The distributionally robust optimization model for a remanufacturing system under cap-and-trade policy: a newsvendor approach

Abstract

This paper considers a remanufacturing system in which the remanufacturer uses both the fresh raw material and recycled material to produce new products. Only partial information of the demand distribution, including the mean and variance, is available in this system. The operational activities of the remanufacturer are major contributors to carbon emissions, and a government agency imposes a cap-and-trade (carbon emission trading) policy on the remanufacturer. In the context of the distributionally robust newsvendor model, a maxmin approach is used to solve the optimal joint collection and production quantities. To study whether implementation of a cap-and-trade policy improves the remanufacturer’s expected profit and reduces corresponding carbon emissions, the case where the government agency does not impose cap-and-trade policy is considered and compared. Finally, numerical analysis is conducted to illustrate and complement the analytical results and to investigate the influences of several main parameters on the distributionally robust newsvendor model for the remanufacturing system under a cap-and-trade policy.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. Alfares, H. K., & Elmorra, H. H. (2005). The distribution-free newsboy problem: Extensions to the shortages penalty case. International Journal of Production Economics, 93–94, 465–477.

    Article  Google Scholar 

  2. Amin, S. H., & Zhang, G. (2012). An integrated model for closed-loop supply chain configuration and supplier selection: Multi-objective approach. Expert Systems with Applications, 39, 6782–6791.

    Article  Google Scholar 

  3. Amin, S. H., & Zhang, G. (2013). A three-stage model for closed-loop supply chain configuration under uncertainty. International Journal of Production Research, 51(5), 1405–1425.

    Article  Google Scholar 

  4. Amin, S. H., Zhang, G., & Akhtar, P. (2017). Effects of uncertainty on a tire closed-loop supply chain network. Expert Systems with Applications, 73, 82–91.

    Article  Google Scholar 

  5. Arikan, E., & Jammernegg, W. (2014). The single period inventory model under dual sourcing and product carobn ffotprint constraint. International Journal of Production Economics, 157, 15–23.

    Article  Google Scholar 

  6. Atasu, A., Beril Toktay, L., & Van Wassenhove, L. N. (2013). How collection cost structure drives a manufacturer’s reverse channel choice. Production and Operations Management, 22(5), 1089–1102.

    Google Scholar 

  7. Bai, Q., & Chen, M. (2016). The distributionally robust newsvendor problem with dual sourcing under carbon tax and cap-and-trade regulations. Computers and Industrial Engineering, 98, 260–274.

    Article  Google Scholar 

  8. Berk, E., Gürler, Ü., & Leine, R. A. (2007). Bayesian demand updating in the lost sales newsvendor problem: A two-moment approximation. European Journal of Operational Research, 182, 256–281.

    Article  Google Scholar 

  9. Canan Savaskan, R., Bhattacharya, S., & Van Wassenhove, L. N. (2004). Closed-loop supply chain models with product remanufacturing. Management Science, 50(2), 239–252.

    Article  Google Scholar 

  10. Chen, X., & Wang, X. (2016). Effects of carbon emission reduction policies on transportation model selections with stochastic demand. Transportation Research Part E: Logisitic and Transportation Review, 90, 196–205.

    Article  Google Scholar 

  11. Chen, X., Wang, X., Kumar, V., & Kumar, N. (2016). Low carbon warehouse management under cap-and-trade policy. Journal of Cleaner Production, 139, 894–904.

    Article  Google Scholar 

  12. Choi, T. M. (2013). Local sourcing and fashion quick response system: The impacts of carbon footprint tax. Transportation Research Part E: Logisitic and Transportation Review, 55, 43–54.

    Article  Google Scholar 

  13. Chuang, C. H., Wang, C. X., & Zhao, Y. (2014). Closed-loop supply chain models for a high-tech product under alternative reverse channel and collection cost structures. International Journal of Production Economics, 156, 108–123.

    Article  Google Scholar 

  14. Du, S., Ma, F., Fu, Z., Zhu, L., & Zhang, J. (2015). Game-theoretic analysis for an emission-dependent supply chain in a ‘cap-and-trade’ system. Annals of Operations Research, 228, 135–149.

    Article  Google Scholar 

  15. Du, S., Qian, J., Liu, T., & Hu, L. (2018). Emission allowance allocation mechanism design: A low-carbon operations persepective. Annals of Operations Research. https://doi.org/10.1007/s10479-018-2922-z.

  16. Fu, Q., Sim, C. K., & Teo, C. P. (2018). Profit sharing agreements in decentralized supply chains: A distributionally robust approach. Operations Research, 66(2), 500–513.

    Article  Google Scholar 

  17. Gallego, G., & Moon, I. (1993). The distribution free newsboy problem: Review and extensions. Journal of the Operational Research Society, 44(8), 825–834.

    Article  Google Scholar 

  18. Govindan, K., Soleimani, H., & Kannan, D. (2015). Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future. European Journal of Operational Research, 240, 603–626.

    Article  Google Scholar 

  19. He, Y. (2017). Supply risk sharing in a closed-loop supply chain. International Journal of Production Economics, 183, 39–52.

    Article  Google Scholar 

  20. Kamburowski, J. (2015). The distribution-free newsboy problem and the demand skew. International Transaction in Operational Research, 22, 929–946.

    Article  Google Scholar 

  21. Kevork, I. S. (2010). Estimating the optimal order quantity and the maximum expected profit for single-period inventory decisions. Omega, 38, 218–227.

    Article  Google Scholar 

  22. Kwon, K., & Cheong, T. (2014). A minimax distribution-free procedure for a newsvendor problem with free shipping. European Journal of Operational Research, 232, 234–240.

    Article  Google Scholar 

  23. Lee, C. M., & Hsu, S. L. (2011). The effect of advertising on the distribution-free newsboy problem. International Journal of Production Economics, 129, 217–224.

    Article  Google Scholar 

  24. Liao, Y., Banerjee, A., & Yan, C. (2011). A distribution-free newsvendor model with balking and lost sales penalty. International Journal of Production Economics, 133, 224–227.

    Article  Google Scholar 

  25. Li, J., Choi, T. M., & Cheng, T. C. E. (2014). Mean variance analysis of fast fashion supply chains with returns policy. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 44(4), 422–434.

    Article  Google Scholar 

  26. Li, X., Li, Y., & Cai, X. (2016). On core sorting in RMTS and RMTO systems: A newsvendor framework. Decision Sciences, 47(1), 60–93.

    Article  Google Scholar 

  27. Liu, B., Holmbom, M., Segerstedt, A., & Chen, W. (2015). Effects of carbon emission regulations on remanufacturing decisions with limited information of demand distribution. International Journal of Production Research, 53(2), 532–548.

