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Production and coordination decisions in a closed-loop supply chain with remanufacturing cost disruptions when retailers compete

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Abstract

This paper studies the production and coordination decisions in a closed-loop supply chain (CLSC) with one manufacturer and two competing retailers when remanufacturing costs are disrupted. We find that the pricing and production quantity decisions have certain robust regions when facing disruptions; and the more intense of competition between retailers, the smaller the robustness regions. Moreover, the manufacturer would like to adjust the production decisions when facing large size of positive or negative disruptions, while the retailers only would like to adjust them when facing large size of negative disruptions. Finally, we find that the revenue-sharing contracts can effectively coordinate the CLSCs, whereas more profits are required by retailers under large positive disruptions.

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Acknowledgments

This work was supported by the Natural Science Foundation of China (No. 71101032) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20104420120008).

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Correspondence to Xiaohua Han.

Appendix: Proofs of propositions

Appendix: Proofs of propositions

Proof of Proposition 1

We can discuss the optimal solutions of model (1) in two scenarios, \(\delta >0\) and \(\delta <0\), respectively. When \(\delta <0\), the model (1) can be simplified as the model (17).

$$\begin{aligned} \mathop {\hbox {max}}\limits _{\tilde{p}_1 ,\tilde{p}_2 ,\tilde{\tau }_1 ,\tilde{\tau }_2} \mathop {\tilde{\prod }}\nolimits _\mathrm{S}^\mathrm{c}= & {} (\phi -\tilde{p}_1 +\beta \tilde{p}_2 )(\tilde{p}_1 -c_m +(\Delta -\delta )\tilde{\tau }_1 )\nonumber \\&+(\phi -\tilde{p}_2 +\beta \tilde{p}_1 )(\tilde{p}_2 -c_m +(\Delta -\delta )\tilde{\tau }_2 ) \nonumber \\&-h\mathop {\tilde{\tau }}\nolimits _1^2 -h\mathop {\tilde{\tau }}\nolimits _2^2 -\mu _1 (\tilde{Q}-Q^{\mathrm{c}{*}}),\, \nonumber \\&s.t. \tilde{Q}-Q^{\mathrm{c}{*}}\ge 0.\, \end{aligned}$$
(17)

It can be verified that \(\tilde{\prod }_\mathrm{S}^\mathrm{c} \) is jointly concave in \(\tilde{p}_1 \) and \(\tilde{p}_2 \), as well as \(\tilde{\tau }_1 \) and \(\tilde{\tau }_2 \). To solve the constrained optimization problem, we introduce the Lagrange multiplier \(\lambda \ge 0\) and relax the constraint \(\tilde{Q}-Q^{\mathrm{c}{*}}\ge 0\). From solving the Kuhn–Tucker condition of model (17), we can obtain the optimal solutions of model (17) when \(A\mu _1 /C\le \delta (\delta -2\Delta )\) and \(0<\delta (\delta -2\Delta )<A\mu _1 /C\), respectively. Similarly, we can derive the optimal solutions of model (1) when \(\delta >0\). We obtain Proposition 1 by combining the scenarios above.

Proof of Proposition 2

Similar to proof of Proposition 1, we can discuss the optimal solutions of model (6) in two scenarios, \(\delta >0\) and \(\delta <0\), respectively. When \(\delta <0\), the model (6) can be simplified as the model (18).

$$\begin{aligned} \mathop {\hbox {max}}\limits _{\tilde{\varpi },\tilde{\tau }_1 ,\tilde{\tau }_2} \mathop {\tilde{\prod }}\nolimits _\mathrm{M}^\mathrm{d}= & {} (\phi -\tilde{p}_1 +\beta \tilde{p}_2 )(\tilde{\varpi }-c_m +(\Delta -\delta )\tilde{\tau }_1 )\nonumber \\&+(\phi -\tilde{p}_2 +\beta \tilde{p}_1 )(\tilde{\varpi }-c_m +(\Delta -\delta )\tilde{\tau }_2 )\nonumber \\&-h\mathop {\tilde{\tau }}\nolimits _1^2-h\mathop {\tilde{\tau }}\nolimits _2^2-\mu _1 (\tilde{Q}-Q^{\mathrm{d}{*}}), \nonumber \\&s.t. \left\{ {\begin{array}{l} \tilde{p}_1 \in \hbox {argmax}\tilde{\prod }_{\mathrm{R1}}^\mathrm{d} , \\ \tilde{p}_2 \in \hbox {argmax}\tilde{\prod }_{\mathrm{R2}}^\mathrm{d} , \\ \tilde{\prod }_{\mathrm{R1}}^\mathrm{d} =(\phi -\tilde{p}_1 +\beta \tilde{p}_2 )(\tilde{p}_1 -\tilde{\varpi }), \\ \tilde{\prod }_{\mathrm{R2}}^\mathrm{d} =(\phi -\tilde{p}_2 +\beta \tilde{p}_1 )(\tilde{p}_2 -\tilde{\varpi }), \\ \tilde{Q}-Q^{\mathrm{d}{*}}\ge 0. \\ \end{array}} \right. \nonumber \\ \end{aligned}$$
(18)

The functions \(\tilde{\prod }_{\mathrm{R1}}^\mathrm{d} \) and \(\tilde{\prod }_{\mathrm{R2}}^\mathrm{d} \) can be easily verified to be concave in \(\tilde{p}_1 \) and \(\tilde{p}_2 \), respectively. Hence, there are the unique best responses to the decision of the retailers. Substitute them into the profit function of the manufacturer, we can verify that \(\mathop {\tilde{\prod }}\nolimits _\mathrm{M}^\mathrm{d} \) is jointly concave in \(\tilde{\tau }_1 \), \(\tilde{\tau }_2 \), and \(\tilde{\varpi }\). To solve the constrained optimization model, we introduce the Lagrange multiplier \(\lambda \ge 0\) and relax the constraint \(\tilde{Q}-Q^{\mathrm{d}{*}}\ge \hbox {0}\). From solving the Kuhn–Tucker condition of model (18), we can obtain the optimal solutions of model (18) when \(\delta (\delta -2\Delta )\ge B\mu _1 /C\) and \(0<\delta (\delta -2\Delta )<B\mu _1 /C\), respectively. Similarly, we can derive the optimal solutions of model (6) when \(\delta >0\). Combining these solutions, we can obtain Proposition 2.

Proof of Proposition 3

We can verify that \({\tilde{\prod }}_{\mathrm{R1}}^\mathrm{r} \) and \({\tilde{\prod }}_{\mathrm{R2}}^\mathrm{r} \) are concave in \(\tilde{p}_1 \) and \(\tilde{p}_2 \), respectively. Therefore, the unique best response can be derived by the first-order optimal conditions of the retailers. Following Cachon (2003), a revenue-sharing contract can coordinate the SC when \(p_\mathrm{1}^{\mathrm{r}{*}}=p_\mathrm{2}^{\mathrm{r}{*}}=p_\mathrm{1}^{\mathrm{c}{*}}=p_\mathrm{2}^{\mathrm{c}{*}}\). \(\tilde{\varpi }^{\mathrm{r}{*}}\) is derived by solving the above equation. The revenue share of retailers is derived from the third and fourth constraint conditions of model (13). Therefore, we can obtain Proposition 3.

Proof of Proposition 4

The proof is similar to the proof of Proposition 3, we thus omit it here.

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Wu, H., Han, X., Yang, Q. et al. Production and coordination decisions in a closed-loop supply chain with remanufacturing cost disruptions when retailers compete. J Intell Manuf 29, 227–235 (2018). https://doi.org/10.1007/s10845-015-1103-z

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