Game theoretic analysis of a three-stage interconnected forward and reverse supply chain

Abstract

The dynamic economic scenario of today ensures that industrial and environmental policies that contribute to greener supply chain are incorporated. This paper considers an interconnected three-stage forward and reverse supply chain, which provides green products to a green conscious market. The procurement of raw materials is responsible for the first stage of the supply chain; the second manufacturing/remanufacturing process; and the third stage of marketing the products to the consumer. There is one supplier, one manufacturer, and one retailer in the forward supply chain. New raw materials are used in this supply chain, and new products are manufactured and sold. There is also a market for remanufactured products, and in this market, the same retailer also sells remanufactured products. There is one collector, one remanufacturer, and one retailer in the reverse supply chain. From consumers, the collector collects used products, processes and sells the remanufacturable ones to the remanufacturer. If the raw materials supplied by the collector are not adequate to satisfy the demand, the remanufacturer purchases the remainder from the seller. Both the manufacturer and the remanufacturer use green manufacturing processes. Two models, namely centralized model and decentralized model, are formulated. A numerical example is taken to illustrate the two model and perform sensitivity analysis.

Appendices

Appendix 1

The Hessian matrix of the objective function $\pi _{\mathrm{SC}}(p_1, p_2,\theta _1, \theta _2)$ of the centralized model given in Eq. (7) is calculated as

$$\begin{aligned} H(p_1, p_2,\theta _1, \theta _2) = \begin{bmatrix} \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_1^2} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_1 \partial p_2} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_1 \partial \theta _1} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_1 \partial \theta _2}\\ \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_2 \partial p_1} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_2^2} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_2 \partial \theta _1} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial p_2 \partial \theta _2}\\ \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _1 \partial p_1} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _1 \partial p_2} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _1^2} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _1 \partial \theta _2}\\ \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _2 \partial p_1} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _2 \partial p_2} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _2 \partial \theta _1} &{}\quad \frac{\partial ^2 \pi _{\mathrm{SC}}}{\partial \theta _2^2} \end{bmatrix} =\begin{bmatrix} -2 b &{}\quad 0 &{}\quad d_1 &{}\quad 0\\ 0 &{}\quad -2 b &{}\quad 0 &{}\quad d_2\\ d_1 &{}\quad 0 &{}\quad -2 I &{}\quad 0\\ 0 &{}\quad d_2 &{}\quad 0 &{}\quad -2 I \end{bmatrix} \end{aligned}$$

The leading principal minors $\Delta _k$ of $H(p_1, p_2,\theta _1, \theta _2)$ of order k are given by

$$\begin{aligned} \Delta _1 = -2b, \Delta _2 = 4 b^2, \Delta _3 = - 2 b (4 b I - d_1^2), \Delta _4 = (4 b I - d_1^2) (4 b I - d_2^2) \end{aligned}$$

From assumptions 3(i) and 3(ii), we get $4 b I - d_1^2>0$ and $4 b I - d_2^2>0$. Therefore, $(- 1)^k\Delta _k>0$ for all leading principle minors of $H(p_1, p_2,\theta _1, \theta _2)$. Thus, $H(p_1, p_2,\theta _1, \theta _2)$ is negative definite and $\pi _{\mathrm{SC}}(p_1, p_2,\theta _1, \theta _2)$ is strictly concave. Using the first order conditions of Eq. (7) simultaneously, we get the unique global optimal solution to the centralized model as given in Eqs. (8)–(11). Putting the Eqs. (7)–(10) into Eq. (1), we get the optimal values of demands for the new and remanufactured products as given in Eqs. (12)–(13). Finally, putting the Eqs. (8)–(10) into Eq. (7), we get the optimal profit of the entire supply chain as given in Eq. (14).

