A Supply Chain Inventory Model for Deteriorating Products with Carbon Emission-Dependent Demand, Advanced Payment, Carbon Tax and Cap Policy

The first phase focuses on degradation inventory problems, with more and more scholars constantly expanding inventory models for deteriorating goods to reflect more authentic inventory features. The supply chain for deteriorating commodities is critical in the storing industry to stay competitive. Most of the fresh or trendy items fade and deteriorate with time because of evaporation, expiration, spoilage, and depreciation, among other factors. Ghare and Schrader [10] proposed an EOQ model with a constant deterioration rate in a pioneering publication. Constant degradation rate to a two-parameter Weibull distribution by Covert and Philip [11]. From constant demand to a linearly growing demand pattern, Dave and Patel 1981 developed the model for deteriorating products. For deteriorating goods, we refer to the investigations by Dye [12]; Singh et al. [13]; Dave et al. [14] and Singh et al. [15]. They developed model for deteriorating goods and also explained how technology investments impact decaying products. For deteriorating items, we’re talking about Pahl and Voß’s [16] research from a few years ago (2014). From the above review, we have a firm idea that deteriorating items with finite planning horizons along with carbon emission policies have been ignored.

The second phase focuses on carbon emission policies with deterioration. Furthermore, one of the most significant economic benefits of reducing carbon emissions and deterioration is the reduction of emission of carbon during the entire supply chain process Carbon emissions and deterioration from economic sectors are causing serious rising temperatures. One of the most pressing issues today is supply chain management for deteriorating goods with carbon reduction regulations, which is becoming a serious concern for urban areas. As a result, some national or international agencies, governments, and businesses are increasingly under pressure to reduce carbon emissions. Manufacturers can lower their carbon footprint by using modernizing carbon-reducing techniques. In today’s emerging economy, increasing rates of environmental degradation and deterioration of inventory are major challenges. Carbon dioxide emissions and degradation of inventory mostly occur due to different human activities. these human activities are directly or indirectly responsible for the degradation of the environment and our deteriorating rate cannot be avoided in the supply chain system which is a main key parameter of the Supply Chain in an inventory system. In inventory control, deterioration can be reduced only by reducing global warming and carbon emission only which is a very difficult task. For that, we have to take the help of carbon emission regulations. We have learned from here that reducing carbon emissions and deterioration will have a positive impact on global warming as well as profit. Reducing carbon emissions in the supply chain is an effective way to reduce greenhouse emissions. It will only be achievable if we implement a carbon tax and carbon cap approach. We can only effectively increase carbon emission reduction activities if we tackle the problem of carbon emissions reduction collaboration and coordination across supply chain firms. Otherwise, achieving the aim of reducing carbon emissions will be challenging. The government of any country and some regulatory agencies considered carbon emission schemes to decrease carbon emission. For instance, carbon tax and cap, carbon tax are the main regulation policies. The majority of the existing research on inventory models, on the other hand, has focused on maximizing profit or lowering cost. Only a few of them consider environmental issues, such as minimizing carbon emissions and deterioration. As Dye and Yang [17] examine a deteriorating inventory system under various carbon emission regulations, as well as the influence of trade credit risk, in this work. Researchers show how environmental restrictions may be included in a decision-making issue for a deteriorating item that involves both trade credit and inventory replenishment. Rout et al. [2] explained a carbon cap and tax scheme, a government or intergovernmental body sets an aggregate legal limit on emissions (the cap) for a specified period and after that limit, a tax will be imposed. A carbon tax is the most important policy for limiting and ultimately eliminating the use of fossil fuels, which are damaging and destroying our environment. A. Carbon dioxide and other greenhouse gas emissions are altering the atmosphere. A carbon tax places a price on such emissions, encouraging individuals, companies, and governments to generate less. Mishra et al. [18] explain a sustainable carbon tax and cap-based production inventory model with three cases; Sustainable carbon tax and cap-based production inventory model without shortages; with partial backorder; and with full back-ordering without and with green technology investment. According to Mishra et al. [4], it has been explained how carbon emissions and deteriorating items can be controlled in a sustainable supply chain model and what are their effect. Price Dependent Linear and Non-Linear Demand Used here to Reduce Carbon Emissions and deterioration rate. Back ordering and non-back ordering are both cases considered. We have learned from here that reducing carbon emissions and deterioration will have a positive impact on global warming as well as profit. Carbon emission-reducing policies like carbon cap and carbon tax have been considered in this sustainable inventory control model to control carbon emissions and maximize profits. Sepehri et al. [19] introduced a price-dependent demand model for deterioration commodities, including a carbon cap and trade, as well as allowed late payments for the buyer to manage inventories and build demand. It has been pointed out that there must be a trade-off between the investment in carbon emission reduction technologies and the profit generated by lowering emissions. With this study, we established the nonavailability of research considering deteriorating items with finite planning horizons along with carbon tax and cap.

