An interval-valued green production inventory model under controllable carbon emissions and green subsidy via particle swarm optimization

Abstract

It is a challenging task to develop appropriate realistic production inventory models due to the presence of imprecision in the data available in the current market situation. In addition, increasing environmental concerns around the world have led to a shift toward green products. There are subsidy programs implemented by governments in several countries for green products. In a real-life production system, the various inventory cost and carbon emission parameters are imprecise in nature. Interval number theory is an efficient tool for handling such impreciseness. This study investigates the effects of a simultaneous investment in greening innovation (GI) and emission reduction technologies (ERTs) in a green production inventory model where the various inventory cost components and the carbon emission parameters are interval-valued. Using the cap-and-trade carbon regulation policy, the optimal inventory decision is investigated based on a price and greenness level sensitive demand. Two models are developed depending on whether the manufacturer wishes or not to invest in ERTs. Both models consider the chances of imperfect production and their reworking process. The resulting interval optimization problems are solved using the quantum-behaved particle swarm optimization technique to derive the interval-valued optimum profit. Numerical illustrations for both models are presented. Several managerial insights are identified through a sensitivity analysis over the optimal solution regarding the main inventory parameters. The result shows that the greenness level of the product increases with the intensity of the subsidy offered by the government. Again, it is found that the manufacturing company of the green product as well as our environment benefit from joint investment in GI and ERT.

Appendices

Appendix A: Proof of Theorem 1

The first- and second-order partial derivatives of ${Prof}^{1}_L$ with respect to $t_1$ gives

$$\begin{aligned} \frac{\partial {Prof}^{1}_L}{\partial t_1}= & {} \frac{a-bp+c\alpha }{Pt_1^2}\left\{ {c_{s}}_R+c_t{e_{s}}_R+\frac{G\alpha ^2}{2}\right\} \nonumber \\{} & {} +\,(a-bp+c\alpha )\Big \{{c_{h}}_RA +c_t{e_{h}}_RA\Big \}-{c_{h}}_R\frac{P}{2}\nonumber \\{} & {} -\,c_t{e_{h}}_R\frac{P}{2} \end{aligned}$$

(21)

and $\dfrac{\partial ^2 {Prof}^{1}_L}{\partial {t_1}^2}= \frac{-2(a-bp+c\alpha )}{Pt_1^3}\left\{ {c_{s}}_R+c_t{e_{s}}_R+\frac{G\alpha ^2}{2}\right\} <0.$ So, ${Prof}^{1}_L$ is concave in $t_1$.

Similarly, ${Prof}^{1}_R$ is concave in $t_1$.

Appendix B: Proof of Theorem 2

The first- and second-order partial derivatives of ${Prof}^{1}_L$ with respect to p gives

$$\begin{aligned} \frac{\partial {Prof}^{1}_L}{\partial p}= & {} a-bp+c\alpha -b(p+s\alpha -{c_{p}}_R\nonumber \\{} & {} -\,{c_{r}}_Rx)+\frac{bc_s}{Pt_1}-bt_1{c_{h}}_RA+c_t{e_{s}}_R\frac{b}{Pt_1}\nonumber \\{} & {} +\,c_t({e_{p}}_R+{e_{r}}_Rx)b-c_t{e_{h}}_RAbt_1+\frac{Gb\alpha ^2}{2Pt_1} \end{aligned}$$

(22)

and $\dfrac{\partial ^2 {Prof}^{1}_L}{\partial p^2}= -b-b=-2b<0.$ So, ${Prof}^{1}_L$ is concave in p.

Similarly, ${Prof}^{1}_R$ is concave in p.

Appendix C: Proof of Theorem 3

The first- and second-order partial derivatives of ${Prof}^{1}_L$ with respect to $\alpha$ gives

$$\begin{aligned} \frac{\partial {Prof}^{1}_L}{\partial \alpha }= & {} (a-bp+c\alpha )s+c(p+s\alpha -{c_{p}}_R-{c_{r}}_Rx)\nonumber \\{} & {} -\,{c_{s}}_R\frac{c}{Pt_1}+ct_1{c_{h}}_RA-c_t{e_{s}}_R\frac{c}{Pt_1}\nonumber \\{} & {} -\,c_t({e_{p}}_R+{e_{r}}_Rx)c+c_t{e_{h}}_RAct_1\nonumber \\{} & {} -\,\frac{G}{2Pt_1}\left( 2a\alpha -2bp\alpha +3c\alpha ^2\right) \end{aligned}$$

(23)

and $\dfrac{\partial ^2 {Prof}^{1}_L}{\partial \alpha ^2}= 2sc-\frac{G}{Pt_1}\left( a-bp+3c\alpha \right)$

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

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

So, ${Prof}^{1}_L$ is concave in $\alpha$ provided the given condition is satisfied.

Similarly, ${Prof}^{1}_R$ is concave in $\alpha$ under the given condition.

