## 4 Model Formulation
### Model Formulation
\begin{array}{rl}&\bigg{[}\bigg{(}d_{e}\theta_{e}+\sum\limits_{i=1}^{n}d_{ i}\theta_{i}\bigg{)}+\Big{(}\beta_{0}+\frac{\beta_{1}}{p_{e}}+\beta_{2}p_{e} \Big{)}+\frac{r^{*}d_{w}}{a-bp_{w}+\delta s^{*}}\bigg{]}p_{n}\\ &=-\bigg{\{}\frac{h_{e}[p_{e}-(a-bp_{w}+\delta s^{*})]}{p_{e}(a-bp_{w}+\delta s ^{*})}-\bigg{[}\sum\limits_{i=1}^{n}h_{i}\Big{(}\frac{1}{p_{e}}-\frac{1}{p_{i }}\Big{)}\bigg{]}\bigg{\}}{p_{n}}^{2}t_{n}.\end{array}  \tag{16}\]  
Because of the left-hand side of Equation (16)  
\[\bigg{[}\bigg{(}d_{e}\theta_{e}+\sum\limits_{i=1}^{n}d_{i}\theta_{i}\bigg{)}+ \Big{(}\beta_{0}+\frac{\beta_{1}}{p_{e}}+\beta_{2}p_{e}\bigg{)}+\frac{r^{*}d_ {re}}{a-bp_{w}+\delta s^{*}}\bigg{]}p_{n}>0,\]  
then we have \(\Delta>0\), where  
\[\Delta\equiv\bigg{\{}\left[\sum\limits_{i=1}^{n}h_{i}\Big{(}\frac{1}{p_{e}}- \frac{1}{p_{i}}\bigg{)}\right]-\frac{h_{e}[p_{e}-(a-bp_{w}+\delta s^{*})]}{p_{ e}(a-bp_{w}+\delta s^{*})}\bigg{\}}{p_{n}}^{2}t_{n}.\]  
Next, taking the first-order derivative of \(G(t_{n})\) with respect to \(t_{n}\), we obtain  
\[\frac{dG(t_{n})}{dt_{n}}=\bigg{\{}\left[\sum\limits_{i=1}^{n}h_{i}\Big{(} \frac{1}{p_{e}}-\frac{1}{p_{i}}\bigg{)}\right]-\frac{h_{e}[p_{e}-(a-bp_{w}+ \delta s^{*})]}{(a-bp_{w}+\delta s^{*})p_{e}}\bigg{\}}{p_{n}}^{2}=\frac{\Delta }{t_{n}}>0. \tag{17}\]  
Therefore, \(G(t_{n})\) is a strictly increasing function \(t_{n}\in(0,\infty)\). **Theorem 1**. : _For any given \(t_{n}\geq 0\), we consider the interval \(t_{n}\in(0,\infty)\),_  
_(a) If \(G(t_{n})<0\), then the solution \((s^{*},t_{n}^{*},r^{*})\) which maximizes \(TP(s,t_{n},r)\) not only exists but also is unique, and \(t_{n}^{*}\in(0,\infty)\)._  
_(b) If \(G(t_{n})\geq 0\), then the optimal value of \(t_{n}\) is \(t_{n}^{*}\to 0\). The production system should not be opened._  
**Proof. **  
(a) Firstly, we consider the interval \(t_{n}\in(0,\infty)\). Because \(\lim_{t_{n}\rightarrow\infty}G(t_{n})=\infty\) and \(\lim_{t_{n}\to 0}G(t_{n})<0\), from the Intermediate Value Theorem, we can find a unique solution \(t_{n}^{*}\in(0,\infty)\) such that \(G(t_{n}^{*})=0\). Substituting \(t_{n}^{*}\) into Equations (8)-(10), the corresponding \(s^{*}\) and \(r^{*}\) can be determined. Furthermore, we also calculate that  
\[\mathbf{H}_{11}=\left.\frac{\partial^{2}TP(s,r|t_{n})}{\partial s^{2}}\right| _{(s,r)=(s^{*},r^{*})}=2\Bigg{\{}\bigg{[}\frac{\delta^{2}p_{n}t_{n}(h_{e}p_{n} t_{n}+rd_{re})}{(a-bp_{w}+\delta s)^{3}}\bigg{]}-(\beta+\delta)\Bigg{\}},\]\[\mathbf{H}_{22}=\left.\frac{\partial^{2}TP(s,r|t_{n})}{\partial r^{2}}\right|_{(s \prime)=(s^{*},r^{*})}=-2\alpha<0,\]  
and  
\[\mathbf{H}_{12}=\mathbf{H}_{21}=\left.\frac{\partial^{2}TP(s,\ r|t_{n})}{ \partial s\partial r}\right|_{(s\prime)=(s^{*},r^{*})}=0.\]  
Therefore, the determinant of the Hessian matrix at the stationary point \((s^{*},r^{*})\) is  
\[ \begin{array}{ll}\text{det}(H)&=\mathbf{H}_{11}\times\mathbf{H}_{22}-\mathbf{ H}_{12}\times\mathbf{H}_{21}\\ &=-4\alpha\bigg{\{}\bigg{[}\frac{\partial^{2}p_{n}t_{n}(h_{n}p_{n}t_{n}+d_{n ^{\prime}})}{(a-bp_{w}+\delta)^{3}}\bigg{]}-(\beta+\delta)\bigg{\}}<0.\end{array} \]  
(b) From Equation (9), we obtain that \(\frac{\partial TP(s,t_{n},r)}{\partial s}<0\), which implies that \(t_{n}\) causes a higher value \(TP(s,t_{n},r)\). Hence, the maximum value \(TP(s,t_{n},r)\) occurs at the point \(t_{n}^{*}\to 0\). It seems reasonable to conclude that the production system will not be opened. This completes the proof. \(\square\)  
Summarizing the above results, the Algorithm 1 was used to obtain the optimal solution to our problem. ```
Step 1:  Start with \(i=1\) and \(t_{i}\to t_{n}\). Step 2:  Put \(t_{i}\) into Equation (12) to obtain the corresponding value of \(t_{n}\), i.e., \(t_{n}\)\({}^{\prime}\). If \(G(t_{n}\)\({}^{\prime})<0\), go to Step 4. Otherwise, set the optimal solutions \(r^{*}=0,s^{*}=0\), \(t_{n}^{*}=0\) (i.e., the manufacturer will not adopt CSR initiatives), then stop the Algorithm. Step 4:  Find the optimal value \(t_{n}\)\({}^{\prime}\) such that \(G(t_{n}\)\({}^{\prime})=0\).