The third phase focuses on the Preliminary payment, cash, and Post payment. An essential component of inventory management is an advance payment. Due to difficult economic conditions and customer uncertainty, it is common for suppliers to demand something of advance payment from their retailers. The manufacturer makes the advance payment to make sure that the order can be delivered on time. In exchange for the advance payment, the supplier offers a reduction in price, a credit facility, or some other type of opportunity to encourage business. Similarly, Cash and Post payments are also beneficial for both retailers and suppliers. Manna et al. investigated an inventory model for defective items in which the producer provides free transportation to the retailer in return for advance payment. Production and transportation decisions from producer to retailer are linked to carbon emissions. Due to environmental rules, the manufacturer is required to pay a carbon emission tax according to this article by Zhang [20] who established the optimal advanced payment approach for saving money and time by making a larger amount of payment in advance. Taleizadeh [21] created an advance-cash payment for an evaporating commodity with partial backordering. this article is slightly different in comparison to Zhang [20]. Simultaneously, Zhang et al. [22] investigated an advance payment schedule in which the retailer pays a proportion of the purchase price in advance to gain a discount, and then takes sufficient time to pay the remaining balance. Teng et al. [23] modified the EOQ model to include advance payments and the fact that a product’s expiration date affects the demand rate. Most advance payment researchers, on the other hand, assumed that the product could be marketed indefinitely. They failed to account for the fact that numerous products (such as baked products, meat, milk, vegetables, fruits, and so on) cannot be sold once they have passed their expiration dates. An EOQ inventory model for perishable products with expiration dates and advance-cash-credit payment systems was described by Wu et al. [24]. Barman studied A multi-cycle vendor buyer supply chain production inventory model with carbon emission regulations such as carbon tax is proposed in this study. The system’s products are assumed to deteriorate at a constant rate. The buyer paid the vendor the purchase cost in advance in some installments before the order quantity was replenished. Mishra et al. [18] have created an EOQ inventory coordination model in which the seller provides three payment options to the buyer: cash, advance, and installment payment, or credit payment, as well as also set a Carbon taxation policy that used to reduce carbon emissions. Mashud et al. [25] non-instantaneous deterioration, advance payments, and partial backorder approaches are all considered in this research article. This study provides appropriate green technology investment and preservation technology to reduce both carbon emissions and product deterioration, as well as illustrating the impacts of deterioration and carbon emissions on total inventory model profit. Wu et al. [24] Researchers introduce an inventory control model with deterioration and carbon emission rate that can be controlled under carbon emission policies such as carbon tax and carbon cap. Various payment approaches can be considered for fully, partially, and no backlogging, where demand is based on price and trade credit. This research examined how greenhouse operators enhance they are investing in preservation and green technologies, as well as introducing trade credit to enhance their earnings. From the above study, it is clear that the need of discussion on the items that are deteriorating in the nature have not been covered for carbon tax and cap policies with different payment options under finite planning horizons.

The fourth phase focuses on carbon-dependent demand. Li [26] investigates a model that explains the impact of carbon emissions and pricing on-demand, as well as supplier-retailer coordination. This article’s main goal is to reduce carbon emissions. Pang et al. [27] examines the centralized decision process and determines the criteria for carbon emission reduction and order quantity management in the supply chain. Second, it presents a supply chain cooperation and coordination based on a revenue-sharing contract based on an economic order quantity policy. Furthermore, the study presents techniques for determining the best order amount and the ideal level of carbon emissions through model optimization, considering that market demand is influenced by buyer environmental consciousness. Finally, it also looks into the effects of carbon trading prices on supply chain carbon emissions. Aliabadi et al. [7] established an EOQ model to lower the risk of default by reducing emissions through a carbon tax policy. The demand rate is concerned about the number of carbon emissions, the credit duration, and the selling price, which is used as a trade credit for deteriorating commodities with backlogging. Lu et al. [28] explained a sustainable counter-productive model was investigated in this study, which also considered carbon tax law and joint investment in carbon emission reduction technology. The level of raw material inventory and price-dependent demand were also recognized. In this phase, we identified the necessity of examining an inventory model taking carbon tax and cap policy and carbon dependent demand for deteriorating items and different payment options under finite planning horizon.

