknowledgements

The authors are sincerely thankful to the esteemed reviewers for their comments and suggestions based on which the manuscript has been improved. Research support from Department of Science and Technology, Government of India (IF160066) is gratefully acknowledged.

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Correspondence to B. C. Giri.

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Appendices

Appendix 1

Time t can be determined from the Wright’s learning curve (Zanoni et al. 2012). Time to produce quantity \(Q_{i}\) is \(t_{i}=\int _{0}^{Q_{i}}T_{i1}x^{-b}=\frac{T_{i1}}{1-b}Q_{i}^{1-b}\). Replacing \(t_{i}\) by t, we get

$$\begin{aligned} t=\frac{T_{i1}}{1-b}Q(t)^{1-b} \end{aligned}$$

where Q(t) represents the production quantity which is a function of time.

Appendix 2

Area of the rectangles under the small triangles i.e \(\triangle ABC\), \(\triangle DEF\) etc.:

$$\begin{aligned}&{\text {Area of}} \,\square AWPS+ \square DVWC+ \square \hbox {XUVF}+ \cdots \\&\quad =(n-1)(Q-Dt)t+(n-2)(Q-Dt)t+(n-3)(Q-Dt)t+\cdots +(Q-Dt)t\\ & \quad =\frac{n(n-1)}{2}(Q-Dt)t\\ \end{aligned}$$

Appendix 3

In policy II, number of consecutive batches shipped is m. After that, production continues for some time and then stops followed by a large shipment of size \((nQ-(m-1)Dt)\) delivered once at a time.

$$\begin{aligned} {\text {The area of rectangles}}\,&\square ADCB+ \square HEDJ+ \square GFEI+ \cdots \\&=(m-1)(Q-Dt)t+(m-2)(Q-Dt)t+(m-3)(Q-Dt)t+\cdots +(Q-Dt)t\\&=\frac{m(m-1)}{2}(Q-Dt)t\\ \end{aligned}$$

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Giri, B.C., Masanta, M. A closed-loop supply chain model with uncertain return and learning-forgetting effect in production under consignment stock policy. Oper Res Int J 22, 947–975 (2022). https://doi.org/10.1007/s12351-020-00571-9

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