Remanufacturing Models Under Technology Licensing with Consideration of Environmental Issues

Abstract

Recently, the environmental and economic benefits of remanufacturing result in increasing interest of this concept. Due to supreme importance of this issue, in this paper, a closed-loop supply chain (CLSC) including a manufacturer, a distributor, and third-party logistics provider is considered. Three multi-level leader-follower Stackelberg game models are presented in order to investigate whether a manufacturer in addition to manufacturing new products should do remanufacturing or sets a fee for technology licensing of distributors and cooperate with them in remanufacturing. Furthermore, we conduct a sensitivity analysis to show how government can mitigate the bad economic, environmental, and social impacts of remanufacturing by setting an emissions tax price and evaluation of impacts of changing remanufacturing parameters on the profit of supply chain members on the three proposed models.

Appendices

Appendix 1.The proofs of the propositions of the Model 1

Proof of the proposition 1.

$$ \frac{\partial {\pi}_2^{d1}}{\partial {P}_r^2}=\alpha -2\beta {P}_r^2+\beta {P}_w^2 $$

(47)

$$ \frac{\partial^2{\pi}_2^{d1}}{\partial {P}_r^{2^2}}=-2\beta $$

(48)

As −2β < 0, the distributor profit function in the second period is concave in ${P}_r^2.$

Proof of the proposition 2. As the second period profit function of the distributor is concave in ${P}_r^2$, we can obtain retail prices for the new and remanufactured products in the second period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}_2^{d1}}{\partial {P}_r^2}=\alpha -2\beta {P}_r^2+\beta {P}_w^2=0 $$

(49)

Proof of the proposition 3.

$$ \frac{\partial {\pi}^{d1}}{\partial {P}_r^1}=\alpha -2\beta {P}_r^1+\beta {P}_w^1 $$

(50)

$$ \frac{\partial^2{\pi}^{d1}}{\partial {P}_r^{1^2}}=-2\beta $$

(51)

As −2β < 0, the distributor profit function in the two-period is concave in ${P}_r^1.$.

Proof of the proposition 4.

As the two-period profit function of the distributor is concave in ${P}_r^1$, we can obtain retail prices for the new products in the first period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}^{d1}}{\partial {P}_r^1}=\alpha -2\beta {P}_r^1+\beta {P}_w^1=0 $$

(52)

Proof of the proposition 5.

$$ \frac{\partial {\pi}_2^{m1}}{\partial {P}_w^2}=\frac{\alpha -2\beta {P}_w^2}{2}+\frac{\beta \left(\delta +{C}_n\right)}{2} $$

(53)

$$ \frac{\partial {\pi}_2^{m1}}{\partial {C}_c}=\left(\alpha -\beta {P}_r^1\right)\left[\frac{\theta \left(-{C}_r+{C}_n+\delta \left(1-\lambda \right)\right)-{C}_c}{2\sqrt{d{C}_c}}-\sqrt{\frac{C_c}{d}}\right] $$

(54)

$$ {H}_m^1=\left[\begin{array}{cc}{\partial}^2{\pi}_2^{m1}/\partial {P}_w^{2^2}& {\partial}^2{\pi}_2^{m1}/\partial {P}_w^2\partial {C}_c\\ {}{\partial}^2{\pi}_2^{m1}/\partial {C}_c\partial {P}_w^2& {\partial}^2{\pi}_2^{m1}/\partial {C}_c^2\end{array}\right]=\left[\begin{array}{cc}-\beta & 0\\ {}0& -\left(\alpha -\beta {P}_r^1\right)\left[\frac{3{C}_c+\theta \left({C}_n+\delta -{C}_r-\lambda \delta \right)}{4{C}_c\sqrt{d{C}_c}}\right]\end{array}\right] $$

(55)

As $-\beta <0\ \mathrm{and}\ \left|{H}_m^1\right|>0$, so the manufacturer’s profit function in the second period ${\pi}_2^{m1}$ is jointly concave in ${P}_w^2$ and C_c.

