# A Discount Technique-Based Inventory Management on Electronics Products Supply Chain
## 4 Theoretical Derivations
\begin{array}{l}(\alpha s-2\beta)sT+s^{2}\beta T^{2}-s^{3}\big{(}T _{vc}+C_{p}\big{)}\big{(}b\delta e^{s_{1}}-be^{sT}-b\delta+b\big{)}\\ -\big{(}\big{(}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{)}\\ -C_{h}\big{[}(bT-b\delta t_{1})s^{2}+sb\Big{(}\delta\big{(}1-e^{s_{1}}\big{)}+ e^{s_{1}}-e^{s(T-t_{1})}\Big{)}\Big{]}-\\ \big{(}1-e^{s_{1}}\big{)}\left[\begin{array}{l}\big{(}\big{(}a+\alpha T+\beta T ^{2}\big{)}s^{2}-(\alpha+2\beta T-bC_{h})s+2\beta\big{)}e^{sT}\\ +(2\alpha-C_{h}b(1-\delta))s-b\delta C_{h}se^{s_{1}}-2 as^{2}-4\beta\end{array} \right]\end{array} \right]}{2 bs^{2}\big{(}e^{s(T+t_{1})}-\delta e^{2 st_{1}}+3 \delta e^{s_{1}}-e^{sT}-2e^{s_{1}}+e^{s(T-t_{1})}-2\delta+1\big{)}} \tag{15}\]  
Again, differentiate Equation (14) with respect to \(p\)  
\[\frac{\partial^{2}\omega}{\partial p^{2}}=-\frac{2b}{sT}\Big{(}e^{sT}\big{(}e ^{s_{1}}-1\big{)}+\Big{(}e^{sT-st_{1}}-2e^{st_{1}}-\delta e^{2 st_{1}}\Big{)}+ \delta\big{(}3e^{s_{1}}-2\big{)}+1\Big{)} \tag{16}\]  
Since \(e^{s(T-t_{1})}>2e^{s_{1}}+\delta e^{2s_{1}}\), easy to say that, \(\frac{\partial^{2}\omega}{\partial p^{2}}<0\) for any value of \(p\). That implies the objective function is a concave function. The critical point \(p\) becomes the optimal point \(p^{*}\). **Proposition 2**. : _The objective function \(\omega(p,T)\) presented in Equation (12) demonstrates the concavity in terms of cycle time \(T\) when the product selling price \(p\) is considered as constants._  
Proof. : Similar to Proposition 1. To avoid redundancy, proof has been omitted. **Proposition 3**. : _The objective function \(\omega(p,T)\) presented in Equation (12) demonstrates the concavity in terms of the product selling price \(p\) as well as cycle time \(T\) with the condition_  
\[\left[ \begin{array}{l}2b\Omega_{1}+\Omega_{4}\\ -2 be^{sT}\end{array} \right]\left[\left( \begin{array}{l}\Omega_{3}+pse^{sT} \\ -(sp-C_{h})\Omega_{1}\end{array} \right)\left( \begin{array}{l}\Omega_{2}+bps \\ +2\beta T+\alpha\end{array} \right)-C_{h}(2\beta T+\alpha)\right]>\left[ \begin{array}{l}\Omega_{1}\Omega_{2}+bC_{h}(\Omega_{1}-1)\\ +\Omega_{3}b-\Omega_{2}e^{sT}\end{array} \right]^{2}\]  
Proof. : The profit function per unit time can be written as by using Lemma 1,  
\[\omega(p,T)=\frac{\varphi(p,T)}{\psi(p,T)} \tag{17}\]  
where,  
\[\varphi(p,T)=\left[ \begin{array}{l}(sp-C_{h})\left[\begin{array}{l}\Big{(} \frac{bp-a}{s}+\frac{a}{s^{2}}-\frac{2\delta}{s^{3}}\Big{)}T+\Big{(}\frac{2 \delta}{s^{2}}-\frac{a}{s}\Big{)}\frac{T^{2}}{2}-\frac{b\delta pt_{1}}{s}- \frac{\beta T^{3}}{3s}\\ +\Big{(}\frac{b\delta p-bp+a}{s^{2}}-\frac{a}{s^{3}}+\frac{2\delta}{s^{4}} \Big{)}\big{(}1-e^{s_{1}}\big{)}\\ +\big{(}\frac{bp-a-T-\delta T^{2}}{s^{2}}+\frac{a+2\delta T}{s^{3}}-\frac{2 \delta}{s^{4}}\Big{)}\Big{(}1-e^{s(T-t_{1})}\Big{)}\end{array}\right]\\ +(a-bp)pT+\frac{apT^{2}}{2}+\frac{apT^{3}}{3}+bp^{2}\delta t_{1}-C-T_{fc}-\\ \left(\begin{array}{l}T_{vc}+C_{p}+\\ \big{(}1-e^{s_{1}}\big{)}\Big{(}p-\frac{C_{h}}{s}\Big{)}\end{array}\right) \left[\begin{array}{l}\frac{bbp}{s}e^{s_{1}}-\Big{(}\frac{bbp-bp+a}{s}-\frac{ a}{s^{2}}+\frac{2\delta}{s^{3}}\Big{)}\\ -\Big{(}\frac{bp-a-T-\delta T^{2}}{s}+\frac{a+2\delta T}{s^{2}}-\frac{2\delta}{ s^{3}}\Big{)}e^{sT}\end{array}\right]\end{array} \right]\]  
\[\psi(p,T)=T\]  
Since \(\psi(p,T)>0\) is a linear function of \(p\), \(T\). For showing \(\omega(p,T)\) is a concave function, it is enough to show \(\varphi(p,T)\) is a concave function. The first order partial derivatives of \(\varphi(p,T)\) with respect to \(p\) and \(T\) are as follows:  
\[\frac{\partial\omega}{\partial p}=\frac{1}{s^{3}}\left[\begin{array}{l}(s \alpha s-2\beta)sT+s^{2}\left(\beta T^{2}-sbpT+sbp\delta t_{1}\right)-s^{3}\left( T_{\rm{vc}}+C_{p}\right)\left( \begin{array}{l}b\delta e^{st_{1}}-be^{sT}\\ -b\delta+b\end{array}