## 3 Results
### Optimal Construction of Aviation Health Systems  
The design and implementation of aviation health systems present a complex optimization challenge, balancing the initial creation costs, operational expenses, and the paramount importance of flight safety. This section delves into a framework aimed at guiding the optimal construction of both health monitoring and health management systems within the aviation sector.  
At the core of this approach is the development of objective functions that encapsulate the multifaceted goals of minimizing costs while maximizing system effectiveness and ensuring unwavering adherence to safety standards.  
Health monitoring system optimization  
Objective function--minimize the total cost of the health monitoring system, including the costs of creation and operation, while maximizing the system's effectiveness in monitoring health and safety.  
\[min\ F_{HM}=C_{c,HM}+C_{o,HM}-\gamma_{HM}E_{HM}\]  
where the following terms are used:  
\(F_{HM}\) is the objective function for the health monitoring system.  
\(C_{c,HM}\) represents the initial costs associated with developing and implementing the health monitoring system.  
\(C_{o,HM}\) includes the ongoing operational costs (maintenance, data analysis, etc.).  
\(E_{HM}\) quantifies the effectiveness of the health monitoring system in terms of its ability to detect potential issues, contributing to overall safety.  
\(\gamma_{HM}\) is a weighting factor that values the system's effectiveness.  
Constraints:  
1. The system must meet or exceed a predefined effectiveness threshold to ensure adequate safety monitoring.
2. Budget constraints for both creation and operational costs.
2. Health management system optimization  
Objective function--minimize the total cost of the health management system, factoring in the costs of system creation and operation and the benefits of predictive maintenance and enhanced safety.  
\[\min\ F_{HMGT}= C_{c,HMGT}+C_{o,HMGT}+C_{PM}+\alpha C_{F}-\beta S_{HMGT}-\gamma_{ HMGT}E_{HMGT}\]  
where the following terms are used:  
\(F_{HMGT}\) is the objective function for the health management system.  
\(C_{c,HMGT}\) and \(C_{o,HMGT}\) represent the initial and ongoing costs, respectively.  
\(C_{PM}\) is the cost of performing predictive maintenance.  
\(C_{F}\) is the cost associated with failures, including direct and indirect costs.  
\(S_{HMGT}\) quantifies the safety benefits derived from the health management system.  
\(E_{HMGT}\) represents the system's overall effectiveness in managing aircraft health.  
\(\alpha\), \(\beta,\gamma_{HMGT}\) are weighting factors for the costs of failures, safety benefits, and system effectiveness.  
Constraints:  
1. The system must achieve a certain level of effectiveness and safety benefits.
2. Budget constraints for creation, operation, and maintenance actions.  
The depth of integration involves the degree to which the system is embedded within aircraft operations and maintenance processes. A deeper integration implies a more comprehensive system that can offer greater benefits but at potentially higher costs.  
To account for depth, one can modify the objective functions to include terms that specifically assess the impact of system depth on costs and benefits, for instance, adding a factor that represents the depth of integration and adjusting the weighting factors to reflect how the depth influences costs and effectiveness.  
The presented general description of the problems of optimal construction of aviation health systems offers a structured methodology for evaluating and optimizing the myriad factors that influence the effectiveness and efficiency of health monitoring and management systems and provides a basis for future research in this direction.