# A Discount Technique-Based Inventory Management on Electronics Products Supply Chain
## 4 Theoretical Derivations
### Numerical Illustration:  
**Example 1**.: _We consider some parameters for an electronics enterprise to run its business incorporate with our inventory management technique. Order placement cost \(C=\$700/\mathrm{cycle}\), ordering quantity \(W=$2800\,\mathrm{unit}/\mathrm{year}$\), purchased cost of the items \(C_{p}=\$200/\mathrm{unit}\), holding cost of the items per unit time \(C_{h}=$50.3/\mathrm{unit}$\), initial demand rate \(a=300\), decreasing demand rate on price \(b=0.8\), discount rate \(\delta=0.4\), increasing demand rate on time \(\alpha=0.1\), the rate of changes of rate on time in the demand rate itself \(\beta=0.05\), increasing demand rate on stock \(s=0.2\), time after the discount closes \(t_{1}=0.5\) months, fixed cost for transportation \(T_{fc}=$80/\mathrm{Cycle}$\), variable cost for transportation \(T_{vc}=$\$0.2/\mathrm{unit}$\)._  
We obtain the optimal solutions: selling price \(p^{*}=$372.9512\), cycle time \(T^{*}=18.56568\) months, profit per unit time \(\psi^{*}=$12115.16\) using Lingo 18.0 software with the aid of an exact optimization approach.  
The graphical view of the profit margin under various investments and timeframes is displayed through the following 3D graphs. Figure 3 delineates the concavity of the profit function subject to the decision variables p and T.  
**Example 2**.: _We intend to omit the discount consideration. Only the discount rate is modified as \(\delta=0\) in Example 1 and rest of the parameters remain unchanged. We obtain the optimal solutions: selling price \(p^{*}=$369.559\), cycle time \(T^{*}=18.027\) months, profit per unit time \(\psi^{*}=$14462.310\)._  
**Example 3**.: _When the case of different discount rate. We tale the same parametric values mentioned in Example 1 except the discount rate \(\delta\)._ Table 2 _shows the optimal solutions for altered values of \(\delta\)._  
Figure 2: An electronics shop. (Source: [[https://lh3.ggpht.com/p/AF1](https://lh3.ggpht.com/p/AF1) QipMf1 FVqbSnGOjLJswWWY_TNs_rlQlbQ40 IX4 mb=s0]([https://lh3.ggpht.com/p/AF1](https://lh3.ggpht.com/p/AF1) QipMf1 FVqbSnGOjLJswWWY_TNs_rlQlbQ40 IX4 mb=s0)). (accessed on 15 July 2021).  
The optimal decision variables \(p^{*}\) and \(T^{*}\) increase as we increase the discount up to \(\delta=2\) and after that the optimal solution declines up to \(\delta=3\) while the solution converted to infeasible for \(\delta\geq 3.5\) values. This is true in the real case also because if a retailer started to provide discounts to the customers that are less than the market selling price, he needs to lose some sales revenue. To manage the warehouse and for the recovery of liquid money, he can do it for a certain time period and for a certain discount rate but after that time passes, he will face extreme loss in profit. As the stock depletes rapidly during the discount process, he needs to sell his products at high selling prices at a high discount rate \(\delta=2\). Moreover, this situation will not last for a long time and soon the losses will be extreme due to the high discount rate and the solution system provides no-optimal solution after \(\delta\geq 3.5\).