Proof of the proposition 6.

As the second period profit function of the manufacturer is concave in C_c and ${P}_w^2$, we can obtain the optimal acquisition price of a unit of a used product which is returned from consumers and wholesale prices for new and remanufactured products in the second period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}_2^{m1}}{\partial {C}_c}=\frac{\theta \left({C}_n+\delta -{C}_r-\lambda \delta \right)-{C}_c-2d{C}_c}{2d\sqrt{\frac{C_c}{d}}}=0 $$

(56)

$$ \frac{\partial {\pi}_2^{m1}}{\partial {P}_w^2}=\frac{\alpha -2\beta {P}_w^2}{2}+\frac{\beta \left(\delta +{C}_n\right)}{2}=0 $$

(57)

Proof of the proposition 7.

$$ \frac{\partial {\pi}^{m1}}{\partial {P}_w^1}=\frac{\alpha -2\beta {P}_w^1+\beta \left({C}_n+\delta \right)}{2} $$

(58)

$$ \frac{\partial^2{\pi}^{m1}}{\partial {P}_w^{1^2}}=-\beta $$

(59)

As −β < 0, the manufacturer profit function in the two-period is concave in ${P}_w^1.$.

Proof of the proposition 8.

As the two-period profit function of the manufacturer is concave in ${P}_w^1$, we can obtain wholesale prices for the new products in the first period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}^{m1}}{\partial {P}_w^1}=\frac{\alpha -2\beta {P}_w^1+\beta \left({C}_n+\delta \right)}{2}=0 $$

(60)

Appendix 2. The proofs of the propositions of the Model 2

Proof of the proposition 9.

$$ \frac{\partial {\pi}_2^{d2}}{\partial {P}_r^2}=\alpha -2\beta {P}_r^2+\beta {P}_w^2 $$

(61)

$$ \frac{\partial {\pi}_2^{d2}}{\partial {C}_c}=\left(\alpha -\beta {P}_r^1\right)\left(\frac{C_d\theta }{2\sqrt{C_cd}}-1.5\sqrt{\frac{C_c}{d}}\right) $$

(62)

$$ {H}_d^2=\left[\begin{array}{cc}{\partial}^2{\pi}_2^{d2}/\partial {P}_r^{2^2}& {\partial}^2{\pi}_2^{d2}/\partial {P}_r^2\partial {C}_c\\ {}{\partial}^2{\pi}_2^{d2}/\partial {C}_c\partial {P}_r^2& {\partial}^2{\pi}_2^{d2}/\partial {C}_c^2\end{array}\right]=\left[\begin{array}{cc}-2\beta & 0\\ {}0& -\frac{\alpha -\beta {P}_r^1}{\sqrt{C_cd}}\end{array}\right] $$

(63)

As −2β < 0 and$\left|{H}_d^2\right|>0,$ so the distributor’s profit function in the second period ${\pi}_2^{m2}$ is jointly concave in ${P}_r^2$ and C_c.

Proof of the proposition 10. As the second period profit function of the distributor is concave in C_c and ${P}_r^2$, we can obtain the optimal acquisition price of a unit of a used product which is returned from consumers and retail prices for new and remanufactured products in the second period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}_2^{d2}}{\partial {P}_r^2}=\alpha -2\beta {P}_r^2+\beta {P}_w^2=0 $$

(64)

$$ \frac{\partial {\pi}_2^{d2}}{\partial {C}_c}=\left({C}_d\theta -{C}_c\right)\sqrt{\frac{C_c}{d}}=0 $$

(65)

Proof of the proposition 11.

$$ \frac{\partial {\pi}^{d2}}{\partial {P}_r^1}=\alpha -2\beta {P}_r^1+\beta {P}_w^1-\frac{2{C}_d\theta \beta}{3}\sqrt{\frac{C_d\theta }{3d}} $$

(66)

$$ \frac{\partial^2{\pi}^{d2}}{\partial {P}_r^{1^2}}=-2\beta $$

(67)

As −2β < 0, the distributor profit function in the two periods is concave in ${P}_r^1.$.

