Trade Credit and Preservation Technologies: An Inventory Replenishment Model for a Sustainable Supply Chain

In today's competitive environment, awareness of sustainable supply chain management is rapidly growing. Sustainable development is crucial for the enterprise's progress. Global climate is getting worse due to carbon emissions. The emphasis of the world today is on environmental issues because of the large increase in CO₂ emissions led by global industrialization. Most developed nations are actively working to reduce carbon emissions through implementing innovative technology and regulatory measures such as carbon taxes and cap policies [1]. To reduce global warming and avert its disastrous impacts on mankind, the Kyoto Protocol, which first looked into effect on February 16, 2005, established emission caps for developed countries. This protocol proposed three adaptable emission reduction approaches. Enterprises are granted the right to emit or discharge a certain amount of the designated pollutant in the form of emissions permits (emission caps) by a central authority or governmental agency. Once the limit or cap on emissions has been set, they will be taxed for any additional emissions [2, 3].

In a supply chain, collaboration enables individuals to generate and benefit from increased profits, lower costs, and reduced carbon emissions. This collaboration may be influenced by the trade credit period, a vital component of the industry. In today’s more competitive environment, both retailers and suppliers can profit through the trade credit period. Suppliers should offer a certain credit period (permissible delay) to retailers, and in return, they can enhance their market sales and generate additional profit. Furthermore, retailers will never face the problem of being out of stock; they can even earn interest during the delay in payment. Collaboration in supply chains has emerged as a key strategy for enhancing profitability, reducing costs, and mitigating carbon emissions.

Nowadays, organizations and governments emphasize the importance of building and operating supply chains with a specific focus on reducing carbon emissions. Therefore, it is necessary to adopt emission-reduction technologies, such as carbon offset, while considering supply chain emission trading regulations in the study [4].

As we are aware, everything gradually undergoes deterioration over time. Pospı́šil et al. [5] emphasizes the commercial and economic significance of various materials, including metals, polymer blends, and organic biomaterials. This underscores the importance of investigating their deteriorating behavior concerning emission concerns. Therefore, we cannot ignore deterioration, which is a significant aspect of daily life. Natural processes, including the deterioration of goods, lead to a decrease in a product's usefulness.

The majority of deteriorating goods, such as vegetables, flowers, and metals, gradually decline with time. Deterioration of materials can be controlled through preservation techniques. Additionally, investments in green technology can be made to lower carbon emissions. Furthermore, by investing in high-quality and fast-moving preservation approaches (PRA) and green approaches (GRA), inventory degradation and greenhouse gas emissions can be reduced [6].

Define the research gap and research problem

While existing research has considered models with certain features, such as demand dependence on time, pricing, and trade credit duration, this study addresses a significant gap. Several researchers have developed models incorporating features such as demand dependence on time, price, and trade credit duration, along with pricing with default risk, backorders, green approaches (carbon offset), preservation approaches, and different payment options in the study [7].

Most researchers have focused on developing inventory models with various parameters under infinite planning. However, the investigation of demand dependence on time with carbon emission policies, preservation technology under a finite planning horizon, and different replenishment cycle times has not been explored in previous studies. The purpose of the proposed research is to develop a model that considers preservation and carbon emission reduction policies for planning scenarios with finite planning horizons and unequal replenishment lengths—a unique approach not explored in previous studies.

The subsequent sections of this research article are organized as follows: Section 2 reviews the literature, identifying a research gap; Section 3 describes the assumptions and abbreviations/notations. Section 4 discusses the derivation of the mathematical model, while Section 5 presents a numerical example, examines the findings, and analyzes the sensitivity analysis. Finally, Section 6 explains the study's conclusion, along with minor drawbacks, managerial implications, and innovative future suggestions.

Before the preliminary stock level reaches zero, the retailer places an order for replenishment stock with the supplier. The retailer's purchase will be instantly replenished by the supplier. Therefore, no shortages or lost sales are considered in this study. The following differential equation will represent the change in stock levels during the (i+1)th cycle:

$\frac{\left(I L_{i+1}(t)\right)}{d t}=-D(t)-(1-P(\Psi)) \theta$  (1)

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

 $I L_{i+1}(t)=e^{-\theta(1-P(\Psi)) t} \int_t^{t_{i+1}} D(t) e^{\theta(1-P(\Psi)) u} d u$    (2)

where, Boundary conditions are given below.

