# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021141
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## Design of an environmental contract under trade credits and carbon emission reduction

 1 School of Management, Nanjing University of Posts and Telecommunications, Nanjing, 210003 Jiangsu, China 2 School of Economics & Managements, Southeast University, Nanjing, 210089 Jiangsu, China

* Corresponding author: Yaxian Wang

Received  March 2021 Revised  May 2021 Early access August 2021

Fund Project: The paper is supported by the 1311 Programs Foundation of Nanjing University of Posts and Telecommunications, the Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY220212), the Key Research Base of Philosophy and Social Sciences in Jiangsu–Information Industry Integration Innovation and Emergency Management Research Center

Most of the previous literatures proposed a single coordination contract to increase the total profit of the supply chain, while this paper focuses on how to design environmental contracts to increase economic and environmental performance in the context of sustainable development. This paper designs the environmental contract based on cap-and-trade mechanism and trade credits which has rarely been studied before, especially the impact of trade credit on environmental performance. We consider a green supply chain, assuming that the demand rate is linear with retail prices, joint carbon emission reduction efforts and trade credit. Two models, a decentralized one and a centralized one, are compared; four contracts are proposed. Via numerous examples and sensitivity analysis, we gain some insight into how to select supply chain contracts to better improve environmental performance. The results reveal that the manufacturer sharing the retailer's revenue and cost contract obtains the highest profit. While revenue sharing contract between both parties is the optimal environmental contract, but it is difficult to increase the profit of supply chain. Furthermore, it is found that trade credit works well in protecting the environment and plays a significant role in achieving coordination.

