Inventory Models Under Carbon Tax and Cap-and-Trade Policies: A Comparative Analysis of Decentralized and Centralized Approaches

Inventory management is all about keeping track of supplies and making sure they're in the right place at the right time. But nowadays, with carbon rules in place, things have gotten a bit more complicated. In this section, we'll look at what researchers have been saying about managing inventory under these new carbon regulations.

Kung et al. [8] have delved into the collaborative efforts between manufacturers and retailers to reduce their carbon footprint. Their study aimed to understand how a carbon tax influences these joint actions within the supply network, and its broader impacts on the economy and environmental sustainability. They found that businesses working together can make a significant difference in reducing carbon emissions, especially when guided by carbon policies.

Other studies [9, 10] have focused on the nitty-gritty of managing inventory in the face of additional costs due to carbon emissions. Their research developed various inventory management systems designed to cope with these extra expenses imposed by carbon policies. By implementing strategies like carbon cost policies and gradual emission tax policies, businesses can navigate through these challenges while maintaining efficient supply chains.

Cheng et al. [11] explored how different types of supply chains-like traditional retail versus digital-adapt to carbon regulations. They investigated the implications of carbon cap regulations on both centralized and decentralized supply chain channels. By analyzing various scenarios, they aimed to understand how these regulations impact the management and flow of goods in different supply chain contexts.

The studies [12, 13] delved into innovative strategies for reducing emissions while managing inventory effectively. They experimented with approaches such as cap-and-trade initiatives and carbon offset strategies to find optimal production and inventory policies. By exploring a range of emission reduction strategies, these studies shed light on practical solutions for businesses striving to balance cost-effectiveness with environmental responsibility.

The studies [14, 15] focused on optimizing inventory management practices to align with sustainability goals. They developed mathematical models to assess the impact of carbon policies, including carbon taxes and cap-and-trade regimes, on supply chain operations. By integrating environmental considerations into inventory management decisions, these studies highlighted pathways for businesses to achieve both economic and environmental objectives.

Lastly, Hua et al. [16] examined the financial implications of carbon regulations on trade credit-a crucial aspect of business transactions. Their study investigated how carbon constraints influence optimal trade credit preferences under different policy scenarios. By understanding the financial dynamics shaped by carbon rules, businesses can adapt their financial strategies to mitigate risks and capitalize on opportunities in a carbon-constrained environment.

As we reflect on the breadth of research in this area, it's evident that managing inventory under carbon regulations poses multifaceted challenges and opportunities for businesses. The main findings of these researchers are summarized in Table 1, highlighting key insights into how various carbon policies impact inventory management practices. In the subsequent sections, we'll delve deeper into the gaps identified in the existing literature and outline the research objectives aimed at addressing these gaps comprehensively.

Based on the study of the literature, it is possible to conclude that in recent decades, many researchers have included different carbon policies in enhancing the inventory replenishment model. Nevertheless, many researchers have concentrated on a single carbon policy, with just a few quantitative articles considering many policies at the same time. Additionally, research investigating various carbon policies has mainly focused on deterministic demand. Until now, no analysis has evaluated several carbon policies concurrently under a finite scheduling horizon for uneven replenishment cycle length. Demand is influenced by both time and inventory levels.

The lack of research on inventory modeling that examines demand that is influenced by both time and inventory levels in the context of two significant carbon regulations over a finite planning horizon is examined in this research study. Earlier literature has not investigated these policies under an FPH, making this study the first to present such approaches. As a result, the research aims to cover a gap by providing a novel approach to gain new insights.

Table 1. Comparison of literature review

The formulation of carbon emissions is essential for accurately quantifying the environmental impact of the activities under investigation. We have carefully selected specific terms and defined parameters to ensure the robustness and relevance of our carbon emissions model. The terms included in the carbon emissions formulation are derived from established models and equations widely recognized in the field of environmental science and sustainability. These terms reflect various factors influencing carbon emissions, such as production processes, transportation methods, and energy consumption.