$\begin{aligned} & \mathrm{IL}_{\mathrm{i}+1}\left(\mathrm{t}_{\mathrm{i}+1}\right)=0 \text { and } \mathrm{IL}_{\mathrm{i}+1}\left(\mathrm{t}_{\mathrm{i}}\right)=\mathrm{OQ}_{\mathrm{i}+1} \\ & \quad Q_{i+1}=I L_{i+1}\left(t_i\right) \\ & \quad=e^{-\theta(1-P(\Psi)) t_i} \int_{t_i}^{t_{i+1}} D(t, M) e^{\theta(1-P(\Psi)) t} d t \\ & Q_{i+1}=I L_{i+1}\left(t_i\right)=\int_{t_i}^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))\left(t-t_i\right)} d t\end{aligned}$   (3)

The cost of replenishment of an order is

$\mathrm{n} \times O_r$   (4)

Purchasing cost: $\sum_{i=0}^{n-1} P_r \times Q_{i+1}$

$\sum_{i=0}^{n-1} P_r \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))\left(t-t_i\right)} d t$   (5)

Cost of hold and stock:

$\begin{gathered}\sum_{i=0}^{n-1} h_r \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}} D(t) e^{\theta(1-P(\Psi))(u-t)} d u\,\, d t \\ \sum_{i=0}^{n-1} h_r \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b u) e^{\theta(1-P(\Psi))(u-t)} d u \,\,d t\end{gathered}$     (6)

Cost of deteriorating inventory:

$\begin{gathered}(1-P(\Psi)) \sum_{i=0}^{n-1} \theta d_r \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a \\ +b t) e^{\theta(1-P(\Psi))(u-t)} d u d t \\ e^{-\lambda \Psi} \sum_{i=0}^{n-1} \theta d_r \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))(u-t)} d u d t\end{gathered}$   (7)

The implementation of green technology is crucial for achieving sustainability and reducing carbon emissions, as emphasized by the United Nations Environment Programme (UNEP). Carbon offsets serve as compensation for emissions rather than a replacement. The variable ĉ represents the fixed carbon emissions related to order placement, including transportation emissions. Meanwhile, the dynamic carbon emissions for each unit ordered are denoted by $\hat{P}_r$, and the carbon emissions associated with refrigeration during warehousing are represented by $\widehat{h}_r$. These concepts have been explored in various studies [31-35]. Thus, the following equation calculates the total carbon emissions for each replenishment cycle:

Amount of carbon emission during holding, placing an order and transportation:

$\begin{aligned} & C e=\sum_{i=0}^{n-1} \hat{c}+\hat{P}_r * Q_{i+1} \\ & +\widehat{h_r} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))(u-t)} d u d t\end{aligned}$

The following presented for investment in the green approach.

$\left(1-\emptyset\left(1-e^{-m G}\right)\right) C e$

The per year carbon emission cost for a cycle is $C T=$ $\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) C e$

$\begin{gathered}=\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \sum_{i=0}^{n-1} c^{\wedge}+\hat{P}_r * Q_{i+1} \\ +\widehat{h_r} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a \\ +b t) e^{\theta(1-P(\Psi))(u-t)} d u d t\end{gathered}$ (8)

In general, buyers adhere to market norms by acquiring goods or materials from suppliers and, in return, make payments. Trade Credit periods are commonly provided by a significant number of businesses. In such arrangements, the supplier extends a period to the retailer, allowing a permissible delay for payment. This practice ensures that neither the supplier nor the retailer incurs losses. Additionally, suppliers may incentivize early payments by offering various discounts. Consequently, the retailer is granted a credit period $M_{i+1}=\delta\left(t_{i+1}\right)$ by the seller. The supplier approves trade credit for the retailer based on multiple criteria, including the order amount placed, creditworthiness, previous records, the nature of the items, and the history of cooperation between the supplier and retailer. The duration of credit is often extended based on the order amount, with larger orders resulting in a more extended credit period.