    Article  Google Scholar 

  28. Manikas, A. S., & Kroes, J. R. (2015). A newsvendor approach to compliance and production under cap and trade emissions regulation. International Journal of Production Economics, 159, 274–284.

    Article  Google Scholar 

  29. Ma, X., Wang, J., Bai, Q., & Wang, S. (2020). Optimizaiton of a three-echelon chain considering freshness-keeping efforts under cap-and-trade regulation in Industry 4.0. International Journal of Production Economics, 220, 107457.

  30. Moon, I., & Choi, S. (1995). The distribution free newsboy problem with balking. Journal of Operational Research Society, 46, 537–542.

    Article  Google Scholar 

  31. Moon, I., & Silver, E. A. (1995). The multi-item newsvendor problem with a budget constraint and fixed ordering costs. Journal of Operational Research Society, 51, 602–608.

    Article  Google Scholar 

  32. Mostard, J., Koster, R., & Teunter, R. (2005). The distribution-free newsboy problem with resale returns. International Journal of Production Economics, 97, 329–342.

    Article  Google Scholar 

  33. Mutha, A., Bansal, S., & Guide, V. D. R. (2016). Managing demand uncertainty through core acquisition in remanufacturing. Production and Operations Management, 25(8), 1449–1464.

    Article  Google Scholar 

  34. Ninh, A., Hu, H., & Allen, D. (2019). Robust newsvendor problems: Effect of discrete demands. Annals of Operations Research, 275, 607–621.

    Article  Google Scholar 

  35. Radhi, M., & Zhang, G. (2016). Optimal configuration of remanufacturing supply newwork with return quality decision. International Journal of Production Research, 54(5), 1487–1502.

    Article  Google Scholar 

  36. Radhi, M., & Zhang, G. (2019). Optimal cross-channel return policy in dual-channel retailing systems. International Journal of Production Economics, 210, 184–198.

    Article  Google Scholar 

  37. Raza, S. A., & Rathinam, S. (2017). A risk tolerance analysis for a joint price differentiation and inventory decisions problem with demand leakage effect. International Journal of Production Economics, 183, 129–145.

    Article  Google Scholar 

  38. Ren, J., Chen, X., & Hu, J. (2020). The effect of production- versus consumption-based emission tax under demand uncertainty. International Journal of Production Economics, 219, 82–98.

    Article  Google Scholar 

  39. Rosiĉ, H., & Jammernegg, W. (2013). The conomic and environmental performance of dual sourcing: A newsvendor approach. International Journal of Production Economics, 143, 109–119.

    Article  Google Scholar 

  40. Rowe, P., Eksioglu, B., & Eksioglu, S. (2017). Recycling procurement strategies with variable yield suppliers. Annals of Operations Research, 249, 215–234.

    Article  Google Scholar 

  41. Sarkar, B., Zhang, C., Majumder, A., Sarkar, M., & Seo, Y. W. (2018). A distribution free newsvendor mdoel with consignment policy and retailer’s royalty reduction. International Journal of Production Research, 56(15), 5025–5044.

    Article  Google Scholar 

  42. Savaskan, R. C., Bhattacharya, S., & Van Wassenhove, L. N. (2004). Closed-loop supply chain models with product remanufacturing. Management Science, 50(2), 239–252.

    Article  Google Scholar 

  43. Scarf, H. (1958). A min-max solution of an inventory problem. In Studies in the mathematical theory of inventory and production. Palo Alto: Stanford University Press.

  44. Shi, J., & Zhang, G. (2010). Multi-product budget-constrained acquisition and pricing with uncertain demand and supplier quantity discount. International Journal of Production Economics, 128, 322–331.

    Article  Google Scholar 

  45. Shi, J., Zhang, G., & Sha, J. (2011). Optimal production planning for a multi-product closed loop system with uncertain demand and return. Computers and Operations Research, 38, 641–650.

    Article  Google Scholar 

  46. Song, S., Govindan, K., Xu, L., Du, P., & Qiao, X. (2017). Capacity and production planning with carbon emission constraints. Transportation Research Part E: Logisitic and Transportation Review, 97, 132–150.

    Article  Google Scholar 

  47. Wang, L., Cai, G., Tsay, A. A., & Vakharia, A. J. (2017). Design of the reverse channel for remanufacturing: Must profit-maximization harm the environment? Production and Operations Management, 26(8), 1585–1603.

    Article  Google Scholar 

  48. Wang, C., Chen, J., & Chen, X. (2017). Pricing and order decisions with option contracts in the presence of customer returns. International Journal of Production Economics, 193, 422–436.

    Article  Google Scholar 

  49. Wang, Y., Chen, W., & Liu, B. (2017). Manufacturing/remanufacturing decisions for a capital-constrained manufacturer considering carbon emission cap and trade. Journal of Cleaner Production, 140, 1118–1128.

    Article  Google Scholar 

  50. Wang, Q., Li, J., Yan, H., & Zhu, S. X. (2016). Optimal remanufacturing strategies in name-your-own-price auctions with limited capcity. International Journal of Production Economics, 181, 113–129.

    Article  Google Scholar 

  51. Wei, C., Rodríguez, R. M., & Martínez, L. (2018). Uncertainty measures of extednd hesitant fuzzy linguistic term set. IEEE Transactions on Fuzzy Systems, 26(3), 1763–1768.

    Article  Google Scholar 

  52. Xu, J., Bai, Q., Xu, L., & Hu, T. (2018). Effects of emission reduction and partial demand information on operational decisions of a newsvendor problem. Journal of Cleaner Production, 188, 825–839.

    Article  Google Scholar 

  53. Zhang, G. (2008). Combining acquisition planning with inventory management under uncertain demand. INFOR: Information Systems and Operational Research, 46(2), 129–135.

    Google Scholar 

  54. Zhang, G. (2010). The multi-product newsboy problem with supplier quantity discounts and a budget constraint. European Journal of Operational Research, 206, 350–360.

    Article  Google Scholar 

  55. Zhang, G., & Ma, L. (2009). Optimal acquisition policy with quantity discounts and uncertain demands. International Journal of Production Research, 47(9), 2409–2425.

    Article  Google Scholar 

  56. Zhang, F., & Zhang, R. (2018). Trade-in remanufacturing, customer purchasing behavior, and governmetn policy. Manufacuring and Service Operations Management, 20(4), 601–616.

