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For carbon emission, this research study investigates a Mathematical inventory model with time, advertising, and inventorydependent demand patterns. The main objective of this research study is to keep the total cost of retailers as well as suppliers and carbon emissions as low as possible. With collaboration and without collaboration, two cases are discussed in this proposed model. Within the first case, retailers and suppliers are not regarded as collaborators, whereas in the second case, collaboration is recognized. The optimality of the planned inventory management model is explained mathematically and theoretically in both situations. The algorithm of the mathematical solution was also properly discussed and the effects of altering various parameters are numerically studied to conduct a sensitivity analysis with the help of Mathematica software version 12. To demonstrate this model, a mathematical illustration, and a tabular and graphical representation, have been also provided. Ultimately, this model reaches a flourishing managerial suggestion and conclusion.
collaboration, advertising demand, and carbon emission
Global temperature rise is the effect of greenhouse gas emissions and even some living beings. Kaya identity also plays a suitable role in the direction of Climate change that had received a great deal of attention in past few years. A worldwide awareness about environmental conservation and protection is encouraging many more researchers, organizations, and other government agencies to create and maintain an ecofriendly as well as a negligible emission management system for supply chains. Kaya identity plays a role to calculate the impact of different factors of the supply chain on carbon emission [1]. Therefore, we rely on the kaya identity equation to calculate the amount of carbon emission.
$\mathrm{Co}_2=\left(\frac{G D P}{\text { Population }}\right)\left(\frac{E C}{G D P}\right)\left(\frac{\mathrm{Co}_2}{E C}\right) *$ Population
The entire value of all products and services generated in a country in a given year is known as the gross domestic product (GDP). GDP quantifies the monetary worth of final goods and services produced in a country over a specified duration, it may be quarterly or yearly.
Everyday activities likeHeating, cooling, electricity supply, and transportation are all dependent on efficient and reasonable energy services. All economic sectors, from business and industry to agriculture, rely on energy to function properly and effectively. Energy intensity is a ratio of the volume of energy required to produce one unit of GDP and can be used to estimate a country's energy efficiency.
The total energy consumed (EC) by endusers, such as households and businesses, industry, and modern agriculture, is referred to as final energy consumption in cooling, heating, and fossil fuel. It is the energy that achieves by the final consumer, except for energy used by the energy sector that produced emissions in large quantities. Our research work mainly focuses on carbon emission during supply chain processes such as holding (heating, refrigeration), and carbon emission due to transportation.
In inventory management, Khan et al. [2] advertising a new product or modified goods plays an extremely important role because it raises product awareness for customers to buy something, retailer, supplier, and even manufacturer, all advertise their new products, and also when a modified product from an old to new, they also advertise it. In this way, the advertisement plays an important role in increasing the demand for any product and from which we can say that the demand for any product on the day depends absolutely on its advertisement. Therefore, the advertisement will increase the product demand and the product will be sold out very soon. For retailers and suppliers to enhance customer demands is a challenging feat. That idea would be great in the case of perishable items, fashionable products and those items that are reaching their expiring date, where their life span is short. Due to carbon emissions and advertisements, demand for products will undoubtedly be influenced. Advertisement is one of the most effective promotional approaches to raise awareness about a product's popularity among all classes of consumers. There are many different ways to do advertising, in today's new technology, people advertise their new products or old to new through mass media to stimulate a larger group of customers to buy their products.
In this regard they aware customer about new products, about their price and extra information about their quality. All the players in supply chain management advertise in different ways to increase the demand for their products, for this, they advertise through different media like newspapers, posters, and television. Moreover, in the advertisement, they show the quality of the product. and at the same time, the good quality attracts the customer to buy the item very much. In the case of carbon emission, it has become very important to advertise carbonfree products, so that customers buy the carbonfree product and through which we can reduce the amount of carbon.