3.1 First case

Now, we consider two cases: first, where $M_{i+1}$ lies within the cycle length (t_i, t_i+1), indicating that the credit duration provided by the seller does not exceed the inventory replenishment length $T_{i+1}$. In this scenario, the retailer accrues interest on their sales revenue up to the credit duration length, while also incurring interest charges on items already stocked.

Referring to Figure 1, the interest charges $M_{i+1}$ should be less than or equal to $T_{i+1}$ like the $M_{i+1} \leq T_{i+1}$. In this case $M_{i+1}=\delta\left(t_{i+1}\right)$ lies into the interval $t_i \leq \mathrm{t} \leq t_{i+1}$.       

tu_pian_1.jpg

Figure 1. Inventory model diagram for $M_{i+1} \leq T_{i+1}$

So, interest earned by the retailer is as follows:

$\sum_{i=0}^{n-1} I_e * s \int_{t_i}^{t_i+\delta\left(t_{i+1}-t_i\right)}(a+b t)\left[t_i+\delta\left(t_{i+1}-t_i\right)-t]dt\right.$  (9)

Interest payable by the retailer is given by:

$\begin{gathered}\sum_{i=0}^{n-1} I_c * W \int_{t_i+\left(t_{i+1}-t_i\right) \delta}^{t_{i+1}}(a+b t)\left[t-t_i-\left(t_{i+1}\right.\right.  \left.\left.-t_i\right) \delta\right] d t\end{gathered}$   (10)

The total cost of the retailer is given below:

Cost of placing an order + holding cost + purchasing cost + carbon preservation cost + deterioration preservation technology cost + Interest charges - Interest Earned.

$\begin{gathered}T_{R e t}=\mathrm{n} * O_r+\sum_{i=0}^{n-1} h_r \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b u) e^{\theta(1-P(\Psi))(u-t)} d u d t+\sum_{i=0}^{n-1} P_r * Q_{i+1} \\ +\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \sum_{i=0}^{n-1}\left(c^{\wedge}+\hat{P}_r * Q_{i+1}+\widehat{h_r} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b u) e^{\theta(1-P(\Psi))(u-t)} d u d t\right) \\ +e^{-\lambda \Psi} \sum_{i=0}^{n-1} \theta d_r \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b u) e^{\theta(1-P(\Psi))(u-t)} d u d t \\ +\sum_{i=0}^{n-1} I_e * s \int_{t_i}^{t_i+\left(t_{i+1}-t_i\right) \delta}(a+b t)\left[t_i+\left(t_{i+1}-t_i\right) \delta-t\right] d t \\ -\sum_{i=0}^{n-1} I_c * W \int_{t_i+\delta\left(t_{i+1}-t_i\right)}^{t_{i+1}}(a+b t)\left[t-t_i-\left(t_{i+1}-t_i\right) \delta\right] d t\end{gathered}$

$\begin{gathered}T_{R e t}=\mathrm{n} * O_r+\sum_{i=0}^{n-1} h_r \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b u) e^{\theta(1-P(\Psi))(u-t)} d u d t \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \sum_{i=0}^{n-1} c^{\wedge}+\hat{P}_r * Q_{i+1} \\ +\widehat{t_{i+1}} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}}(a+b u) e^{\theta(1-P(\Psi))(u-t)} d u d t+e^{-\lambda \Psi} \sum_{i=0}^{n-1} \theta d_r \int_{t_i}^{t_{i+1}} \int_{t}^{t_{i+1}}(a+b u) e^{\theta(1-P(\Psi))(u-t)} d u d t \\ +\sum_{i=0}^{n-1} s * I_e \int_{t_i}^{t_i+\left(t_{i+1}-t_i\right) \delta}(a+b t)\left[t_i+\left(t_{i+1}-t_i\right) \delta-t\right] d t-\sum_{i=0}^{n-1} I_c * W \int^{t_{i+1}}_{t_i+\left(t_{i+1}-t_i\right)\delta}(a+b t)\left[t-t_i-\left(t_{i+1}-t_i\right) \delta\right] d t\end{gathered}$