    Article  Google Scholar 

  57. Zhao, S., & Zhu, Q. (2017). Remanufacturing supply chain coordination under the stochastic remanufacturability rate and the random demand. Annals of Operations Research, 257, 661–695.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the three anonymous referees for their valuable comments and suggestions which have significantly improved the quality of the paper. The research is supported in part by the National Natural Science Foundation of China under Grants 71771138, 71702087, 71620107002 and 11771251, Humanities and Social Sciences Youth Foundation of Ministry of Education of China under Grant 17YJC630004, Natural Science Foundation of Shandong Province, China under Grant ZR2017MG009, Special Foundation for Taishan Scholars of Shandong Province, China under Grant tsqn201812061, and Science and Technology Research Program for Higher Education of Shandong Province, China under Grant 2019KJI006.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Qingguo Bai.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendices

Proof of Lemma 2

We prove the property of \(f_{s}(Q)\) shown in Lemmas 2 by considering the following two observations: (i) \(f_{s}(Q)\) is an increasing function of Q, and (ii) \(Q_{t_{s}}\) is a unique solution of \(f_{s}(Q)=0\).

Firstly, we take the first partial derivative of f(Q) with respect to Q and have

$$\begin{aligned} \frac{\partial f_{s}(Q)}{\partial Q}= & {} \frac{c_{L}e_{v}(Q-\mu )}{\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}+c_{L}(2e_{n}+e_{v})\nonumber \\= & {} \frac{c_{L}(2e_{n}+e_{v})\sqrt{\sigma ^{2}+(Q-\mu )^{2}}+c_{L}e_{v}(Q-\mu )}{\sqrt{\sigma ^{2}+(Q-\mu )^{2}}} \end{aligned}$$
(A.1)

Using \(\sqrt{\sigma ^{2}+(Q-\mu )^{2}}>\mu -Q\), we have \(\frac{\partial f(Q)}{\partial Q}>o\), implying that \(f_{s}(Q)\) is an increasing function of Q.

Secondly, rearranging equation \(f_{s}(Q)=0\), we have

$$\begin{aligned} c_{L}e_{v}\sqrt{\sigma ^{2}+(Q-\mu )^{2}}=\varOmega _{s}-c_{L}(2e_{n}+e_{v})Q \end{aligned}$$
(A.2)

Equation (A.2) yields

$$\begin{aligned} Q<\frac{\varOmega _{s}}{c_{L}(2e_{n}+e_{v})}. \end{aligned}$$
(A.3)

Simplifying Eq. (A.3), we further have

$$\begin{aligned} 4c^{2}_{L}e_{n}(e_{n}+e_{v})Q^{2}+2c_{L}Q[c_{L}\mu e^{2}_{v}-\varOmega _{s}(2e_{n}+e_{v})]+\varOmega _{s}^{2}-c^{2}_{L}e^{2}_{v}(\mu ^{2}+\sigma ^{2})=0. \end{aligned}$$
(A.4)

Solving Eq. (A.4) yields that there exist two roots \(Q_{t_{s}}\) and \(Q_{t_{0}}\) satisfying \(f_{s}(Q)=0\), where

$$\begin{aligned} Q_{t_{s}}=\frac{(2e_{n}+e_{v})\varOmega _{s}-c_{L}\mu e^{2}_{v}-e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2} +4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}, \end{aligned}$$
(A.5)

and

$$\begin{aligned} Q_{t_{0}}=\frac{(2e_{n}+e_{v})\varOmega _{s}-c_{L}\mu e^{2}_{v}+e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2} +4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}.\nonumber \\ \end{aligned}$$
(A.6)

We prove that \(Q_{t_{0}}\) is not a feasible root of \(f_{s}(Q)=0\) by considering the following two equations:

$$\begin{aligned} \sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v}) \sigma ^{2}}>c_{L}\mu (2_{n}+e_{v})-\varOmega _{s}, \end{aligned}$$
(A.7)

and

$$\begin{aligned} (2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}>e_{v}[c_{L}\mu (2_{n}+e_{v})-\varOmega _{s}]. \end{aligned}$$
(A.8)

From Eq. (A.8), we have

$$\begin{aligned}&\varOmega _{s}(2e_{n}+e_{v})^{2}-c_{L}\mu (2e_{n}+e_{v}) e^{2}_{v} \nonumber \\&\qquad +\,e_{v}(2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}\nonumber \\&\quad >4\varOmega _{s} e_{n}(e_{n}+e_{v}), \end{aligned}$$
(A.9)

and

$$\begin{aligned} Q_{t_{0}}>\frac{\varOmega _{s}}{c_{L}(2e_{n}+e_{v})}, \end{aligned}$$
(A.10)

which contradicts with Eq. (A.3).

Similar analysis to Eq. (A.8), we easily have

$$\begin{aligned} (2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}>e_{v}[\varOmega _{s}-c_{L}\mu (2_{n}+e_{v})]. \end{aligned}$$
(A.11)

Rearranging Eq. (A.11), we have

$$\begin{aligned}&\varOmega _{s}(2e_{n}+e_{v})^{2}-c_{L}\mu (2e_{n}+e_{v}) e^{2}_{v} \nonumber \\&\qquad -\,e_{v}(2e_{n}+e_{v})\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}\nonumber \\&\quad <4\varOmega _{s} e_{n}(e_{n}+e_{v}), \end{aligned}$$
(A.12)

and

$$\begin{aligned} Q_{t_{s}}<\frac{\varOmega _{s}}{c_{L}(2e_{n}+e_{v})}. \end{aligned}$$
(A.13)

which means that \(Q_{t_{s}}\) satisfies Eq. (A.3).

In the following, we prove \(Q_{t_{s}}>0\). Recalling \(C> \max \{{\underline{C}}_{1},{\underline{C}}_{2}\}\ge {\underline{C}}_{1}\), we have

$$\begin{aligned} C>\frac{c_{L}e_{v}(\sqrt{\sigma ^{2}+\mu ^{2}}-\mu )-\delta _{e}(\delta _{c}-c_{A}+p_{s}\delta _{e})}{2c_{L}}, \end{aligned}$$
(A.14)

and

$$\begin{aligned} \varOmega _{s}>c_{L}e_{v}\sqrt{\sigma ^{2}+\mu ^{2}}. \end{aligned}$$
(A.15)

Equation (A.15) yields

$$\begin{aligned} \varOmega _{s}^{2}-\mu ^{2}e^{2}_{v}c^{2}_{L}>c^{2}_{L}e^{2}_{v}\sigma ^{2}, \end{aligned}$$
(A.16)

and

$$\begin{aligned}{}[\varOmega _{s} (2e_{n}+e_{v})-c_{L}\mu e^{2}_{v}]^{2}>e^{2}_{v}\{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}\}.\nonumber \\ \end{aligned}$$
(A.17)

Using \(\varOmega _{s} (2e_{n}+e_{v})>c_{L}\mu e^{2}_{v}\), from Eq. (A.17), we have

$$\begin{aligned} \varOmega _{s} (2e_{n}+e_{v})-c_{L}\mu e^{2}_{v}>e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}},\nonumber \\ \end{aligned}$$
(A.18)

which means that \(Q_{t_{s}}\) shown in Eq. (A.5) is higher than zero, i.e., \(Q_{t_{s}}>0\). Using Eq. (A.13), we further have that \(Q_{t_{s}}\) is a unique feasible root of \(f_{s}(Q)=0\). Using the monotonicity of \(f_{s}(Q)\), we have that \(f_{s}(Q)\le 0\) if and only if \(Q\le Q_{t_{s}}\), and \(f_{s}(Q)>0\) if and only if \(Q>Q_{t_{s}}\).