Therefore carbon emission with advertisement demand has become an integral part of our proposed work.
Our proposed model is a way to reduce carbon footprint and optimize the total cost of suppliers and retailers. As a result of this model, we have driven the average credit rate, the total cost of the supplier, and the total cost of the retailer are very sensitive to the deterioration rate. Another important outcome is that the average credit rate, the total cost of the supplier, and the total cost of the retailer are highly sensitive to the advertisementdependent demand.
The vast majority of the rest of this manuscript is structured as follows. A review of the existing literature has been carried out in Section 1. The Assumptions and notations of this manuscript are presented in Section 2, and the Mathematical study and their calculating work are provided in Section 3. The sensitivity study in the graphical and tabular form is described in Section 4. Section 5, contains the managerial findings and conclusion of SCM in the context of carbon taxes.
Kaya's identity, initially expressed by Kaya [1] and eventually formally accepted by some others is the combination of four factors: emissions per unit of energy expenditure; population; per capita income and energy intensity per unit of Manufacturing output Kaya, 1989 also can be presented as follow.
$\mathrm{Co}_2=\left(\frac{G D P}{\text { Ppopulation }}\right)\left(\frac{E C}{G D P}\right)\left(\frac{\mathrm{Co}_2}{E C}\right) *Population$
Since supply chain processes account for the majority of carbon emissions, so many governments have enacted carbon taxes to encourage energy efficiency and emission reduction. In this context, the effects of carbon taxes on supply chains become topics of discussion. Zhou et al. [3] traditionally supply chains have concentrated exclusively on economic gain only. but modern technology is focusing on sustainability. Supply chains are greatly influenced by carbon taxes. Because of this need to take into account the local, national, and worldwide ecosystems as well as the widening environmentalism of the general populace. According to Liu et al. [4], producers were compelled to select lowemitting techniques after the carbon tax was elevated to a certain level. Carbon taxes may also be advantageous for manufacturers, suppliers and retailers as well as for improved environmental processes. Benjaafar et al. [5] studied that a carbon tax implemented as part of a supply chain was able to accomplish the two main goals of lowering costs and emissions. Because of this, studies have discovered that an appropriate carbon tax technique can help to achieve supply chain social, economic, and ecologic collaboration. In a twolevel supply chain with a carbon tax, Cheng et al. [6] presented a collaborative model approach in which the retailer and manufacturer freely decided on collaborative carbon reduction activities. On the whole, it was determined that the carbon tax encouraged supply chain participants' joint efforts and enhanced the environment and economic performance. Park et al. [7] explained that a carbon tax was more impactful in balancing the exchange between public welfare, environmental conservation, and economic growth. From the above literature to study of carbon tax become an integral part of our proposed model.
Khan et al. [2] investigated that the advertisement of an item is directly associated with the demand. For this purpose, Goyal and Gunasekaran [8], Bhunia and Maiti [9], Shah and Pandey [10], Bhunia and Shaikh [11], Shah et al. [12], Bhunia and Shaikh [13], Shaikh et al. [14], Mishra [15], Khan et al. [2], Panda et al. [16], and many others established inventory models that took into account the effect of advertising on sales. So the retailer, supplier, and manufacturer must determine the advertisement process before the sales period. Understanding from this review of research articles, it is found that no researcher has considered an advertisement, time, and inventorydependent demand in the finite period horizon which is unique in itself.
The supply chain covers the entire process of producing and selling essential goods. Manufacturers, suppliers, transports, warehouses, retailers, and customers are all part of the supply chain. A process by which natural resources or raw materials are turned into finished items and then sold to endusers or customers. Moreover, the majority of the available conventional literature did not analyze credit terms coordination and collaboration. However, supply chain participants such as suppliers, retailers, and manufacturers are all affected by one another. A collaborative approach to supply chain management is necessary for beneficial and effective supply chain management. So, in past few years, few such as Wu and Zhao [17], Singh et al. [18], Kim and Sarkar [19], and Barman et al. [20] researchers have begun concentrating their attention on supply chain collaboration with credit terms as a beneficial framework for retailers and supplier. In today's global world, an institution's or business owner's primary aim is to optimize overall value simultaneously and also emphasize carbon emission reduction that can be achieved by coordinating different strategies among supply chain participants. To extend the entire supply chain, a lot of work has been conducted. Multiple parameters are used to enhance supply chain efficiency. Therefore, the most pressing issues today are supply chain management for coordination as well as without coordination and carbon emission for Advertisement, time, and inventory demand underneath finite planning.