$\begin{aligned} T_{R e t}=\mathrm{n} * O_r+ & \sum_{i=0}^n\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h_r}}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\} \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))\left(t-t_i\right)} d t \\ & +\sum_{i=0}^n\left\{P_r+\hat{P}_r * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right)\right\} \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))\left(t-t_i\right)} d t+c^{\wedge} * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \\ & -\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h_r}}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}\left(a H+0.5 * b * H^2\right) \\ & +\sum_{i=0}^{n-1} s * I_e \int_{t_i}^{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}(a+b t)\left[t_i+\left(\delta * t_{i+1}-\delta * t_i\right)-t\right] d t \\ & -\sum_{i=0}^{n-1} I_c * W \int_{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}^{t_{i+1}}(a+b t)\left[t-t_i-\left(\delta * t_{i+1}-\delta * t_i\right)\right] d t\end{aligned}$

$\begin{aligned} {\left[T_{R e t}=\mathrm{n} * O_r+\right.} & \sum_{i=0}^n\left(\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h_r}}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}+\left\{P_r+\hat{P}_r * €\left(1-\emptyset\left(1-e^{-m G}\right)\right)\right\}\right) \int_{t_i}^{t_{i+1}}(a \\ & +b t) e^{\theta(1-P(\Psi))\left(t-t_i\right)} d t+c^{\wedge} * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \\ & -\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h_r}}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}\left(a H+0.5 * b * H^2\right) \\ & +\sum_{i=0}^{n-1} s * I_e \int_{t_i}^{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}(a+b t)\left[t_i+\left(\delta * t_{i+1}-\delta * t_i\right)-t\right] d t \\ & \left.-\sum_{i=0}^{n-1} I_c * W \int_{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}^{t_{i+1}}(a+b t)\left[t-t_i-\left(\delta * t_{i+1}-\delta * t_i\right)\right] d t\right]\end{aligned}$

$\begin{aligned} & \frac{\partial\left(T_{R e t}\right)}{\partial t_i}=\left(\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h_r}}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}+\left\{P_r+\hat{P}_r * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right)\right\}\right)((a \\ & \left.\left.+b t_i\right)\left(e^{\theta(1-P(\Psi))\left(\mathrm{t}_i-t_{i-1}\right)}-1\right)-\theta(1-P(\Psi)) \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t\right)+s \\ & * I_e \int_{t_i}^{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}(a+b t)(1-\delta) d t-\left(a+b t_i\right)\left(\delta * t_{i+1}-\delta * t_i\right)+ \\ & \text { * } W \int_{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}^{t_{i+1}}(a+b t)(1-\delta) d t \\ & \end{aligned}$ (11)

$\begin{aligned} & \frac{\partial\left(T_{R e t}\right)}{\partial t_i}=\left(\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h}_r}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}\right. \\ & \left.+\left\{P_r+\hat{P}_r * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right)\right\}\right)\left(\left(a+b t_i\right)\left(e^{\theta\left(\mathrm{t}_i-t_{i-1}\right)(1-P(\Psi))}-1\right)+\theta(P(\Psi)\right. \\ & \left.\text {-1) } \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t\right)+s \\ & * I_e\left\{\int_{t_{i-1}}^{t_{i-1}+\left(\delta * t_{i+1}-\delta * t_i\right)}(a+b t) \delta d t+\int_{t_i}^{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}(a+b t)(1-\delta) d t-(a\right. \\ & \left.\left.+b t_i\right)\left(\delta * t_{i+1}-\delta * t_i\right)\right\}-I_c \\ & \text { * W\{ } \int_{t_{i-1}+\left(\delta * t_i-\delta * t_{i-1}\right)}^{t_i}(a+b t) \delta d t+\int_{t_i+\left(\delta * t_{i+1}-\delta * t_i\right)}^{t_{i+1}}(a+b t)(1-\delta) d t-\left(a+b t_i\right)(1 \\ & \left.-\delta)\left[t_i-t_{i-1}\right]\right\} \\ & \end{aligned}$   (12)

By taking a partial derivative of $T_{\text {Ret }}$ w.r.t to $t_i$ equal to zero, we can find the value of $t_i$.