Appendix 2

In decentralized model, first we consider retailer’s profit function. The Hessian matrix of the objective function $\pi _\mathrm{R}(p_1, p_2)$ given in Eq. (2) is calculated as

Table 4 Change of unit selling prices and market demand w.r.t. % change of $a, b,\ldots$

Table 5 Change of unit selling prices and market demand w.r.t. % change of $d_1, d_2$

Table 6 Change of different profit functions w.r.t. % change of $a, b,\ldots$

Table 7 Change of different profit functions w.r.t. % change of $d_1, d_2$

$$\begin{aligned} H(p_1, p_2) =\begin{bmatrix} \frac{\partial ^2 \pi _\mathrm{R}}{\partial p_1^2} &{}\quad \frac{\partial ^2 \pi _\mathrm{R}}{\partial p_1 \partial p_2 } \\ \frac{\partial ^2 \pi _\mathrm{R}}{\partial p_2 \partial p_1 } &{}\quad \frac{\partial ^2 \pi _\mathrm{R}}{\partial p_2^2} \end{bmatrix} =\begin{bmatrix} -2 b &{}\quad 0\\ 0 &{}\quad -2 b \end{bmatrix} \end{aligned}$$

The leading principal minors $\Delta _k$ of $H(p_1, p_2)$ of order k are given by,

$$\begin{aligned} \Delta _1 = -2 b, \Delta _2 = 4 b^2 \end{aligned}$$

Therefore, $(- 1)^k\Delta _k>0$ for all leading principle minors of $H(p_1, p_2)$. Thus, $H(p_1, p_2)$ is negative definite and $\pi _\mathrm{R}(p_1, p_2)$ is strictly concave. Using the first order conditions of Eq. (2) with respect to $p_1$ and $p_2$, we get the retailer’s optimum response functions as given in Eqs. (15)–(16).

Next, we consider the manufacturer’s and the remanufacturer’s profit functions. We first substitute the retailer’s optimum response function given in Eq. (15) into the manufacturer’s profit function given in Eq. (3). The Hessian matrix of the objective function $\pi _\mathrm{M}(\theta _1, w_1)$ given in Eq. (3) is calculated as

$$\begin{aligned} H(\theta _1, w_1) =\begin{bmatrix} \frac{\partial ^2 \pi _\mathrm{M}}{\partial \theta _1^2} &{}\quad \frac{\partial ^2 \pi _\mathrm{M}}{\partial \theta _1 \partial w_1} \\ \frac{\partial ^2 \pi _\mathrm{M}}{\partial w_1 \partial \theta _1} &{}\quad \frac{\partial ^2 \pi _\mathrm{M}}{\partial w_1^2} \end{bmatrix} =\begin{bmatrix} -2 I &{}\quad \frac{d_1}{2}\\ \frac{d_1}{2} &{}\quad -b \end{bmatrix} \end{aligned}$$

The leading principal minors $\Delta _k$ of $H(\theta _1, w_1)$ of order k are given by,

$$\begin{aligned} \Delta _1 = -2 I, \Delta _2 = 2 b I - \frac{d_1^2}{4} \end{aligned}$$

From assumptions 3(i), we get $2 b I - \frac{d_1^2}{4}>0$. Therefore, $(- 1)^k\Delta _k>0$ for all leading principle minors of $H(\theta _1, w_1)$. Thus, $H(\theta _1, w_1)$ is negative definite and $\pi _\mathrm{M}(\theta _1, w_1))$ is strictly concave. Using the first order conditions of Eq. (3) with respect to $\theta _1$ and $w_1$, we get the manufacturer’s optimum response functions as given in Eqs. (17)–(18).