According to Daryanto et al. [21], transportationrelated carbon output is determined by the amount of fuel used by the vehicle, the amount of fuel discharged, and the distance travelled. Also, discuss that fuel consumption energy is affected by transportation and emission strategy side by side total cost and amount of emission are also affected. In inventory control reducing global warming and carbon emission only through green farming is a very difficult task. For that, we have to take the help of advertisement technology and carbon regulations. advertisement technology and carbon regulations can help us to reduce total costs, carbon emissions, and global warming. We have learned from here that reducing carbon emissions and the total cost will have a positive impact on global warming as well as profit. facing these challenges, there has been very little study before this that has described carbon emission in the supply chain for advertisement, time and carbondependent demand underneath finite planning horizon. Hence this research provides a joint decision on inventory control with carbon emission. Various parameters such as Advertisement, time and inventorydependent demand, and carbon emission due to refrigeration and transportation technology are considered to design a longterm supply chain.
Table 1. Comparison of literature
Article 
Inventory dependent demand 
Advertisement dependent demand 
Collaboration 
Carbon Emissions cost 
Transportation cost 
Finite planning horizon 
Benjaafar et al. [5] 
× 
× 
√ 
√ 
× 
× 
Cheng et al. [6] 
× 
× 
× 
√ 
× 
× 
Khan et al. [2] 
× 
√ 
× 
√ 
× 
× 
Mishra [15] 
× 
√ 
× 
√ 
√ 
× 
Park et al. [7] 
× 
× 
× 
√ 
× 
× 
Wu and Zhao [17] 
√ 
× 
√ 
× 
× 
√ 
Singh et al. [18] 
√ 
× 
√ 
× 
× 
√ 
Daryanto et al. [21] 
× 
× 
× 
√ 
√ 
× 
This Paper 
√ 
√ 
√ 
√ 
√ 
√ 
Research problem:
This article considers linear timedependent, inventorydependent, and advertisementdependent demand with carbon emission regulation. The main objective is to properly analyze the economic as well as market scenario for some specific goods that are newly launched products, clothes, permissible items, fashionable items, and fastmoving consumer goods near deterioration or expiration date. Their demand rises or diminished according to time, inventory and advertisement. The tradecredit period offered by the supplier to its retailers for his purchases is considered under the Finite planning horizon (FPH). The collaboration of suppliers and retailers with carbon regulation is being considered. Based on the above literature, we have an idea that this will be a novel work under a finite planning horizon. Nobody has taken this problem in a finite planning horizon.
Notations:
The following are now the relevant factors only for the retailer and supplier:
O_{r} – Total cost for placing an order (\$/order)
a  Initial demand for marketing (per yr.)
b  Demand that depends upon a time (per yr.)
h_{c} – Cost for managing inventory in the warehouse (\$/un.)
$\Theta$ – Demand that is dependent upon inventory
W_{p} – Total wholesale price of items (\$/unit)
Q_{i }– Total amount of shipments per replenishment process (units)
$T_{I N D}^r$ – The total cost of the retailer when no coordination occurs (\$/time unit)
$T_{J T}^r$ – The total cost of the retailer when coordination occurs (\$/time unit)
$T_{I N D}^S$ – The total cost of the retailer when no coordination occurs (\$/time unit)
$T_{J T}^S$ – The total cost of the retailer when coordination occurs (\$/time unit)
The following are now the relevant factors only for carbon emission:
$\hat{c}$  Fixed amount of carbon emission
$\hat{h}_c$  Amount of carbon emission associated with the refrigerator consumed energy during the holding
$\hat{P}_r$  Varying carbon emissions due to the managing of each unit
ρ  Demand depend upon the amount of carbon emission
CO_{2 } Amount of carbon emission
τ  Carbon emission price conducted by the government
S_{c} Total setup cost of supplier
C_{p} its purchase price per unit ($/unit)
h  It denotes the capital cost ($/unit/yr.)
δ  the rate of the credit period.
Now, the following are the decision variables:
n  No replenishment cycles
T  Time planning horizon (Yr.)
T_{i}  Length of each cycle during placing an order (time/ unit)
t_{i}  A timeframe for placing an order (time unit)
A  Advertising and marketing frequency
$\alpha$  marketing and advertising demand function fluctuation
Transportation variables:
v_{c } The variable cost of fuel consumed by the retailer during transportation depends upon the fuel consumption (\$/litre)
e_{2}  Retailer’s additional (refrigeration) carbon emission cost from transporting one unit item (\$/unit/km)
e_{1}  Charge of carbon dioxide emissions from retail transportation (\$/km)
C_{1}  Fuel usage of a retailer's vehicle when it has been empty (litres per kilometre)
C_{2}  Retailer's additional (refrigeration and vehicle services) transportation energy consumption for every per ton of payload (litre/km/ton)
d  Distance covered from supplier to retailer (km)
F_{c}  Transportation fixed cost when order is placed by the retailer (\$)
In this Phase, a mathematical explanation with no shortages of this manuscript for a finite planning period is formulated. Using the assumptions and notations mentioned in the previous phase, the following basic inventory details over the finite planning horizon. Demand is considered advertisement, inventory, carbon and timedependent. The inventory declines eventually, usually to meet the demand and with advertising of the product. The shift in inventory level over time ($t$) during the time interval $\boldsymbol{t}_{\boldsymbol{i}} \leq \boldsymbol{t} \leq \boldsymbol{t}_{i+1}$ is defined by the differential Eq. (1).
$\frac{\mathrm{dI}_i(t)}{\mathrm{dt}}=A^\alpha\left(a+\mathrm{bt}\rho C o_2\right)\theta \mathrm{I}_i(t)$ (1)
where, t_{i}≤t≤t_{i}_{+1}.
Figure 1. Graphical representation of inventory level
Now, Eqns. (2)(5) represent the demand rate, inventory level, order quantity and holding cost given below. Labelled Figure 1 depicts the rise and fall of inventory.
$\mathrm{D}\left(\mathrm{t}, \theta, \mathrm{A}, C o_2\right)=A^\alpha\left(a+\mathrm{bt}\rho C o_2\right)+\theta \mathrm{I}_i(t)$ (2)
where, c > 0, d ≥ 0, and t is positive.
When I_{i}(t_{i}_{+1})=0 and I_{i}(t_{i})=Q_{i.}
$I_i(t)=\int_t^{t_{i+1}} A^\alpha\left(a+\mathrm{b} * \mathrm{u}\rho C o_2\right) e^{\theta(ut)} \mathrm{du}$ (3)
The order quantity for ith cycles:
$Q_i=I_i\left(t_i\right)=\int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{bt}\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t$
Ordering cost:
$O_c=\mathrm{n} * O_r$ (4)
Holding cost:
$H_c$$=\sum_{i=0}^{n1} h_c \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}} A^\alpha\left(a+\mathrm{bt}\rho C o_2\right) e^{\theta(ut)} d u d t$ (5)
Carbon emission cost doesn't include carbon emission due to transportation. According to Benjaafar et al. [5], Xiang and Lawley [22], and Shi et al. [23] and Mishra and Ranu [24], $\hat{c}$ is fixed carbon emissions involved with putting an order (carbon emissions caused by transportation), $\hat{P}_r$ varying carbon emissions due to the management of each unit, and $\widehat{h_c}$ is carbon emissions associated with refrigeratorconsumed energy in the storage of each unit. As a result, the total amount of carbon emissions for each replenishment process is represented by Eq. (6) given below.
$C E=\hat{c}+\sum_{i=0}^{n1} \hat{P}_r * Q_i+\widehat{h_c} \int_{t_i}^{t_{i+1}} \int_t^{t_{i+1}} A^\alpha(a+$$\left.\mathrm{bt}\rho C o_2\right) e^{\theta(ut)} d u d t$ (6)
The Retailer’s transportation cost considers the fixed and variable transportation costs and carbon emissions due to FEC during refrigeration. Therefore Eq. (7) represents the transportation cost given below.
$\begin{gathered}T c=F_c+2 \mathrm{~d} v_c C_1+\mathrm{d}^* v_c C_2 \int_{t_i}^{t_{i+1}} A^a\left(a+\mathrm{bt}\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t+2 \mathrm{~d} e_1+\mathrm{d} \\ e_2 \int_{t_i}^{t_{i+1}} A^a\left(a+\mathrm{bt}\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t\end{gathered}$ (7)
The total individual cost of the retailer, supplier and order quantity in the case of no collaboration is given by the Eqns. (8), (9) and (10).
$\begin{gathered}T_{I N D}^r=n_1 * O_r+\hat{c}+\left\{\frac{\left(h_c+\tau \hat{h}_c\right)}{\theta}+W_p+\hat{P}_r+\mathrm{d} * v_c C_2+\mathrm{d} e_2\right\} \sum_{i=0}^{n_11} \int_{t_i}^{t_{i+1}} A^\alpha(a+\mathrm{bt} \\ \left.\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t\frac{A^\alpha\left(h_c+\tau \hat{h}_c\right)}{\theta}\left(a T+\frac{1}{2} b T^2\right)+F_c+2 \mathrm{~d} v_c C_1+2 \mathrm{~d} e_1\end{gathered}$ (8)
$T_{I N D}^S=n_1^* * S_c+C_p * \sum_{i=0}^{n_1^*1} \int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{bt}\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t$ (9)
$Q_{\text {IND }}=\sum_{i=0}^{n_1^*1} Q_i^*$ (10)
Now, the joint total cost of the retailer, supplier and order quantity in the case of collaboration is given by (11), (12) and (13).
$\begin{gathered}T_{J T}^s=n_2 *\left(S_c+O_r\right)+C_p * \sum_{j=0}^{n_21} \int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{b} \mathrm{t}\rho C O_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t+\left\{\frac{\left(h_c+\tau \hat{h}_c\right)}{\theta}+W_p+\hat{P}_r+\mathrm{d} * v_c C_2+\mathrm{d} e_2\right\} \\ \sum_{i=0}^{n_21} \int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{bt}\rho C o_2\right) e^{\theta\left(tt_i\right)} \mathrm{d} t\frac{A^\alpha\left(h_c+\tau \hat{h}_c\right)}{\theta}\left(a T+\frac{1}{2} b T^2\right)+F_c+2 \mathrm{~d} v_c C_1+2 \mathrm{~d} e_1 \\ T_{I N D}^r\left(n_1^*, t_0, t_1^*, \ldots \ldots \ldots t_{n_1}^*\right)\end{gathered}$ (11)