Proof of the proposition 12. As the two-period profit function of the distributor is concave in ${P}_r^1$, we can obtain the optimal retail prices for new in the first period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}^{d2}}{\partial {P}_r^1}=\alpha -2\beta {P}_r^1+\beta {P}_w^1-\frac{2{C}_d\theta \beta}{3}\sqrt{\frac{C_d\theta }{3d}}=0 $$

(68)

Proof of the proposition 13.

$$ \frac{\partial {\pi}_2^{m2}}{\partial {P}_w^2}=\frac{\alpha -2\beta {P}_w^2+\beta \left({C}_n+\delta \right)}{2} $$

(69)

$$ \frac{\partial^2{\pi}_2^{m2}}{\partial {P}_w^{2^2}}=-\beta $$

(70)

As −β < 0, the manufacturer profit function in the second period is concave in ${P}_w^2.$

Proof of the proposition 14. As the second period profit function of the manufacturer is concave in ${P}_w^2$, we can obtain the optimal wholesale prices for the new and remanufactured products in the second period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}_2^{m2}}{\partial {P}_w^2}=\frac{\alpha -2\beta {P}_w^2+\beta \left({C}_n+\delta \right)}{2}=0 $$

(71)

Proof of the proposition 15.

$$ \frac{\partial {\pi}_2^{m2}}{\partial {P}_w^1}=\frac{\alpha -2\beta {P}_w^1+\beta \left({C}_n+\delta \right)}{2}-\beta \frac{\theta }{2}\sqrt{\frac{C_d\theta }{3d}}\left({C}_n+\delta \left(1-\lambda \right)-{C}_r-{C}_d\right) $$

(72)

$$ \frac{\partial^2{\pi}_2^{m2}}{\partial {P}_w^{1^2}}=-\beta $$

(73)

As −β < 0, the manufacturer profit function in the two-period is concave in ${P}_w^1.$.

Proof of the proposition 16. As the two-period profit function of the manufacturer is concave in ${P}_w^1$, we can obtain the optimal wholesale prices for the new products in the first period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}_2^{m2}}{\partial {P}_w^1}=\frac{\alpha -2\beta {P}_w^1+\beta \left({C}_n+\delta \right)}{2}-\beta \frac{\theta }{2}\sqrt{\frac{C_d\theta }{3d}}\left({C}_n+\delta \left(1-\lambda \right)-{C}_r-{C}_d\right)=0 $$

(74)

Appendix 3. The proofs of the propositions of Model 3

Proof of the proposition 17.

$$ \frac{\partial {\pi}_2^{t3}}{\partial {C}_c}=\frac{\left({C}_t-{C}_c\right)\left(\alpha -\beta {P}_r^1\right)\theta }{2d\sqrt{\frac{C_c}{d}}}-\sqrt{\frac{C_c}{d}}\left(\alpha -\beta {P}_r^1\right)\theta $$

(75)

$$ \frac{\partial^2{\pi}_2^{t3}}{\partial {C}_c^2}=-\frac{\theta \left(\alpha -\beta {P}_r^1\right)\sqrt{C_cd}+\frac{d\theta \left(\alpha -\beta {P}_r^1\right)\left({C}_c-{C}_t\right)}{2\sqrt{C_cd}}}{2{C}_c} $$

(76)

Since ${\partial}^2{\pi}_2^{t3}/\partial {C}_c^2<0$, so the third party’s profit function in the second period ${\pi}_2^{t2}$ is concave in C_c.

