## 4 Model Formulation
### Model Formulation
\begin{array}{rcl}TP(s,t_{n},r)&=&(SR-SC-HC_{e}-HC_{c}-DC-RC-PC)\\ &=&(p_{w}-s+c_{1}r)(a-bp_{w}+\delta s)\\ &&-\bigg{[}\frac{h_{e}[p_{e}-(a-bp_{w}+\delta s)]}{(a-bp_{w}+\delta s)p_{e}}+ \bigg{[}\sum\limits_{i=1}^{n}h_{i}\Big{(}\frac{1}{p_{e}}-\frac{1}{p_{i}} \Big{)}\bigg{]}\bigg{]}\frac{p_{n}^{2}t_{n}^{2}}{2}\\ &&-\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{nd_{w} }{a-bp_{w}+\delta s}\bigg{]}p_{n}t_{n}\\ &&+(d_{re}-c_{1})\sum\limits_{i=1}^{m}t_{ndi}\\ &&-\beta s^{2}-\alpha r^{2}-k.\end{array}  \tag{7}\]  
Aiming to address this nonlinear programming issue, the authors first ignore the restriction and take the first-order derivation of \(TP(s,t_{n},r)\) with respect to \(s,t_{n},r\), respectively. We obtain  
\[ \begin{array}{rl}\frac{\partial TP(s,t_{n},r)}{\partial s}=&[\delta c_{1}r+ (b-\beta)p_{w}+(\beta+\delta)(p_{w}-2s)-a]\\ &&-(p_{n}t_{n}h_{e}+2 rd_{re})\bigg{[}\frac{\delta}{(a-bp_{w}+\delta s)^{2}} \bigg{]}\frac{p_{n}t_{n}}{2},\\ &&\\ \frac{\partial TP(s,t_{n},r)}{\partial t_{n}}=&-\bigg{\{}\frac{h_{e}[p_{e}-(a -bp_{w}+\delta s)]}{(a-bp_{w}+\delta s)p_{e}}+\bigg{[}\sum\limits_{i=1}^{n}h_{ i}\Big{(}\frac{1}{p_{e}}-\frac{1}{p_{i}}\Big{)}\bigg{]}\bigg{\}}{p_{n}}^{2}t_{n} \\ &&-\bigg{[}\bigg{(}d_{e}\theta_{e}+\sum\limits_{i=1}^{n}d_{i}\theta_{i}\bigg{)} +\Big{(}\beta_{0}+\frac{\beta_{1}}{p_{e}}+\beta_{2}p_{e}\bigg{)}+\frac{rd_{re }}{a-bp_{w}+\delta s}\bigg{]}p_{n},\end{array}  \tag{8}\]  
and  
\[\frac{\partial TP(s,t_{n},r)}{\partial r}=c_{1}(a-bp_{w}+\delta s)+(d_{re}-c_{ 1})\sum\limits_{i=1}^{m}t_{ndi}-2\alpha r. \tag{10}\]  
To find the optimal solution of \((s,t_{n},r)\), let \(\partial TP(s,t_{n},r)/\partial s=0\), \(\partial TP(s,t_{n},r)/\partial t_{n}=0\), and \(\partial TP(s,t_{n},r)/\partial r=0\), simultaneously. Solving these three equations, we obtain  
\[\delta c_{1}r+(b+\delta)p_{w}=\Bigg{[}\frac{p_{n}t_{n}h_{e}+2 rd_{re}}{2(a-bp_{ w}+\delta s)^{2}}\bigg{]}\delta p_{n}t_{n}+a+2s(\delta+\beta), \tag{11}\]  
\[ \begin{array}{l}\bigg{\{}\frac{h_{e}[p_{e}-(a-bp_{w}+\delta s)]}{(a-bp_{w}+ \delta s)p_{e}}+\bigg{[}\sum\limits_{i=1}^{n}h_{i}\Big{(}\frac{1}{p_{e}}-\frac {1}{p_{i}}\Big{)}\bigg{]}\bigg{\}}p_{n}^{2}t_{n}+\frac{rd_{w}}{a-bp_{w}+\delta s }p_{n}\\ \\ =-\bigg{[}\bigg{(}d_{e}\theta_{e}+\sum\limits_{i=1}^{n}d_{i}\theta_{i}\bigg{)} -\Big{(}\beta_{0}+\frac{\beta_{1}}{p_{e}}+\beta_{2}p_{e}\bigg{)}\bigg{]}p_{n}, \end{array}  \tag{12}\]  
and  
\[r=\frac{c_{1}(a-bp_{w}+\delta s)+(d_{re}-c_{1})\sum\limits_{i=1}^{m}t_{ndi}}{2 \alpha}. \tag{13}\]  
From Equations (11)-(13), it is clear that \(s\) and \(r\) can be uniquely determined as functions of \(t_{n}\). To analyze the inventory problem, we will show that for any given feasible \((s^{*},r^{*})\), the optimal production run time also exists and is unique. For given \(s^{*}\) and \(r^{*}\), the first-order necessary condition for \(TP(\ t_{n}|s^{*},r^{*})\) to be maximum is \[\frac{dTP(t_{n}|s^{*},r^{*})}{dt_{n}} = -\bigg{\{}\frac{h_{e}[p_{e}-(a-bp_{w}+\delta s)]}{(a-bp_{w}+\delta s )p_{e}}-\bigg{[}\sum\limits_{i=1}^{n}h_{i}\Big{(}\frac{1}{p_{e}}-\frac{1}{p_{i}} \Big{)}\bigg{]}\bigg{\}}{p_{n}}^{2}t_{n} \tag{14}\] \[-\bigg{[}\bigg{(}d_{e}\theta_{e}+\sum\limits_{i=1}^{n}d_{i}\theta _{i}\bigg{)}+\Big{(}\beta_{0}+\frac{\beta_{1}}{p_{e}}+\beta_{2}p_{e}\Big{)}+ \frac{rd_{w}}{a-bp_{w}+\delta s}\bigg{]}p_{n}\] \[= 0.\]  
Let \(G(t_{n})\) be the left-hand side of Equation (14), i.e.,  
\[ \begin{array}{rl}G(t_{n})=&-\bigg{\{}\frac{h_{e}[p_{e}-(a-bp_{w}+\delta s^{* })]}{(a-bp_{w}+\delta s^{*})p_{e}}-\bigg{[}\sum\limits_{i=1}^{n}h_{i}\Big{(} \frac{1}{p_{e}}-\frac{1}{p_{i}}\Big{)}\bigg{]}\bigg{\}}{p_{n}}^{2}t_{n}\\ &-\bigg{[}\bigg{(}d_{e}\theta_{e}+\sum\limits_{i=1}^{n}d_{i}\theta_{i}\bigg{)} +\Big{(}\beta_{0}+\frac{\beta_{1}}{p_{e}}+\beta_{2}p_{e}\Big{)}+\frac{r^{*}d_ {w}}{a-bp_{w}+\delta s^{*}}\bigg{]}p_{n}.\end{array}  \tag{15}\]  
We first rewrite Equation (14) and have  
\[