$Q_{J T}=\sum_{j=0}^{n_2^*1} Q_j^{\prime *}$ (12)
$\begin{gathered}T_{J T}^r=n_2 * O_r+\hat{c}+\left\{\frac{\left(h_c+\tau \hat{h}_c\right)}{\theta}+W_p+\hat{P}_r+\mathrm{d} * v_c C_2+\mathrm{d} e_2\right\} \sum_{i=0}^{n_11} \int_{t_j}^{j_{j+1}} A^\alpha(a+\mathrm{bt} \\ \left.\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_j\right)} \mathrm{d} t\frac{A^\alpha\left(h_c+\tau \hat{h}_c\right)}{\theta}\left(a T+\frac{1}{2} b T^2\right)+F_c+2 \mathrm{~d} v_c C_1+2 \mathrm{~d} e_1 \\ T_{I N D}^r\left(n_1^*, t_0, t_1^*, \ldots \ldots \ldots t_{n_1}^*\right)\end{gathered}$ (13)
The credit period offered by the supplier to the retailer is given by Eqns. (14)(16):
$\delta=T_{J T}^r\left(n_2^{\prime *}, t_0, t_1^{\prime *}, \ldots \ldots \ldots . t_{n_2}^{\prime *}\right)T_{I N D}^r\left(n_1^*, t_0, t_1^*, \ldots . \ldots . . t_{n_1}^*\right) / \sum_{j=0}^{n_2^*1} h\left(t_{\mathrm{j}+1}^{\prime *}t_{\mathrm{j}}^{\prime *}\right) Q_j^{\prime *}$ (14)
$\begin{gathered}\delta_{\max }=T_{I N D}^s\left(n_1^*, t_0, t_1^*, \ldots \ldots \ldots . t_{n_1}^*\right)n_1^* * S_cC_p * \sum_{i=0}^{n_1^*1} \int_{t_i}^{t_{i+1}} A^a(a+ \\ \left.\mathrm{bt}\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t / \sum_{j=0}^{n_2^*1} h\left(t_{\mathrm{j}+1}^{\prime *}t_{\mathrm{j}}^{\prime *}\right) Q_j^{\prime *}\end{gathered}$ (15)
$\delta_{\min }=T_{J T}^r\left(n_2^{\prime *}, t_0, t_1^{\prime *}, \ldots \ldots \ldots . t_{n_2}^{\prime *}\right)T_{I N D}^r\left(n_1^*, t_0, t_1^*, \ldots \ldots \ldots . t_{n_1}^*\right) / \sum_{j=0}^{n_2^*1} h\left(t_{\mathrm{j}+1}^{\prime *}t_{\mathrm{j}}^{\prime *}\right) Q_j^{\prime *}$ (16)
The average of the credit period which is beneficial for both retailer as well as the supplier has given by Eq. (17).
where we know that $\bar{\delta}=\frac{\delta_{\min } \, \,+\delta_{\max }}{2}$ (17)
Final total cost after introducing the average credit period for both retailer as well as supplier and order quantity is given by Eqns. (18)(19).
$\bar{T}_{J T}^r=T_{J T}^r\left(n_2^{\prime *}, t_0, t_1^{\prime *}, \ldots \ldots \ldots . t_{n_2}^{\prime *}\right)\sum_{j=0}^{n_2^*1} \bar{\delta} h\left(t_{\mathrm{j}+1}^{\prime *}t_{\mathrm{j}}^{\prime *}\right) Q_j^{\prime *}$ (18)
$\bar{T}_{J T}^s=n_1^* * S_c+\sum_{i=0}^{n_1^*1} C_p Q_i^*+\sum_{i=0}^{n_2^*1} \bar{\delta} h\left(t_{\mathrm{j}+1}^{\prime *}t_{\mathrm{j}}^{\prime *}\right) Q_j^{\prime *}$ (19)
The value of Hessian matrix as discussed in researches [19] and [25] has to be shown as positive definite after solving, for $T_{I N D}^r$ to be minimum.
Now, replenishment time intervals are obtained with the help of Eq. (20) which is the derivative of Eq. (8).
$\begin{gathered}\frac{\partial T_{I N D}^r}{\partial t_i}=A^\alpha\left(a+\mathrm{b} t_i+\rho C o_2\right)\left(e^{\theta\left(\mathrm{t}_it_{i1}\right)}1\right) \\ \theta \int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{bt}\rho C o_2\right) e^{\theta\left(\mathrm{t}\mathrm{t}_i\right)} \mathrm{d} t\end{gathered}$ (20)
where, i=1,2,3…………n.
Theorem. If $t_i$ satisfy inequations (i) $\frac{\partial^2 T_{I N D}^r}{\partial t_i^2} \geq 0$ (ii) $\frac{\partial^2 T_{I N D}^r}{\partial t_i^2} \geq\left\frac{\partial^2 T_{I N D}^r}{\partial t_i t_{i1}}\right+\left\frac{\partial^2 T_{I N D}^r}{\partial t_i t_{i+1}}\right$ for all i= 1,2 , 3 ………. $\mathrm{n}_1$ then $\nabla^2 T_{I N D}^r$ is positive definite.
$\begin{gathered}\frac{\partial^2 T_{I N D}^r}{\partial t_i^2}=\theta A^\alpha\left(a+\mathrm{b} t_i\rho C o_2\right) e^{\theta T_i}+\theta A^\alpha(a+ \\ \left.\mathrm{b} t_{i+1}+\rho C o_2\right) e^{\theta T_{i+1}}+A^\alpha b\left(e^{\theta T_i}e^{\theta T_{i+1}}\right)\end{gathered}$ (21)
$\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i1}}=\theta A^\alpha\left(a+\mathrm{b} t_i\rho C o_2\right) e^{\theta T_i}$ (22)
Similarly,
$\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i+1}}=\theta A^\alpha\left(a+\mathrm{b} t_i\rho C o_2\right) e^{\theta T_{i+1}}$ (23)
$\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_m}=0$ (24)
for all m ≠i, i+1, i1.
$T_{c r}$ is positive definite if Eqns. (21)(24) satisfy the given inequality.
$\begin{gathered}\frac{\partial^2 T_{I N D}^r}{\partial t_i^2}>\left\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i1}}\right+\left\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i+1}}\right+\left\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i+1}}\right \\ \theta\left(a+\mathrm{b} t_i\rho C o_2\right) e^{\theta T_i}+\theta\left(a+\mathrm{b} t_{i+1}\rho C o_2\right) e^{\theta T_{i+1}}+A^\alpha b\left(e^{\theta T_i}e^{\theta T_{i+1}}\right)> \\ \left\theta\left(a+\mathrm{b} t_i\rho C o_2\right) e^{\theta T_i}\right+\left\theta\left(a+b t_{i+1}\right) e^{\theta T_{i+1}}\right+0 \\ \theta\left(a+\mathrm{b} t_i\rho C o_2\right) e^{\theta T_i}+\theta\left(a+\mathrm{b} t_{i+1}\rho C o_2\right) e^{\theta T_{i+1}}+A^\alpha b\left(e^{\theta T_i}e^{\theta T_{i+1}}\right)> \\ \theta\left(a+\mathrm{b} t_i\rho C o_2\right) e^{\theta T_i}+\theta\left(a+\mathrm{b} t_{i+1}\rho C o_2\right) e^{\theta T_{i+1}}\end{gathered}$