The fifth phase focuses on a finite planning horizon in inventory control and management. Wu and Zhao [29] investigate a model in which the researcher includes time and inventory dependent demand under a finite planning horizon. Singh et al. [13] proposed an EOQ inventory supply chain model with deteriorating items in a finite planning horizon for two scenarios: with and without payment delays. The re-manufacturing of inventories was covered by Singh et al. [15] under the centralized and decentralized planning horizon with deteriorated products. A trade credit policy was also incorporated under finite planning. Xu et al. [30] develop inventory models with partial backlogs for deteriorating items, investigate the effects of carbon emission controls on the inventory system, and also consider time-varying demand in the system because customer demands in a real deteriorating inventory system commonly vary with time. In inventory control and management, a supply chain model has yet to be proposed for deteriorating items, along with carbon tax and cap policy, different payment options, and carbon dependent demand a under finite planning horizon. Table 1 compares the above study to find out the research gap.

Research gap: There has been very little study before this that has described carbon emission in the supply chain for deterioration and carbon emission policies.

To the best of our knowledge, no prior research has taken a supplier-retailer inventory model for deteriorating items, with price or emission-dependent demand under a finite planning horizon. Also, none of the researchers have discussed supplier–retailer inventory coordination under carbon emission policies such as carbon tax and carbon cap for deteriorating products with price-carbon dependent demand and three payment options. This is a huge research gap and the work is also unique in that the problems discussed are relevant to purely economic terms and the environment. This research provides a joint decision on inventory control and carbon emission. Various parameters such as carbon emission demand with carbon emission policies, carbon tax and cap are considered to design a long-term sustainable supply chain.

Defining problem: Therefore, this research work is mainly focused on carbon emissions with Inventory Control and management and deterioration with carbon emission regulations and three payment options: Preliminary, Cash, and Post payment over a finite horizon.

Table 1. Comparison of the above study to find out the research gap

Due to no shortage, before the previous inventory level reaches zero, the retailer orders their products from the supplier. The supplier will then immediately replenish his order, with the same quantity as products ordered. Length of the cycle (T) varies in finite planning horizon. The amount of quantity (R_i+1) ordered is also not constant. The differential equation that represents the changes in inventory level is represented by the figure of inventory level (Figure 1).

1.png

Figure 1. Preliminary payment by Wu and Zhao [29]

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

where, s_i≤t≤s_i+1.

$I L_{i+1}(t)=e^{-\theta t} \int_{t}^{s_{i+1}}(P, G) e^{\theta u} d u$                (2)

See Appendix.

where, IL_i+1(s_i+1)=0 and IL_i+1(s_i)=R_i+1.

$\mathrm{IL}_{\mathrm{i}+1}(\mathrm{t})=\int_{t}^{\mathrm{s}_{\mathrm{i}+1}}(\mathrm{P}, \mathrm{G}) \mathrm{e}^{\theta(\mathrm{u}-\mathrm{t})} \mathrm{du}$              (3)

$\mathrm{IL}_{\mathrm{i}+1}(\mathrm{t})=\frac{\mathrm{D}(\mathrm{P}, \mathrm{G})\left(\mathrm{e}^{\theta\left(\mathrm{s}_{\mathrm{i}+1}-\mathrm{t}\right)}-1\right)}{\theta}$             (4)

$R_{i+1}=I L_{i+1}\left(s_{i}\right)=e^{-\theta s_{i}} \int_{s_{i}}^{s_{i+1}}(P, G) e^{\theta t} d t$                      (5)

$R_{i+1}=I L_{i+1}\left(s_{i}\right)=(P, G)\left. e^{\theta\left(s_{i+1}-s_{i}\right)}-1\right.$           (6)

Ordering cost

$O_{c}=\mathrm{m} 1 * O_{x}$                (7)

