## 3 Materials and Methods
### Spare Parts Management System
There is a spare parts distribution point (management center of the company's service network). From this point, it is necessary to distribute in M regions lots of spare parts of volume \(\mathrm{q_{ij}}\) (\(\mathrm{i=\overline{1,N}}\)), (\(\mathrm{j=\overline{1,M}}\)), where N is the total number of items in the spare parts supply lot. Deliveries are carried out in sets of the i-th spare part, which can be considered a generalized nomenclature. The level of stock in the warehouses of the service center decreases in accordance with the intensity of service requests, which in turn correspond to the value of \(\mathrm{\lambda_{ij}}\)--the failure rate of i-th parts, assemblies, and assemblies in the j-th region of operation. Based on the value of \(\mathrm{\lambda_{ij}}\), the corporate service network control center plans customer requests for the planned period \(\mathrm{t_{P}}\) in the future. Upon reaching the time \(\mathrm{\tau}\)--the point of realization of batches, depending on the distance of the j-th region of operation from the control center of the branded service network, as well as upon reaching the existing stock kij (amount of existing stock of i-th spare part in the j-th region) in the warehouses of service centers Sij--the value of the minimum stock of the i-th spare parts in j-th region, the control center of the branded service network delivers spare parts in the amount of qij with the least cost gij. In conditions of random demand, the model does not include the indicator of the supplies' frequency. Instead, it is more expedient to use the value of the total number of deliveries B for the period (year). Thus, through the planning of customer requests and the calculated failure rate of the i-th parts in the j-th region, the stock of stored spare parts is replenished by the control center of the branded service network, and additional costs are associated with each such replenishment. When minimizing expenses for a period, storage costs and penalties are calculated based on the expected balance (deficit) by the end of the period (year). The penalty is determined by the probability of shortage. First of all, the service center bears the costs hij (storage costs of the i-th spare part in the j-th region) for the storage of the i-th spare parts in their warehouse for the entire period and the delivery of the batch volume gij. Service requests are satisfied until their total volume (since the beginning of the planning period) exceeds the initial stock. All subsequent requests cannot be serviced immediately, causing the customer's vehicle to sit idle while waiting for service. Downtime results in lost profits for the customer and fines for the service center. The penalties for the absence of the required spare parts in the warehouse of the service center include the excess of the costs of emergency delivery over the costs of ordinary and lost profits (percentage of working capital frozen in the service center due to a shortage):  
\[\mathrm{d}=\sum\
olimits_{i=1}^{\mathrm{N}}\sum\
olimits_{j=1}^{\mathrm{M}} \left(\frac{\lambda_{\bar{\mathrm{u}}}\mathrm{B}^{2}}{2\mathrm{g}_{\bar{ \mathrm{ij}}}}-\frac{1}{\mathrm{h}_{\bar{\mathrm{uj}}}}\right).\] (1  
To ensure the minimum amount of costs \(Z\) in the conditions of the smallest shortage and the best values of performance indicators, it is necessary to calculate the optimal value of the delivery point.