$\frac{\partial\left(T_{R e t}\right)}{\partial t_i}=0$

3.2 Second case

When the credit duration is higher than the inventory replenishment length $T_{i+1} . M_{i+1}$ lies outside the cycle length $\left(\mathrm{t}_{\mathrm{i}}, \mathrm{t}_{\mathrm{i}+1}\right)$. In the second case, we assume, $M_{i+1} \geq T_{i+1}$. In this scenario, $M_{i+1}$ lies outside the interval as illustrated in Figure 2 , where $t_i \leq \mathrm{t} \leq t_{i+1}$ represents the time interval. Therefore, the interest earned by the retailer is as follows:      

$\sum_{i=0}^{n-1} I_e * s \int_{t_i}^{t_{i+1}}(a+b t)\left[t_i+\delta\left(t_{i+1}-t_i\right)-t\right] d t$   (13)

tu_pian_2.jpg

Figure 2. Inventory model diagram for $M_{i+1} \geq T_{i+1}$

$\begin{aligned} T_{R e t}=\mathrm{n} * O_r+ & \sum_{i=0}^n\left(\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h_r}}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}+\left\{P_r+\hat{P}_r * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right)\right\}\right) \int_{t_i}^{t_{i+1}}(a \\ & +b t) e^{\theta(1-P(\Psi))\left(t-t_i\right)} d t++\hat{c} * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \\ & -\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h_r}}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}\left(a H+0.5 * b * H^2\right) \\ & +\sum_{i=0}^{n-1} s * I_e \int_{t_i}^{t_{i+1}}(a+b * t)\left[t_i+\left(\delta * t_{i+1}-t_i * \delta\right)-t\right] d t\end{aligned}$

$\begin{aligned} & \frac{\partial\left(T_{R e t}\right)}{\partial t_i}=\left(\left\{\frac{h_r}{\theta(1-P(\Psi))}+\frac{\tau\left(1-\emptyset\left(1-e^{-m G}\right)\right) \widehat{h}_r}{\theta(1-P(\Psi))}+d_r e^{-\lambda \Psi}\right\}\right. \\ &\left.+\left\{P_r+\hat{P}_r * \tau\left(1-\emptyset\left(1-e^{-m G}\right)\right)\right\}\right)\left(\left(a+b t_i\right)\left(e^{\theta(1-P(\Psi))\left(t_i-t_{i-1}\right)}-1\right)-\theta(1\right. \\ &\left.-P(\Psi)) \int_{t_i}^{t_{i+1}}(a+b t) e^{\theta(1-P(\Psi))\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t\right)+s \\ & * I_e\left\{\int_{t_{i-1}}^{t_i}(a+b t) \delta d t+\int_{t_i}^{t_{i+1}}(a+b t)(1-\delta) d t+\left(a+b t_i\right)\left(t_i-t_{i-1}-\delta * t_i-\delta\right.\right. \\ &\left.\left.* t_{i-1}\right)-\left(a+b t_i\right)\left(t_{i+1}-t_i-\delta * t_{i+1}+\delta * t_i\right)\right\}\end{aligned}$   (14)

By taking a partial derivative of $T_{\text {Ret }}$ w.r.t to $t_i$ equal to zero, we can find the value of $t_i$.

$\frac{\partial\left(T_{R e t}\right)}{\partial t_i}=0$

$T_{\text {Sup }}=n^* * S_r+\sum_{i=0}^{n^*-1} C_p \delta\left(t_{i+1}-t_i\right) I_c * Q_i^*$    (15)

$Q_i=\sum_{i=0}^{n^*-1} Q_i^*$   (16)

Step 1: First of all, set the starting value for all parameters.