Proof of the proposition 18. As the second period profit function of the third party is concave in C_c , we can obtain the optimal acquisition price of a unit of a used product which is returned from consumers in the second period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}_2^{t3}}{\partial {C}_c}=\frac{\left({C}_t-{C}_c\right)\left(\alpha -\beta {P}_r^1\right)\theta }{2d\sqrt{\frac{C_c}{d}}}-\sqrt{\frac{C_c}{d}}\left(\alpha -\beta {P}_r^1\right)\theta =0 $$

(77)

Proof of the proposition 19 .

$$ \frac{\partial {\pi}_2^{d3}}{\partial {P}_r^2}=\alpha -2\beta {P}_r^2+\beta {P}_w^2 $$

(78)

$$ \frac{\partial^2{\pi}_2^{d3}}{\partial {P}_r^{2^2}}=-2\beta $$

(79)

Since −2β < 0, so the distributor’s profit function in the second period ${\pi}_3^{d2}$ is concave in ${P}_r^2$.

Proof of the proposition 20. As the second period profit function of the distributor is concave in ${P}_r^2$, we can obtain the optimal retail prices for new and remanufactured products in the second period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}_2^{d3}}{\partial {P}_r^2}=\alpha -2\beta {P}_r^2+\beta {P}_w^2=0 $$

(80)

Proof of the proposition 21.

$$ \frac{\partial {\pi}^{d3}}{\partial {P}_r^1}=\alpha -2\beta {P}_r^1+{P}_w^1\beta -\left[\left({P}_w^2-{C}_r-{C}_t\right)\omega \theta - l\omega \theta \right]\beta \sqrt{\frac{C_t}{3d}} $$

(81)

$$ \frac{\partial^2{\pi}^{d3}}{\partial {P}_r^{1^2}}=-2\beta $$

(82)

As −2β < 0, the distributor profit function in the two-period is concave in ${P}_r^1.$.

Proof of the proposition 22. As the two-period profit function of the distributor is concave in ${P}_r^1$, we can obtain the optimal retail prices for new in the first period by setting the first-order derivatives to zero.

$$ \frac{\partial {\pi}^{d3}}{\partial {P}_r^1}=\alpha -2\beta {P}_r^1+{P}_w^1\beta -\left[\left({P}_w^2-{C}_r-{C}_t\right)\omega \theta - l\omega \theta \right]\beta \sqrt{\frac{C_t}{3d}}=0 $$

(83)

Proof of the proposition 23.

$$ {\displaystyle \begin{array}{l}\frac{\partial {\pi}_2^{m3}}{\partial {P}_w^2}=\frac{\alpha -2\beta {P}_w^2+\beta \left({C}_n+\delta \right)}{2}\\ {}\kern1.92em +\frac{\theta \omega}{2}\sqrt{\frac{C_t}{3d}}\left\{\beta \theta \sqrt{\frac{C_t}{3d}}\left[\left(2\omega -1\right)\left({C}_r+{C}_t+\lambda \delta \right)+{C}_n+\delta -2\omega {P}_w^2\right]-\alpha +\beta {P}_w^1\right\}\end{array}} $$

(84)

$$ \frac{\partial^2{\pi}_2^{m3}}{\partial {P}_w^{2^2}}=-\beta {P}_w^2-{\theta}^2{\omega}^2\beta \frac{C_t}{3d} $$

(85)

As −β − θ²ω²βC_t/3d < 0 , the manufacturer profit function in the second period is concave in ${P}_w^2.$

Proof of the proposition 24. As the second period profit function of the manufacturer is concave in ${P}_w^2$, we can obtain the optimal wholesale prices for the new and remanufactured products in the second period by setting the first-order derivatives to zero.

$$ {\displaystyle \begin{array}{l}\frac{\partial {\pi}_2^{m3}}{\partial {P}_w^2}=\frac{\alpha -2\beta {P}_w^2+\beta \left({C}_n+\delta \right)}{2}\\ {}\kern1.92em +\frac{\theta \omega}{2}\sqrt{\frac{C_t}{3d}}\left\{\beta \theta \sqrt{\frac{C_t}{3d}}\left[\left(2\omega -1\right)\left({C}_r+{C}_t+\lambda \delta \right)+{C}_n+\delta -2\omega {P}_w^2\right]-\alpha +\beta {P}_w^1\right\}=0\end{array}} $$

(86)

Proof of the proposition 25.