With similar analysis, we prove that the property of \(f_{b}(Q)\) shown in Lemmas 2 holds and the detailed proof is omitted.

Proof of Theorem 1

Using \(Q\wedge D=D-(D-Q)^{+}\) and \((Q-D)^{+}=Q-D+(D-Q)^{+}\), and from Eqs. (1) and (2), we simplify \(\varPi _{0}(\tau ,Q)\) and \(J(\tau ,Q)\) as

$$\begin{aligned} \varPi _{0}(\tau ,Q)=(p-v)\mu +[v-c_{n}+(\delta _{c}-c_{A})\tau ]Q-c_{L}\tau ^{2}Q^{2}-(p+c_{s}-v)E(D-Q)^{+},\nonumber \\ \end{aligned}$$
(B.1)

and

$$\begin{aligned} J(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +e_{v}E(D-Q)^{+}. \end{aligned}$$
(B.2)

For model \(M_{1s}\), we substitute Eqs. (B.1) and (B.2) into (7), and have

$$\begin{aligned} \varPi _{1s}(\tau ,Q)= & {} (p-v+p_{s}e_{v})\mu +p_{s}C+[v-c_{n}-p_{s}(e_{n}+e_{v})]Q \nonumber \\&+\,(\delta _{c}-c_{A}+p_{s}\delta _{e})\tau Q-c_{L}\tau ^{2}Q^{2}\nonumber \\&-\,(p+c_{s}-v+p_{s}e_{v})E(D-Q)^{+}. \end{aligned}$$
(B.3)

Using Lemma 1 to solve Eqs. (B.2) and (B.3), we further have

$$\begin{aligned} J^{F}(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +\frac{1}{2}e_{v}[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )], \end{aligned}$$
(B.4)

and

$$\begin{aligned} \varPi ^{F}_{1s}(\tau ,Q)= & {} (p-v+p_{s}e_{v})\mu +p_{s}C+[v-c_{n}-p_{s}(e_{n}+e_{v})]Q \nonumber \\&+\,(\delta _{c}-c_{A}+p_{s}\delta _{e})\tau Q-c_{L}\tau ^{2}Q^{2}\nonumber \\&-\,\frac{1}{2}(p+c_{s}-v+p_{s}e_{v})[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )]. \end{aligned}$$
(B.5)

For a fixed \(Q>0\), taking the first and second partial derivatives of \(\varPi ^{F}_{1s}(\tau ,Q)\) with respect to \(\tau \) and from Eq. (B.5), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1s}(\tau ,Q)}{\partial \tau }=(\delta _{c}-c_{A}+p_{s}\delta _{e})Q-2c_{L}\tau Q^{2}, \end{aligned}$$
(B.6)

and

$$\begin{aligned} \frac{\partial ^{2}\varPi ^{F}_{1s}(\tau ,Q)}{\partial \tau ^{2}}=-2c_{L} Q^{2}<0. \end{aligned}$$
(B.7)

Equation (B.7) yields that \(\varPi ^{F}_{1s}(\tau ,Q)\) is concave in \(\tau \) for a fixed Q. Solving \(\frac{\partial \varPi ^{F}_{1s}(\tau ,Q)}{\partial \tau }=0\) and from Eq. (B.6), we have

$$\begin{aligned} \tau ^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}Q}. \end{aligned}$$
(B.8)

Substituting Eqs. (B.8) into (9) yields

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L} \tau _{u}}\le Q\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L} \tau _{l}}. \end{aligned}$$
(B.9)

Substituting Eqs. (B.8) into (B.4) and using the expression of \(f_{s}(Q)\) yield

$$\begin{aligned} J^{F}(\tau ,Q)=\frac{1}{2c_{L}}f_{s}(Q)+C. \end{aligned}$$
(B.10)

Using \(J^{F}(\tau ,Q)\le C\) and from Lemma 2, we have

$$\begin{aligned} 0<Q<Q_{t_{s}}, \end{aligned}$$
(B.11)

where \(Q_{t_{s}}=\frac{(2e_{n}+e_{v})\varOmega _{s}-c_{L}\mu e^{2}_{v}-e_{v}\sqrt{[\varOmega _{s}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}\).

Substituting Eqs. (B.8) into (B.5), we have

$$\begin{aligned} \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)= & {} \frac{1}{2}(p-v-c_{s}+p_{s}e_{v})\mu +p_{s}C+\frac{(\delta _{c}-c_{A}+p_{s}\delta _{e})^{2}}{4c_{L}} +\frac{1}{2}[p+v+c_{s}-2c_{n}\nonumber \\&-\,p_{s}(2e_{n}+e_{v})]Q-\frac{1}{2}(p+c_{s}-c+p_{s}e_{v})\sqrt{\sigma ^{2}+(Q-\mu )^{2}}. \end{aligned}$$
(B.12)

Taking the first and second partial derivatives of \(\varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)\) with respect to Q and from Eq. (B.12), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)}{\partial Q}=\frac{1}{2}[p+v+c_{s}-2c_{n}-p_{s}(2e_{n}+e_{v})]-\frac{(p+c_{s}-v+p_{s}e_{v})(Q-\mu )}{2\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}, \end{aligned}$$
(B.13)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)}{\partial Q^{2}}=-\frac{1}{2}\sigma ^{2}(p+c_{s}-v+p_{s}e_{v})[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{3}{2}}<0. \end{aligned}$$
(B.14)

Equation (B.14) yields that \(\varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)\) is concave in Q. Solving \(\frac{\partial \varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)}{\partial Q}=0\) and from Equation (B.13), we have

$$\begin{aligned} Q_{0_{s}}= & {} \mu +\frac{\sigma }{2}[\sqrt{\frac{p+c_{s}-c_{n}-p_{s}e_{n}}{p_{s}(e_{n}+e_{v})+c_{n}-v}}-\sqrt{\frac{p_{s}(e_{n}+e_{v})+c_{n}-v}{p+c_{s}-c_{n}-p_{s}e_{n}}}]\nonumber \\= & {} \mu +\frac{\sigma (A_{s}-B_{s})}{2\sqrt{A_{s}B_{s}}}. \end{aligned}$$
(B.15)

Using the concavity property of \(\varPi ^{F}_{1s}(\tau ^{*}_{1s},Q)\) and from Eqs. (B.9), (B.11), and (B.15), we solve the optimal robust production quantity of the remanufacturer in problem \(M_{1s}\) by considering the following three cases:

Case 1\(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}\le Q_{t_{s}}\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}\). From Eqs. (B.9) and (B.11), we have that the feasible region of problem \(M_{1s}\) becomes

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q\le Q_{t_{s}}. \end{aligned}$$
(B.16)

When \(Q_{t_{s}}<Q_{0_{s}}\), we have \(Q^{*}_{1s}=Q_{t_{s}}\). When \(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} >Q_{0_{s}}\), we have \(Q^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}\). When \(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q_{0_{s}}\le Q_{t_{s}}\), we have \(Q^{*}_{1s}=Q_{0_{s}}\).

