# Development of Optimal Size Range of Modules for Driving of Automatic Sliding Doors
## 2 Solving the Problem
### Stage 4. Development of a Cost Model  
Production costs \(TS(x_{l_{j}})\) are modeled on the basis of the cost of the building components of the drive modules. The most significant share determining these costs are the following three components: motor, belt, and pulleys. Engineering calculations have been performed to select these components for specific values of the main parameter, load capacity. After calculations for selecting belt drive and motor corresponding to the loading, the components' prices are obtained from selected suppliers [29; 30]. In the following calculations, risk and uncertainty are not considered, i.e., the calculations are under certain conditions and without taking into account risk (chosen concrete suppliers of parts, no change in the economic conditions). At first, from the initial size range \(\overline{Z}\) of one hundred and ten sizes, three elements \(\overline{x}_{1},\overline{x}_{55},\overline{x}_{110}\) are selected. Cost calculations are made for them, and the relevant components have been selected and valued, i.e., \(TS(\overline{x}_{1})\), \(TS(\overline{x}_{55})\), and \(TS(\overline{x}_{110})\) are calculated. These sizes are selected on the basis of equal intervals in the size index range \(l_{j}\in[1;110]\). Todetermine the cost of the remaining one hundred and seven sizes, a method of piecewise interpolation using a cubic Hermite interpolation polynomial, proposed in [31], was used. The selected interpolation polynomial preserves the monotonicity in the interpolated data and does not skip points if the data do not describe a smooth function. The result of the interpolation at the three selected points is shown in Figure 4. The values for the load capacity of the modules in kg are plotted on the abscissa, and the production costs in \(\upxi\) on the ordinate. The points of the input data are depicted as solid points connected by a curve depicting the interpolation polynomial. Then, the interpolation polynomial was calculated for data with two added points to the original three from Figure 4, i.e., the cost was calculated for five sizes, and the corresponding polynomial was built on this basis--in this case, the production costs for sizes \(\overline{x}_{1},\overline{x}_{28},\ \overline{x}_{55},\overline{x}_{82}, \overline{x}_{110}\). The result is shown in Figure 5.  
Comparing the results obtained from Figures 4 and 5, it is evident that there are significant differences; therefore, five more points were added to the calculated five points, \(\overline{x}_{1},\overline{x}_{14},\overline{x}_{28},\ \overline{x}_{41}, \overline{x}_{55},\overline{x}_{68},\overline{x}_{75},\overline{x}_{82}, \overline{x}_{96},\overline{x}_{110}\), and the interpolation polynomial was generated again (Figure 6). Figure 4: Interpolated production cost data at three calculated points. Figure 5: Interpolated production cost data at five calculated points. Analyzing the results of Figures 5 and 6, some differences in the interpolation polynomial are visible, but to a significant extent it retains its form. The final interpolation polynomial used for the production cost model is given in Figure 7. The difference between it and the one shown in Figure 6 is in one added point, i.e., for Figure 7, \(\overline{x}_{1},\overline{x}_{14},\overline{x}_{28},\)\(\overline{x}_{41},\overline{x}_{55},\overline{x}_{68},\overline{x}_{75},\overline{x}_{82}, \overline{x}_{96},\overline{x}_{103},\overline{x}_{110}\). A consideration for stopping further addition of interpolation points is the limited variety of motor sizes and belt and pulley sizes available on the market (also subject to size ranges), i.e., differences in production costs upon further granulation would arise from different belt lengths. Since the lengths of the door wings increase relatively smoothly, the change in belt length will have a smaller impact on production costs when comparing adjacent sizes. Therefore, it is considered that the interpolation polynomial depicted in Figure 7 interpolates \(TS\Big{(}x_{l_{j}}\Big{)}\) with sufficient accuracy.