$$ {\displaystyle \begin{array}{l}\frac{\partial {\pi}^{m2}}{\partial {P}_w^1}=\frac{\alpha -2\beta {P}_w^1+\beta \left({C}_n+\delta \right)}{2}+B\frac{\alpha -\beta \left(2A+2B{P}_w^1\right)+\beta \left({C}_n+\delta \right)}{2}\\ {}+\beta \frac{\theta }{2}\left[\omega \theta B\frac{\theta {C}_t}{3d}-\sqrt{\frac{C_t}{3d}}\right]\left[{C}_n+\delta -{C}_r-{C}_t-\lambda \delta -\omega \left(A+B{P}_w^1-{C}_r-{C}_t-\lambda \delta +l\right)\right]\\ {}-\omega B\left[\frac{\theta }{2}\sqrt{\frac{C_t}{3d}}\left(\alpha -\beta {P}_w^1+\beta \omega \theta \left[A+B{P}_w^1-{C}_r-{C}_t-l\right]\sqrt{\frac{C_t}{3d}}\right)\right]\end{array}} $$

(87)

$$ \frac{\partial^2{\pi}^{m2}}{\partial {P}_w^{1^2}}=-\beta -\beta B{}^2-\omega \theta B\beta \sqrt{\frac{C_t}{3d}}\left[\omega {\theta}^2B\sqrt{\frac{C_t}{3d}}-1\right] $$

(88)

If $\omega {\theta}^2B\sqrt{C_t/3d}\ge 1,$ then ${\partial}^2{\pi}^{m2}/\partial {P}_w^{1^2}<0$; otherwise, if $\omega {\theta}^2B\sqrt{C_t/3d}<1,$ then $-1\le \omega {\theta}^2B\sqrt{C_t/3d}-1<0$ and in the pessimistic case if $\omega {\theta}^2B\sqrt{C_t/3d}-1=-1,$ then as $1+{B}^2>\omega {\theta}^2B\sqrt{C_t/3d},{\partial}^2{\pi}^{m2}/\partial {P}_w^{1^2}<0.$ So, the manufacturer profit function in the two periods is concave in ${P}_w^1$.

Proof of the proposition 26. As the two-period profit function of the manufacturer is concave in $\kern0.5em {P}_w^1$, we can obtain the optimal wholesale prices for the new products in the first period by setting the first-order derivatives to zero.

$$ {\displaystyle \begin{array}{l}\frac{\partial {\pi}^{m2}}{\partial {P}_w^1}=\frac{\alpha -2\beta {P}_w^1+\beta \left({C}_n+\delta \right)}{2}+B\frac{\alpha -2\beta \left(A+B{P}_w^1\right)+\beta \left({C}_n+\delta \right)}{2}\\ {}+\beta \frac{\theta }{2}\left[\omega \theta B\frac{\theta {C}_t}{3d}-\sqrt{\frac{C_t}{3d}}\right]\left[{C}_n+\delta -{C}_r-{C}_t-\lambda \delta -\omega \left(A+B{P}_w^1-{C}_r-{C}_t-\lambda \delta +l\right)\right]\\ {}-\omega B\left[\frac{\theta }{2}\sqrt{\frac{C_t}{3d}}\left(\alpha -\beta {P}_w^1+\beta \omega \theta \sqrt{\frac{C_t}{3d}}\left[A+B{P}_w^1-{C}_r-{C}_t-l\right]\right)\right]=0\end{array}} $$

The results of the sensitivity analysis

The results of the sensitivity analysis by changing the tax rate are provided in Tables 4, 5, and 6.

Table 4 displays the effect of changing the tax rate on the profit of the manufacturer and distributor in each model and period.

Table 4 Effect of changing the tax rate on the profit of the manufacturer and distributor

Table 5 Effect of changing the tax rate on unit retail price

Table 6 Effect of changing the tax rate on the demand

Table 7 Effect of changing the licensing fee on the profit of the manufacturer and distributor

Table 8 Effect of changing ω on the profit of the manufacturer and distributor