## 4 A Case Study  
### Problem Description  
The description of the primary building materials of a distributor in Vietnam is shown in Figure 3. There are four main material products for coating in construction works: Spectite CW100 (a), Spectite WS (b), Spectite HP600 (c), and Speccoat PE145 (d). The statistical summary of historical sale data (2017-2019), and the component costs of materials are demonstrated in Tables 2 and 3, respectively. Figure 1: The experimental runs of the central composite design (CCD) and Box–Behnken design (BBD). Figure 3: List of the main materials group of the distributor. Figure 2: A computational framework. Currently, the company places orders from a supplier every three or four months. Afterward, the transportation freight charge for 20 ft and 40 ft containers that are used for shipments directly to a warehouse. For inventory management, each unit of products (set, paid, or bag) is charged for holding cost per unit per month. Over-storage cost per unit is charged when ending inventory excesses the warehouse capacity. Shortage cost per unit is charged if satisfied demand is less than the actual demand. Key performances are measure periodically, which are the total cost, calculated by the sum of holding inventory, shortage cost, over-storage cost, and transportation cost, as presented in Equation (3); and the customer service level, which is determined by the accumulation of the total demand satisfied, as shown in Equation (4). Subsequently, the company is facing a problem in which should setup different factors (i.e., referred to independent variables in this paper) that affect the performance of the inventory system. Therefore, designing a computational framework to find the feasible values of the optimal setting of levels for independent variables that strongly affect the minimum total cost of holding cost, shortage cost, over-storage cost, transportation cost, and customer service level is needed. Total cost (TC) can be calculated from the following Equation (3):  
\[TC=C_{h}+C_{s}+C_{o}+C_{t} \tag{3}\]  
where \(C_{h}\) is holding cost, \(C_{s}\) is shortage cost, \(C_{o}\) is over-storage cost, and \(C_{t}\) is transportation costs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**2017** & **Specific CW100** & **Specific WS** & **Specific HP600** & **Speccoat PE145** \\ \hline Max & 1663 & 131 & 224 & 40 \\ Min & 215 & 0 & 0 & 1 \\ Avg & 1044.58 & 45 & 36.33 & 5.83 \\ SD & 470.18 & 47.13 & 63.58 & 11.81 \\ \hline
**2018** & **Specific CW100** & **Specific WS** & **Specific HP600** & **Speccoat PE145** \\ \hline Max & 2140 & 965 & 100 & 114 \\ Min & 411 & 0 & 0 & 2 \\ Avg & 1093.50 & 198.17 & 21.42 & 31.83 \\ SD & 538.73 & 265.93 & 35.64 & 34.91 \\ \hline
**2019** & **Specific CW100** & **Specific WS** & **Specific HP600** & **Speccoat PE145** \\ \hline Max & 2290 & 359 & 75 & 83 \\ Min & 219 & 0 & 0 & 0 \\ Avg & 1387.75 & 142.42 & 32.67 & 30.17 \\ SD & 649.19 & 124.85 & 30.79 & 24.88 \\ \hline \hline \end{tabular} Note: calculated by the researchers.  
\end{table} 
Table 2: Statistical summary of historical sale data added up to 36 months (2017–2019). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **Specific CW100** & **Specific WS** & **Specific HP600** & **Speccoat PE145** & **Unit** \\ \hline Packing size & 22 kg/set & 25 kg/bag & 20 L/pail & 15 L/pail & - \\ Wholesale price & 11 & 20 & 78 & 49.5 & USD/unit \\ Receipt cost & 12.93 & 22.12 & 84.73 & 54.70 & USD/unit \\ Selling price & 18.42 & 32.89 & 116.22 & 87.72 & USD/unit \\ Holding cost & 0.075 & 0.129 & 0.494 & 0.319 & USD/unit/month \\ Shortage cost & 0.553 & 0.987 & 3.487 & 2.632 & USD/unit \\ Over-storage cost & 0.162 & 0.277 & 1.059 & 0.684 & USD/unit/month \\ \hline \hline \end{tabular} Note: calculated by the researchers.  
\end{table} 
Table 3: Component costs of materials.