$\mathrm{a}=0.0001$ unt., $O_r=20 \$ /$ setup/year, $\mathrm{b}=1000$ units, $\mathrm{M}=$ 0.5 year, $h_r=0.4 \$ /$ unt. /annually, $\tau=0.6 \$ / \mathrm{kg} /$ annually, $\mathrm{m}=$ 0.5 unt., $\varnothing=0.4$ unt., $\theta=4, \alpha=0.02$ unt. $, \Psi=0.5, \lambda=0.8$ unt., $P_r=2 \$$ /unt. $/$ Year, $\mathrm{c}^{\wedge}=10 \mathrm{~kg} /$ year, $\hat{P}_r=40 \mathrm{~kg} /$ order/ annually, $\widehat{h_r}=8 \mathrm{~kg} /$ annually, $I_e=0.08 \$ /$ unt. $/$ Annually, $I_c=0.1 \$ /$ unt. $/$ Year, $\mathrm{s}=50 \$ /$ unt., $\mathrm{S}_{\mathrm{s}}=120 \$ / \mathrm{setup} /$ annually.

Step 2: Find the Root $\left[\frac{\partial\left(T_{R e t}\right)}{\partial t_i}=0\right]$.

Step 3: The final results are optimal solutions $t_i$.

Step 4: Insert optimal values of $t_i$ into the equations to find the value of total cost and order quantity.

5.1 Tables and figures

Figure 3 shows the optimal level of the total cost of the retailer. at n=5 replenishment cycle, the total optimal value of the total cost of the retailer is 49743.41, similarly, for different values of ‘a’, we can see in Table 2 and Table 3. Optimal level of retailer total cost. For the different values of ‘a’ optimal value arrived at the 5th replenishment cycle. Table 3 shows the replenishment time, optimal value of retailer, supplier and order quantity.

Table 2. Total cost for the retailer when $M_{i+1} \leq T_{i+1}$ for five different combinations of ' $\mathrm{a}$ '

$\downarrow \mathbf{a}$	$\rightarrow \mathbf{n}$	1	2	3	4	5	6	7	8
0.001		807408.58	212851.20	101532.77	65367.62	49743.41	74762.86	155208.69	516831.09
0.0008		807408.73	212851.23	101532.79	65367.62	55494.86	93719.13	216591.70	942996.74
0.0009		807408.65	212851.22	101532.78	65367.62	50832.96	83209.24	181501.34	675694.18
0.0011		807408.50	212851.19	101532.76	65367.61	49743.41	67850.95	134956.30	409364.16
0.0012		807408.42	212851.17	101532.76	65367.61	49743.41	62106.30	118989.62	333520.31

Table 3. Replenishment time for $T_{R e t}, T_{\text {sup }}$, and $Q_{n t}$ when $M_{i+1} \leq T_{i+1}$

$\downarrow \mathbf{a}$	$\rightarrow \mathbf{t}_{\mathbf{i}}$	t0		t1	t2	t3	t4	t5	n	$T_{\text {Ret }}$	$T_{\text {sup }}$	Tc	$Q_{n t}$
0.001			0	1.4259	2.0827	2.6329	3.1244	4	5	49743.41	23977.87	73721.28	18092.3
0.0008			0	1.4719	2.1499	2.7178	3.2251	4	5	55494.86	29425.50	84920.36	20168.3
0.0009			0	1.435	2.096	2.6496	3.1443	4	5	50832.96	25604.07	76437.03	18486.1
0.0011			0	1.4259	2.0827	2.6329	3.1244	4	5	49743.41	23085.97	72829.38	18092.3
0.0012			0	1.4259	2.0827	2.6329	3.1244	4	5	49743.41	22324.44	72067.85	18092.3

Table 4. Total cost for retailer for different values of '$\mathrm{a}$ ' when $M_{i+1}>T_{i+1}$

$\downarrow \mathbf{a}$	$\rightarrow \mathbf{n}$	1	2	3	4	5	6	7	8
0.001		762631	179074.2	74765.77	43266.14	30942.01	46862.71	102241.4	388303.8
0.0008		762631.2	179074.2	74765.79	43266.15	34815.61	60016.82	148908.5	744379.7
0.0009		762631.1	179074.2	74765.78	43266.15	31670.30	52660.24	121929.2	525295.8
0.0011		762630.9	179074.2	74765.76	43266.14	30942.01	42201.82	87445.92	298127.1
0.0012		762630.9	179074.1	74765.75	43266.14	30942.01	38389.75	76045.01	236187.3

Table 5. Replenishment time for $T_{R e t}, T_{\text {sup }}$, and $Q_{n t}$ when $M_{i+1}>T_{i+1}$