that is true for all i = 1, 2, ..., n.
The following steps are available to find the solution:
Step 1: First of all, Consider a new and unique setup of parameters: $\mathrm{O}_{\mathrm{c}}$, $\mathrm{D}_{\mathrm{t}}$, $\mathrm{H}_{\mathrm{c}}$, $\mathrm{S}_{\mathrm{s}}$, $\mathrm{A}_{\mathrm{c}}$, $\mathrm{C}_{\mathrm{c}}$, $\mathrm{I}_{\mathrm{c}}$, $\mathrm{I}_{\mathrm{e}}$, $\mathrm{P}_{\mathrm{r}}$, $\hat{c}$, $\hat{P}_r$, $\hat{h}_c$, $\mathrm{~N}$, $\mathrm{r}$, $\mathrm{n}$, $\mathrm{M}_{\mathrm{c}}$, $\theta$, $\ldots \ldots \ldots \ldots \ldots$, $\tau$, $\mathrm{C}_{\mathrm{e}}$, $\mathrm{T}_{\mathrm{i}+1}$, $\mathrm{~T}$ etc. And then assigning values to all Parameters Considered from already existing literature.
Step 2: We will determine the values of $t_j$ for this purpose. Keep looking for it in the following ways.
a) Then, start with considered $\mathrm{n}=1$. Then $\mathrm{t}_0=0$ and $\mathrm{t}_1=\mathrm{T}$
b) If we take $n=2$, then then we assumed $\mathrm{t}_0=0$ and $\mathrm{t}_2=T$. Then, to find $\mathrm{t}_1$ arranged the partial derivative of the function $T_{I N D}^r$ in terms of $t_j$ is equal to zero. It's identical to $\frac{\partial T_{I N D}^r}{\partial t_i}=0$
c) Afterwards, by using values of $t_2$, get $t_3$ and then $t_4 \ldots \ldots$.so on. Similarly, by this above process, we found all $t_j$.
d) After degerming the $t_i$ values, verify the result $\left\frac{\partial^2 T_{I N D}^r}{\partial \mathrm{t}_{\mathrm{i}}^2}\right \geq$ $\left\frac{\partial^2 T_{I N D}^r}{\partial\left(\mathrm{t}_{\mathrm{i}}\right)\left(\mathrm{t}_{\mathrm{i}1}\right)}\right+\left\frac{\partial^2 T_{I N D}^r}{\partial\left(\mathrm{t}_{\mathrm{i}}\right)\left(\mathrm{t}_{\mathrm{i}+1}\right)}\right$ that prove the convexity of function. Based on this convexity we can calculate the optimal value of $t_i$.
Step 3: Based on optimal $t_i$, the total cost of the retailer $T_{\text {IND }}^r$ can be calculated by Eq. (8). The following steps are available to find the $T_{I N D}^r$.
a) We Start by considering $\mathrm{n}=1$ and $T_{I N D}^r(\mathrm{n} 1) \leq$ $T_{I N D}^r(\mathrm{n} 1+1)$, then $T_{I N D}^r(\mathrm{n} 1)=T_{I N D}^{r o}(\mathrm{n} 1)$ and stop here.
b) we set $\mathrm{n} \geq 2$ and $T_{I N D}^r(\mathrm{n} 1) \leq T_{I N D}^r(\mathrm{n} 1$ 1) and $T_{I N D}^r(\mathrm{n} 1) \leq T_{I N D}^r(\mathrm{n} 1+1)$ then $T_{I N D}^r(\mathrm{n} 1)=$ $T_{I N D}^{r o}(\mathrm{n} 1)$ and stop. If $\mathrm{n} 1=\mathrm{n} 1+1$ then go to the step 2 (b).
Step 4: Using the values of $t_i$ obtained in the previous step, by which we can determine the optimal replenishment order quantity. $\mathrm{Q}_{(\mathrm{i}+1)}=\sum_{\mathrm{i}=0}^{\mathrm{n}_1^*1} \mathrm{Q}_{\mathrm{i}+1}^*$.
Step 5: Subsequently, by Eq. (28) we find $T_{I N D}^s$ (Supplier total cost) which depends upon the retailer's ordering process.
Step 6: Similarly, the same process is in the case of coordination.
Numerical illustration to understand this problem:
With their appropriate units, the following parametric fundamental values are permitted to enter: $O_r=$ $80$ \$/order, $h_c=1 $ \$/unit/time, $\Theta=3$, $W_p=3$, $\mathrm{a}=50,60,70$, $\mathrm{b}=15$, $\mathrm{a} S_s=100$, $\hat{c}$, $\hat{P}_r=0.02$, $\mathrm{Co}_2=0.02$, $\tau=6 S_c$, $C_p=0.01$, $\delta=0.2$, $\mathrm{n}$, $\alpha=1.3$, $\mathrm{v}_{\mathrm{c}}=8$, $\mathrm{e}_2=2.31 \times 10^{6}$, $\mathrm{e}_1=0.043$, $\mathrm{C}_1=25$, $\mathrm{C}_2=0.36$, $\mathrm{~d}=25$, $\mathrm{~F}_{\mathrm{c}}=0.01$, $\mathrm{A}=2$, $\mathrm{T}=2$, $\mathrm{h}=60$. The nonlinear partially derivative of Eq. (20) is solved with the help of numerical Iterative software Mathematica (version 12.0).
For no collaboration, Table 1, Table 2, Figure 2 and Figure 3 show optimal retailer’s overall cost for a= 40,50,60,70,80 are \$124188, \$135541, \$144159, \$151166 and $158282 are reach its minimum at 7, 8, 8 and 8 optimal ordering cycles respectively. After reaching its minimum at n1=7, 7, 8, 8 and 8 then again started increasing gradually for all upcoming cycles. Table 1 shows the convexity of the retailer’s cost.
This is also discussed in this section with the help of graphical illustrations.
In the case of collaboration, Table 3, Table 4 and Figure 4 give us the optimal supplier’s overall cost for different value of a= 40, 50, 60, 70, 80 are \$896, \$946, \$1103, \$1150 and \$ at 7, 8, 8 and 8 optimal ordering cycles respectively. Again, after attaining its minimum at n_{2}=7, 7, 8, 8 and 8 then again started increasing gradually for all next upcoming cycles. Table 3, Table 5, Table 6, Figure 2, Figure 3, and Figure 4 provided us with information on convexity whenever supplier and retailer are in with no collaboration and with collaboration.
Table 1. The total cost of the retailer with no collaboration with the supplier for different values of a and n_{1}
↓a 
→ n 
1 
2 
3 
4 
5 
6 
7 
8 
9 
40 
1382870 
367427 
203246 
152401 
132840 
125552 
124188 
126095 
129983 