Furthermore, the parameters defining these terms are chosen to capture the essential aspects of the system under study. We have based our parameter definitions on empirical data, theoretical models, and industry standards to ensure accuracy and reliability. Specifically, the parameters are sourced from authoritative literature, including works [6, 20], among others, which provide valuable insights into carbon emissions estimation methodologies.

The shifts in the quantity of stock $I_{n_{(i+1)}}(t)$ over time can be derived from the solution to the given mathematically differential Eq. (1) below, which corresponds to the (i+1)^th replenishment cycle:

$I_{n_{(i+1)}}(t)=e^{-\theta t} \int_t^{t_{i+1}}(a+b u) e^{\theta * u} \mathrm{dt}$               (1)

where, $t_i<\mathrm{t}<t_{i+1}$

$\begin{gathered}I_{n_{(i+1)}}(t)=0 \\ Q_{i+1}=I_{i+1}\left(t_i\right)=\int_{t_{i+1}}^{t_i}(a+b u) e^{\theta *(u-t)} \mathrm{dt} \\ I_{n(i+1)}(t)=\int_t^{t_{i+1}}(a+b u) e^{\theta(u-t)} \mathrm{du}\end{gathered}$                (2)

$\begin{aligned} & I_{n_{(i+1)}}(t)=\left[\frac{(a+b u) e^{\theta(u-t)}}{\theta}-\frac{b}{\theta^2} e^{\theta(u-t)}\right]_t^{t_{i+1}} \\ & I_{n_{(i+1)}}(t)=\frac{\left(a+b t_{i+1}\right) e^{\theta\left(t_{i+1}-t\right)}}{\theta}-\frac{b}{\theta^2} e^{\theta\left(t_{i+1}-t\right)} \frac{(a+b t)}{\theta}+\frac{b}{\theta^2}\end{aligned}$ 

 The order quantity for i^th cycles

$\begin{gathered}Q_{i+1}=I_{i+1}\left(t_i\right)=\int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(t-t_i\right)} \mathrm{dt} \\ Q_{i+1}=I_{i+1}\left(t_i\right)=\left[\frac{(a+b u) e^{\theta\left(u-t_i\right)}}{\theta}-\frac{b}{\theta^2} e^{\theta\left(u-t_i\right)}\right]_{t_i}^{t_{i+1}}\end{gathered}$

 $\begin{gathered}Q_{i+1}=I_{i+1}\left(t_i\right)=\frac{\left(a+b t_{i+1}\right) e^{\theta\left(t_{i+1}-t_i\right)}}{\theta}- \frac{b}{\theta^2} e^{\theta\left(t_{i+1}-t_i\right)}-\frac{\left(a+b t_i\right)}{\theta}+\frac{b}{\theta^2}\end{gathered}$            (3)

According to the studies [6, 9], the proposed study includes the $\hat{c}$ fixed quantity emission, $\hat{P}$ connected with inventory replenishment, and $\widehat{h_r}$ associated with inventory holding or management (refrigeration effect).

$A m t_{\text {carbon }}=\hat{c}+\hat{P}_r * Q_{i+1}+\widehat{h_r} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b u) e^{\theta(u-t)} d u \, d t$

4.1 Decentralized case

In decentralized decision-making within the supply chain, each entity, including suppliers, manufacturers, and distributors, operates autonomously, making decisions independently without coordination. This lack of coordination means that each party pursues its own interests and objectives without considering the broader implications for the entire supply chain. Consequently, the total cost incurred in the decentralized supply chain encompasses several components, including ordering cost, holding cost, purchasing cost, and the cost of carbon tax.