$\downarrow \mathbf{a}$	$\rightarrow \mathbf{n}$	t0	t1	t2	t3	t4	t5	n	$T_{\text {Ret }}$	$T_{\text {sup }}$	Tc	$Q_{n t}$
0.001		0	1.4259	2.0827	2.6329	3.1244	4	5	30942.01	15428.31	46370.32	18092.29
0.0008		0	1.4719	2.1499	2.7178	3.2251	4	5	34815.61	18896.41	53722.02	20168.36
0.0009		0	1.435	2.096	2.6496	3.1443	4	5	31670.30	16463.58	48133.88	18486.1
0.0011		0	1.4259	2.0827	2.6329	3.1244	4	5	30942.01	14860.51	45802.52	18092.3
0.0012		0	1.4259	2.0827	2.6329	3.1244	4	5	30942.01	14375.68	45317.69	18092.3

Table 6. Represent the following facts of sensitivity analysis for main parameters

Parameters	%Changes	Optimal Replenish Cycle	Total Order Quantity $Q_{n t}$	Retailer's Total Cost  $T_{\text {Ret }}$	Supplier's Total Cost  $T_{\text {sup}}$
a	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	5 5 5 5 5	18092.3 18092.3⁠ 18092.3 20168.3⁠ 18486.1	49743.41 49743.41 49743.41 55494.86 50832.9	22324.44 23977.87 23085.97 29425.50 25604.07
b	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	4 4 4 4 4	28206.8 25856.2 23505.6 21155.1 18804.5	78286.61 71789.39 65292.17 58794.94 52297.7	3218901.1 2950659.02 2682416.88 2794658.59 2575774.22
$\theta$	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	5 5 5 5 5	24276.9 20870.7 18092.2 15812.2 13929.1	68347.01 58057.29 49668.03 42789.90 37118.60	4267545.71 3583507.45 3033108.27 2588149.43 2226602.26
Ob	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	5 6 6 5 5	211.42 211.42 211.42 211.42 211.42	859.78 834.78 809.78 781.85 751.85	1860.18 1860.18 1860.18 1860.18 1860.18
Ss	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	4 4 4 4 4	211.42 211.42 211.42 211.42 211.42	809.78 809.78 809.78 809.78 809.78	723.10 663.10 603.10 543.10 483.10

Table 7. Represent the following facts of sensitivity analysis for the main parameters

Parameters	%Changes	Optimal Replenish Cycle	Total Order Quantity $Q_{n t}$	Retailer's Total Cost  $T_{\text {Ret }}$	Supplier's Total Cost  $T_{\text {sup}}$
a	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	6 6 6 6 6	246961.56 229788.07 214125.64 199814.24 186711.34	290818.70 271067.58 253278.14 237205.04 222635.75	37108.96 32219.83 28003.36 24358.73 21201.25
b	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	4 4 4 4 5	627510476 575278343 523043328 470804689 418864929	186533.33 171006.99 155479.80 1399515.29 124512.83	723033.14 662817.30 602686.98 603342.10 542508.00
$\theta$	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	4 4 4 4 4	221117363 107218244 52304332 25694820 12726308	662621004 320461137 155479805 75525839 36553824	254786691720 123541769573 60268698111 29607391885 14664173720
Ob	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	6 6 6 6 6	214125.64 214125.64 214125.64 214125.64 214125.64	253374.14 253278.14 253230.14 253326.14 253182.14	14830218.5 14830218.5 14830218.5 14830218.5 14830218.5
Ss	$\left\{\begin{array}{c}+20 \\ +10 \\ 0 \\ -10 \\ -20\end{array}\right.$	6 6 6 6 6	214125.64 214125.64 214125.64 214125.64 214125.64	253278.14 253182.14 253230.14 253326.14 253374.14	28111.36 28057.36 28003.36 27949.36 27895.36

3.png

Figure 3. Optimal level of total cost of retailer $M_{i+1}<T_{i+1}$

4.png

Figure 4. Optimal level of the total cost of the retailer when $M_{i+1}>T_{i+1}$

Figure 4 shows the optimal level of the total cost of retailer for $M_{i+1}>T_{i+1}$. At $\mathrm{n}=5$ replenishment cycle, the total optimal value of the total cost of the retailer is 30942.01 . Similarly, as seen in Tables 4 and 5, the impact of varying the parameter 'a' is illustrated. Optimal level of retailer total cost. For the different values of ' $a$ ' optimal value arrived at the $5^{\text {th }}$ replenishment cycle. Table 5 shows the replenishment time, optimal value of retailer, supplier, and order quantity.