50 
160150 
421552 
230447 
170745 
147239 
137958 
135541 
136910 
140572 

60 
185790 
484481 
260902 
189908 
160987 
148596 
144179 
144159 
146801 

70 
2107720 
545702 
290439 
208455 
174276 
158872 
152523 
151166 
152830 

80 
2359060 
607653 
320403 
227303 
187791 
169324 
161006 
158282 
158945 
Table 2. Optimal retailer’s cost and optimal replenishment cycles with no collaboration different value of a and n_{1}
↓a 
→ t_{i} 
t_{0} 
t_{1} 
t_{2} 
t_{3} 
t_{4} 
t_{5} 
t_{6} 
t_{7} 
t_{8} 
t_{9} 
$T_{I N D}^r$ 
40 
0 
0.5688 
1.1105 
1.6303 
2.1318 
2.6177 
3.0900 
3.5503 
4. 

124188 

50 
0 
0.5590 
1.0957 
1.6138 
2.1158 
2.6040 
3.0799 
3.5449 
4. 

135541 

60 
0 
0.4939 
0.9717 
1.4354 
1.8869 
2.3274 
2.7581 
3.1799 
3.5937 
4. 
144159 

70 
0 
0.4884 
0.9628 
1.4249 
1.8760 
2.3173 
2.7496 
3.1737 
3.5903 
4. 
151166 

80 
0 
0.4840 
0.9557 
1.4164 
1.86721 
2.3090 
2.74255 
3.16854 
3.58752 
4. 
158282 
Table 3. The total cost of the supplier in the case of collaboration for different values of a and n_{2}
↓a 
→n2 
1 
2 
3 
4 
5 
6 
7 
8 
9 
40 
1259810 
243811 
79609 
28827 
9359351 
2159 
896 
2906 
6900 

50 
1467320 
286674 
95520 
35871 
12449 
3263 
946 
2420 
6189 

60 
1715340 
341075 
117420 
46470 
17629 
5332 
1017 
1103 
3854 

70 
1958350 
395364 
140004 
58057 
23956 
8647 
2400 
1150 
2925 

80 
2202780 
450279 
162908 
69839 
3040 
12029 
3815 
1200 
1974 
Table 4. Optimal supplier’s cost and optimal replenishment cycles with the collaboration of different values a and n_{2}
↓a 
→ti 
t0 
t1 
t2 
t3 
t4 
t5 
t6 
t7 
t8 
t9 
$T_{J T}^s$ 
40 
0 
0.5689 
1.1105 
1.6303 
2.1318 
2.617 
3.0901 
3 4 
4. 

896 

50 
0 
0.5591 
1.0958 
1.6138 
2.1159 
2.6041 
3.0800 
3.5449 
4. 

946 

60 
0 
0.4940 
0.9717 
1.4355 
1.8869 
2.3274 
2.7582 
3.180 
3.5937 
4. 
1103 

70 
0 
0.4884 
0.9628 
1.4250 
1.8761 
2.3173 
2.7496 
3.1737 
3.5903 
4. 
1150 

80 
0 
0.4840 
0.9557 
1.4164 
1.8672 
2.309 
2.7425 
3.1685 
3.5875 
4. 
3815 
Table 5. Optimality with collaboration and without collaboration different values of a and n_{2}


with collaboration without collaboration 

Parameters 
$n_1^*$ 
$T_{I N D}^r$ 
$T_{I N D}^s$ 
$Q_{I N D}$ 
$\delta_{\min }$ 
$\delta_{\max }$ 
$n_2^*$ 
$T_{J T}^r$ 
$T_{J T}^s$ 
$Q_{J T}$ 
$\% T_{I N D}^r$ 
$\% T_{I N D}^r$ 
40 
7 
124188 
137511 
1378.8 
0.00168 
0.00160 
7 
127440 
140763 
1378.8 
2.61 
2.36 
50 
7 
135541 
172653 
1580.2 
0.00171 
0.00182 
7 
130399 
167512 
1580.2 
3.7 
2.9 
60 
8 
144159 
242246 
1633.0 
0.00178 
0.00234 
7 
115525 
213612 
1781.3 
19.8 
11.8 
70 
8 
151166 
280562 
1817.2 
0.00042 
0.00239 
8 
36393 
165791 
1817.2 
75.9 
40.9 
80 
8 
158282 
320269 
2001.2 
0.00043 
0.00252 
8 
26203 
188190 
2001.2 
83.4 
41.2 
Table 6. Sensitivity findings and analysis by variation in different parameters
%Change in Parameters 
a 
b 
$\theta$ 
$O_r$ 
CO_{2} 
A 
$S_s$ 
α 
$h_c$ 
$\tau$ 
$W_p$ 