Appendix D: Proof of Theorem 4

The first two-order partial derivatives of ${Prof}^{2}_L$ with respect to $t_1$ yield

$\dfrac{\partial {Prof}^{2}_L}{\partial t_1}= \frac{a-bp+c\alpha }{Pt_1^2}\Bigg \{{c_{s}}_R+c_t{e_{s}}_R\left( 1-\theta +\theta e^{-mk}\right) +\frac{G\alpha ^2}{2}+k\Bigg \}+(a-bp+c\alpha )\Big \{{c_{h}}_RA+c_t{e_{h}}_RA\big (1-\theta +\theta e^{-mk}\big )\Big \}$ $-{c_{h}}_R\frac{P}{2}-c_t{e_{h}}_R\frac{P}{2}\big (1-\theta +\theta e^{-mk}\big )$ and $\dfrac{\partial ^2 {Prof}^{2}_L}{\partial {t_1}^2}= \dfrac{-2(a-bp+c\alpha )}{Pt_1^3}\Big \{{c_{s}}_R+c_t{e_{s}}_R\big (1-\theta +\theta e^{-mk}\big )+\frac{G\alpha ^2}{2}+k\Big \}$ $<0$. So, ${Prof}^{2}_L$ is concave in $t_1$.

Similarly, ${Prof}^{2}_R$ is concave in $t_1$.

Appendix E: Proof of Theorem 5

The first two-order partial derivatives of ${Prof}^{2}_L$ with respect to p yield $\dfrac{\partial {Prof}^{2}_L}{\partial p}=-b(p+s\alpha -{c_{p}}_R-{c_{r}}_Rx)+a-bp+c\alpha +\frac{bc_s}{Pt_1}-bt_1{c_{h}}_RA+c_t{e_{s}}_R\left( 1-\theta +\theta e^{-mk}\right) \frac{b}{Pt_1}+c_t({e_{p}}_R+{e_{r}}_Rx)b\left( 1-\theta +\theta e^{-mk}\right) -c_t{e_{h}}_RAb\left( 1-\theta +\theta e^{-mk}\right)$ $t_1+\frac{Gb\alpha ^2}{2Pt_1}+\frac{kb}{Pt_1}$ and $\dfrac{\partial ^2 {Prof}^{2}_L}{\partial p^2}= -b-b=-2b<0$. So, ${Prof}^{2}_L$ is concave in p.

Similarly, ${Prof}^{2}_R$ is concave in p.

Appendix F: Proof of Theorem 6

The first two-order partial derivatives of ${Prof}^{2}_L$ with respect to $\alpha$ yield $\dfrac{\partial {Prof}^{2}_L}{\partial \alpha }= (a-bp+c\alpha )s+c(p+s\alpha -{c_{p}}_R-{c_{r}}_Rx)-{c_{s}}_R\frac{c}{Pt_1}+ct_1{c_{h}}_RA -c_t{e_{s}}_R\left( 1-\theta +\theta e^{-mk}\right) \frac{c}{Pt_1}-c_t({e_{p}}_R+{e_{r}}_Rx)\left( 1-\theta +\theta e^{-mk}\right) c+c_t{e_{h}}_RA\left( 1-\theta +\theta e^{-mk}\right) ct_1-\frac{kc}{Pt_1}-\frac{G}{2Pt_1}(2a\alpha -2bp\alpha +3c\alpha ^2)$ and $\dfrac{\partial ^2 {Prof}^{2}_L}{\partial \alpha ^2}= 2sc-\frac{G}{Pt_1}\left( a-bp+3c\alpha \right)$.

Therefore, by following similar procedure as in Appendix C, we can show that ${Prof}^{2}_L$ is concave in $\alpha$ under the given condition.

Similarly, ${Prof}^{2}_R$ is concave in $\alpha$ under the given condition.

Appendix G: Proof of Theorem 7

The first two-order partial derivatives of ${Prof}^{2}_L$ with respect to k yield $\dfrac{\partial {Prof}^{2}_L}{\partial k}=\frac{a-bp+c\alpha }{Pt_1}c_t{e_{s}}_R\theta m$ $e^{-mk}+c_t({e_{p}}_R+{e_{r}}_Rx)(a-bp+c\alpha )\theta me^{-mk}-c_t{e_{h}}_RAt_1(a-bp+c\alpha )\theta me^{-mk}+c_t{e_{h}}_R\frac{P}{2}t_1\theta me^{-mk}-\frac{a-bp+c\alpha }{Pt_1}$ and $\dfrac{\partial ^2 {Prof}^{2}_L}{\partial k^2} = -\frac{a-bp+c\alpha }{Pt_1}c_t{e_{s}}_R\theta m^2e^{-mk}-c_t({e_{p}}_R+{e_{r}}_Rx)(a-bp+c\alpha )\theta m^2e^{-mk}+c_t{e_{h}}_RAt_1(a-bp+c\alpha )\theta m^2e^{-mk}-c_t{e_{h}}_R\frac{P}{2}t_1\theta m^2e^{-mk}$.

So, $\dfrac{\partial ^2 {Prof}^{2}_L}{\partial k^2} < 0 \Rightarrow \theta m^2e^{-mk}\left[ (a-bp+c\alpha )\left\{ c_t{e_{h}}_RAt_1 -\frac{c_t{e_{s}}_R}{Pt_1}-c_t({e_{p}}_R+{e_{r}}_Rx)\right\} -c_t{e_{h}}_R\frac{P}{2}t_1\right]$ $<0 \Rightarrow (a-bp+c\alpha )\left\{ c_t{e_{h}}_RAt_1-\frac{c_t{e_{s}}_R}{Pt_1} -c_t({e_{p}}_R+{e_{r}}_Rx)\right\} -c_t{e_{h}}_R\frac{P}{2}t_1<0.$

So, ${Prof}^{2}_L$ is concave with respect to k under the stated condition.

Similarly, ${Prof}^{2}_R$ is concave with respect to k under the stated condition.