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A reverse Logistics Inventory Model with Consideration of Carbon Tax Policy, Imperfect Production, and Partial Backlogging Under a Sustainable Supply Chain

Process Integration and Optimization for Sustainability Aims and scope Submit manuscript

Abstract

The awareness of consumers for reused or recycled items and environmentally friendly practices has grown over the past few decades, which has encouraged businesses to integrate sustainability development into their operations. As a result, industry’s supply chains are focusing on product recycling, waste reduction initiatives, and proper disposal of expired products. Sustainable development has benefited a variety of industries, including inventory management; it helps to sustain the environment as well as the economy. In this study, the effect of carbon emission on a reverse logistics inventory model is investigated when manufacturing and remanufacturing with multiple retailers are taken into account. In an effort to minimize the carbon emission, many countries are imposing strict limits on carbon emitters. Carbon tax policy is one of the laws that has been successfully implemented in many countries. In this paper, carbon emissions are released from various supply chain activities like product manufacturing, remanufacturing, storage, waste disposal, transportation, and deterioration. To control the excess release of carbon emission, carbon tax regulation is incorporated. During remanufacturing, due to some error, the machine switches from being “in control” to an “out of control” state and generates some imperfect quantities that are instantly inspected and reworked at some cost. Shortages are permitted with partial backlogging, and backlogging rate is constant. The aim of this study is to determine the optimal total cost under a forwards and backwards supply chain system. The model is demonstrated using several numerical examples, followed by a sensitivity analysis that seeks to investigate the impact of key parameters on the optimal solution. Convexity is depicted graphically, and significant features of the results are discussed to extract some management knowledge. The results indicate that the manufacturing cost, remanufacturing cost, and the rate of remanufacturing have a significant impact on the integrated cost of the inventory system. The obtained results also explain that parameters involved in carbon emission activity produce noticeable changes in the total cost. The findings are very managerial and instructive for organizations and businesses seeking environmentally conscious production systems.

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Data Availability

All datasets generated or analyzed during this study are included in this published article. All other relevant datasets are available from the corresponding author on reasonable request.

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Acknowledgements

The authors express their gratitude to the dear editor and the anonymous reviewers for their comments and valuable suggestions to improve the quality of this paper.

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Correspondence to Chandni Katariya.

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The authors declare no competing interests.

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Appendices

Appendix 1

Differential equations representing the behavior of system for manufacturing cycle (see Fig. 4) with respect to time are given by as follows:

$$\begin{array}{cc}\frac{{dI}_{{m}{\prime}}}{dt}=-\phi \alpha & 0\le t\le u\end{array}$$
(39)
$$\begin{array}{cc}\frac{{dI}_{{m}^{^{\prime\prime} }}}{dt}=\left(x-1\right)\alpha & u\le t\le v\end{array}$$
(40)
$$\begin{array}{cc}\frac{{dI}_{{{{m}{\prime}}{\prime}}{\prime}}}{dt}+\theta {I}_{{{{m}{\prime}}{\prime}}{\prime}}=\left(x-1\right)\left(\alpha +\delta t\right)& v\le t\le w\end{array}$$
(41)
$$\begin{array}{cc}\frac{{dI}_{{{{{m}{\prime}}{\prime}}{\prime}}{\prime}}}{dt}+\theta {I}_{{{{{m}{\prime}}{\prime}}{\prime}}{\prime}}=-\left(\alpha +\delta t\right)& w\le t\le y\end{array}$$
(42)

Boundary equation of the system is as follows:

$$\begin{array}{l}I_{m'}\left(0\right)=0,I_{m''}\left(v\right)=0,I_{m'''}\left(v\right)=0,I_{m''''}\left(y\right)=0\\\mathrm{Also},\mathrm{at}\;t=u,I_{m'}\left(u\right)=I_{m''}\left(u\right)\end{array}$$
(43)

Solutions of Eqs. (39), (40), (41), and (42) are given by

$$\begin{array}{cc}{I}_{{m}{\prime}}\left(t\right)=-\alpha \phi t& 0\le t\le u\end{array}$$
(44)
$$\begin{array}{cc}{I}_{{m}^{{\prime}{\prime}}}\left(t\right)=\left(x-1\right)\alpha \left(t-v\right)& u\le t\le v\end{array}$$
(45)
$$\begin{array}{cc}{I}_{{{{m}{\prime}}{\prime}}{\prime}}\left(t\right)=\left(x-1\right)\frac{1}{{\theta }^{2}}\left[\left(\left(\alpha +\delta t\right)\theta -\delta \right)-\left(\left(\alpha +\delta v\right)\theta -\delta \right){e}^{\theta \left(v-t\right)}\right]& v\le t\le w\end{array}$$
(46)
$$\begin{array}{cc}{I}_{{{{{m}{\prime}}{\prime}}{\prime}}{\prime}}\left(w\right)=\frac{1}{{\theta }^{2}}\left[\left(\left(\alpha +\delta y\right)\theta -\delta \right){e}^{\theta \left(y-t\right)}-\left(\left(\alpha +\delta t\right)\theta -\delta \right)\right]& w\le t\le y\end{array}$$
(47)