$\begin{gathered}\mathrm{Tc}_{\mathrm{Ret}}\left(n_1, t_i\right)=n_1 * O_r+H_r \sum_{i=0}^{n_1} \int_{t_i}^{t_{i+1}} I_{n_{(i+1)}}(t) \mathrm{dt}+ P_r \sum_{i=1}^{n_1} Q_{i+1}+\sum_{i=0}^{n_m} \tau\left(\hat{c}+\hat{P}_r *\right. \left.Q_{i+1}+\hat{h}_r \int_{t_i}^{t_{i+1}} I_{n(i+1)}(t) \mathrm{dt}\right)\end{gathered}$                    (4)

$\begin{gathered}T c_{R e t}\left(n_1, t_i\right)=n_1 * O_r+H_r \sum_{i=1}^{n_1} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+ \\ b u) e^{\theta(u-t)} \mathrm{du} d t+P_r \sum_{i=1}^{n_1} \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(t-t_i\right)} \mathrm{dt}+ \\ \sum_{i=0}^{n_1-1} \tau\left(\hat{c}+\hat{P}_r * \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(t-t_i\right)} \mathrm{dt}+\widehat{h_r} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+\right.  \left.b u) e^{\theta(u-t)} du \,\, dt\right)\end{gathered}$

$\begin{gathered}T c_{\text {Ret }}\left(n_1, t_i\right)=n_1 * O_r+\left(H_r+\tau * \widehat{h_r}\right) \sum_{i=1}^{n_1} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+ \\ b u) e^{\theta(u-t)} \mathrm{du}\, d t+\tau * \hat{c}+\sum_{i=0}^{n_1-1}\left(\left(P_r+\hat{P}_r * \tau\right) \int_{t_i}^{t_{i+1}}(a+\right. \\ \left.b t) e^{\theta\left(t-t_i\right)} \mathrm{dt}\right)\end{gathered}$

$\begin{gathered}T c_{R e t}\left(n_1, t_i\right)=n_1 * O_r+\tau * \hat{c}+\left(\frac{H_r+\tau * h_r}{\theta}+\left(P_r+\widehat{P}_r *\right.\right. \\ \tau) \sum_{i=1}^{n_1} \int_{t_i}^{t_i}(a+\mathrm{bt}) e^{\theta\left(t-t_i\right)} \mathrm{dt}-\frac{H_r+\tau * \hat{h}_r}{\theta}\left(a * H+\frac{1}{2} * b^* H^2\right)\end{gathered}$                    (5)

where, $\mathrm{H}=t_{n_1}=t_{i+1}-t_i$

$T c_{\text {Sup }}\left(n_1, t_i\right)=n_1^* * S_r+\sum_{i=0}^{n^*-1} C_p * Q_{i+1}^*$                    (6)

$T c_{S u p}\left(n_1, t_i\right)=n_1{ }^* * S_r+C_p \sum_{i=0}^{n_1{ }^*-1} \int_{t_i}^{t_{i+1}}(a+\mathrm{bt}) e^{\theta\left(t-t_i\right)} \mathrm{dt}$

    $Q_{i+1}=\sum_{i=0}^{Q_{i+1}}=\sum_{i=0}^{n_1^*-1} Q_{i+1}^{n_1-1}(a+b t) e^{\theta\left(t-t_i\right)} \mathrm{dt}$                (7)

   $\begin{gathered}T c_{\text {System }}\left(n_1, t_i\right)=n_1 *\left(O_r+S_r\right)+\tau * \hat{c}+ \\ \left(\frac{H_r+\tau * \hat{h}_r}{\theta}+\left(P_r+\hat{P}_r * \tau+C_p\right)\right) \sum_{i=1}^{n_1} \int_{t_i}^{t_{i+1}}(a+ \\ b t) e^{\theta\left(t-t_i\right)} d t-\frac{H_r+\tau * \hat{h}_r}{\theta}\left(a^* H+\frac{1}{2} * b^* H^2\right)\end{gathered}$                 (8)