Holding cost

$H_{c}=\sum_{i=0}^{m_{1}-1} h_{c} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u d t$               (8)

Deterioration cost

$\mathrm{Dc}=\sum_{i=0}^{m_{1}-1} \theta d_{c} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u d t$            (9)

Carbon emission cost

$C e=\sum_{i=0}^{m_{1}-1} \mathrm{c}^{\hat{r}}+\hat{P}_{r} * R_{i+1}+\widehat{h_{c}} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u d t$                (10)

4.1 Model for Preliminary payment under finite planning horizon

As demonstrated in Figure 2, retailer prepays the acquisition cost in n equal installments. Figure 2 displays one cycle out of several cycles in a finite planning horizon.

2.png

Figure 2. Preliminary payment by Shi et al. [9]

Acquisition cost

$\sum_{i=0}^{m_{1}-1}(1-\mathrm{r}) * P_{r} \int_{s_{i}}^{s_{i+1}}(P, G) e^{\theta\left(t-s_{i}\right)} d t$                   (11)

Capital cost

$C c_{1}=\sum_{i=0}^{m_{1}-1} \frac{\mathrm{I} * \mathrm{~N}(\mathrm{n}+1)(1-\mathrm{r}) * P_{r}}{2 n} \int_{s_{i}}^{s_{i+1}}(P, G) e^{\theta\left(t-s_{i}\right)} d t$                    (12)

Interest charges

$I c_{1}=\sum_{i=0}^{m_{1}^{-1}} I(1-r) P_{r} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u$              (13)

Retailers overall cost

Total costTCR₁=Ordering cost+ holding Cost + deterioration cost + Aquation cost + capital cost+ charges intrest + carbon cost.

$T C R_{1}=m_{1} * O_{c}+(1-r) * P_{r} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}}(P, G) e^{\theta\left(t-s_{i}\right)} d t$

$+\sum_{i=0}^{m_{1}-1} \frac{\mathrm{I} * \mathrm{~N}(\mathrm{n}+1)(1-\mathrm{r}) * P_{r}}{2 n} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}}(P, G) e^{\theta\left(t-s_{i}\right)} d t$

$+\sum_{i=0}^{m_{1}-1} I(i-r) P_{r} * \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u d t$

$+\sum_{i=0}^{m_{1}-1} h_{c} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u d t$

$+\sum_{i=0}^{m_{1}-1} \theta d_{c} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u d t+\sum_{i=0}^{m_{1}-1} \tau\left(Z-\left(\mathrm{c}^{\wedge}+\hat{P}_{r}\right.\right.$

$\left.\left.* R_{i+1}+\widehat{h_{c}} \int_{s_{i+1}}^{s_{i}} \int_{t}^{s_{i+1}}(P, G) e^{\theta(u-t)} d u d t\right)\right)$            (14)

$\frac{\partial T C R_{1}}{\partial s_{i}}=\left\{\left\{(1-r) * P_{r}+\frac{\mathrm{I} * \mathrm{~N}(\mathrm{n}+1)(1-\mathrm{r}) * P_{r}}{2 n}-\tau \hat{P}_{r}\right\}\right.$

$\left.+\frac{\left\{I(1-r) P_{r}-\widehat{h_{c}} \mathrm{\tau}+h_{c}+\theta d_{c}\right\}}{\theta}\right\} D(P, G) \sum_{i=0}^{m_{1}-1}\left\{e^{\theta\left(s_{i}-s_{i-1}\right)}\right.\left.-e^{\theta\left(s_{i+1}-s_{i}\right)}\right\}$             (15)

$\frac{\partial^{2} T C R_{1}}{\partial s_{i}{ }^{2}}=\left\{\left\{(1-r) * P_{r}+\frac{\mathrm{I} * \mathrm{~N}(\mathrm{n}+1)(1-\mathrm{r}) * P_{r}}{2 n}-\tau \hat{P}_{r}\right\}\right.$

$\left.+\frac{\left\{I(1-r) P_{r}-\widehat{h}_{c} \tau+h_{c}+\theta d_{c}\right\}}{\theta}\right\} D(P, G) \theta \sum_{i=0}^{m_{1}-1}\left\{e^{\theta\left(s_{i}-s_{i-1}\right)}\right.\left.+e^{\theta\left(s_{i+1}-s_{i}\right)}\right\}$                     (16)

