# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020078

## Optimal production and emission reduction policies for a remanufacturing firm considering deferred payment strategy

 School of Economics and Management, Southeast University, Nanjing, P.R. China

* Corresponding author: Weida Chen

Received  June 2019 Revised  December 2019 Published  April 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (NSFC) under Grants 71571042, 71971058

Carbon emission reduction is regarded as an effective way to protect the environment, which requires a large amount of capital. Thus, for a remanufacturing firm with limited initial capital, trade credits act as an effective financing method in supporting production and emission reductions. In this study, under the cap-and-trade and government's subsidy policies, a joint decision on recycling, remanufacturing and emission reduction by a financially constrained remanufacturer with considering deferred payment to a third-party recycler is analyzed. On the basis, optimization models are established to derive the optimal recycling quantity, carbon reduction rate and government subsidy rate by using a backward induction. Furthermore, an analytical comparison is provided between the cases of base model, carbon abatement investment model and deferred payment model. Numerical experiment results indicate that the remanufacturer can always make use of the investment option to further decrease its carbon emissions and gain more profit. We also find that deferred payment can effectively mitigate carbon emissions only when the degree of emission efforts is more than a certain critical value, and it also plays a positive role in the third-party recycler's revenue, especially for the case with higher initial capital. Some other managerial implications are further discussed.