Lemma1: $t_i$ increase where $\mathrm{i}=1,2,3 \ldots \ldots \ldots \ldots \mathrm{n}-1$ strictly monotonic increase function of last replenishment cycle $t_n$.

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

Theorem 1: The unique solution only exists for the non-linear system of Eq. (11) is the optimal replenishment period for a fixed replenishment cycle n.

The Hessian matrix of $T_{R e t}$ must be positive definite for $t_i$ to be minimum for a fixed $\mathrm{n}$.

Therefore, in Appendix, the theorem establishes that $T_{\text {Ret }}$ is positive definite. As a result, the loptimum value of $t_i$ for a given fixed $\mathrm{n}+\mathrm{ve}$ integer can be computed by using the numerical iterative technique and Mathematica programs version 12.0. On the basis, of the optimal value of $t_i$, the tota cost function also will be optimal.

Theorem 2: If $t_i$ satisfy in equations (i) $\frac{\partial^2 T_{R e t}}{\partial t_i^2} \geq 0$ (ii) $\frac{\partial^2 T_{R e t}}{\partial t_i^2} \geq\left|\frac{\partial^2 T_{R e t}}{\partial t_i t_{i-1}}\right|+\left|\frac{\partial^2 T_{R e t}}{\partial t_i t_{i+1}}\right|$ for all $\mathrm{i}=1,2,3 \ldots \ldots \ldots \mathrm{n}_1$ then $\nabla^2 T_{\text {Ret }}$ is positive definite.

Now, we will discuss about the comparison between existing literature and proposed model:

Upon validation of our comparative study against the benchmarks set forth in study [28], we have rigorously evaluated and contrasted our research outcomes with those posited in the referenced work. As depicted in Figure 5, our analysis entailed a thorough comparison with the existing body of literature. The results indicate that our study not only aligns with but also surpasses the solutions presented in study [28]. The graphical representation in Figure 5 clearly illustrates the enhancements and progressive strides our research has made over the foundational work of study [28].

5.png

Figure 5. Comparison between current study and existing literature

Precisely evaluating the unique trade credit period supplied by suppliers to retailers, which considerably impacts the overall findings and conclusions, is an important aspect of our research. This unique feature extensively affects inventory management, financial strategy, and supply chain sustainability.

The demand rate in this article is influenced by the time and edit period. We analyse, how variations in demand over time have a direct influence on holding costs, ordering patterns, and total replenishment techniques. To identify the optimality of the supply chain inventory problem, an algorithm is prepared, and sensitivity analysis is conducted. This transparency enhances the credibility of our findings.

The following aspects are contained in the proposed research study, which distinguishes it from most other inventory models.

• Demand is dependent on time and has a linear relationship with time.
• Preservation technology to reduce the deterioration of materials.
• Green technology investment.
• Supplier offers the retailer a special trade credit period.
• The finite planning horizon.
• Discussed two different trade credit phases.

According to the findings, continuous technology investments that reduce the deterioration of materials and preserve the environment can significantly enhance inventory management and control accuracy. To prevent the rate of deterioration of materials and environmental pollutants, both retailers and suppliers need to make a preservation investment.

The following outcomes of this research study were achieved:

• The research yields a unique solution set, demonstrating that replenishment time cycles are convex functions that provide a single optimal solution, thereby achieving optimum cost.

The proposed approach may be modified for future study by various approaches. For some examples, it may extend into a new approach by considering some other kinds of demand functions, including exponential and quadratic, time-varying demand, carbon, and inventory-dependent demand, etc. In addition, we may further consider the instantaneous deterioration of materials, shortages, and some carbon regulations, Therefore, future research can appraise the effects of each of these extra aspects.