   $\begin{gathered}\frac{\partial}{\partial t i} t_{t_i} \mathrm{Tc}_{\text {systend }}\left(n_1, t_i\right)=\left(\frac{H_r+\tau^* \hat{h}_r}{\theta}+\left(P_r+\right.\right. \\ \left.\left.\hat{P}_r^* \tau\right)\right)\left\{\left(a+b t_i\right)\left(e^{\theta\left(t_i-t_{-1}\right)}-1\right)-\theta \int_{t_i}^{t_{i+1}}(a+\right. \\ \left.b t) e^{\theta\left(t-t_i\right)} d t\right\}\end{gathered}$                 (9)

    $\begin{gathered}\frac{\partial}{\partial t_i} \mathrm{Tc}_{\text {system }}\left(n_1, t_i\right)=\left(\frac{H_r+\tau * \hat{h}_r}{\theta}+\left(P_r+\hat{P}_r *\right.\right. \\ \tau))\left\{\left(a+\mathrm{bt}_i\right)\left(e^{\theta\left(t_i-t_{i-1}\right)}\right)-(a+\right. \\ \left.\left.\mathrm{bt}_{i+1}\right) e^{\theta\left(t_{i+1}-t_i\right)}+\frac{b}{\theta} e^{\theta\left(t_{i+1}-t_i\right)}-\frac{b}{\theta}\right\}\end{gathered}$               (10)

4.2 Centralized case

Decisions in centralised supply chain management scenarios are made by collaborative efforts that benefit the entire system. As a result, we're looking into introducing a emissions trading scheme to lower the overall cost of the supply chain system as a whole. Firms that emit low amounts of carbon can sell their carbon credits to firms that emit large levels of carbon under this system of trading. The emissions trading programme is not only ecologically and environmentally, but also environmentally and socially conscious.

Optimal cycles for replenishment, as determined through analysis, is at n₁=4. Once the minimum threshold has been reached at n₁=4. Subsequently, it gradually ascends in subsequent cycles. Similarly, for the centralized (cap and trade) case, total cost of system is 112.21$ which reach its optimal level at n₁=4. Tables 2 and 3 exhibit a convex pattern of the system’s cost function. This observation is further supported by graphical representations.

$\begin{gathered}T c_{\text {System }}\left(n_1, t_i\right)=n_1 *\left(O_r+S_r\right)+H_r \sum_{i=1}^{n_1} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+ \\ b u) e^{\theta(u-t)} \mathrm{du} d t+P_r \sum_{i=1}^{n_1} \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(t-t_i\right)} \mathrm{dt}+ \\ \sum_{i=0}^{n_1-1} \delta\left(\hat{c}+\hat{P}_r * \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(t-t_i\right)} \mathrm{dt}+\widehat{h_r} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+\right. \\ \left.b u) e^{\theta(u-t)} d u d t-CO_{2_{C a p}}\right)\end{gathered}$

$\begin{gathered}T c_{\text {System }}\left(n_1, t_i\right)=n_1 *\left(O_r+S_r\right)+\delta * \hat{c}+ \\ \left(\frac{H_r+\delta * \hat{h}_r}{\theta}+\left(P_r+\widehat{P}_r * \delta+c_p\right)\right) \sum_{i=1}^{n_1} \int_{t_i}^{t_{i+1}}(a+ \\ \text { bt) } e^{\theta\left(t-t_i\right)} \mathrm{dt}-\frac{H_r+\delta * \hat{h}_r}{\theta}\left(a * H+\frac{1}{2} * b^* H^2\right)-\delta * \\ \operatorname{CO}_{2_{\text {cap }}}\end{gathered}$             (11)

Preposition 1:

$\left(a+b t_{i+1}\right) e^{\theta\left(\mathrm{T}_{i+1}\right)}<b\left(e^{\theta\left(\mathrm{T}_i\right)}-1\right) / \theta+(a+b t) e^{\theta\left(\mathrm{T}_i\right)}$