$\frac{\partial^{2} T C R_{1}}{\partial\left(s_{i}\right)\left(s_{i-1}\right)}=\left\{\left\{(1-r) * P_{r}+\frac{\mathrm{I} * \mathrm{~N}(\mathrm{n}+1)(1-\mathrm{r}) * P_{r}}{2 n}-\tau \hat{P}_{r}\right\}\right.$

$\left.+\frac{\left\{I(1-r) P_{r}-\widehat{h_{c}} \mathrm{\tau}+h_{c}+\theta d_{c}\right\}}{\theta}\right\} D(P, G) \sum_{i=0}^{m_{1}-1}-\theta\left\{e^{\theta\left(s_{i}-s_{i-1}\right)}\right\}$                                  (17)

$\frac{\partial^{2} T C R_{1}}{\partial\left(s_{i}\right)\left(s_{i+1}\right)}=\left\{\left\{(1-r) * P_{r}+\frac{\mathrm{I} * \mathrm{~N}(\mathrm{n}+1)(1-\mathrm{r}) * P_{r}}{2 n}-\tau \hat{P}_{r}\right\}\right.$

$\left.+\frac{\left\{I(1-r) P_{r}-\widehat{h_{c}} \mathrm{\tau}+h_{c}+\theta d_{c}\right\}}{\theta}\right\} D(P, G) \sum_{i=0}^{m_{1}-1} \theta\left\{e^{\theta\left(s_{i+1}-s_{i}\right)}\right\}$               (18)

$\frac{\partial^{2} T C R_{1}}{\partial\left(s_{i}\right)\left(s_{n}\right)}=0$ f or all $\mathrm{n}$ is $\mathrm{i}-1, \mathrm{i}+1, \quad$ and $\mathrm{I}$          (19)

By Singh et al. [13, 15] and Sarkar et al. [31] the fact that the Hessian matrix of TCR is positive definite is sufficient for a total cost to be min as given in Figure 3.

3.png

Figure 3. Hesian matrix Sarkar et al. [31]

Condition to check positive definite:

$\begin{aligned} &\left|\frac{\partial^{2} T C R_{1}\left(m_{1}, s_{0}, s_{1} \ldots \ldots \ldots \ldots \ldots s_{m}\right)}{\partial s_{i}^{2}}\right| \\ &\geq\left|\frac{\partial^{2} T C R_{1}\left(m_{1}, s_{0}, s_{1} \ldots \ldots \ldots \ldots \ldots s_{m}\right)}{\partial\left(s_{i}\right)\left(s_{i-1}\right)}\right| \\ &+\left|\frac{\partial^{2} T C R_{1}\left(m_{1}, s_{0}, s_{1} \ldots \ldots \ldots \ldots \ldots s_{m}\right)}{\partial\left(s_{i}\right)\left(s_{i+1}\right)}\right| \end{aligned}$

$\left|\frac{\partial^{2} T C R_{1}\left(m_{1}, s_{0}, s_{1} \ldots \ldots \ldots \ldots \ldots s_{m}\right)}{\partial s_{i}^{2}}\right|-$

$\left|\frac{\partial^{2} T C R_{1}\left(m_{1}, s_{0}, s_{1} \ldots \ldots \ldots \ldots \ldots . \ldots s_{m}\right)}{\partial\left(s_{i}\right)\left(s_{i-1}\right)}\right|-$

$\left|\frac{\partial^{2} T C R_{1}\left(m_{1}, s_{0}, s_{1} \ldots \ldots \ldots \ldots \ldots s_{m}\right)}{\partial\left(s_{i}\right)\left(s_{i+1}\right)}\right| \geq 0$                     (20)

Model for cash payment under finite planning horizon.

