## 3 Detailed Operators
### Subsystem Weight Considering Development Cost Based on Interval Expert Evaluation  
This process corresponds to Stage IV in Figure 1, and is described in Section 2. The subsystem weight considering the development cost \(w^{1}\) is influenced by many factors, and its calculation is a multi-criteria decision making (MCDM) process. What should be emphasized are the difficulty and impracticality in description of the relationship between different subsystems considering different factors which have an influence on the subsystem weight considering the development cost, and there are many uncertainties unavoidable in this process. Therefore, fuzzy expert evaluation based on the interval numbers is adopted in this section. Assume the compromised variable on subsystem weight considering the development cost for \(i\)th subsystem be an interval number, \(Z_{i}\), \(i=1,2\),..., \(n\) and \(n\) is the total number of subsystems. Six factors are selected to enhance the feasibility of the expert evaluation.  
\(\bullet\) Complexity (\(K\))  
Complexity can be assessed according to the number of components in a subsystem. The subsystem weight considering the development cost for a complex subsystem needs to be improved for cost saving. Thus, if a subsystem is highly complex, its weight considering the development cost tends to be high.  
Hence, \(Z_{i}\propto K_{i}\), where \(K_{i}\) is the complexity factor for the \(i\)th subsystem.  
\(\bullet\) State-of-the-art (S) factorThe novelty or technical maturity of a subsystem affects its development cost. The high technology maturity of a subsystem corresponds to low development cost.  
Hence, \(Z_{i}\propto 1/S_{i}\), where \(S_{i}\) is the state-of-the-art factor for the \(i\)th subsystem.  
* Criticality of failure (\(C_{r}\))  
The development cost of a subsystem with critical failure effects should be improved as much as possible to prevent severe failure.  
Hence, \(Z_{i}\propto C_{ri}\), where \(C_{ri}\) is the criticality of the failure factor for the \(i\)th subsystem.  
* Performance time (\(T\))  
If a subsystem is designed for a long performance time, its development cost is expected to be high to avoid frequent failures and maintenance in the process.  
Hence, \(Z_{i}\propto T_{i}\), where \(T_{i}\) is the performance time factor for the \(i\)th subsystem.  
* Maintainability (\(M\))  
The maintainability of a subsystem is inversely proportional to its development cost because low reliability in a highly maintainable subsystem or components reduces product cost.  
Hence, \(Z_{i}\propto 1/M_{i}\), where \(M_{i}\) is the maintainability factor for \(i\)th subsystem.  
* Environmental harshness (\(E\))  
The harshness degree of the working environment of a subsystem significantly affects subsystem development cost. The higher degree of environmental harshness of a subsystem, the higher development cost is needed to reduce frequent failures.  
Hence, \(Z_{i}\propto E_{i}\), where \(E_{i}\) is the working environment factor for the \(i\)th subsystem.  
Therefore, the equation for interval values (\(Z_{i}\)) is expressed as follows:  
\[Z_{i}=(K_{i}\times Cr_{i}\times T_{i}\times E_{i})/(M_{i}\times S_{i}) \tag{9}\]  
where \(K_{i}\), \(M_{i}\), \(S_{i}\), \(Cr_{i}\), \(T_{i}\) and \(E_{i}\) are the interval assessment values of the \(i\)th subsystem for the aforementioned factors. These variables are represented by interval numbers within [1; 2; 3; 4; 5; 6; 7; 8; 9] and translated into crisp numbers with the interval algorithm [36].  
(\(x_{1}\), \(y_{1}\)) and (\(x_{2}\), \(y_{2}\)) are two interval numbers, where \(x_{1}\) and \(x_{2}\), and \(y_{1}\) and \(y_{2}\) represent left-end and right-end points, respectively. Four fuzzy fundamental operations are shown in Equations (5)-(8), and the defuzzified value \(G\) of the interval number \(G^{\#}=[x,y]\) is obtained by Equations (9),  
\[[x_{1},y_{1}]+[x_{2},y_{2}]=[x_{1}+x_{2},y_{1}+y_{2}] \tag{10}\]  
\[[x_{1},y_{1}]-[x_{2},y_{2}]=[x_{1}-y_{2},y_{1}-x_{2}] \tag{11}\]  
\[[x_