## 2 Basic Theory
### A Mathematical Product Option Selection Optimization Model for Civil Aircraft  
The model formulation comprehensively takes reliability, maintainability, availability, economy and other operational constraints into consideration, making the optimized option selection decision-making scheme under the multi-objective cycle iteration more in line with the economic needs of operators. Taking reliability parameters as input variables and maintainability as a link, the relationship connected with the AFMC or availability is established. The general form of an option selection mathematical model for multi-objective optimization is:  
\[ \begin{array}{ll}\text{min}&\text{f}_{\text{i}}\ \ (\text{x}_{1},\text{x}_{2}, \text{x}_{3},\ldots\text{x}_{\text{n}})\\ \text{s.t.}&\text{x}_{\text{n}}\geq\text{x}_{\text{n}}{}^{\prime}(\text{i}=1, 2,\ldots\text{n})\end{array}  \tag{4}\]  
When operated in daily life, the maintenance staff or airlines use the typical systems decided by ATA in the Reliability Monthly Report, unless a certain product breaks down frequently and will be tracked for a period with special attention. This classification is followed in the thesis to maximize the overall availability and minimize the related AFMC on the whole system level. The objective functions are expressed as the form of summing the objective function of a single product. In addition, it is also necessary to  
Figure 1: Physical meaning diagram of availability. MTIR—mean time to repair, MSPT—mean shop processing time, MTBF—mean time between failures. uniformly minimize the model, which takes the availability function reciprocally. Finally, a mathematical optimization model for products option selection can be expressed as:  
\[ \begin{split}\min&\quad\text{f}_{1}\ \left(x_{1},x_{2},x_{3}, \ldots,x_{n}\right)=1/(\sum\limits_{i=1}^{n}\frac{x_{i}-\text{MSPT}\times h_{p}- \sum\limits_{i}\frac{x_{i}}{\text{MSPT}\times h}}{x_{i}})\\ \min&\quad\text{f}_{2}\ \left(x_{1},x_{2},x_{3}, \ldots,x_{n}\right)=\sum\limits_{i=1}^{n}\left(\frac{x_{i}}{x_{i}^{*}}\times \text{PMC}_{i}+\frac{x_{i}^{*}}{x_{i}}\times\text{UPMC}_{i}\right)\\ &\quad\text{s.t.}\quad\text{x}_{i}\ \geq x_{i}{}^{\prime}\ \ \ \ \ \ (i=1,2,3, \ldots,n)\end{split}  \tag{5}\]  
The above model uses a unified AFMC calculation without paying attention to the nature of planned and unplanned costs. The PMC is directly proportional to MTBF, and the UPMC is inversely proportional to MTBF. The astringency for objective functions is not consistent with the PMC or UPMC. In order to quantify the impact or make decisions for different costs, AFMC can also be divided into two objective functions. \[ \begin{split}\min&\quad\text{f}_{1}\ \left(x_{1},x_{2},x_{3}, \ldots,x_{n}\right)=1/(\sum\limits_{i=1}^{n}\frac{x_{i}-\text{MSPT}\times h_{p}- \sum\limits_{i}\frac{x_{i}}{\text{MSPT}\times h}\times h}{x_{i}})\\ \min&\quad\text{f}_{2}\ \left(x_{1},x_{2},x_{3}, \ldots,x_{n}\right)=\sum\limits_{i=1}^{n}\left(\frac{x_{i}}{x_{i}^{*}}\times \text{PMC}_{i}\right)\\ \min&\quad\text{f}_{3}\ \left(x_{1},x_{2},x_{3}, \ldots,x_{n}\right)=\sum\limits_{i=1}^{n}\left(\frac{x_{i}^{*}}{x_{i}}\times \text{UPMC}_{i}\right)\\ &\quad\text{s.t.}\quad\text{x}_{i}\ \geq x_{i}{}^{\prime}\ \ \ \ \ (i=1,2,3, \ldots,n)\end{split}  \tag{6}\]  
The performance of the two models is compared by solving them, respectively. Thus, we can analyze whether the algorithm can maintain uniform convergence for different linear relationships. External maintenance can also ensure product performance, such as the increase or decrease in maintenance tasks and MTBM. However, they do not directly improve the performance of the product directly through the selection and optimization. By directly optimizing the product itself in the design or operation stage, we can find a balance between planned and unplanned costs, so that cost control can be controlled within a controllable range. This is the focus of this study and the hypothesis of building the model.