## 2 Equipment Group Maintenance Planning Model Based on Service Age Regression Model
### Optimization Objective  
#### 2.2.1 The Probability of the Fleet to Satisfy the Requirement in the Next Mission  
The mission satisfied capability of the fleet refers to the health condition that the fleet can keep during the mission. In the series-parallel system, as long as the failure probability of the equipment system during the mission is lower than the required level, the equipment system can be considered to have achieved the health condition required by the mission after maintenance. The performance of a component, subsystem or system can be described in a variety of ways. In the research aimed at maintenance planning, such performance can often be measured by a series of indicators, such as reliability, failure rate or mean time between failures. In this paper, it is assumed that the component lifetime model follows the two-parameter Weibull distribution in Equation (1) :  
\[f(x)=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}exp\left[-\left( \frac{t}{\eta}\right)^{\beta}\right] \tag{1}\]  
where \(\beta\) is the shape parameter and \(\eta\) is the scale parameter. [PERSON] et al. proposed the concept of conditional survival probability to describe the probability that a component does not fail in the duration of the mission [25]; that is, the probability that the components will still work normally when the task ends if the components work normally at the beginning of the task:  
\[r_{ij}(n)=P\left(X_{ij}^{-}=1|X_{ij}^{+}=1\right) \tag{2}\]  
In this chapter, reliability is used to represent the probability that components work normally. When the task duration is \(\tau_{k}\), Equation (2) becomes  
\[r_{ij}(n)=R\left(V_{ij}^{+}(n)|V_{ij}^{-}(n)\right)=R\left(V_{ij}^{+}(n)\right) /R\left(V_{ij}^{-}(n)\right) \tag{3}\]  
In the virtual service age model, the component failure rate is determined by the virtual service age rather than the actual working time. According to the system mission reliability model in Figure 1, the conditional survival probability of the fourth equipment in the equipment cluster can be expressed as  
\[P_{e}=\prod_{i=1}^{s}\prod_{j=1}^{N_{i}}R_{e,ij}\left(V_{ij}^{+}(k)\right)/ \prod_{i=1}^{s}\prod_{j=1}^{N_{i}}R_{e,ij}\left(V_{ij}^{-}(k)\right) \tag{4}\]  
The mission satisfied capability of a single equipment system can be evaluated according to its reliability. However, the health condition of the fleet cannot be simply described by reliability, which can only be comprehensively evaluated according to the attributes of each equipment unit. In this section, a k-out-of-n system model is used to describe the mission satisfied capability of the fleet. It is believed that at any stage in the mission process, as long as the amount of failed equipment in the fleet remains below a certain level, the fleet can be regarded as able to successfully complete the mission objective. Therefore, the probability of the the amount of failed equipment kept at the required level can be regarded as the mission satisfied capability of the fleet. In this section, the k-out-of-n system model is combined with the survival probability of the equipment system mentioned above, and the relationship between the component health condition and the mission satisfied capability of the fleet is described by the method of universal generating function. In the fleet described in this chapter, the state (failure or nonfailure) of component \(N_{ij}^{e}\) in the mission process is expressed by the binary variable \(X_{ij}^{e}\). According to the survival probability defined in Equation (5), the mission performance of the fleet can be expressed by the following structure function:  
\[\phi(X_{1},X_{2},\cdots,X_{N})= \begin{cases}1&\text{if}\quad\sum_{i=1}^{N}X_{ i}\geq k\\ 0&\text{if}\quad\sum_{i=1}^{N}X_{i}\leq k\end{cases}  \tag{5}\]  
Among them, the binary variable \(\phi\) represents whether the health state of the equipment cluster meets the mission requirements. The structure function of the system indicates the relationship between the state of the system and the state of its components.