## 4 Model Formulation and Solution
### Retailer's Total Profit with Carbon Tax  
The retailer's inventory level of finished goods at time \(t\) during the replenishment cycle changes due to a price-dependent demand and the deterioration of the finished goods, as represented by the differential equation:  
\[dI_{R}(t)/dt+\theta_{2}I_{R}(t)=-D(p),\ \ 0\leq t\leq T_{b}. \tag{1}\]  
Assuming the boundary condition \(I_{R}(T_{b})=0\), Equation (1) transforms into:  
\[I_{R}(t)=\frac{D(p)}{\theta_{2}}\Big{[}e^{\theta_{2}(T_{b}-t)}-1\Big{]},\ 0\leq t \leq T_{b}, \tag{2}\]  
where the quantity shipped by the manufacturer to the retailer per shipment \(q\) can be described by:  
\[q=I_{R}(0)=\frac{D(p)}{\theta_{2}}\Big{(}e^{\theta_{2}T_{b}}-1\Big{)} \tag{3}\]  
The retailer's total profit per unit time is determined by sales revenue, ordering cost, purchasing cost, transportation cost, holding cost, and carbon emission reduction investment, which are evaluated as follows:  
1. The sales revenue per unit time is \(PD(P)\)
2. The ordering cost per unit time is \(A_{R}/T_{b}\)
3. The retailer's purchasing cost per unit time is: \[\frac{vq}{T_{b}}=\frac{vD(p)}{\theta_{2}T_{b}}\Big{(}e^{\theta_{2}T_{b}}-1\Big{)}\]
4. The retailer's transportation cost consists of fixed and variable costs per unit time and is given by: \[\frac{C_{T}+C_{t}q}{T_{b}}=\frac{C_{T}}{T_{b}}+\frac{C_{t}D(p)}{\theta_{2}T_{ b}}\Big{(}e^{\theta_{2}T_{b}}-1\Big{)}\]
5. The retailer's holding cost per unit time is: \[\frac{h_{b}}{T_{b}}\int_{0}^{T_{b}}I_{R}(t)dt=\frac{h_{b}}{T_{b}}\int_{0}^{T_{ b}}\frac{D(p)}{\theta_{2}}\Big{[}e^{\theta_{2}(T_{b}-t)}-1\Big{]}dt=\frac{h_{b}D(p)} {\theta_{2}^{2}T_{b}}\Big{(}e^{\theta_{2}T_{b}}-\theta_{2}T_{b}-1\Big{)}\]
6. The technological investment in carbon emission reduction is \(\omega\). Because the technology investment is shared between the retailer and manufacturer, with \(\beta\) (\(0\leq\beta\leq 1\)) being the proportion of the retailer's investment, the investment reduces the carbon emission per unit time for the retailer as indicated by \(\beta\omega\). Considering the points above, the retailer's total profit per unit time can be formulated as: \[TP_{b}(p,\ T_{b})=pD(p)-\frac{1}{T_{b}}\Bigg{[}A_{R}+C_{T}+\frac{(C_{t}+v)D(p )}{\theta_{2}}\Big{(}e^{\theta_{2}T_{b}}-1\Big{)}+\frac{h_{b}D(p)}{\theta_{2} ^{2}}\Big{(}e^{\theta_{2}T_{b}}-\theta_{2}T_{b}-1\Big{)}\Bigg{]}-\beta\omega\] (4)  
Subsequently, the amount of carbon emissions produced by the retailer per unit time is relative to the ordering, transportation, purchase, and holding inventory; thus, it can be reduced through investment in carbon emission technology. At a reduction rate of the total carbon emissions \(m(\omega)\), the amount of carbon emissions produced by the retailer per unit time is defined as:  
\[E_{b}(p,\ T_{b})=\frac{[1-m(\omega)]}{T_{b}}\Bigg{[}\hat{A}_{R}+\hat{C}_{T}+ \frac{\Big{(}\hat{C}_{t}+\vartheta\Big{)}D(p)}{\theta_{2}}\Big{(}e^{\theta_{2} T_{b}}-1\Big{)}+\frac{\hat{h}_{b}D(p)}{\theta_{2}^{2}}\Big{(}e^{\theta_{2}T_{b}}- \theta_{2}T_{b}-1\Big{)}\Bigg{]}. \tag{5}\]  
With the carbon tax regulation, the retailer pays a \(C\) amount of money for each unit of carbon emitted [37]. Hence, the retailer's total profit per unit time with carbon tax is represented by:  
\[\begin{array}{ll}&TP_{CTb}(p,\ T_{b})=TP_{b}(p,\ T_{b})-CE_{b}(p,\ T_{b})\\ =pD(p)-\frac{1}{T_{b}}&\Big{\{}(A_{R}+C_{T})+[1-m(\omega)]\Big{(}\hat{A}_{R}+ \hat{C}_{T}\Big{)}\\ &+\frac{\big{[}(C_{t}+v)+[1-m(\omega)]\big{(}\hat{C}_{t}+\hat{\phi}\big{)}D(p )}{\theta_{2}}\Big{(}e^{\theta_{2}T_{b}}-1\Big{)}\\ &+\frac{\big{[}h_{b}+[1-m(\omega)]h_{b}\big{]}D(p)}{\theta_{2}^{2}}\Big{(}e^{ \theta_{2}T_{b}}-\theta_{2}T_{b}-1\Big{)}\Big{\}}-\beta\omega.\end{array} \tag{6}\]