# A Discount Technique-Based Inventory Management on Electronics Products Supply Chain
## 4 Theoretical Derivations
\right)\\ +\big{(}\big{(}bp-a-\alpha T-\beta T^{2}\big{)}s^{2}+(\alpha+2\beta T)s-2\beta \big{)}\Big{(}1-e^{s(T-t_{1})}\big{)}\\ +(sp-C_{h}\big{)}\Big{[}(bT-b\delta t_{1})s^{2}+sb\left(\delta(1-e^{s_{1}})+e^ {st_{1}}-e^{s(T-t_{1})}\right)\Big{]}-\\ \left(1-e^{st_{1}}\right)\left[ \begin{array}{l}(2b\delta ps^{2}-b\delta C_{ h}s)e^{s_{1}}-\left(\begin{array}{l}(2b\delta ps^{2}-b\delta C_{h}s)e^{s_{1}} -\left(\begin{array}{l}(2b\delta ps^{2}-b\delta C_{h}s)e^{s_{1}}-\left( \begin{array}{l}(\alpha+2\beta T-bC_{h})s-2\beta\\ +(\alpha+2\beta T-bC_{h})s-2\beta\end{array}\right)e^{sT}\\ +(3 bp(1-\delta)-2a)s^{2}+(2\alpha-C_{h}b(1-\delta))s-4\beta\end{array}\right] \end{array}\right]\end{array}  \tag{18}\]  
\[\frac{\partial\varphi}{\partial T}=\frac{1}{s}\left[ \begin{array}{l}(sp-C_{ h})e^{s(T-t_{1})}+(sp-C_{h})e^{s(T+t_{1})}\\ -\big{(}\big{(}C_{p}+T_{\rm{vc}}+p\big{)}s-C_{h}\big{)}e^{sT}+C_{h}\end{array}  \right]\left(\beta T^{2}+\alpha T-bp+a\right) \tag{19}\]  
The second order partial derivatives of \(\varphi(p,T)\) with respect to \(p\) and \(T\) are as follows:  
\[\frac{\partial^{2}\omega}{\partial p^{2}}=-\frac{2b}{s}\Big{(}e^{sT}\big{(}e^ {st_{1}}-1\big{)}+\Big{(}e^{sT-st_{1}}-2e^{st_{1}}-\delta e^{2 st_{1}}\Big{)}+ \delta\big{(}3e^{st_{1}}-2\big{)}+1\Big{)} \tag{20}\]  
\[\frac{\partial^{2}\varphi}{\partial T^{2}}=-\frac{1}{s}\left[ \begin{array}{ l}\left(\begin{array}{l}\big{(}(C_{p}+T_{\rm{vc}}+p)s-C_{h}\big{)}e^{sT}\\ -(sp-C_{h})e^{s(T-t_{1})}-(sp-C_{h})e^{s(T+t_{1})}\end{array}\right)\left( \beta T^{2}+\alpha T-bp+a\right)s\\ +\left(\begin{array}{l}\big{(}(C_{p}+T_{\rm{vc}}+p)s-C_{h}\big{)}e^{sT}-C_{h }-\\ (sp-C_{h})e^{s(T-t_{1})}-(sp-C_{h})e^{s(T+t_{1})}\end{array}\right)\left(2\beta T +\alpha\right)\end{array} \right] \tag{21}\]  
\[\frac{\partial^{2}\varphi}{\partial p\partial T}=\frac{1}{s}\left[ \begin{array}[ ]{l}\big{(}\big{(}T^{2}\beta+T\alpha-2 bp+a\big{)}s+bC_{h}\big{)}\Big{(}e^{s( T-t_{1})}+e^{s(T+t_{1})}\big{)}-bC_{h}\\ +\big{(}\big{(}\big{(}T_{\rm{vc}}+2p+C_{p}\big{)}b-T^{2}\beta-T\alpha-a\big{)} s-bC_{h}\big{)}e^{sT}\end{array} \right] \tag{22}\]  
The Hessian matrix for \(\varphi(p,T)\) is as follows:  
\[H=\left| \begin{array}{cc}\frac{\partial^{2}\varphi}{\partial p^{2}}&\frac{ \partial^{2}\varphi}{\partial T\partial p}\\ \frac{\partial^{2}\varphi}{\partial p\partial T}&\frac{\partial^{2}\varphi}{ \partial T^{2}}\end{array} \right| \tag{23}\]  
The first principal diagonal minor  
\[D_{1} =\frac{\partial^{2}\varphi}{\partial p^{2}}\] \[=-\frac{2b}{s}\big{(}e^{sT}\big{(}e^{st_{1}}-1\big{)}+\big{(}e^ {sT-st_{1}}-2e^{st_{1}}-\delta e^{2 st_{1}}\big{)}+\delta\big{(}3e^{st_{1}}-2 \big{)}+1\big{)}\]  
From Proposition 1, \(D_{1}\) is always negative. The second principal diagonal minor  
\[D_{2}=\Big{(}\frac{\partial^{2}\varphi}{\partial p^{2}}\Big{)} \Big{(}\frac{\partial^{2}\varphi}{\partial T^{2}}\Big{)}-\Big{(}\frac{\partial^ {2}\varphi}{\partial p\partial T}\Big{)}\] \[=\frac{2b}{s^{2}}\left[ \begin{array}{l}e^{sT}\big{(}e^{st_{1}}-1 \big{)}+\delta\big{(}3e^{st_{1}}-2\big{)}\\ +\big{(}e^{sT-st_{1}}-2e^{st_{1}}-\delta e^{2 st_{1}}\big{)}\\ +1\end{array} \right]\left[\begin{array}{l}\left( \begin{array}{l}\big{(}(C_{p }+T_{\rm{vc}}+p)s-C_{h}\big{)}e^{sT}-\\ (sp-C_{h}\big{)}\Big{(}e^{s(T-t_{1})}+e^{s(T+t_{1})}\big{)}\\ -C_{h}(2\beta T+\alpha\big{)}\end{array} \right)\left( \begin{array}{l}s\beta T ^{2}+s\alpha T-sbp\\ +as+2\beta T+\alpha\end{array} \right)\\ -\frac{1}{s^{2}}\left[ \begin{array}{l}\big{(}(T^{2}\beta+T\alpha-2 bp+a\big{)}s +bC_{h}\big{)}\Big{(}e^{s(T-t_{1})}+e^{s(T+t_{1})}\big{)}-bC_{h}\\ +\big{(}\big{(}\big{(}T_{\rm{vc}}+2p+C_{p}\big{)}b-T^{2}\beta-T\alpha-a\big{)} s-bC_{h}\big{)}e^{sT}\end{array} \right]^{2}\]  
For any value of \(p\) and \(T\) the second principal diagonal minor \(D_{2}\) is positive if where,  
\[