Citation: Chong Zhang, Yaxian Wang, Haiyan Wang. Design of an environmental contract under trade credits and carbon emission reduction. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021141
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##### References:
Effects of ${W_e}$ on optimal solutions
Effects of $\varepsilon$ on optimal solutions
Summary of Related Literature
 Decision variables Demand dependency Contract (ⅰ) (ⅱ) (ⅲ) (ⅳ) (ⅰ) (ⅱ) (ⅲ) (ⅳ) (ⅴ) (ⅵ) [38] $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\surd$ $\times$ Krishnan et al.[25] $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ Tsao and Sheen [43] $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\surd$ $\times$ Ma et al.[28] $\times$ $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ $\surd$ Xu et al. [48] $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\times$ $\surd$ $\times$ Ji et al.[19] $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\times$ Bai et al.[3] $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ Heydari et al.[13] $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ Zhou and Ye [53] $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Yang et al.[50] $\times$ $\surd$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ Hosseini-Motlagh et al.[15] $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\times$ Pakhira et al.[29] $\surd$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ $\times$ Phan et al.[32] $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ Tsao[42] $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ Ranjan and Jha[35] $\times$ $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ $\times$ This paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$
 Decision variables Demand dependency Contract (ⅰ) (ⅱ) (ⅲ) (ⅳ) (ⅰ) (ⅱ) (ⅲ) (ⅳ) (ⅴ) (ⅵ) [38] $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\surd$ $\times$ Krishnan et al.[25] $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ Tsao and Sheen [43] $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\surd$ $\times$ Ma et al.[28] $\times$ $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ $\surd$ Xu et al. [48] $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\times$ $\surd$ $\times$ Ji et al.[19] $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\times$ Bai et al.[3] $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ Heydari et al.[13] $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ Zhou and Ye [53] $\times$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ Yang et al.[50] $\times$ $\surd$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ Hosseini-Motlagh et al.[15] $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\times$ Pakhira et al.[29] $\surd$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ $\times$ Phan et al.[32] $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ Tsao[42] $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\surd$ Ranjan and Jha[35] $\times$ $\times$ $\surd$ $\surd$ $\times$ $\times$ $\surd$ $\surd$ $\surd$ $\times$ This paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\times$ $\surd$
Symbols in this paper
 Parameters $C$ The production cost per unit $W$ The wholesale price per unit $I$ The market interest rate $v$ The effectiveness of carbon reduction efforts on carbon emissions for making a unit, $v > 0$ $\varepsilon$ The manufacturer's carbon emissions for making a product without efforts on carbon reduction ${C_M}$ The manufacturer's carbon pollution limit or cap ${W_e}$ The trading price of carbon emissions ${\eta _1}$ The investment cost coefficient on carbon reduction efforts, ${\eta _1} > 0$ ${\eta _2}$ The investment cost coefficient on promotional efforts, ${\eta _2} > 0$ $a$ Market scale parameter, $a > 0$ $b$ Price influence on the demand rate, $b > 0$ ${\gamma _1}$ Effect of carbon reduction effort on demand, ${\gamma _1} > 0$ ${\gamma _2}$ Effect of promotional effort on demand, ${\gamma _2} > 0$ Decision variables $s$ The retailer's promotional efforts $P$ The retail price per unit $M$ The length of the trade credit period offered by the manufacturer in years. $M > 0$ represents a credit payment, $M = 0$ represents an advance payment, and $M < 0$ represents a cash payment ${\theta}$ The manufacturer's carbon reduction efforts Functions ${\pi _i}$ The supply chain member i's profit per year; $i = s, r, sc$ refer to the manufacturer, the retailer and the supply chain, respectively $*$ Represents the optimal value of a decision variable