Appendix 2

Assuming that \(k\) is the rate of deterioration and demand of the system is the function of stock, then the level of inventory (see Fig. 5) at any time t is given as follows:

$$\begin{array}{cc}\frac{{dI}_{x}}{dt}+k{I}_{x}=-\left(\gamma +\mu {I}_{x}\left(t\right)\right)& 0\le t\le y\end{array}$$
(48)

Boundary equations are given by as follows:

$${I}_{x}\left(y\right)=0$$
(49)

Solution of Eq. (48) is given by

$$\begin{array}{cc}{I}_{x}\left(t\right)=\left(\frac{\gamma }{\left(k+\mu \right)}\left({e}^{\left(k+\mu \right)\left(y-t\right)}-1\right)\right)& 0\le t\le y\end{array}$$
(50)

Appendix 3

Differential equations representing the behavior of system for remanufacturing cycle (see Fig. 6) with respect to time are given by as follows:

$$\begin{array}{cc}\frac{{dI}_{{R}{\prime}}}{dt}+{\theta }_{r}{I}_{{R}{\prime}}=\left({P}_{r}-{D}_{r}\right)& 0\le t\le z\end{array}$$
(51)
$$\begin{array}{cc}\frac{{dI}_{{R}^{{\prime}{\prime}}}}{dt}+{\theta }_{r}{I}_{{R}^{{\prime}{\prime}}}=-{D}_{r}& z\le t\le T\end{array}$$
(52)

Boundary equation of the system is as follows:

$${I}_{{R}{\prime}}\left(0\right)=0,{I}_{{R}^{{\prime}{\prime}}}\left(T\right)=0$$
(53)

Solutions of Eqs. (51) and (52) are given by

$$\begin{array}{cc}{I}_{{R}{\prime}}\left(t\right)=\left(\frac{\left({P}_{r}-{D}_{r}\right)}{{\theta }_{r}}\left(1-{e}^{-{\theta }_{r}t}\right)\right)& 0\le t\le z\end{array}$$
(54)
$$\begin{array}{cc}{I}_{{R}^{{\prime}{\prime}}}\left(t\right)=\left(\frac{{D}_{r}}{{\theta }_{r}}\left({e}^{{\theta }_{r}\left(T-t\right)}-1\right)\right)& z\le t\le T\end{array}$$
(55)

Appendix 4

Assuming that \({k}_{r}\) is the rate of deterioration and demand of the system is the function of stock, then the level of inventory (see Fig. 7) at any time t is given as follows:

$$\begin{array}{cc}\frac{{dI}_{r}}{dt}+{k}_{r}{I}_{r}=-\left(\eta +\tau {I}_{r}\left(t\right)\right)& 0\le t\le T\end{array}$$
(56)

Boundary equations are given as follows:

$${I}_{r}\left(T\right)=0$$
(57)

Solutions of Eq. (56) are given by

$$\begin{array}{cc}{I}_{r}\left(t\right)=\left(\frac{\eta }{\left({k}_{r}+\tau \right)}\left({e}^{\left({k}_{r}+\tau \right)\left(T-t\right)}-1\right)\right)& 0\le t\le T\end{array}$$
(58)

