## 4 Model Formulation and Solution
### Manufacturer's Total Profit with Carbon Tax  
Figure 2 presents the manufacturer's inventory level of raw materials in a complete production cycle. In the cycle, upon receipt of the retailer's order (\(Q\) units), the manufacturer purchases raw materials from an original material supplier. Assuming that one unit of the finished product requires \(r\) units of raw materials and the materials deteriorate during storage, the manufacturer's inventory level fluctuates, with respect to the use of materials for production and to the deterioration of the materials during the time interval \([0,T_{s}]\). As such, the inventory level changes at time \(t\) within \([0,T_{s}]\), which can be represented in the differential equation:  
\[dI_{M}(t)/dt+\theta_{1}I_{M}(t)=-rP,\ \ \ 0\leq t\leq T_{s} \tag{7}\]  
Assuming the boundary condition \(I_{M}(T_{s})=0\), the manufacturer's inventory level of materials per production cycle is given by:  
\[I_{M}(t)=\frac{rP}{\theta_{1}}\Big{[}e^{\theta_{1}(T_{s}-t)}-1\Big{]},\ \ \ 0\leq t\leq T_{s}, \tag{8}\]  
from which the total amount of raw materials per production cycle \(q_{M}\) is:  
\[q_{M}=I_{M}(0)=\frac{rP}{\theta_{1}}\Big{(}e^{\theta_{1}T_{s}}-1\Big{)}. \tag{9}\]  
To comply with JIT, the manufacturer delivers products to the retailer immediately at the start of the production cycle while producing \(q=Q/n\) units of the finished products. Moreover, regular shipments of fixed quantity (\(q\)units) are made in succession every \(T_{b}\) until the \(n\) number of shipments per production cycle is fulfilled (Figure 2). Accordingly, the manufacturer's inventory level of finished goods changes due to the production and deterioration of items within \([0,T_{s}]\). Because the production rate exceeds the demand rate, the manufacturer stops the production after a certain inventory level \(I_{max}\) is achieved and then gradually decreases with the deterioration of finished goods during \([T_{s},T_{v}]\). On the other hand, the manufacturer delivers products to the retailer for \(n\) times throughout the time period \([0,T_{v}]\). The manufacturer's and retailer's cumulative inventory of finished goods is presented in Figure 3.  
The manufacturer's inventory level of finished goods at time \(t\) during the time interval \([0,T_{s}]\) is governed by the differential equation:  
\[dl_{P}(t)/dt+\theta_{2}I_{p}(t)=P,\ \ \ 0\leq t\leq T_{s} \tag{10}\]  
Assuming \(I_{p}(0)=0\), the manufacturer's inventory level of finished goods becomes:  
\[I_{P}(t)=\frac{P}{\theta_{2}}\Big{(}1-e^{-\theta_{2}t}\Big{)},\ \ \ 0\leq t\leq T_{S} \tag{11}\]  
After completing the first \(q\) units at time \(T_{p}\), the manufacturer immediately ships the products to the retailer. This is governed by the relation \(I_{p}\Big{(}T_{p}\Big{)}=q=\frac{D(p)}{\theta_{2}}\Big{(}e^{\theta_{2}T_{b}}-1 \Big{)}\) from Equation (3), implying that:  
\[T_{p}=\frac{1}{\theta_{2}}\ln\left[\frac{P}{P-D(p)(e^{\theta_{2}T_{b}}-1)}\right] \tag{12}\]  
Thus, at \([T_{s},T_{p}]\), the manufacturer no longer produces, which results in a drop in the inventory level due to material deterioration. At a certain time \(t\), the inventory level of the finished goods is governed by the rule:  
\[dl_{d}(t)/dt+\theta_{2}I_{d}(t)=0,\ \ \ T_{s}\leq t\leq T_{v} \tag{13}\]  
Assuming the boundary condition \(I_{d}(T_{v})=nq=\frac{nD(p)}{\theta_{2}}\Big{(}e^{\theta_{2}T_{b}}-1\Big{)}\), Equation (13) can be solved to determine the inventory level during the interval \([T_{s},T_{v}]\), which is given by:  
\[I_{d}(t)=\frac{nD(p)}{\theta_{2}}\Big{(}e^{\theta_{2}T_{b}}-1\Big{)}e^{\theta_ {2}(T_{v}-t)},\ T_{s}\leq t\leq T_{v} \tag{14}\]  
Figure 3: Manufacturer’s and retailer’s cumulative inventory of finished goods.