Citation: Qianru li, Weida chen, Yongming zhang. Optimal production and emission reduction policies for a remanufacturing firm considering deferred payment strategy. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020078
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##### References:
Framework of the remanufacturing system with limited capital
Framework of the game model of Model Ⅱ
The sequence of the events
Comparison of emission differences under different models for varying values of $\xi$ when $c_{m} = 4$
Comparison of profit differences of the remanufacturer under different models for varying values of $\xi$ when $c_{m} = 4$
Comparison of profit differences of the third-party recycler under different models for varying values of $\xi$ when $c_{m} = 4$
Comparison of emission differences under different models when $c_{m} = 8$
Comparison of profit differences of the third-party recycler under different models when $c_{m} = 8$
Comparison of profit differences of the third-party recycler under different models when $c_{m} = 8$
Carbon abatement investment related research
 Research object Carbon abatement investment method Carbon emission regulation Tax or subsidy Consumer environmental awareness references Supply chain Offset investment Technology investment Cap-and-offset, carbon tax, cap-and-trade policy cap-and-trade policy No Benjaafar et al.(2013) Dong et al. (2016), Yang et al. (2017) Individual firm — — Yes No Qiu and Tao (1998), Wang et al. (2018) — — Yes Yes Yu et al. (2016) Offset investment Cap-and-offset, carbon tax, cap-and-trade policy No Chen et al.(2013) Technology investment — No No Chen et al. (2017), Cap-and-trade Luo et al. (2016) Carbon cap, carbon tax, cap-and-trade policy Jiang and Klabjan (2012), Toptal et al.(2014) — Yes Yes Yalabik and Fairchild (2011) Remanufacturing firm Technology investment Cap-and-trade Yes Yes Our paper
 Research object Carbon abatement investment method Carbon emission regulation Tax or subsidy Consumer environmental awareness references Supply chain Offset investment Technology investment Cap-and-offset, carbon tax, cap-and-trade policy cap-and-trade policy No Benjaafar et al.(2013) Dong et al. (2016), Yang et al. (2017) Individual firm — — Yes No Qiu and Tao (1998), Wang et al. (2018) — — Yes Yes Yu et al. (2016) Offset investment Cap-and-offset, carbon tax, cap-and-trade policy No Chen et al.(2013) Technology investment — No No Chen et al. (2017), Cap-and-trade Luo et al. (2016) Carbon cap, carbon tax, cap-and-trade policy Jiang and Klabjan (2012), Toptal et al.(2014) — Yes Yes Yalabik and Fairchild (2011) Remanufacturing firm Technology investment Cap-and-trade Yes Yes Our paper
 Research object Financing method Supply chain coordination Default risk references Supply chain Bank loan & trade credit No No Chod (2016) Yes Yes Xiao et al. (2017), Yang and Birge (2017) Trade credit No No Peura et al. (2017), Tunca and Zhu (2017) No Yes Babich and Tang (2012), Rui and Lai (2015) Yes No Tsao and Yu-Chung (2017), Lee and Rhee (2011) Yes Yes Devalkar and Krishnan (2014), Yang and Birge (2011) Remanufact-uring system Bank loan No No Li (2018), Sun et al. (2017), Wang et al. (2017), Wang and Chen (2017) Trade credit Our paper
 Research object Financing method Supply chain coordination Default risk references Supply chain Bank loan & trade credit No No Chod (2016) Yes Yes Xiao et al. (2017), Yang and Birge (2017) Trade credit No No Peura et al. (2017), Tunca and Zhu (2017) No Yes Babich and Tang (2012), Rui and Lai (2015) Yes No Tsao and Yu-Chung (2017), Lee and Rhee (2011) Yes Yes Devalkar and Krishnan (2014), Yang and Birge (2011) Remanufact-uring system Bank loan No No Li (2018), Sun et al. (2017), Wang et al. (2017), Wang and Chen (2017) Trade credit Our paper
Notation
 Indices $i$ Index for the case of Model $j, i=$ 1, 2, 3 for Model I, II, III respectively. $j$ Index for model, $j=$ I, II, III. Parameters $c$ Unit remanufacturing cost. $c_{0}$ Unit disposal cost. $h$ Unit stock-holding cost. $s$ Unit shortage cost. $v_{t}$ Unit acquisition price by the third-party recycler. $a$ Unit recycling price by the remanufacturer. $p$ Unit sales price, $p>s>c.$ $\xi$ Remanufacturing rate. $D$ Stochastic demand with support on [0, $+\infty$), CDF $F$ (), PDF $f$ (). Suppose it obey the uniform distribution on [$\alpha, \beta$], $F (\alpha) =0$, $F (\beta) =1$. $q_{i}$ Production quantity for Model $j$, $q_{i} =R_{i} \ast \xi$. $t$ Consumers' low carbon preference coefficient. $\delta$ Environmental benefit coefficient. $c_{m}$ Unit carbon trading price. $e_{0}$ Initial unit carbon emissions. $\Delta e_{i}$ Unit carbon emission reduction for Model $j$. $m$ Cost coefficient of emission reduction. $E_{g}$ Carbon emission quota. $B$ Initial capital by the remanufacturer. $T_{1}, T_{2}$ Recycling period and production period. $M$ Sales period/credit period. $k$ Sensitivity to deferred payment length. $\pi_{ri}, \pi_{ti}$ The profit by the remanufacturer and the third-party recycler for Model $j,$ respectively. Decision variables $R_{i}$ Recycling quantity of Model $j$ by the remanufacturer. $\tau_{i}$ Carbon reduction rate of Model $j$, $\tau_{i} =\Delta e_{i}/e_{0}$. $f_{i}$ Government subsidy rate for carbon emission reduction of Model $j$.