Acquisition cost

$A c_{2}=\sum_{i=0}^{m_{1}-1} P_{r} \int_{s_{i}}^{s_{i+1}}(P, G) e^{\theta\left(t-s_{i}\right)} d t$               (21)

Interest charges

$I c_{2}=\sum_{i=0}^{m_{1}-1} I * P_{r} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}} D(P, G) e^{\theta(u-t)} d u d t$                  (22)

Retailer’s overall cost

Total cost TCR₂=Ordering cost + holding Cost + deterioration cost + Aquation cost + charges interest + carbon cost

$T C R_{2}=m_{1} * O_{c}+P_{r} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} D(P, G) e^{\theta\left(t-s_{i}\right)} d t+I$

$* P_{r} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} I L_{i+1} d t$

$+h_{c} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} I L_{i+1} d t \theta d_{c} \sum_{i=0}^{m_{1}-1{ }_{s_{i+1}}^{s_{i}}} I L_{i+1} d t \sum_{i=0}^{m_{1}-1} \tau(Z \left.-\left(\mathrm{c}^{\hat{n}}+\hat{P}_{r} * R_{i+1}+\widehat{h_{c}} \int_{s_{i}} I L_{i+1}(t) d t\right)\right)$               (23)

$T C R_{2}=m_{1} * O_{c}+\left(P_{r}-P_{r} \tau\right) \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} D(P, G) e^{\theta\left(t-s_{i}\right)} d t$

$+\left\{I * P_{r}+h_{c}+\theta d_{c}\right.$

$\left.+\widehat{h_{c}} \tau\right\} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1} s_{i+1}} \int_{t} D(P, G) e^{\theta(u-t)} d u d t+Z * \tau-\hat{c} \tau$                  (24)

$\begin{aligned} \frac{\partial T C R_{2}}{\partial s_{i}}=\left\{P_{r}+\right.& \frac{\mathrm{I} * P_{r}-\widehat{h_{c}} \tau+h_{c}+\theta d_{c}}{\theta}\left.-\tau \widehat{P}_{r}\right\} D(P, G) \sum_{i=0}^{m_{1}-1}\left\{e^{\theta\left(s_{i}-s_{i-1}\right)}\right.\left.-e^{\theta\left(s_{i+1}-s_{i}\right)}\right\} \end{aligned}$               (25)

4.2 Model for post payment under finite planning horizon

In general, according to the market norm, the simplest method is to purchase a product and pay for it. Yet, Trade Credit is a practical choice for the majority of firms. In this case, the supplier gives the retailer a set amount of time to pay for the purchases. within this permissible limit, there are no losses for the supplier. Sometimes, to reduce the risk of default, the supplier usually offers a discount to promote early payment. Therefore, the retailer is given a specified credit period M_rby the supplier to the retailer in this situation.

The realtor’s acquisition cost per replenishment cycle time T_i+1 shown clearly by:

Acquisition cost

$A c_{3}=\sum_{i=0}^{m_{1}-1} P_{r} \int_{s_{i}}^{s_{i+1}}(P, G) e^{\theta\left(t-s_{i}\right)} d t$               (26)

Interest charges

$I c_{3}=\sum_{i=0}^{m_{1}-1} I * P_{r} \int_{M}^{s_{i+1}} \int_{t}^{s_{i+1}} D(P, G) e^{\theta(u-t)} d u d t$            (27)

Sub-case 1 of M_c ≤ T ^s

Since M_c≤T, the retailer in this sub-case must pay interest on the products in stock after time M_c

The interest charged per cycles:

$\mathrm{TCR}_{3} I E_{31}=I * I_{e} * S * D(P, G) M_{C}^{2}$               (28)

$T C R_{3}=m_{1} * O_{c}+\left(P_{r}-P_{r} \tau\right) \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{i+1} D(P, G) e^{\theta\left(t-s_{i}\right)} d t$

$+\left\{I * P_{r}+h_{c}+\theta d_{c}\right.$

$\left.-\widehat{h_{c}} \tau\right\} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} \int_{t}^{s_{i+1}} D(P, G) e^{\theta(u-t)} d u d t+Z * \tau -\mathrm{c}^{\wedge} \tau-\left\{\mathrm{I} * I_{e} * \mathrm{~s} * \mathrm{D}(\mathrm{P}, \mathrm{G}) * M^{2}\right\}$                      (29)

4.2.1 Model for Preliminary payment under finite planning horizon

The interest charged per cycles:

$I E_{32}=\sum_{i=0}^{m_{1}-1} I_{e} * S * D(P, G) T_{i+1}\left\{M_{c}-\frac{1}{2} T_{i+1}\right\}$                 (30)

Interest charges $\left(I C_{32}\right)=0$                     (31)

Retailers overall cost

Total cost (TCR₃₂) =Ordering Cost+ Holding Cost + Deterioration Cost  + Aquation cost+ Charges Interest- Interest Earn+ Carbon Cost