In a similar way, we substitute the retailer’s optimum response function given in Eq. (16) into the remanufacturer’s profit function given in Eq. (4). The Hessian matrix of the objective function $\pi _{\mathrm{RM}}(\theta _2, w_2)$ given in Eq. (4) is calculated as

$$\begin{aligned} H(\theta _2, w_2) = \begin{bmatrix} \frac{\partial ^2 \pi _{\mathrm{RM}}}{\partial \theta _2^2} &{} \quad \frac{\partial ^2 \pi _{\mathrm{RM}}}{\partial \theta _2 \partial w_2 } \\ \frac{\partial ^2 \pi _{\mathrm{RM}}}{\partial w_2 \partial \theta _2 } &{}\quad \frac{\partial ^2 \pi _{\mathrm{RM}}}{\partial w_2^2} \end{bmatrix} =\begin{bmatrix} -2 I &{}\quad \frac{d_2}{2}\\ \frac{d_2}{2} &{}\quad -b \end{bmatrix} \end{aligned}$$

The leading principal minors $\Delta _k$ of $H(\theta _2, w_2)$ of order k are given by,

$$\begin{aligned} \Delta _1 = -2 I, \Delta _2 = 2 b I - \frac{d_2^2}{4} \end{aligned}$$

From assumptions 3(ii), we get $2 b I - \frac{d_2^2}{4}>0$. Therefore, $(- 1)^k\Delta _k>0$ for all leading principle minors of $H(\theta _2, w_2)$. Thus, $H(\theta _2, w_2)$ is negative definite and $\pi _{\mathrm{RM}}(\theta _2, w_2))$ is strictly concave. Using the first order conditions of Eq. (4) with respect to $\theta _2$ and $w_2$, we get the remanufacturer’s optimum response functions as given in Eqs. (19)–(20).

In the last step, we consider the supplier’s and the collector’s profit functions. We first substitute the optimum response functions of the retailer, manufacturer, remanufacturer given in Eqs. (15)–(20) into the supplier’s profit function given in Eq. (5). Then, the second order derivative of the supplier’s objective function $\pi _{\mathrm{S}}(s_1)$ given in Eq. (5) with respective to $s_1$ is calculated as

$$\begin{aligned} \frac{\partial ^2 \pi _{\mathrm{S}}}{\partial s_1^2} = - \frac{4 b^2 I}{8 b I - d_1^2} - \frac{4 b^2 I (1 - \beta )^2}{8 b I - d_2^2} \end{aligned}$$

From assumptions 3(i) and 3(ii), we get $8 b I - d_1^2>0$ and $8 b I - d_2^2>0$. And since $I>0$, it can be seen that $- \frac{4 b^2 I}{8 b I - d_1^2} - \frac{4 b^2 I (1 - \beta )^2}{8 b I - d_2^2}<0$. Therefore, $\frac{\partial ^2 \pi _{\mathrm{S}}}{\partial s_1^2}<0$, i.e. $\pi _{\mathrm{S}}(s_1)$ is a strictly concave function.

In a similar way, we first substitute the optimum response functions of the retailer, remanufacturer given in Eqs. (16), (19) and (20) into the collector’s profit function given in Eq. (6). Then, the second order derivative of the collector’s objective function $\pi _{\mathrm{C}}(s_2)$ given in Eq. (5) with respective to $s_2$ is calculated as

$$\begin{aligned} \frac{\partial ^2 \pi _{\mathrm{C}}}{\partial s_2^2} = - \frac{4 b^2 I \beta ^2}{8 b I - d_2^2} \end{aligned}$$

From assumptions 3(ii), we get $8 b I - d_2^2>0$. And since $I>0$, it can be seen that $- \frac{4 b^2 I \beta ^2}{8 b I - d_2^2}<0$. Therefore, $\frac{\partial ^2 \pi _{\mathrm{C}}}{\partial s_2^2}<0$, i.e. $\pi _{\mathrm{C}}(s_2)$ is a strictly concave function.

Finally, simultaneously solving the first order condition of Eq. (5) with respect to $s_1$ and first order condition of Eq. (6) with respect to $s_2$ gives the optimal unit selling prices of the supplier and the collector as given in Eqs. (21) and (22). Putting the Eqs. (15)–(22) into Eq. (1), we get the optimal values of demands for the new and remanufactured products as given in Eqs. (23)–(24). Finally, putting the Eqs. (15)–(22) into Eq. (7), we get the optimal profit of the overall supply chain as given in Eq. (25).