 Indices $i$ Index for the case of Model $j, i=$ 1, 2, 3 for Model I, II, III respectively. $j$ Index for model, $j=$ I, II, III. Parameters $c$ Unit remanufacturing cost. $c_{0}$ Unit disposal cost. $h$ Unit stock-holding cost. $s$ Unit shortage cost. $v_{t}$ Unit acquisition price by the third-party recycler. $a$ Unit recycling price by the remanufacturer. $p$ Unit sales price, $p>s>c.$ $\xi$ Remanufacturing rate. $D$ Stochastic demand with support on [0, $+\infty$), CDF $F$ (), PDF $f$ (). Suppose it obey the uniform distribution on [$\alpha, \beta$], $F (\alpha) =0$, $F (\beta) =1$. $q_{i}$ Production quantity for Model $j$, $q_{i} =R_{i} \ast \xi$. $t$ Consumers' low carbon preference coefficient. $\delta$ Environmental benefit coefficient. $c_{m}$ Unit carbon trading price. $e_{0}$ Initial unit carbon emissions. $\Delta e_{i}$ Unit carbon emission reduction for Model $j$. $m$ Cost coefficient of emission reduction. $E_{g}$ Carbon emission quota. $B$ Initial capital by the remanufacturer. $T_{1}, T_{2}$ Recycling period and production period. $M$ Sales period/credit period. $k$ Sensitivity to deferred payment length. $\pi_{ri}, \pi_{ti}$ The profit by the remanufacturer and the third-party recycler for Model $j,$ respectively. Decision variables $R_{i}$ Recycling quantity of Model $j$ by the remanufacturer. $\tau_{i}$ Carbon reduction rate of Model $j$, $\tau_{i} =\Delta e_{i}/e_{0}$. $f_{i}$ Government subsidy rate for carbon emission reduction of Model $j$.
The impact of m on the total emission (k = 0.2)
 m 1200 1400 1600 1800 2000 $E_{2}^{\ast}$ 19.9770 41.8060 53.7053 61.0017 65.8504 $E_{3}^{(1)}$ 37.5510 40.3498 42.4490 44.0816 45.3878 $E_{3}^{(2)}$ 29.5592 46.1639 55.1360 60.5937 64.1942 Notes: carbon emission unit: kilo
 m 1200 1400 1600 1800 2000 $E_{2}^{\ast}$ 19.9770 41.8060 53.7053 61.0017 65.8504 $E_{3}^{(1)}$ 37.5510 40.3498 42.4490 44.0816 45.3878 $E_{3}^{(2)}$ 29.5592 46.1639 55.1360 60.5937 64.1942 Notes: carbon emission unit: kilo
The impact of m on the total emission (k = 0.6)
 m 1200 1400 1600 1800 2000 $E_{2}^{\ast}$ 19.9770 41.8060 53.7053 61.0017 65.8504 $E_{3}^{(1)}$ 37.551 40.3498 42.4490 44.0816 45.3878 $E_{3}^{(2)}$ 34.0743 34.1072 33.8873 33.6214 33.3659 Notes: carbon emission unit: kilo
 m 1200 1400 1600 1800 2000 $E_{2}^{\ast}$ 19.9770 41.8060 53.7053 61.0017 65.8504 $E_{3}^{(1)}$ 37.551 40.3498 42.4490 44.0816 45.3878 $E_{3}^{(2)}$ 34.0743 34.1072 33.8873 33.6214 33.3659 Notes: carbon emission unit: kilo
Present values for the base parameters ($vt = 1$)
 Parameter $p$ $c$ $a$ $c_{0}$ $h$ $s$ $v_{t}$ (RMB) (RMB) (RMB) (RMB) (RMB) (RMB) (RMB) Present value 40 8 2 1 2 1.6 1 Parameter $\xi$ $[\alpha, \beta]$ $e_{0}$ $c_{m}$ $\delta$ $t$ $E$ — — (kilo) (RMB) — (RMB) (kilo) Present value 0.8 [0, 100] 2 6 0.6 1 60
 Parameter $p$ $c$ $a$ $c_{0}$ $h$ $s$ $v_{t}$ (RMB) (RMB) (RMB) (RMB) (RMB) (RMB) (RMB) Present value 40 8 2 1 2 1.6 1 Parameter $\xi$ $[\alpha, \beta]$ $e_{0}$ $c_{m}$ $\delta$ $t$ $E$ — — (kilo) (RMB) — (RMB) (kilo) Present value 0.8 [0, 100] 2 6 0.6 1 60
Comparison of the optimal solutions for varying values of $k$ under two different self-owned capital strategies
 $B$ $k$ $R$ $\tau$ $f$ $E$ $\Delta E$ $I$ $\pi_{r}$ $\pi_{t}$ sw (RMB) (RMB) — — (kilo) (kilo) (RMB) (RMB) (RMB) (RMB) $B<\min (B_{1}, B_{2}) , B_{1}=$ 414.2992, $B_{2}=$ 583.7571 200 0.2 23.8095 0.1829 0.0833 31.1292 6.9660 22.9878 543.7939 62.9580 1365.8837 300 0.2 35.7143 0.2743 0.0833 41.4694 15.6735 51.7224 633.6119 94.4371 1365.8837 400 0.2 47.6190 0.3657 0.0833 48.3265 27.8639 91.9510 695.3773 125.9161 1523.7365 400 0.6 47.6190 0.3657 0.0833 48.3265 27.8639 91.9510 292.7556 528.5379 1121.1148 $B>\max (B_{2}, B_{3}), B_{2}=$ 586.0089, $B_{3}=$ 645.6875 700 0.2 69.7630 0.5408 0.0919 51.2557 60.3650 199.2045 737.4973 184.4698 1573.5686 800 0.2 69.7630 0.5408 0.0919 51.2557 60.3650 199.2045 737.4973 184.4698 1573.5686 900 0.2 69.7630 0.5408 0.0919 51.2557 60.3650 199.2045 737.4973 184.4698 1573.5686 900 0.6 26.6087 0.2011 0.0687 34.0100 8.5639 28.2608 342.0076 295.3374 1165.0597
 $B$ $k$ $R$ $\tau$ $f$ $E$ $\Delta E$ $I$ $\pi_{r}$ $\pi_{t}$ sw (RMB) (RMB) — — (kilo) (kilo) (RMB) (RMB) (RMB) (RMB) $B<\min (B_{1}, B_{2}) , B_{1}=$ 414.2992, $B_{2}=$ 583.7571 200 0.2 23.8095 0.1829 0.0833 31.1292 6.9660 22.9878 543.7939 62.9580 1365.8837 300 0.2 35.7143 0.2743 0.0833 41.4694 15.6735 51.7224 633.6119 94.4371 1365.8837 400 0.2 47.6190 0.3657 0.0833 48.3265 27.8639 91.9510 695.3773 125.9161 1523.7365 400 0.6 47.6190 0.3657 0.0833 48.3265 27.8639 91.9510 292.7556 528.5379 1121.1148 $B>\max (B_{2}, B_{3}), B_{2}=$ 586.0089, $B_{3}=$ 645.6875 700 0.2 69.7630 0.5408 0.0919 51.2557 60.3650 199.2045 737.4973 184.4698 1573.5686 800 0.2 69.7630 0.5408 0.0919 51.2557 60.3650 199.2045 737.4973 184.4698 1573.5686 900 0.2 69.7630 0.5408 0.0919 51.2557 60.3650 199.2045 737.4973 184.4698 1573.5686 900 0.6 26.6087 0.2011 0.0687 34.0100 8.5639 28.2608 342.0076 295.3374 1165.0597
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