 Parameters $C$ The production cost per unit $W$ The wholesale price per unit $I$ The market interest rate $v$ The effectiveness of carbon reduction efforts on carbon emissions for making a unit, $v > 0$ $\varepsilon$ The manufacturer's carbon emissions for making a product without efforts on carbon reduction ${C_M}$ The manufacturer's carbon pollution limit or cap ${W_e}$ The trading price of carbon emissions ${\eta _1}$ The investment cost coefficient on carbon reduction efforts, ${\eta _1} > 0$ ${\eta _2}$ The investment cost coefficient on promotional efforts, ${\eta _2} > 0$ $a$ Market scale parameter, $a > 0$ $b$ Price influence on the demand rate, $b > 0$ ${\gamma _1}$ Effect of carbon reduction effort on demand, ${\gamma _1} > 0$ ${\gamma _2}$ Effect of promotional effort on demand, ${\gamma _2} > 0$ Decision variables $s$ The retailer's promotional efforts $P$ The retail price per unit $M$ The length of the trade credit period offered by the manufacturer in years. $M > 0$ represents a credit payment, $M = 0$ represents an advance payment, and $M < 0$ represents a cash payment ${\theta}$ The manufacturer's carbon reduction efforts Functions ${\pi _i}$ The supply chain member i's profit per year; $i = s, r, sc$ refer to the manufacturer, the retailer and the supply chain, respectively $*$ Represents the optimal value of a decision variable
Optimal solutions for the example
 Model $M$ $P$ $\theta$ $s$ $\pi_r$ $\pi_m$ $\pi_{sc}$ $E(D)$ $Centralized$ 0.74 33.694 7.916 3.401 - - 4094.153 171.326 $Decentralized$ -0.39 37.683 3.402 1.389 228.357 762.078 990.434 94.911 $Contract A\left[\rho_{1}, \lambda_{1}\right]$ $[0.1,0.7]$ -0.14 37.961 3.821 5.469 407.49 796.88 1204.37 131.86 $[0.1,0.6]$ -0.02 37.207 3.773 4.573 550.99 816.57 1367.55 156.29 $[0.1,0.5]$ 0.04 36.773 3.735 3.870 650.65 829.39 1480.03 172.24 $[0.1,0.4]$ 0.08 36.501 3.707 3.325 715.62 837.43 1553.04 182.32 $[0.2,0.5]$ 0.49 36.049 3.938 5.004 1724.03 967.01 2691.04 314.26 $[0.3,0.5]$ 0.92 35.660 4.104 6.088 4116.45 1227.64 5344.10 554.05 $Contract B\left[\rho_{2}, \tau_{1}\right]$ $[0.1,0.6]$ -0.73 35.846 3.611 0.804 86.24 740.29 826.54 63.78 $[0.2,0.6]$ -0.25 34.136 3.864 1.387 327.23 788.97 1116.20 122.98 $[0.3,0.6]$ 0.21 33.440 3.985 1.953 953.53 884.78 1838.31 228.22 $[0.4,0.6]$ 0.68 33.087 4.060 2.519 2500.88 1077.13 3578.01 419.39 $[0.2,0.5]$ -0.10 34.401 3.841 1.608 470.48 810.54 1281.02 148.85 $[0.2,0.4]$ 0.05 34.613 3.825 1.827 663.39 837.37 1500.75 180.05 $Contract C\left[\varphi_{1}, \lambda_{2}\right]$ $[0.5,0.5]$ -0.40 43.264 6.599 3.367 198.32 808.30 1006.62 42.36 $[0.5,0.4]$ -0.35 42.287 6.753 2.925 236.66 819.58 1056.24 45.20 $[0.5,0.3]$ -0.32 41.707 6.833 2.562 261.25 826.59 1087.83 46.95 $[0.4,0.2]$ -0.35 41.603 5.377 2.148 247.94 842.09 1090.03 63.22 $[0.5,0.2]$ -0.30 41.341 6.876 2.269 277.01 831.03 1108.03 48.07 $[0.6,0.2]$ -0.22 40.962 9.507 2.473 315.25 795.62 1110.88 9.18 $Contract D\left[\varphi_{2}, \tau_{2}\right]$ $[0.6,0.05]$ -0.28 41.512 9.147 1.899 263.38 772.25 1035.63 13.98 $[0.61,0.05]$ -0.27 41.459 9.511 1.921 266.65 774.34 1040.99 8.23 $[0.62,0.05]$ -0.26 41.404 9.904 1.945 269.71 776.63 1046.34 1.66 $[0.6,0.05]$ -0.28 41.512 9.147 1.899 263.38 772.25 1035.63 13.98 $[0.6,0.04]$ -0.27 41.307 9.232 1.920 277.15 775.09 1052.24 13.03 $[0.6,0.003]$ -0.25 41.105 9.316 1.941 291.41 778.01 1069.42 12.02