$\frac{\triangleleft \overline{T c}_r^{C \rho}}{\overline{T c}_{r o}^{C \rho}} \times 100 \%$ 
$\left\{\begin{array}{c}20 \\ 10 \\ 0 \\ +10 \\ +20\end{array}\right.$ 
120.938 104.812 70.0103 35.064 0. 
5.994 1.773 1.155 0.564 0. 
24.794 12.177 9.003 1.275 0. 
0.164 0. 0. 0. 0. 
2.555 0. 0. 0. 0. 
18.833 10.473 10.712 5.429 0. 
3.485 0.078 0.052 0.026 0. 
13.135 6.790 8.579 4.482 0. 
0.205 0.153 0.102 0.051 0. 
3.215 0.311 0.207 0.103 0. 
4.340 0.574 0.383 0.191 0. 
$\frac{\triangleleft \overline{T c}_s^{c_\rho}}{\overline{T c}_{s o}^{c_\rho}} \times 100 \%$ 
$\left\{\begin{array}{c}20 \\ 10 \\ 0 \\ +10 \\ +20\end{array}\right.$ 
32.186 24.402 16.274 8.139 0. 
18.438 15.536 10.412 5.232 0. 
878.38 1140.69 501.20 616.63 0. 
5.142 0. 0. 0. 0. 
7.392 0. 0. 0. 0. 
115.770 144.072 21.002 10.644 0. 
2.245 0.596 0.397 0.198 0. 
176.399 193.323 16.307 8.520 0. 
0.174 0.130 0.087 0.043 0. 
2.677 0.259 0.172 0.086 0. 
1.471 1.168 0.778 0.389 0. 
$\frac{\triangleleft \bar{\delta}}{\bar{\delta}_0} \times 100 \%$ 
$\left\{\begin{array}{c}20 \\ 10 \\ 0 \\ +10 \\ +20\end{array}\right.$ 
15.196 11.028 6.945 3.290 0. 
4.162 4.936 3.216 1.572 0. 
38692.43 20318.62 2929.29 1477.71 0. 
5.204 0. 0. 0. 0. 
7.335 0. 0. 0. 0. 
288.762 277.491 0.127 0.057 0. 
2.820 0.202 0.134 0.067 0. 
322.398 309.718 0 0 0 
0.176 0.132 0.088 0.044 0. 
2.713 0.262 0.175 0.087 0. 
1.491 1.184 0.789 0.394 0. 
$n_1^*$ 
$\left\{\begin{array}{c}20 \\ 10 \\ 0 \\ +10 \\ +20\end{array}\right.$ 
8 8 8 8 8 
8 8 8 8 8 
7 7 8 8 9 
8 8 8 8 8 
9 9 9 9 9 
7 7 8 8 8 
8 8 8 8 8 
7 7 8 8 8 
8 8 8 8 8 
8 8 8 8 8 
8 8 8 8 8 
$n_2^*$ 
$\left\{\begin{array}{c}20 \\ 10 \\ 0 \\ +10 \\ +20\end{array}\right.$ 
8 8 8 8 8 
8 8 8 8 8 
7 7 8 8 9 
8 8 8 8 8 
9 9 9 9 9 
7 7 8 8 8 
8 8 8 8 8 
7 7 8 8 8 
8 8 8 8 8 
8 8 8 8 8 
8 8 8 8 8 
Figure 2. Pictorial presentation for Table 1
Figure 3. Convexity representation for Table 1
Figure 4. Pictorial presentation for Table 3
The level of sensitivity of each parameter in the Example is illustrated in Table 6. Sensitivity findings and analysis were attained by varying the value of each component that is used in the Example by $10 \%,20 \%, 20 \%$, and $10 \%$. As all of us fully understand, the lack of certainty and the unpredictable nature of real economic conditions can cause fluctuation in the values of some variables in a decisionmaking scenario. Therefore, it's indeed essential to examine the resulting changes besides altering the different parameters. This seems to be possible through comprehensive sensitivity analysis, which illustrates the effects of altering variables. Each of the parameters $O_r$, $h_c$, $\Theta$, $W_p$, $\mathrm{a}$, $\mathrm{b}$, $S_s$, $\hat{c}$, $\hat{P}_r$, $\mathrm{Co}_2$, $\tau$, $C_p$, $\delta$, $\alpha$, $\mathrm{v}_{\mathrm{c}}$, $\mathrm{e}_2$, $\mathrm{e}_1$, $\mathrm{C}_1$, $\mathrm{C}_2$, $\mathrm{~d}$, $\mathrm{~F}_{\mathrm{c}}$, $\mathrm{A}$, $\mathrm{T}$, and $\mathrm{h}$ are changed one by one %. Only one parameter value is changeable slightly at a time remains the parameters keep fixed. Sensitivity Analysis and observations for $\overline{T c}_r^{C \rho}$, $\overline{T c}_s^{C \rho}$, $\bar{\delta}$, $\mathrm{n} 1 *$ and $\mathrm{n} 2 *$ by $10 \%,20 \%$, $+20 \%$, and $+10 \%$, concentrating attention on one component at a time and keeping the remaining values fixed. This process of observation was completed with the succour of "MATHEMATICA" numerical iterative computation application version12. This is analysed in detail by using Table 6. As a result of the study, the value of $n 1^*$ and $n 2^*$ is shown in Table 6 to be highly reactive for the parameters like A, $\alpha$, and, $\Theta$ and insensitive to other left parameters such as $S_s, \mathrm{Co}_2, \tau, C_p, O_r, h_c, W_p, \mathrm{a}$, and $\mathrm{b}$.
1. The average credit $\bar{\delta}$ given by the supplier to the retailer is very sensitive to the parameters $\Theta$, and $A$, moderately sensitive to $\tau, h_c$, and practically insensitive to $S_s, \mathrm{Co}_2, C_p, O_r, W_p$, and b.
2. The total cost of the retailer's $\overline{T c}_r^{C_\rho}$ is very sensitive to the parameters, a and $A$, moderately sensitive to $\Theta$ and practically insensitive to $S_s, \mathrm{Co}_2, \tau, C_p, O_r, h_c, W_p$, and b.
3. The total cost of the supplier's $\overline{T c} c_s^{C \rho}$ is very sensitive to the parameters, a, $\Theta$, and A. moderately sensitive to $S_s, \tau, C_p, h_c, W_p$, and b. and practically insensitive to $\mathrm{Co}_2$, and $\mathrm{O}_r$.
We can apply the findings of this article in a variety of ways to build better deals. First, we can illustrate our approach to determining effective terms such as time, and no. of items and advertisement of new launched or carbonfree products, and conditions in existing markets. Second, manufacturers can make better use of their current resources by making a major contribution to a more sustainable environment (Table 6) by lowering carbon emissions caused by the manufacturing process, supply, or service based on the availability of carbon footprint measurements and identified analysis. They can determine how much of each product to generate to keep inside a GHG emissions cap decided by the government or some regulatory (policymaking) agencies. Third, producers can do a sensitivity analysis to evaluate the optimum cost (Figure 2) of a product enhancement to the longterm profit obtained from that investment. This information will also help manufacturers to reduce costs, manage and control inventory, enhance working capacity and increase customer demand for their products. Moreover, this research increases flexibility throughout the manufacturing process, leading to better resource usage and, as a result, the possibility of increased profit.
This article introduces an optimization technique for identifying the optimal ordering replenishment policies for supply chain management with time, carbon, and advertisement demands. The framework includes an optimization module that is based on an algorithm with and without coordination. A single retailer and supplier with a singleitem chain system were thoroughly examined in this study. To best our research knowledge, this is the first study of both centralized and decentralized controls for an inventory system using an advertisement demandbased optimization setting.in collaboration cases, the supplier's main purpose reduces their cost provide. Therefore, credit is offered by the supplier to the retailer. As a result of collaboration, the total cost of both suppliers as well as retailers has been reduced than without collaboration.
To avoid the complexity of the model, we have limited consideration of single supplierretailer collaboration for a single item with no shortages. But in future, we will extend this by adding some extra parameters, such as multiechelon, and carbon offset. it also can be developed by taking shortages in the future. In Future, this research study may look into the multiitem and multiechelon models with time, inventory, and advertisingdependent demand. Some other logical extension is the reworking process during the production of the defective item.
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