$T C R_{32}=m_{1} * O_{c}+P_{r} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} D(P, G) e^{\theta\left(t-s_{i}\right)} d t+I *$

$P_{r} \sum_{i=0}^{m_{1}-1} \int_{M_{c}}^{s_{i+1}} I L_{i+1} d t+$

$h_{c} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} I L_{i+1} d t+\theta d_{c} \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} I L_{i+1} d t+\sum_{i=0}^{m_{1}-1} \tau(Z-$

$\left.\left(c^{\wedge}+\hat{P}_{r} * R_{i+1}+\widehat{h_{c}} \int_{s_{i}}^{s_{i+1}} I L_{i+1}(t) d t\right)\right)-\sum_{i=0}^{m_{1}-1} I_{e} * s *$

$D(P, G) T_{i+1}\left\{M_{c}-\frac{1}{2} T_{i+1}\right\}$                  (32)

$\begin{aligned} T C R_{32}=m_{1} * O_{c}+\left(P_{r}\right.&\left.-P_{r} \tau\right) \sum_{i=0}^{m_{1}-1} \int_{s_{i}}^{s_{i+1}} D(P, G) e^{\theta\left(t-s_{i}\right)} d t \\ &+\left\{h_{c}+\theta d_{c}-\widehat{h_{c}} \tau\right\} \\ &+\sum_{i=0}^{m_{1}-1} \int_{s_{i} s_{i+1} s_{i+1}} \int_{t} D(P, G) e^{\theta(u-t)} d u d t \\ &+Z * \tau-c^{n} \tau \\ &-\sum_{i=0}^{m_{1}-1} I_{e} * s \\ & * D(P, G)\left\{s_{i+1}-s_{i}\right\}\left\{M_{c}\right. \left.-\frac{1}{2}\left\{s_{i+1}-s_{i}\right\}\right\} \end{aligned}$           (33)

$R_{(i+1)}=\sum_{i=0}^{m_{1}^{*}-1} R_{i+1}^{*}$                   (34)

Supplier total cost is given by the following equation.

$T C S=m_{1}^{*} S_{c}+\sum_{i=0}^{m_{1}^{*}-1} P_{r} * R_{i+1}^{*}$              (35)

Numerical data: This section give an information about numerical data; we’ll look at a numerical data to illustrate how various factors affect the overall cost of retailer’s and suppliers. Let a=0.15, n=10, r=0.5, h_d=10, b=0.000001, c=0.0001, W=4, I=0.1, S_r=350, S_s=400, θ=2, cˆ=1, h_c=2, hˆc =0.2, Cc=5, P=1, G=1, N=0.17, M_r=0.17, I_e=0.08, s=50, τ= 0.5, g= 0.00001, d_c=1, Z = 0.002, P_r= 0.02, Pˆr=10 with the appropriate units. We will use Mathematica software(version-12) to solve Non-linear Eqns. (20) and (24). we can see tabulation and graphically presentation of the overall cost of suppliers and retailers and optimal quantity all of the different parameters and replacement cycles, i.e. for m₁−1=1, 2, 3, and so on.

4.png

Figure 4. Graph presentation for Table 2

5.png

Figure 5. Graph presentation for Table 3

66.png

Figure 6. 3-D Graph presentation for Table 3 in the case of preliminary cost

7.jpg

Figure 7. Graph presentation for Table 3

8.png

Figure 8. Graph presentation for Table 4

The government has a major role to play in encouraging the creation and management of the circular economy throughout the supply chain. The government’s actions have a significant impact on supply chain partners’ decision- making.

In our proposed model, we have focused on three different models that used carbon tax and cap regulations, which support the government in assigning carbon emission caps to industries to encourage them to reduce carbon emissions.

Our findings will help the government create carbon reduction regulations to help companies reduce carbon emissions and promote a sustainable business strategy.

Comparison study

To compare the proposed model to that of Shi et al. [9], in which the parameter values are the same as in Shi et al. [9] model (Table 5). Furthermore, the values of all of the additional parameters are introduced in Example. 1 of the current model.

In this comparison study, we can conclude that our proposed model tries to minimize retailer, supplier and carbon emission costs. This occurrence opposes Shi's observations. Therefore, the proposed model is more effective in comparison to the Shi model which is economical as well as environmentally sensitive.

Table 5. Comparison chart