A Collaborating Supply Chain Inventory Model Including Linear Time-Dependent, Inventory, and Advertisement-Dependent Demand Considering Carbon Regulations

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 low-emitting 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 two-level 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 inventory-dependent 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 end-users 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], transportation-related 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 carbon-dependent 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 inventory-dependent demand, and carbon emission due to refrigeration and transportation technology are considered to design a long-term supply chain.

Table 1. Comparison of literature

Research problem:

This article considers linear time-dependent, inventory-dependent, and advertisement-dependent 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 fast-moving consumer goods near deterioration or expiration date. Their demand rises or diminished according to time, inventory and advertisement. The trade-credit 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.

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 time-dependent. 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.

1.png

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.

$I_i(t)=\int_t^{t_{i+1}} A^\alpha\left(a+\mathrm{b} * \mathrm{u}-\rho C o_2\right) e^{\theta(u-t)} \mathrm{du}$      (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}^{n-1} 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(u-t)} 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 refrigerator-consumed 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}^{n-1} \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(u-t)} 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).

$\begin{gathered}T_{I N D}^r=n_1 * O_r+\hat{c}+\left\{\frac{\left(h_c+\tau \hat{h}_c\right)}{\theta}+W_p+\hat{P}_r+\mathrm{d} * v_c C_2+\mathrm{d} e_2\right\} \sum_{i=0}^{n_1-1} \int_{t_i}^{t_{i+1}} A^\alpha(a+\mathrm{bt}- \\ \left.\rho C o_2\right) e^{\theta\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t-\frac{A^\alpha\left(h_c+\tau \hat{h}_c\right)}{\theta}\left(a T+\frac{1}{2} b T^2\right)+F_c+2 \mathrm{~d} v_c C_1+2 \mathrm{~d} e_1\end{gathered}$       (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).

$\begin{gathered}T_{J T}^s=n_2 *\left(S_c+O_r\right)+C_p * \sum_{j=0}^{n_2-1} \int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{b} \mathrm{t}-\rho C O_2\right) e^{\theta\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t+\left\{\frac{\left(h_c+\tau \hat{h}_c\right)}{\theta}+W_p+\hat{P}_r+\mathrm{d} * v_c C_2+\mathrm{d} e_2\right\} \\ \sum_{i=0}^{n_2-1} \int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{bt}-\rho C o_2\right) e^{\theta\left(t-t_i\right)} \mathrm{d} t-\frac{A^\alpha\left(h_c+\tau \hat{h}_c\right)}{\theta}\left(a T+\frac{1}{2} b T^2\right)+F_c+2 \mathrm{~d} v_c C_1+2 \mathrm{~d} e_1- \\ T_{I N D}^r\left(n_1^*, t_0, t_1^*, \ldots \ldots \ldots t_{n_1}^*\right)\end{gathered}$     (11)

$Q_{J T}=\sum_{j=0}^{n_2^*-1} Q_j^{\prime *}$       (12)

$\begin{gathered}T_{J T}^r=n_2 * O_r+\hat{c}+\left\{\frac{\left(h_c+\tau \hat{h}_c\right)}{\theta}+W_p+\hat{P}_r+\mathrm{d} * v_c C_2+\mathrm{d} e_2\right\} \sum_{i=0}^{n_1-1} \int_{t_j}^{j_{j+1}} A^\alpha(a+\mathrm{bt}- \\ \left.\rho C o_2\right) e^{\theta\left(\mathrm{t}-\mathrm{t}_j\right)} \mathrm{d} t-\frac{A^\alpha\left(h_c+\tau \hat{h}_c\right)}{\theta}\left(a T+\frac{1}{2} b T^2\right)+F_c+2 \mathrm{~d} v_c C_1+2 \mathrm{~d} e_1- \\ T_{I N D}^r\left(n_1^*, t_0, t_1^*, \ldots \ldots \ldots t_{n_1}^*\right)\end{gathered}$     (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)

$\begin{gathered}\delta_{\max }=T_{I N D}^s\left(n_1^*, t_0, t_1^*, \ldots \ldots \ldots . t_{n_1}^*\right)-n_1^* * S_c-C_p * \sum_{i=0}^{n_1^*-1} \int_{t_i}^{t_{i+1}} A^a(a+ \\ \left.\mathrm{bt}-\rho C o_2\right) e^{\theta\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t / \sum_{j=0}^{n_2^*-1} h\left(t_{\mathrm{j}+1}^{\prime *}-t_{\mathrm{j}}^{\prime *}\right) Q_j^{\prime *}\end{gathered}$     (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)

gong_.png

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).

$\begin{gathered}\frac{\partial T_{I N D}^r}{\partial t_i}=A^\alpha\left(a+\mathrm{b} t_i+\rho C o_2\right)\left(e^{\theta\left(\mathrm{t}_i-t_{i-1}\right)}-1\right)- \\ \theta \int_{t_i}^{t_{i+1}} A^\alpha\left(a+\mathrm{bt}-\rho C o_2\right) e^{\theta\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t\end{gathered}$      (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) $\frac{\partial^2 T_{I N D}^r}{\partial t_i^2} \geq\left|\frac{\partial^2 T_{I N D}^r}{\partial t_i t_{i-1}}\right|+\left|\frac{\partial^2 T_{I N D}^r}{\partial t_i t_{i+1}}\right|$ 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)

$\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i-1}}=-\theta A^\alpha\left(a+\mathrm{b} t_i-\rho C o_2\right) e^{\theta T_i}$      (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, i-1.

$T_{c r}$ is positive definite if Eqns. (21)-(24) satisfy the given inequality.

$\begin{gathered}\frac{\partial^2 T_{I N D}^r}{\partial t_i^2}>\left|\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i-1}}\right|+\left|\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i+1}}\right|+\left|\frac{\partial^2 T_{I N D}^r}{\partial t_i \partial t_{i+1}}\right| \\ \theta\left(a+\mathrm{b} t_i-\rho C o_2\right) e^{\theta T_i}+\theta\left(a+\mathrm{b} t_{i+1}-\rho C o_2\right) e^{\theta T_{i+1}}+A^\alpha b\left(e^{\theta T_i}-e^{\theta T_{i+1}}\right)> \\ \left|-\theta\left(a+\mathrm{b} t_i-\rho C o_2\right) e^{\theta T_i}\right|+\left|-\theta\left(a+b t_{i+1}\right) e^{\theta T_{i+1}}\right|+|0| \\ \theta\left(a+\mathrm{b} t_i-\rho C o_2\right) e^{\theta T_i}+\theta\left(a+\mathrm{b} t_{i+1}-\rho C o_2\right) e^{\theta T_{i+1}}+A^\alpha b\left(e^{\theta T_i}-e^{\theta T_{i+1}}\right)> \\ \theta\left(a+\mathrm{b} t_i-\rho C o_2\right) e^{\theta T_i}+\theta\left(a+\mathrm{b} t_{i+1}-\rho C o_2\right) e^{\theta T_{i+1}}\end{gathered}$

that is true for all i = 1, 2, ..., n.

The 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 decision-making 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 version-12. 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 $\mathrm{O}_r$.