## 5 Numerical Examples  
In this section, we use several numerical examples by referring to previous literature (such as [17; 22; 36]) and try to apply reasonable data to obtained the results from the previous section and perform sensitivity analysis for the parameters considered in the proposed model:  
**Example 1**. : _Consider an inventory situation where \(P=5000\) units/year, \(A_{R}=\$200\)order, \(A_{M}=\$300\)order, \(S=\$500\) setup, \(c_{1}=\$3\)/unit, \(c_{2}=\$10\)/unit, \(v=\$50\)/unit, \(h_{b}=\$0.5\)/unit/year, \(h_{p}=\$0.3\)/unit/year, \(h_{m}=\$0.3\)/unit/year, \(C_{T}=\$50\)/ship, \(C_{t}=\$3\)/unit, \(\theta_{1}=0.05\), \(\theta_{2}=0.1\), \(r=\)1/unit, \(\hat{A}_{R}=30\) kg/order, \(\hat{A}_{M}=50\) kg/order, \(\hat{S}=100\) kg/setup, \(\hat{c}_{1}=0.3\) kg/unit, \(\hat{c}_{2}=0.5\) kg/unit, \(\hat{v}=1\) kg/unit, \(\hat{h}_{b}=0.05\) kg/unit/year, \(\hat{h}_{p}=0.03\) kg/unit/year, \(\hat{h}_{m}=0.01\) kg/unit/year, \(\hat{C}_{T}=3\) kg/ship, \(\hat{C}_{t}=0.05\)/unit, \(C=\$0.5\)/unit, \(\beta=0.5\), and \(D(p)=(a-bp)\) units/year, with \(a=1000\) and \(b=8\). Based on the above more realistic parameters, the purpose of this example is to show the manufacturer's optimal production, shipping and investing polices and the retailer's pricing and ordering polices when Stackelberg equilibrium is achieved. The solution procedure of the proposed model is shown as in Table 2. Applying the above algorithm, the manufacturer's optimal numbers of shipment \(n^{\star\star}=3\), shipping quantity \(q^{\star\star}=162.385\), and technology investment for reducing carbon emissions \(\omega^{\star}=39.5397\), and the retailer's optimal selling price per unit \(p^{\star\star}=90.0145\) and order quantity \(Q^{\star\star}=487.155\), when achieving a Stackelberg equilibrium with the carbon tax regulation. Furthermore, the manufacturer's total profit \(TP^{\star\star}_{\text{CT}\eta}\) is \(\$93975.44\), whereas the retailer's total profit \(TP^{\star\star}_{\text{CT}\eta}\) is \(\$9308.23\)._  
**Example 2**. : _To explore the impact of changes in carbon emissions from various operating activities or carbon tax on the optimal equilibrium solutions, this example outlines the effects of changes in carbon parameters \(\hat{A}_{R}\), \(\hat{v}\), \(\hat{h}_{b}\), \(\hat{C}_{T}\), \(\hat{C}_{t}\), \(\hat{A}_{M}\), \(\hat{S}\), \(\hat{c}_{1}\), \(\hat{c}_{2}\), \(\hat{h}_{v}\), \(\hat{h}_{m}\), and \(C\) on the optimal solutions. The sensitivity analysis is performed by changing each of the parameters by \(+20\%\), \(+10\%\), \(-10\%\), and \(-20\%\), taking one parameter at a time and keeping the remaining parameters unchanged. Changes in carbon emission-related parameters are relatively insensitive to the optimal equilibrium solutions or profits. The computational results are presented in Table 3, showing these observations._  
1. _Except for carbon taxes, changes in carbon emission-related parameters are relatively insensitive to the optimal equilibrium solutions or profits. Nevertheless, for the sustainable development of the enterprise, it is necessary to take carbon emission parameters into consideration of the proposed model._
2. _The optimal equilibrium technology investment for carbon emission reduction increases with an increase in the values of the carbon emission parameters, except for the amounts of fixed carbon emissions per order or per shipment for the retailer and the amount of carbon emission to hold finished goods inventory for the manufacturer._
3. _The retailer may increase the optimal equilibrium selling price to respond to an increase in the carbon emission parameters, amount of carbon emission to hold finished goods inventory for the manufacturer, and carbon tax. When the manufacturer's other carbon parameters increase, the retailer's optimal equilibrium selling price decreases._
4. _The manufacturer's optimal equilibrium shipping quantity and the retailer's optimal equilibrium order quantity decrease with an increase in the value of the carbon emission parameters, except for the amounts of