# Subway Station Capacity Maintained by Optimizing a Maintenance Schedule of Key Equipment
## Maintenance Schedule Optimization  
The key equipment maintenance schedule optimization of subway station aims at keeping the reliable operation of equipment in a certain range with the lowest maintenance cost. For the key equipment, the failure rates of different components of the equipment are different. In order to repair the equipment more pertinently, improve the maintenance efficiency and reduce the maintenance cost, the equipment can be divided into n parts, and different maintenance schedules are adopted respectively. The maintenance schedule of the \(i\)th part is \(T_{i}\), the corresponding reliability function is \(R_{i}(t)\), and the corresponding maintenance cost is \(C_{i}(T_{i})\), \(i\in\{0,1,2,\ldots,n\}\). Therefore, the total maintenance cost of equipment is \(C_{e}(T_{1},T_{2},\cdots,T_{n})\). Therefore, with the constraint of equipment reliability, the average maintenance cost per unit maintenance schedule is taken as the optimization objective, and the maintenance schedule optimization model is established. In consideration of the impact of equipment failure on the normal operation of the station, the maintenance cost not only refers to the cost of equipment maintenance and labor, but also includes the loss of equipment downtime [21]. In addition, the maintenance cost data can be obtained from the maintenance account records. The maintenance schedule optimization model can be formulated as follows:  
\[\min(C_{e}(T_{1},T_{2},\cdots,T_{n}))=\min(C_{1}(T_{1})+C_{2}(T_{2})+\cdots C_ {n}(T_{n})) \tag{12}\]  
\[C(T_{i})=\frac{c_{pi}R_{i}(T_{i})+c_{fi}\int_{0}^{T_{i}}f_{i}(t)dt}{\int_{0}^{ T_{i}}R_{i}(t)dt} \tag{13}\]  
\[R_{1}(T_{1})*R_{2}(T_{2})*\cdots*R_{n}(T_{n})\geq R_{e} \tag{14}\]  
\[T_{1},T_{2},\ldots T_{n}\in N^{*} \tag{15}\]  
where \(C(T_{i})\) is the expected cost per unit time of normal operation of \(i\)th part of equipment within a single maintenance schedule, \(c_{p_{i}}\) is the average total cost of \(i\)th part of equipment within a single periodic maintenance including preventive replacement cost for non-repairable components and repair maintenance cost for repairable components, \(c_{f_{i}}\) is the average total cost of \(i\)th part of equipment failure including replacement cost, repair maintenance cost and the loss of equipment downtime, \(R_{i}(t)\) is the reliability function of \(i\)th part of the equipment, \(f_{i}(t)\) is the fault probability density function of \(i\)th part of the equipment, \(T_{i}\) is maintenance schedule of \(i\)th part which is the decision variable, \(\int_{0}^{T}R(t)dt\) is the expected time of normal operation of equipment with a single maintenance schedule, \(R_{e}\) is lower bound of equipment reliability which is determined by sensitivity analysis based on the equipment reliability-expected station capacity curve. The enumeration method is adopted to solve the model, and the algorithm flow chart is shown in Figure 4. Firstly, the reliability function \(R_{i}(t)\) and the fault probability density function \(f_{i}(t)\) of each part of the equipment and the lower bound of equipment reliability \(R_{e}\) are given before the optimization. Secondly, the minimum value of maintenance schedule \(T_{i}\) for each part of the equipment which cannot meet the equipment reliability requirement are determined by loop statement, since the reliability of the equipment components decreases as service time goes by. Then the preliminary maintenance schedule collections \(T_{c}\) for all of the equipment are obtained by combining the available maintenance schedules of the equipment components. Thirdly, the final maintenance schedule collections \(T_{s}\) which meet the equipment reliability requirement for the entire equipment is selected from \(T_{c}\). Finally, for each maintenance schedule collection of \(T_{s}\), the total cost of the equipment \(C_{e}\) within a single maintenance schedule is calculated. Afterwards, the minimum value is selected, and the optimal maintenance schedule for each part of the equipment is obtained.