Proof: $F\left(t_i\right)-F\left(t_{i+1}\right)<\frac{F^{\prime}\left(t_i\right)}{F\left(t_i\right)} \int_{t_i}^{t_{i+1}} F(t) d t$

Let $\mathrm{F}(\mathrm{t})=(a+b t) e^{\theta\left(\mathrm{t}-\mathrm{t}_i\right)}$ is a log convex function [21]. By putting the value of F(t) in the above equation. we have

$\begin{aligned} & \left(a+b t_{i+1}\right) e^{\theta\left(t_{i+1}-t_i\right)}-\left(a+b t_i\right)<\left(\frac{b}{\left(a+b t_i\right)}+\right. \\ & \theta) \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(t-t_i\right)} d t \\ & \frac{\partial T c_{\text {system }}\left(n_1, t_i\right)}{\partial t_i}=\left(a+b t_i\right)\left(e^{\theta\left(t_i-t_{i-1}\right)}-1\right) \\ & -\theta \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(t-t_i\right)} d t=0 \\ & \end{aligned}$

By Eq. (10) we have

$\begin{gathered}\frac{\left(a+b t_i\right)\left(e^{\theta\left(\mathrm{t}_i-t_{i-1}\right)}-1\right)}{\theta}=\int_{t_i}^{t_{i+1}}(a+b t) e^{\theta\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t \\ \left(a+b t_{i+1}\right) e^{\theta\left(T_{i+1}\right)}-\left(a+b t_i\right)<\left(\frac{b}{\left(a+b t_i\right)}+\theta\right) \\ \frac{\left(a+b t_i\right) e^{\theta\left(T_i\right)}-\left(a+b t_i\right)}{\theta} \\ \left(a+b t_{i+1}\right) e^{\theta\left(\mathrm{T}_{i+1}\right)}<\frac{b\left(e^{\theta\left(\mathrm{T}_i\right)}-1\right)}{\theta}+\left(a+b t_i\right) e^{\theta\left(\mathrm{T}_i\right)}\end{gathered}$

Lemma1:

$t_i$ strictly monotonic increase function of the last replenishment cycle $t_n$ where i=1,2,3…………. n₁-1.

Proof: See Appendix A.

This lemma highlights the connection between the replenishment time, last replenishment time, length of replenishment time, and the time horizon.

$T_{i+1}=t_{i+1}-t_i$ and $t_n=\mathrm{H}-T_n$

Theorem 1: The optimal replenishment period for a fixed replenishment cycle is the unique solution that exists for the nonlinear system represented by Eq. (8).

The Hessian matrix obtained by the partial differentiation of $T c_{\text {System }}\left(n_1, t_i\right)$ it is necessary for it to be positive definite for t_i to be minimum for a fixed n.

As a result, see Appendix B, the theorem establishes that $T c_{\text {System }}\left(n_1, t_i\right)$ is positive definite. Therefore, the optimum value of t_i obtained using numerical iterative technique for a given fixed positive integer with mathematical programs constructed by Mathematica software version-12.0. Based on the optimal value of t_i, Total cost function also will minimize.

Theorem 2: For a finite horizon planning H, the number of replenishment cycles exhibits convex behaviour of the function $T c_{\text {System }}\left(n_1, t_i\right)$.

Proof: See Appendix C.

6.1 Parametric analysis in the case of cap-and-trade mechanisms

We focused to conduct sensitivity studies for several factors under the cap-and-trade framework in the discussion below. Our discussion has been confined to two factors that are directly connected, namely the carbon cap (C) and the rate of trade credit $\delta$. These variables have major impacts on the number of replenishment cycles, replenishment time, carbon emissions, and total cost.

Total cost, total amount of carbon, no replenishment cycles, and total quantity with changing $\delta$

Figures 4 and 5, and Table 4 demonstrate the relationship between the $\delta$ the order quantity, number of replenishment cycles, overall system cost, and amount of carbon emission. According to the results shown, increasing $\delta$ leads to a substantial increase in the order quantity and the count of replenishment cycles, resulting in a decrease in the overall the cost of the system and the level of carbon emissions.

