# Technology Identification, Evaluation, Selection, and Optimization of a HALE Solar Aircraft
## 4 General Methodology
### Determination of System Feasibility/Viability
#### 4.5.2 Desirability Function for Optimization  
Optimization of HALE solar aircraft was performed using desirability function based on the results of response surface methodology. The desirability function is one of the most popular performance metrics for simultaneous optimization of multiple continuous response variables [29]. The individual responses are first estimated through regressions, and then each estimated response is converted to the desirability value. The individual desirabilities are then combined using the geometric mean to get the overall desirability [29]. The desirability functions are categorized as nominal-the-better (NTB), smaller-the-better (STB), and larger-the-better (LTB), as shown in Equations (29)-(31) as follows [12]:  
\[NTB\ :d_{i}=\left\{\begin{array}{ll}\left(\frac{\hat{y}_{i}-y_{i}^{L}}{T_{i}-y_ {i}^{L}}\right)^{s}&if\ y_{i}^{L}\leq\hat{y}_{i}\leq T_{i}\\ \left(\frac{\hat{y}_{i}-y_{i}^{L}}{T_{i}-y_{i}^{L}}\right)^{t}&if\ T_{i}<\hat{y}_{i} \leq y_{i}^{U}\\ 0&if\ \hat{y}_{i}>y_{i}^{U}\end{array}\right. \tag{29}\]  
\[STB\ :d_{i}=\left\{\begin{array}{ll}1&if\ \hat{y}_{i}<T_{i}\\ \left(\frac{\hat{y}_{i}-y_{i}^{L}}{T_{i}-y_{i}^{L}}\right)^{s}&if\ T_{i}\leq \hat{y}_{i}\leq y_{i}^{U}\\ 0&if\ \hat{y}_{i}>y_{i}^{U}\end{array}\right. \tag{30}\]  
\[LTB\ :d_{i}=\left\{\begin{array}{ll}0&if\ \hat{y}_{i}<y_{i}^{L}\\ \left(\frac{\hat{y}_{i}-y_{i}^{L}}{T_{i}-y_{i}^{L}}\right)^{s}&if\ y_{i}^{L}< \hat{y}_{i}<T_{i}\\ 1&if\ \hat{y}_{i}>T_{i}\end{array}\right. \tag{31}\]  
where \(d_{i}\) is the desirability value, \(\hat{y}_{i}\) is the desired response, \(y_{i}^{L}\) is the lower value, and \(y_{i}^{U}\) is the upper value, \(T_{i}\) is the target value. The exponents \(s\) and \(t\) are the shape constants of the desirability function, in general, they are chosen in the range from 0.01 to 10. The \(y_{i}^{L}\) and \(y_{i}^{U}\) are respectively the lower and upper specification limit for each type response variable with a target value \(T_{i}\)[30].  
The goal of the desirability function approach is to find the combination of design variables by which the geometric mean is maximized for each desirability function. The geometric mean is defined as shown in Equation (32). The function shape can be determined by changing the variables. Once these variables are defined as greater than 1, they approach the target values [29]:  
\[D=\left(d_{1}\times d_{2}\times\cdots\times d_{n}\right)^{\frac{1}{n}} \tag{32}\]  
In this study, the smaller-the-better (STB) optimization was used to optimize the design objective parameters, wing loading, power to weight, maximum take-off weight defined in Table 5. Lift to drag ratio was optimized using the larger-the-better (LTB) optimization through JMP \"desirability function.\" In Figure 19, the \(x\)-axis represents the independent design variables and the \(y\)-axis represents the response variables. The desirability characteristic is presented by the slope on the right. The smaller-the-better (STB) has a negative slope whereas the larger-the-better (LTB) has a positive slope.  
As shown in Table 13, the optimized independent design variable values are converted to actual values, and these values can be applied to the solar aircraft design framework to derive design objective parameter values as shown in Table 14.