Case 2\(Q_{t_{s}}>\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}\). From Eqs. (B.9) and (B.11), we have that the feasible region of problem \(M_{1s}\) becomes

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}. \end{aligned}$$
(B.17)

When \(Q_{0_{s}}>\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}\), we have \(Q^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}\). When \(Q_{0_{s}}<\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}\), we have \(Q^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}\). When \(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}} \le Q_{0_{s}}\le \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{l}}\), we have \(Q^{*}_{1s}=Q_{0_{s}}\).

Case 3\(Q_{t_{s}}<\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}\). From Eqs. (B.9) and (B.11), we have that the feasible region of problem \(M_{1s}\) is empty set.

Consequently, from Cases 1 and 2, we have that when \(Q_{t_{s}}\ge \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}\), i.e., \(\tau _{u}\ge \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}Q_{t_{s}}}\), the optimal robust production quantity \(Q^{*}_{1s}\) and collection rate \(\tau ^{*}_{1s}\) are \(Q^{*}_{1s}=\max \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}, \min \{Q_{0_{s}},\min \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}},Q_{t_{s}}\}\}\}\) and \(\tau ^{*}_{1s}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}Q^{*}_{1s}}\).

Proof of Corollary 1

  1. (i)

    Using the expression of \(C_{0}\) to simplify \(C\ge C_{0}\) yields

    $$\begin{aligned} C\ge \frac{e_{v}}{2}\left[ \sqrt{\sigma ^{2}+(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu )}-\mu \right] +\frac{(\delta _{c}-c_{A}+p_{s}\delta _{e}) (2e_{n}-e_{v}-2\delta _{e}\tau _{u})}{4c_{L}\tau _{u}}, \end{aligned}$$
    (C.1)

    and

    $$\begin{aligned} 2c_{L}C\ge c_{L}e_{v}[\sqrt{\sigma ^{2}+(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu )}-\mu ]+\frac{(\delta _{c}-c_{A}+p_{s}\delta _{e}) (2e_{n}+e_{v}-2\delta _{e}\tau _{u})}{2\tau _{u}}. \end{aligned}$$
    (C.2)

Using the expressions of f(Q) and \(\varOmega \) to rearrange Eq. (C.2), we have

$$\begin{aligned} 0\ge c_{L}e_{v}\sqrt{\sigma ^{2}+\left( \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu \right) }+\frac{c_{L}(2e_{n} +e_{v})(\delta _{c}-c_{A}+p_{s}\delta _{e})}{2c_{L}\tau _{u}}-\varOmega _{s}, \end{aligned}$$
(C.3)

and

$$\begin{aligned} f_{s}(Q_{t_{s}})\ge f(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}). \end{aligned}$$
(C.4)

Recalling that \(f_{s}(Q)\) is an increasing function of Q, from Eq. (C.4), we further have

$$\begin{aligned} Q_{t_{s}}\ge \frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}. \end{aligned}$$
(C.5)

Using \(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}>0\), \(f_{s}(0)=c_{L}e_{v}\sqrt{\sigma ^{2}+\mu ^{2}}-\varOmega \), and the monotonicity of \(f_{s}(Q)\), we have

$$\begin{aligned} e_{v}\sqrt{\sigma ^{2}+(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}-\mu )^{2}} +\frac{(2e_{n}+e_{v})(\delta _{c}-c_{A}+p_{s}\delta _{e})}{2c_{L}\tau _{u}}>e_{v}\sqrt{\sigma ^{2}+\mu ^{2}}.\nonumber \\ \end{aligned}$$
(C.6)

Equation (C.6) yields \(C_{0}>{\underline{C}}\). From Eq. (C.5) and Theorem 1, we have that when \(C\ge C_{0}\), there exist the optimal robust collection rate and production quantity such that the expected of the remanufacturer is maximized.

Similarly, simplifying \(C<C_{0}\), we have \(f_{s}(Q_{t_{s}})< f(\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}})\) and \(Q_{t_{s}}<\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}\tau _{u}}\). From Theorem 1, we have that when \(C<C_{0}\), there does not exist the robust collection rate and production quantity such that the expected of the remanufacturer is maximized.

  1. (ii)

    From Lemma 2 and using the monotonicity of \(f_{s}(Q)\), we have that if \(Q^{*}_{1s}=Q_{t_{s}}\), then \(J^{F}(\tau ^{*}_{1s}, Q^{*}_{1s})=C\). Otherwise, \(J^{F}(\tau ^{*}_{1s}, Q^{*}_{1s})<C\).

Proof of Theorem 2

With similar analysis to problem \(M_{1s}\), using Lemma1 to solve problem \(M_{1b}\), we simplify the expected profit and carbon emissions of the remanufacturer in the worst demand scenario as

$$\begin{aligned} \varPi ^{F}_{1b}(\tau ,Q)= & {} (p-v+p_{b}e_{v})\mu +p_{b}C+[v-c_{n}-p_{b}(e_{n}+e_{v})]Q\nonumber \\&+\,(\delta _{c}-c_{A}+p_{b}\delta _{e})\tau Q-c_{L}\tau ^{2}Q^{2}\nonumber \\&-\,\frac{1}{2}(p+c_{s}-v+p_{b}e_{v})[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )], \end{aligned}$$
(D.1)

and

$$\begin{aligned} J^{F}(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +\frac{1}{2}e_{v}[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )]. \end{aligned}$$
(D.2)

For a fixed \(Q>0\), taking the first and second partial derivatives of \(\varPi ^{F}_{1b}(\tau ,Q)\) with respect to \(\tau \) and from Eq. (D.1), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1b}(\tau ,Q)}{\partial \tau }=(\delta _{c}-c_{A}+p_{b}\delta _{e})Q-2c_{L}\tau Q^{2}, \end{aligned}$$
(D.3)

and

$$\begin{aligned} \frac{\partial ^{2}\varPi ^{F}_{1b}(\tau ,Q)}{\partial \tau ^{2}}=-2c_{L} Q^{2}<0. \end{aligned}$$
(D.4)

