A Production Inventory Model for Green Products with Emission Reduction Technology Investment and Green Subsidy

Abstract

Increasing concerns over environmental pollution across the globe have encouraged us to replace some conventional products with green products. The production cost for green products being higher, the governments in various countries have initiated subsidy policies for green product manufacturers. The carbon regulatory authorities in different nations have started carbon taxation policies to lower the emission. Investment in emission reduction technologies can control the emission of carbon from a manufacturing firm. This paper explores the impacts of joint investment in greening innovation and emission reduction technology in a green production inventory model and provides some better insights to the real-life practitioner. Assuming a selling price and greenness level dependent demand, the optimal inventory decisions are examined under the cap and trade carbon regulatory policy. The model also considers the possibility of defective production and their repairing process. The aim is to find the optimal selling price, the optimal degree of greenness, optimal emission reduction technology investment, and optimal production run time that maximizes the optimal profit. Numerical illustration is presented to validate the model. Sensitivity analysis of the optimal solutions concerning the key inventory parameters is conducted for identifying several managerial implications. It is found that higher subsidy intensity increases the degree of greenness of the product. It is also seen that the simultaneous investment in greening innovation and emission reduction technology is beneficial for the green product manufacturer and the environment.

Appendices

Appendix 1. Proof of Theorem 1

The first- and second-order partial derivatives of Prof with respect to $t_1$ give $\frac{\partial {Prof}}{\partial t_1}= \frac{a-bp+c\alpha }{Pt_1^2}\left\{ c_s+c_te_s\left( 1-\theta +\theta e^{-mk}\right) +\frac{G\alpha ^2}{2}+k\right\} +(a-bp+c\alpha )\Big \{c_hA+c_te_hA\left( 1-\theta +\theta e^{-mk}\right) \Big \}$ $-c_h\frac{P}{2}-c_te_h\frac{P}{2}\left( 1-\theta +\theta e^{-mk}\right)$ and $\frac{\partial ^2 {Prof}}{\partial {t_1}^2}= \frac{-2(a-bp+c\alpha )}{Pt_1^3}\left\{ c_s+c_te_s\left( 1-\theta +\theta e^{-mk}\right) +\frac{G\alpha ^2}{2}+k\right\}$ $<0$. So, Prof is concave in $t_1$.

Appendix 2. Proof of Theorem 2

The first two order partial derivatives of Prof with respect to p yield $\frac{\partial {Prof}}{\partial p}=-b(p+s\alpha -c_p-c_rx)+a-bp+c\alpha +\frac{bc_s}{Pt_1}-bt_1c_hA+c_te_s\left( 1-\theta +\theta e^{-mk}\right) \frac{b}{Pt_1}+c_t(e_p+e_rx)b\left( 1-\theta +\theta e^{-mk}\right) -c_te_hAb\left( 1-\theta +\theta e^{-mk}\right)t_1$ $+\frac{Gb\alpha ^2}{2Pt_1}+\frac{kb}{Pt_1}$ and $\frac{\partial ^2 {Prof}}{\partial p^2}= -b-b=-2b<0$. So, Prof is concave in p.

Appendix 3. Proof of Theorem 3

The first two order partial derivatives of Prof with respect to $\alpha$ yield $\frac{\partial {Prof}}{\partial \alpha }= (a-bp+c\alpha )s+c(p+s\alpha -c_p-c_rx)-c_s\frac{c}{Pt_1}+ct_1c_hA -c_te_s\left( 1-\theta +\theta e^{-mk}\right) \frac{c}{Pt_1}-c_t(e_p+e_rx)\left( 1-\theta +\theta e^{-mk}\right) c+c_te_hA\left( 1-\theta +\theta e^{-mk}\right) ct_1-\frac{kc}{Pt_1}-\frac{G}{2Pt_1}(2a\alpha -2bp\alpha +3c\alpha ^2)$ and $\frac{\partial ^2 {Prof}}{\partial \alpha ^2}= 2sc-\frac{G}{Pt_1}\left( a-bp+3c\alpha \right)$.

Now $\frac{\partial ^2 {Prof}}{\partial \alpha ^2}< 0 \Rightarrow 2sc-\frac{G}{Pt_1}\left( a-bp+3c\alpha \right) <0 \Rightarrow \alpha >\frac{1}{3c}\left\{ \frac{2scPt_1}{G}-a+bp\right\} .$

Replacing $\alpha$ by $\alpha _{\text {min}}+\beta (\alpha _{\text {max}}-\alpha _{\text {min}})$, we get $\alpha _{\text {min}}+\beta \left( \alpha _{\text {max}}-\alpha _{\text {min}}\right)>\frac{1}{3c}\left\{ \frac{2scPt_1}{G}-a+bp\right\} \text { i.e. } \beta >\frac{1}{\alpha _{\text {max}}-\alpha _{\text {min}}}\left\{ \frac{1}{3c}\left( \frac{2scPt_1}{G}-a+bp\right) -\alpha _{\text {min}}\right\}$.

So, Prof is concave in $\alpha$ under the given condition.

Appendix 4. Proof of Theorem 4

The first two order partial derivatives of Prof with respect to k yield $\frac{\partial {Prof}}{\partial k}=\frac{a-bp+c\alpha }{Pt_1}c_te_s\theta me^{-mk}+c_t(e_p+e_rx)(a-bp+c\alpha )\theta me^{-mk}-c_te_hAt_1(a-bp+c\alpha )\theta me^{-mk}+c_te_h\frac{P}{2}t_1\theta me^{-mk}-\frac{a-bp+c\alpha }{Pt_1}$ and $\frac{\partial ^2 {Prof}}{\partial k^2} = -\frac{a-bp+c\alpha }{Pt_1}c_te_s\theta m^2e^{-mk}-c_t(e_p+e_rx)(a-bp+c\alpha )\theta m^2e^{-mk}+c_te_hAt_1(a-bp+c\alpha )\theta m^2e^{-mk}-c_te_h\frac{P}{2}t_1\theta m^2e^{-mk}$.

So, $\frac{\partial ^2 {Prof}}{\partial k^2}< 0 \Rightarrow \theta m^2e^{-mk}\left[ (a-bp+c\alpha )\left\{ c_te_hAt_1-\frac{c_te_s}{Pt_1}-c_t(e_p+e_rx)\right\} -c_te_h\frac{P}{2}t_1\right]<0 \Rightarrow (a-bp+c\alpha )\left\{ c_te_hAt_1-\frac{c_te_s}{Pt_1}-c_t(e_p+e_rx)\right\} -c_te_h\frac{P}{2}t_1<0.$

So, Prof is concave with respect to k under the stated condition.