Total cost, total amount of carbon, no replenishment cycles, and total quantity with changing ${{CO}_2}_{Cap}$

Table 5 demonstrates that increasing the carbon cap increases the order quantity, and the number of replenishment cycles, and decreases the overall system expenses and the quantity of carbon discharge. (As seen in Figures 6 and 7) because of adjustment on provided cap. It means that decision making agencies cannot directly reduce emissions by establishing more rigorous carbon restrictions, but it may dissuade enterprises from emitting more carbon by causing considerable cost increases.

A strict cap raises the economic burden on the organization, but if the emission cap is freely allocated, the organization can minimize the overall cost by selling an unutilized quota of carbon. Figures 6 and 7 show the shifting trends of order quantity, replenishment number, the aggregate system expenditure and emission quantity.

Total cost, total amount of carbon, no replenishment cycles, and total quantity with changing $\boldsymbol\tau$

Table 6 presents the results of sensitivity analysis conducted by varying the tax paid on each unit of carbon emitted (τ). As the tax rate increases from 0.020 to 0.024, we observe changes in the optimal replenishment time (ti), the number of replenishment cycles (n1), the replenishment quantity (Qnt), and the total system cost (Tc_System).

With a tax rate of 0.020, the optimal replenishment time is distributed over eight periods, resulting in a total system cost of $150.47. As the tax rate increases to 0.021 and 0.022, the optimal replenishment time decreases, leading to a reduction in the number of replenishment cycles and the total system cost. However, beyond a tax rate of 0.022, further increases in the tax rate lead to a significant decrease in the optimal replenishment time and the total system cost.

The sensitivity analysis reveals that higher carbon taxes drive firms to optimize replenishment strategies, reducing emissions and minimizing tax burdens while maintaining efficiency. Aligning supply chain decisions with carbon regulations is crucial for cost savings and environmental sustainability.

4.png

Figure 4. The effect of change in $\delta$ on overall optimal total cost and amount of total emission of the system

5.png

Figure 5. The effect of change in $\delta$ on overall optimal ordered quantity and total number of restocking operations conducted of the system

Table 4. The most economical and optimal system cost, number of replenishment cycles, amount of emission and replenishment quantity have been calculated with changing $\delta$

Table 5. The most economical and optimal system cost, total number of replenishment cycles carried out, amount of emission and amount for replenishment have been calculated with changing carbon cap

Table 6. The most economical and optimal system cost, total number of replenishment cycles carried out, amount of emission and amount for replenishment have been calculated with changing tax paid on each unit of carbon emitted

6.png

Figure 6. The impact of change in cap on overall best possible total cost and amount of total emission of the system

7.png

Figure 7. The effect of altering in emission cap on overall best possible ordered quantity and no of replenishment cycles of the system

The sensitivity analysis results offer valuable insights into how variations in parameters impact key aspects of the system under the cap-and-trade framework. By examining the effects of changes in variables such as the carbon cap and trade credit rate, we gain a deeper understanding of their influence on critical performance metrics such as system cost, carbon emissions, replenishment cycles, and order quantities.

These insights can help decision-makers optimize their strategies for managing carbon emissions while balancing economic considerations. For example, our analysis may reveal trade-offs between minimizing system costs and reducing environmental impact, highlighting the importance of carefully selecting parameter values to achieve desired outcomes.

In comparison to past studies, our findings may align with established trends or provide new perspectives on the dynamics of carbon policies and inventory management. By identifying similarities and differences, we can enrich our understanding of the factors driving system behavior and inform future research directions.

Overall, the sensitivity analysis results contribute valuable insights that can enhance decision-making and policy formulation in the context of carbon management and inventory optimization.