# A Discount Technique-Based Inventory Management on Electronics Products Supply Chain
## 3 Mathematical Formulation for Proposed Electronics Product Inventory Model
### The Total Cost per Unit Time per Cycle  
(a) Ordering cost per cycle = \(C\)  
(b) Holding cost (HC) = \(C_{h}\left[\int\limits_{0}^{t_{1}}I_{1}(t)\,dt+\int\limits_{t_{1}}^{T}I_{2}(t) \,dt\right]\) i.e.,  
\[C_{h}\left[\begin{array}{l}\left(\frac{bp-a}{s}+\frac{a}{s^{2}}-\frac{2\beta} {s^{3}}\right)T+\left(\frac{2\beta}{s^{2}}-\frac{a}{s}\right)\frac{T^{2}}{2}+ \left(\frac{w}{s}+\frac{b\delta p-bp+a}{s^{2}}-\frac{a}{s^{3}}+\frac{2\beta}{ s^{4}}\right)\left(1-e^{st_{1}}\right)+\\ \left(\frac{bp-a-nT-\beta T^{2}}{s^{2}}+\frac{a+2\beta T}{s^{3}}-\frac{2\beta} {s^{4}}\right)\left(1-e^{s(T-t_{1})}\right)-\frac{b\delta p_{1}}{s}-\frac{ \beta T^{3}}{3s}\end{array}\right]\] (6) (c) Purchase cost (PC) = \(C_{p}\)\({}^{*}\)\(w\)  
\[C_{p}\left[\frac{b\delta p}{s}e^{st_{1}}-\left(\frac{bp-a-\alpha T-\beta T^{2}}{s }+\frac{\alpha+2\beta T}{s^{2}}-\frac{2\beta}{s^{3}}\right)e^{sT}-\left(\frac{ b\delta p-bp+a}{s}-\frac{\alpha}{s^{2}}+\frac{2\beta}{s^{3}}\right)\right]\] (7) (d) Transportation cost (TC) = \(T_{fc}+T_{\rm rc}\)\({}^{*}\)\(w\)  
\[T_{fc}+T_{\rm rc}\left[\begin{array}{l}\frac{b\delta p}{s}e^{st_{1}}-\left( \frac{bp-a-\alpha T-\beta T^{2}}{s}+\frac{\alpha+2\beta T}{s^{2}}-\frac{2\beta }{s^{3}}\right)e^{sT}\\ -\left(\frac{b\delta p-bp+a}{s}-\frac{a}{s^{2}}+\frac{2\beta}{s^{3}}\right) \end{array}\right]\] (8) (e) Sales revenue (SR) = \(p^{*}\left[\int_{0}^{t_{1}}D\,dt+\int\limits_{t_{1}}^{T}D\,dt\right]\)  
\[=p^{*}\left[\int_{0}^{t_{1}}\left(a-b\,p\left(1-\delta\right)+\alpha\,t+ \beta\,t^{2}+s\,I_{1}(t)\right)\,dt+\int\limits_{t_{1}}^{T}\left(a-b\,p\,+ \alpha\,t+\beta\,t^{2}+s\,I_{2}(t)\right)\,dt\right]\]  
\[=\left(a-bp\right)pT+\frac{apT^{2}}{2}+\frac{bpT^{3}}{3}+bp^{2}\delta t_{1}+\]  
\[sp\left[\begin{array}{l}\left(\frac{bp-a}{s}+\frac{a}{s^{2}}-\frac{2\beta} {s^{3}}\right)T+\left(\frac{2\beta}{s^{2}}-\frac{a}{s}\right)\frac{T^{2}}{2}+ \left(\frac{w}{s}+\frac{b\delta p-bp+a}{s^{2}}-\frac{a}{s^{3}}+\frac{2\beta}{ s^{4}}\right)\left(1-e^{st_{1}}\right)+\\ \left(\frac{bp-a-\alpha T-\beta T^{2}}{s^{2}}+\frac{a+2\beta T}{s^{3}}-\frac{2 \beta}{s^{4}}\right)\left(1-e^{st_{1}-t_{1}}\right)-\frac{b\delta p_{1}}{s}- \frac{\beta T^{3}}{3s}\end{array}\right] \tag{9}\]  
Now, the total profit per unit time one can write as  
\[\omega(p,T)=\frac{1}{T}(SR-C-HC-PC-TC)\]  
\[\omega(p,T)=\frac{1}{T}\left(\begin{array}{l}\left(sp-C_{h}\right)\left[ \begin{array}{l}\left(\frac{bp-a}{s}+\frac{a}{s^{2}}-\frac{2\beta}{s^{3}} \right)T+\left(\frac{2\beta}{s^{2}}-\frac{a}{s}\right)\frac{T^{2}}{2}-\frac{b \delta p_{1}}{s}-\frac{\beta T^{3}}{3s}\\ +\left(\frac{w}{s}+\frac{b\delta p-bp+a}{s^{2}}-\frac{a}{s^{3}}+\frac{2\beta} {s^{4}}\right)\left(1-e^{st_{1}}\right)\\ +\left(\frac{bp-a-\alpha T-\beta T^{2}}{s^{2}}+\frac{a+2\beta}{s^{3}}-\frac{2 \beta}{s^{4}}\right)\left(1-e^{s(T-t_{1})}\right)\end{array}\right]\\ +(a-bp)pT+\frac{apT^{2}}{2}+\frac{bT^{3}}{3}+bp^{2}\delta t_{1}-C-T_{fc}-\\ \left(T_{\rm rc}+C_{p}\right)\left[\begin{array}{l}\frac{b\delta p}{s}e^{st_{1 }}-\left(\frac{bp-a-\alpha T-\beta T^{2}}{s}+\frac{a+2\beta T}{s^{2}}-\frac{2 \beta}{s^{3}}\right)e^{sT}\\ -\left(\frac{b\delta p-bp+a}{s}-\frac{\alpha}{s^{2}}+\frac{2\beta}{s^{3}} \right)\end{array}\right]\end{array}\right) \tag{10}\]  
with  
\[\left.\begin{array}{l}D=a-b\,p\left(1-\delta\right)+\alpha\,t+\beta\,t^{2}+s \,I_{1}(t)>0;\\ p<\frac{\delta}{s};\\ T>0;\\ p>C_{p};\end{array}\right\} \tag{11}\]