Appendix 5

$$T.{E}_{MM}=\frac{1}{y}\left(\begin{array}{l}{H}_{{T}_{e}}{E}_{e}\left(\begin{array}{c}\left(\frac{(w-v)}{2{\theta }^{2}}\{2\alpha \theta +\delta \theta (w+v)-2\delta \}+\frac{1}{{\theta }^{3}}\{(\alpha +\delta v)\theta -\delta \}\{{e}^{\theta (v-w)}-1\}\right)\\ +\left(\frac{(w-y)}{2{\theta }^{2}}\{2\alpha \theta +\delta \theta (w+y)-2\delta \}+\frac{1}{{\theta }^{3}}\{(\alpha +\delta y)\theta -\delta \}\{{e}^{\theta (y-w)}-1\}\right)\end{array}\right)+\left({m}_{p}{E}_{e}x\left(\alpha (v-u)+\alpha (w-v)+\frac{\delta }{2}({w}^{2}-{v}^{2})\right)\right)\\ +{B}_{{d}_{e}}\theta \left(\begin{array}{c}\left(\frac{(w-v)}{2{\theta }^{2}}\{2\alpha \theta +\delta \theta (w+v)-2\delta \}+\frac{1}{{\theta }^{3}}\{(\alpha +\delta v)\theta -\delta \}\{{e}^{\theta (v-w)}-1\}\right)\\ +\left(\frac{(w-y)}{2{\theta }^{2}}\{2\alpha \theta +\delta \theta (w+y)-2\delta \}+\frac{1}{{\theta }^{3}}\{(\alpha +\delta y)\theta -\delta \}\{{e}^{\theta (y-w)}-1\}\right)\end{array}\right)+{w}_{a}{E}_{{w}_{1}}x\left(\alpha (v-u)+\alpha (w-v)+\frac{\delta }{2}({w}^{2}-{v}^{2})\right)\\ +{B}_{{S}_{e}}{n}_{1}k\left(\frac{\gamma }{{(\mu +k)}^{2}}\left({e}^{(k+\mu )y}-(k+\mu )y-1\right)\right)+{H}_{{S}_{e}}{E}_{e}{n}_{1}\left(\frac{\gamma }{{(\mu +k)}^{2}}\left({e}^{(k+\mu )y}-(k+\mu )y-1\right)\right)\\ +\left(2\lambda {F}_{{e}_{1}}+\lambda {F}_{{e}_{1}}\left(\alpha v+\alpha (y-v)+\frac{\delta }{2}({y}^{2}-{v}^{2})\right)\right)\end{array}\right)$$

Appendix 6

$$T.{E}_{RR}=\frac{1}{T}\left[\begin{array}{l}{R}_{{S}_{e}}{M}_{e}\left(\frac{({P}_{r}-{D}_{r})}{{\theta }_{r}^{2}}\left(z{\theta }_{r}+{e}^{-{\theta }_{r}z}-1\right)+\frac{{D}_{r}}{{\theta }_{r}^{2}}\left((z-T){\theta }_{r}+{e}^{{\theta }_{r}(T-z)}-1\right)\right)+\left(2{\lambda }_{2}{F}_{{e}_{2}}+{\lambda }_{2}{F}_{{e}_{2}}{D}_{r}T\right)\\ +{U}_{{S}_{e}}{\theta }_{r}\left(\frac{({P}_{r}-{D}_{r})}{{\theta }_{r}^{2}}\left(z{\theta }_{r}+{e}^{-{\theta }_{r}z}-1\right)+\frac{{D}_{r}}{{\theta }_{r}^{2}}\left((z-T){\theta }_{r}+{e}^{{\theta }_{r}(T-z)}-1\right)\right)\\ {G}_{{S}_{e}}{M}_{e}{n}_{2}\left(\frac{\eta }{{({k}_{r}+\tau )}^{2}}\left({e}^{({k}_{r}+\tau )T}-({k}_{r}+\tau )T-1\right)\right)+{i}_{p}{M}_{e}\left(z{P}_{r}\right)+{w}_{b}{E}_{{w}_{2}}\left(z{P}_{r}\right)\\ +{L}_{{S}_{e}}{k}_{r}{n}_{2}\left(\frac{\eta }{{({k}_{r}+\tau )}^{2}}\left({e}^{({k}_{r}+\tau )T}-({k}_{r}+\tau )T-1\right)\right)\\\\ \end{array}\right]$$

Appendix 7

Validation of numerical example (1) through sufficient condition is given by as follows:

$$\begin{array}{cc}\left(\frac{{\partial }^{2}T.A.{C}_{MM}}{\partial {v}^{2}}\right)=4.1693>0, & \left(\frac{{\partial }^{2}T.A.{C}_{MM}}{\partial {w}^{2}}\right)=15.2635>0\end{array}$$
(59)
$$\left(\left(\frac{{\partial }^{2}T.A.{C}_{MM}}{\partial {v}^{2}}\right)\left(\frac{{\partial }^{2}T.A.{C}_{MM}}{\partial {w}^{2}}\right)-{\left(\frac{{\partial }^{2}T.A.{C}_{MM}}{\partial v\partial w}\right)}^{2}\right)=62.3655>0$$
(60)

The above result shows that the objective function \(T.A.{C}_{MM}\) is convex.

Appendix 8

Validation of numerical example (2) through sufficient condition is given by as follows:

$$\left(\frac{{\partial }^{2}T.A.{C}_{RR}}{\partial {z}^{2}}\right)=64.6901>0$$
(61)

The above result shows that the objective function \(T.A.{C}_{RR}\) is convex.

Appendix 9

SI unit symbols, t for a ton, km for a kilometer, CO2 for Co2, and kWh for KWH are used in the present study.

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Handa, N., Singh, S.R. & Katariya, C. A reverse Logistics Inventory Model with Consideration of Carbon Tax Policy, Imperfect Production, and Partial Backlogging Under a Sustainable Supply Chain. Process Integr Optim Sustain (2023). https://doi.org/10.1007/s41660-023-00381-4

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