From Eq. (D.4), we have that \(\varPi ^{F}_{1b}(\tau ,Q)\) is a concave function of \(\tau \) for any fixed \(Q>0\). Solving \(\frac{\partial \varPi ^{F}_{1b}(\tau ,Q)}{\partial \tau }=0\) and from Eq. (D.3), we have

$$\begin{aligned} \tau ^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}Q}. \end{aligned}$$
(D.5)

Substituting Eqs. (D.5) into (13), we have

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}\le Q\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}. \end{aligned}$$
(D.6)

Substituting Eqs. (D.5) into (D.2) and from Lemma 2, we rearrange \(J^{F}(\tau ^{*}_{1b},Q)\ge C\) as

$$\begin{aligned} Q\ge Q_{t_{b}}, \end{aligned}$$
(D.7)

where \( Q_{t_{b}}=\frac{(2e_{n}+e_{v})\varOmega _{b}-c_{L}\mu e^{2}_{v}-e_{v}\sqrt{[\varOmega _{b}-c_{L}\mu (2e_{n}+e_{v})]^{2}+4c^{2}_{L}e_{n}(e_{n}+e_{v})\sigma ^{2}}}{4c_{L}e_{n}(e_{n}+e_{v})}\).

Taking the first and second partial derivatives of \(\varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)\) with respect to Q and from Eqs. (D.1) and (D.5), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)}{\partial Q}=\frac{1}{2}[p+v+c_{s}-2c_{n}-p_{b}(2e_{n}+e_{v})]-\frac{(p+c_{s}-v+p_{b}e_{v})(Q-\mu )}{2\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}, \end{aligned}$$
(D.8)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)}{\partial Q^{2}}=-\frac{1}{2}\sigma ^{2}(p+c_{s}-v+p_{b}e_{v})[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{3}{2}}<0. \end{aligned}$$
(D.9)

Equation (D.9) implies that \( \varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)\) is concave in Q. Solving \(\frac{\partial \varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)}{\partial Q}=0\) and from Eq. (D.8), we have

$$\begin{aligned} Q_{0_{b}}= & {} \mu +\frac{\sigma }{2}[\sqrt{\frac{p+c_{s}-c_{n}-p_{b}e_{n}}{p_{b}(e_{n}+e_{v}) +c_{n}-v}}-\sqrt{\frac{p_{b}(e_{n}+e_{v})+c_{n}-v}{p+c_{s}-c_{n}-p_{b}e_{n}}}\nonumber \\= & {} \mu +\frac{\sigma (A_{b}-B_{b})}{2\sqrt{A_{b}B_{b}}}. \end{aligned}$$
(D.10)

Using the concavity property of \(\varPi ^{F}_{1b}(\tau ^{*}_{1b},Q)\) and from Eqs. (D.6), (D.7), and (D.10), we obtain the optimal robust production quantity of the remanufacturer by considering the following three cases:

Case 1\( Q_{t_{b}}< \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}\). From Eqs. (D.5) and (D.6), we have that the feasible region of problem \(M_{1s}\) becomes

$$\begin{aligned} \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}} \le Q\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}. \end{aligned}$$
(D.11)

When \(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}<Q_{0_{b}}\), we have \(Q^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}\). When \(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}>Q_{0_{b}}\), we have \(Q^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}\). When \(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}}\le Q_{0_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}\), we have \(Q^{*}_{1b}=Q_{0_{b}}\).

Case 2\(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{u}} \le Q_{t_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}\). From Eqs. (D.5) and (D.6), we have that the feasible region of problem \(M_{1s}\) becomes

$$\begin{aligned} Q_{0_{b}} \le Q\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}. \end{aligned}$$
(D.12)

When \(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}<Q_{0_{b}}\), we have \(Q^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}\). When \(Q_{t_{b}}>Q_{0_{b}}\), we have \(Q^{*}_{1b}=Q_{t_{b}}\). When \(Q_{t_{b}}\le Q_{0_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}\), we have \(Q^{*}_{1b}=Q_{0_{b}}\).

Case 3\(Q_{t_{b}}>\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}\tau _{l}}\). From Eqs. (D.6) and (D.7), we have that the feasible region of problem \(M_{1b}\) is empty set.

Consequently, from Cases 1 and 2, we have that when \(Q_{t_{b}}\le \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}\), i.e., \(\tau _{l}\ge \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}Q_{t_{b}}}\), the optimal robust production quantity \(Q^{*}_{1b}\) and collection rate \(\tau ^{*}_{1b}\) are \(Q^{*}_{1b}=\min \{\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}, \max \{Q_{0_{b}},\max \{\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}},Q_{t_{b}}\}\}\}\) and \(\tau ^{*}_{1b}=\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}Q^{*}_{1b}}\).

Proof of Theorem 4

Using \(Q\wedge D=D-(D-Q)^{+}\) and \((Q-D)^{+}=Q-D+(D-Q)^{+}\) to simplify Eq. (1), and from Lemma 1, we rearrange the expected worst-case profit of the remanufacturer as

$$\begin{aligned} \varPi ^{F}_{0}(\tau ,Q)= & {} \frac{1}{2}\mu (p-v-c_{s})+\frac{1}{2}Q(p+v+c_{s}-2c_{n})+(\delta _{c}-c_{A})\tau Q-c_{L}\tau ^{2}Q^{2} \nonumber \\&-\frac{1}{2}(p+c_{s}-v)\sqrt{\sigma ^{2}+(Q-\mu )^{2}}. \end{aligned}$$
(E.1)

From Eq. (E.1), for a fixed \(Q>0\), taking the first and second partial derivatives of \(\varPi ^{F}_{0}(\tau ,Q)\) with respect to \(\tau \) yields

$$\begin{aligned} \frac{\partial \varPi ^{F}_{0}(\tau ,Q)}{\partial \tau }=(\delta _{c}-c_{A})Q-2\tau c_{L}Q^{2}, \end{aligned}$$
(E.2)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{0}(\tau ,Q)}{\partial \tau ^{2}}=-2c_{L}Q^{2}<0. \end{aligned}$$
(E.3)

From Eq. (E.3), we have that \(\varPi ^{F}_{0}(\tau ,Q)\) is concave in \(\tau \) for a fixed \(Q>0\). Solving \(\frac{\partial \varPi ^{F}_{0}(\tau ,Q)}{\partial \tau }=0\) and from Eq. (E.2), we have

$$\begin{aligned} \tau ^{*}_{0}=\frac{\delta _{c}-c_{A}}{2c_{L}Q}. \end{aligned}$$
(E.4)