 Model $M$ $P$ $\theta$ $s$ $\pi_r$ $\pi_m$ $\pi_{sc}$ $E(D)$ $Centralized$ 0.74 33.694 7.916 3.401 - - 4094.153 171.326 $Decentralized$ -0.39 37.683 3.402 1.389 228.357 762.078 990.434 94.911 $Contract A\left[\rho_{1}, \lambda_{1}\right]$ $[0.1,0.7]$ -0.14 37.961 3.821 5.469 407.49 796.88 1204.37 131.86 $[0.1,0.6]$ -0.02 37.207 3.773 4.573 550.99 816.57 1367.55 156.29 $[0.1,0.5]$ 0.04 36.773 3.735 3.870 650.65 829.39 1480.03 172.24 $[0.1,0.4]$ 0.08 36.501 3.707 3.325 715.62 837.43 1553.04 182.32 $[0.2,0.5]$ 0.49 36.049 3.938 5.004 1724.03 967.01 2691.04 314.26 $[0.3,0.5]$ 0.92 35.660 4.104 6.088 4116.45 1227.64 5344.10 554.05 $Contract B\left[\rho_{2}, \tau_{1}\right]$ $[0.1,0.6]$ -0.73 35.846 3.611 0.804 86.24 740.29 826.54 63.78 $[0.2,0.6]$ -0.25 34.136 3.864 1.387 327.23 788.97 1116.20 122.98 $[0.3,0.6]$ 0.21 33.440 3.985 1.953 953.53 884.78 1838.31 228.22 $[0.4,0.6]$ 0.68 33.087 4.060 2.519 2500.88 1077.13 3578.01 419.39 $[0.2,0.5]$ -0.10 34.401 3.841 1.608 470.48 810.54 1281.02 148.85 $[0.2,0.4]$ 0.05 34.613 3.825 1.827 663.39 837.37 1500.75 180.05 $Contract C\left[\varphi_{1}, \lambda_{2}\right]$ $[0.5,0.5]$ -0.40 43.264 6.599 3.367 198.32 808.30 1006.62 42.36 $[0.5,0.4]$ -0.35 42.287 6.753 2.925 236.66 819.58 1056.24 45.20 $[0.5,0.3]$ -0.32 41.707 6.833 2.562 261.25 826.59 1087.83 46.95 $[0.4,0.2]$ -0.35 41.603 5.377 2.148 247.94 842.09 1090.03 63.22 $[0.5,0.2]$ -0.30 41.341 6.876 2.269 277.01 831.03 1108.03 48.07 $[0.6,0.2]$ -0.22 40.962 9.507 2.473 315.25 795.62 1110.88 9.18 $Contract D\left[\varphi_{2}, \tau_{2}\right]$ $[0.6,0.05]$ -0.28 41.512 9.147 1.899 263.38 772.25 1035.63 13.98 $[0.61,0.05]$ -0.27 41.459 9.511 1.921 266.65 774.34 1040.99 8.23 $[0.62,0.05]$ -0.26 41.404 9.904 1.945 269.71 776.63 1046.34 1.66 $[0.6,0.05]$ -0.28 41.512 9.147 1.899 263.38 772.25 1035.63 13.98 $[0.6,0.04]$ -0.27 41.307 9.232 1.920 277.15 775.09 1052.24 13.03 $[0.6,0.003]$ -0.25 41.105 9.316 1.941 291.41 778.01 1069.42 12.02
The increase rates with different contract coefficients
 Parameter $\pi_r$ $\pi_m$ $\pi_{sc}$ $E(D)$ $Contract A\left[\rho_{1}, \lambda_{1}\right]$ $[0.1,0.7]$ 78.45% 4.57% 21.60% 38.93% $[0.1,0.6]$ 141.28% 7.15% 38.08% 64.67% $[0.1,0.5]$ 184.93% 8.83% 49.43% 81.48% $[0.1,0.4]$ 213.38% 9.89% 56.81% 92.10% $[0.1,0.5]$ 184.93% 8.83% 49.43% 81.48% $[0.2,0.5]$ 654.97% 26.89% 171.70% 231.11% $[0.3,0.5]$ 1702.6% 61.09% 439.57% 483.76% $Contract B\left[\rho_{2}, \tau_{1}\right]$ $[0.2,0.5]$ 106.02% 6.36% 29.34% 56.83% $[0.2,0.4]$ 190.50% 9.88% 51.53% 89.71% $[0.2,0.3]$ 303.66% 14.26% 80.99% 129.38% $[0.2,0.6]$ 43.30% 3.53% 12.70% 29.57% $[0.3,0.6]$ 317.56% 16.10% 85.61% 140.46% $[0.4,0.6]$ 995.15% 41.34% 261.26% 341.89% $Contract C\left[\varphi_{1}, \lambda_{2}\right]$ $[0.5,0.4]$ 3.63% 7.55% 6.64% -52.38% $[0.5,0.3]$ 14.40% 8.46% 9.83% -50.53% $[0.5,0.2]$ 38.05% 4.40% 12.16% -90.32% $[0.6,0.2]$ 38.05% 4.40% 12.16% -90.32% $[0.5,0.2]$ 21.30% 9.05% 11.87% -49.35% $[0.4,0.2]$ 8.58% 10.50% 10.06% -33.39% $Contract D\left[\varphi_{2}, \tau_{2}\right]$ $[0.6,0.05]$ 15.33% 1.34% 4.56% -85.27% $[0.61,0.05]$ 16.77% 1.61% 5.10% -91.33% $[0.62,0.05]$ 18.11% 1.91% 5.64% -98.25% $[0.6,0.05]$ 15.33% 1.34% 4.56% -85.27% $[0.6,0.04]$ 21.37% 1.71% 6.24% -86.27% $[0.6,0.03]$ 27.61% 2.09% 7.98% -87.34%
 Parameter $\pi_r$ $\pi_m$ $\pi_{sc}$ $E(D)$ $Contract A\left[\rho_{1}, \lambda_{1}\right]$ $[0.1,0.7]$ 78.45% 4.57% 21.60% 38.93% $[0.1,0.6]$ 141.28% 7.15% 38.08% 64.67% $[0.1,0.5]$ 184.93% 8.83% 49.43% 81.48% $[0.1,0.4]$ 213.38% 9.89% 56.81% 92.10% $[0.1,0.5]$ 184.93% 8.83% 49.43% 81.48% $[0.2,0.5]$ 654.97% 26.89% 171.70% 231.11% $[0.3,0.5]$ 1702.6% 61.09% 439.57% 483.76% $Contract B\left[\rho_{2}, \tau_{1}\right]$ $[0.2,0.5]$ 106.02% 6.36% 29.34% 56.83% $[0.2,0.4]$ 190.50% 9.88% 51.53% 89.71% $[0.2,0.3]$ 303.66% 14.26% 80.99% 129.38% $[0.2,0.6]$ 43.30% 3.53% 12.70% 29.57% $[0.3,0.6]$ 317.56% 16.10% 85.61% 140.46% $[0.4,0.6]$ 995.15% 41.34% 261.26% 341.89% $Contract C\left[\varphi_{1}, \lambda_{2}\right]$ $[0.5,0.4]$ 3.63% 7.55% 6.64% -52.38% $[0.5,0.3]$ 14.40% 8.46% 9.83% -50.53% $[0.5,0.2]$ 38.05% 4.40% 12.16% -90.32% $[0.6,0.2]$ 38.05% 4.40% 12.16% -90.32% $[0.5,0.2]$ 21.30% 9.05% 11.87% -49.35% $[0.4,0.2]$ 8.58% 10.50% 10.06% -33.39% $Contract D\left[\varphi_{2}, \tau_{2}\right]$ $[0.6,0.05]$ 15.33% 1.34% 4.56% -85.27% $[0.61,0.05]$ 16.77% 1.61% 5.10% -91.33% $[0.62,0.05]$ 18.11% 1.91% 5.64% -98.25% $[0.6,0.05]$ 15.33% 1.34% 4.56% -85.27% $[0.6,0.04]$ 21.37% 1.71% 6.24% -86.27% $[0.6,0.03]$ 27.61% 2.09% 7.98% -87.34%

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