Substituting Eqs. (E.4) into (E.1) yields

$$\begin{aligned} \varPi ^{F}_{0}(\tau ^{*}_{0},Q)= & {} \frac{1}{2}\mu (p-v-c_{s})+\frac{1}{2}Q(p+v+c_{s}-2c_{n}) +\frac{(\delta _{c}-c_{A})^{2}}{4c_{L}}\nonumber \\&-\frac{1}{2}(p+c_{s}-v)\sqrt{\sigma ^{2}+(Q-\mu )^{2}}. \end{aligned}$$
(E.5)

Substituting Eqs. (E.4) into (19) yields

$$\begin{aligned} \frac{\delta _{c}-c_{A}}{2\tau _{u}c_{A}}\le Q\le \frac{\delta _{c}-c_{A}}{2\tau _{l}c_{A}}. \end{aligned}$$
(E.6)

Taking the first and second partial derivatives of \(\varPi ^{F}_{0}(\tau ,Q)\) with respect to Q and from Eq. (E.1), we have

$$\begin{aligned} \frac{\partial \varPi ^{F}_{0}(\tau ^{*}_{0},Q)}{\partial Q}=\frac{1}{2}(p+v+c_{s}-2c_{n})-\frac{(p+c_{s}-v)(Q-\mu )}{2\sqrt{\sigma ^{2}+(Q-\mu )^{2}}}, \end{aligned}$$
(E.7)

and

$$\begin{aligned} \frac{\partial ^{2} \varPi ^{F}_{0}(\tau ^{*}_{0},Q)}{\partial Q^{2}}=-\frac{1}{2}(p+c_{s}-v)\sigma ^{2}[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{3}{2}}<0. \end{aligned}$$
(E.8)

Equation (E.8) yields that \(\varPi ^{F}_{0}(\tau ^{*}_{0},Q)\) is a concave function of Q. Therefore, solving \(\frac{\partial \varPi ^{F}_{0}(\tau ^{*}_{0},Q)}{\partial Q}=0\) and from Eq. (E.7), we have

$$\begin{aligned} Q_{0}=\mu +\frac{\sigma }{2}(\sqrt{\frac{p+c_{s}-c_{n}}{c_{n}-v}}-\sqrt{\frac{c_{n}-v}{p+c_{s}-c_{n}}})=\mu +\frac{\sigma (A_{0}-B_{0})}{2\sqrt{A_{0}B_{0}}}. \end{aligned}$$
(E.9)

Using a constraint condition for Q shown in Eqs. (E.6) and from (E.9), we consider the following three cases to solve the optimal robust production quantity of model \(M_{0}\). (i) When \(\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}<Q_{0}\), we have \(Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}\). (ii) When \(\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}>Q_{0}\), we have \(Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}\). and (iii) When \(\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}\le Q_{0}\le \frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}\), we have \(Q^{*}_{0}=Q_{0}\). Consequently, from Eq. (E.4), we have the optimal robust collection rate and production quantity as \(\tau ^{*}_{0}=\frac{\delta _{c}-c_{A}}{2c_{L}Q^{*}_{0}}\) and \(Q^{*}_{0}=\max \{\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{A}},\min \{\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{A}},Q_{0}\}\}\).

Proof of Theorem 5

  1. (i)

    Define \(f(x)=\mu +\frac{\sigma }{2}(x-\frac{1}{x})\), where \(x>0\). By proving \(\frac{\partial f(x)}{\partial x}=\frac{\sigma }{2}(1+\frac{1}{x^{2}})>0\), we have that f(x) is an increasing function of x. Using \(\frac{A_{0}}{B_{0}}=\frac{p+c_{s}-c_{n}}{c_{n}-v}>\frac{p+c_{s}-c_{n}-p_{s}e_{n}}{c_{n}-v}>\frac{p+c_{s} -c_{n}-p_{s}e_{n}}{p_{s}(e_{n}+e_{v})+c_{n}-v}=\frac{A_{s}}{B_{s}}\) and from Eqs. (16) and (21), we have \(Q_{0}>Q_{0_{s}}\). Similarly, using \(\frac{A_{0}}{B_{0}}=\frac{p+c_{s}-c_{n}}{c_{n}-v}>\frac{p+c_{s}-c_{n}-p_{b}e_{n}}{c_{n}-v}>\frac{p+c_{s} -c_{n}-p_{b}e_{n}}{p_{b}(e_{n}+e_{v})+c_{n}-v}=\frac{A_{b}}{B_{b}}\) and from Eqs. (17) and (21), we have \(Q_{0}>Q_{0_{b}}\). Consequently, we have \(Q_{0}>\max \{Q_{0_{s}},Q_{0_{b}}\}\). Moreover, from the proof of Theorem  1 and 2 , and 4 , we easily have \(\tau ^{*}_{1}Q^{*}_{1}=\frac{\delta \delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}}\) or \(\frac{\delta \delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}}\) and \(\tau ^{*}_{0}Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2c_{L}}\). Using \(\min \{\frac{\delta \delta _{c}-c_{A}+p_{s}\delta _{e}}{2c_{L}},\frac{\delta \delta _{c}-c_{A}+p_{b}\delta _{e}}{2c_{L}}\}>\frac{\delta _{c}-c_{A}}{2c_{L}}\), we have \(\tau ^{*}_{1}Q^{*}_{1}>\tau ^{*}_{0}Q^{*}_{0}\).

  2. (ii)

    We consider the following three cases:

Case 1\(\tau ^{*}_{0}=\tau _{u}\). From the proof of Theorem 4, we have \(Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}\). When \(\tau ^{*}_{1}=\tau _{u}\), from the proofs of Theorems 1 and 2 , we have \(Q^{*}_{1}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}\) or \(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}\). We easily have \(Q^{*}_{1}>Q^{*}_{0}\). When \(\tau ^{*}_{1}=\tau _{l}\), we have \(Q^{*}_{1}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}}\) or \(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}\) and further have \(Q^{*}_{1}>Q^{*}_{0}\) with \(\tau _{l}<\tau _{u}\). When \(\tau _{l}<\tau ^{*}_{1}<\tau _{u}\), from the proofs of Theorems 1 and 2 , we have \(Q^{*}_{1}=Q_{t_{s}}\) or \(Q_{0_{s}}\) under the costraint condition of \(\min \{Q_{t_{s}},Q_{0_{s}}\}>\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}\) and \(Q^{*}_{1}=Q_{t_{b}}\) or \(Q_{0_{b}}\) under the costraint condition of \(\min \{Q_{t_{b}},Q_{0_{b}}\}>\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}\). Using \(\min \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{u}c_{L}}, \frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{u}c_{L}}\}>\frac{\delta _{c}-c_{A}}{2\tau _{u}c_{L}}\) yields \(Q^{*}_{1}>Q^{*}_{0}\).

Case 2\(\tau ^{*}_{0}=\tau _{l}\). From the proof of Theorem 4, we have \(Q^{*}_{0}=\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}\). When \(\tau ^{*}_{1}=\tau _{l}\), from the proofs of Theorems 1 and 2 , we have \(Q^{*}_{1}=\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}}\) or \(\frac{\delta _{c}-c_{A}+p_{b}\delta _{e}}{2\tau _{l}c_{L}}\). Using \(\min \{\frac{\delta _{c}-c_{A}+p_{s}\delta _{e}}{2\tau _{l}c_{L}},\frac{\delta _{c}-c_{A} +p_{b}\delta _{e}}{2\tau _{l}c_{L}}\}>\frac{\delta _{c}-c_{A}}{2\tau _{l}c_{L}}\), we have \(Q^{*}_{1}>Q^{*}_{0}\).

Case 3\(\tau _{l}<\tau ^{*}_{0}<\tau _{u}\). From the proof of Theorem 4, we have \(Q^{*}_{0}=\mu +\frac{\sigma }{2}(\sqrt{\frac{A_{0}}{B_{0}}} -\sqrt{\frac{B_{0}}{A_{0}}})\). When \(\tau _{l}<\tau ^{*}_{1}<\tau _{u}\), from the proofs of Theorems 1 and 2 , we have \(Q^{*}_{1}=Q_{0_{s}}\) or \(Q_{0_{b}}\) or \(Q_{t_{s}}\) satisfying \(Q_{t_{s}}\le Q_{0_{s}}\), or \(Q_{t_{b}}\) satisfying \(Q_{t_{b}}>Q_{0_{b}}\) Using \(Q_{0}>\max \{Q_{0_{s}},Q_{0_{b}}\}\) and from the condition \(Q_{t_{b}}<Q_{0}\), we have \(Q^{*}_{1}<Q^{*}_{0}\).

  1. (iii)

    From Eq. (B.4), we have the worst-case expected carbon emissions as

    $$\begin{aligned} J^{F}(\tau ,Q)=(e_{n}+e_{v}-\delta _{e}\tau )Q-e_{v}\mu +\frac{1}{2}e_{v}[\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-(Q-\mu )],\qquad \end{aligned}$$
    (F.1)

Substituting Eqs. (E.4) into (F.1), we simplify the worst-case expected carbon emissions in model \(M_{0}\) as

$$\begin{aligned} J^{F}(\tau ^{*}_{0},Q)=\frac{1}{2}(2e_{n}+e_{v})Q+\frac{1}{2}e_{v}\sqrt{\sigma ^{2}+(Q-\mu )^{2}}-\frac{1}{2}e_{v}\mu -\frac{\delta _{e}(\delta _{c}-c_{A})}{2c_{L}}.\qquad \end{aligned}$$
(F.2)

Taking the first partial derivative of \(J^{F}(\tau ^{*}_{0},Q)\) with respect to Q, we have

$$\begin{aligned} \frac{\partial J^{F}(\tau ^{*}_{0},Q)}{\partial Q}=\frac{1}{2}[\sigma ^{2}+(Q-\mu )^{2}]^{-\frac{1}{2}} \{(2e_{n}+e_{v})[\sigma ^{2}+(Q-\mu )^{2}]^{\frac{1}{2}}+e_{v}(Q-\mu )\}.\nonumber \\ \end{aligned}$$
(F.3)

Using \([\sigma ^{2}+(Q-\mu )^{2}]^{\frac{1}{2}}>\mu -Q\), we have \(\frac{\partial J^{F}(\tau ^{*}_{0},Q)}{\partial Q}>0\), which implies that \(J^{F}(\tau ^{*}_{0},Q)\) is an increasing function of Q. Therefore, when \(Q^{*}_{0}>Q^{*}_{1}\), we have

$$\begin{aligned}&J^{F}(\tau ^{*}_{0},Q^{*}_{0})>J^{F}(\tau ^{*}_{0},Q^{*}_{1})\nonumber \\&\quad =\frac{1}{2}(2e_{n}+e_{v})Q^{*}_{1} +\frac{1}{2}e_{v}\sqrt{\sigma ^{2}+(Q^{*}_{1}-\mu )^{2}}-\frac{1}{2}e_{v}\mu -\frac{\delta _{e}(\delta _{c}-c_{A})}{2c_{L}}\nonumber \\&\quad >\frac{1}{2}(2e_{n}+e_{v})Q^{*}_{1} +\frac{1}{2}e_{v}\sqrt{\sigma ^{2}+(Q^{*}_{1}-\mu )^{2}}-\frac{1}{2}e_{v}\mu -\frac{\delta _{e}(\delta _{c}-c_{A}+\max \{p_{s},p_{b}\}\delta _{e})}{2c_{L}}\nonumber \\&\quad \ge J^{F}(\tau ^{*}_{1},Q^{*}_{1}). \end{aligned}$$
(F.4)

The last inequality in Eq. (F.4) holds because of Eqs. (B.5), (B.8), and (D.5).

Moreover, from Eq. (7), we have that when \(C>J^{F}(\tau ,Q)\), \(\varPi ^{F}_{1}(\tau ,Q)=\varPi ^{F}_{1s}(\tau ,Q)=\varPi ^{F}_{0}(\tau ,Q)+p_{s}[C-J^{F}(\tau ,Q)]\). From Eqs. (5) and (6), we have that \((\tau ^{*}_{0},Q^{*}_{0})\) is one of the feasible solutions of model \(M_{1}\). We further have \(\varPi ^{F}_{1}(\tau ^{*}_{0},Q^{*}_{0})=\varPi ^{F}_{0}(\tau ^{*}_{0},Q^{*}_{0})+p_{s}[C-J^{F}(\tau ^{*}_{0},Q^{*}_{0})]> \varPi ^{F}_{0}(\tau ^{*}_{0},Q^{*}_{0})\). Using the optimal property of \((\tau ^{*}_{1},Q^{*}_{1})\), we have \(\varPi ^{F}_{1}(\tau ^{*}_{1},Q^{*}_{1})\ge \varPi ^{F}_{1}(\tau ^{*}_{0},Q^{*}_{0})\). Therefore, we have \(\varPi ^{F}_{1}(\tau ^{*}_{1},Q^{*}_{1})\ge \varPi ^{F}_{0}(\tau ^{*}_{0},Q^{*}_{0})\) when \(C>J^{F}(\tau ^{*}_{0},Q^{*}_{0})\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bai, Q., Xu, J. & Zhang, Y. The distributionally robust optimization model for a remanufacturing system under cap-and-trade policy: a newsvendor approach. Ann Oper Res (2020). https://doi.org/10.1007/s10479-020-03642-4

Download citation

Keywords

  • Remanufacturing
  • Collection
  • Cap-and-trade
  • Newsvendor
  • Robust optimization