## 5 Case Study
### Weighting of Robustness Criteria
#### 5.2.2 Objective Weighting  
The objective weights are determined. Nonlinear intercriteria correlations among \(r_{s}\), \(r_{p}\), and \(r_{c}\), are calculated using the Copula function. A total of 200 schedules are generated, and their \(r_{s}\), \(r_{p}\), and \(r_{c}\) are calculated. Their frequency distributions, including skewness and kurtosis, are shown in Figure 5.  
Figure 5 indicates that the frequency distribution curves of the three robustness criteria do not obey normal, Poisson, and other basic distributions. Thus, the distribution function of the three criteria is fitted, which are shown in Figure 6.  
Figure 5: Frequency distributions of \(r_{s}\), \(r_{p}\), and \(r_{c}\).  
Figure 6: Nuclear density estimation results of \(r_{s}\), \(r_{p}\) and \(r_{c}\).  
Then, the results above are applied to the five Copula functions to estimate the unknown parameters of each function by using the maximum likelihood method. The results are shown in Table 3.  
The squared Euclidean distances \(d^{2}\) between the empirical Copula function and the five classical Copula functions are calculated, as shown in Table 4.  
It can be seen from Table 4 that the \(d^{2}\) between the empirical Copula function and the Frank Copula function is the smallest, which suggests that the Frank Copula function fits best. Thus, the Frank Copula function was adopted to calculate the nonlinear intercriteria correlations among robustness criteria. Figure 7 illustrates the Frank Copula functions and their comparisons with the empirical Copula function.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(r_{u},r_{p}\) & **Function** & **Normal Copula** & **t-Copula** & **Gumbel Copula** & **Clayton Copula** & **Frank Copula** \\  & \(d^{2}\) & 0.0118 & 0.0098 & 0.0115 & 0.0348 & 0.0056 \\ \hline \(r_{u},r_{c}\) & **Function** & **Normal Copula** & **t-Copula** & **Gumbel Copula** & **[PERSON]** & **Frank Copula** \\  & \(d^{2}\) & 0.0180 & 0.0128 & 0.0170 & 0.0447 & 0.0064 \\ \hline \(r_{p}\,r_{c}\) & **Function** & **Normal Copula** & **t-Copula** & **Gumbel Copula** & **[PERSON]** & **Frank Copula** \\  & \(d^{2}\) & 0.0108 & 0.0082 & 0.0097 & 0.0267 & 0.0064 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Squared Euclidean distances \(d^{2}\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(r_{u},r_{p}\) & **Function** & **[PERSON]** & **t-Copula** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\  & **Estimated value** & 0.998 & 0.9983 & 24.8429 & 28.0823 & 130.1466 \\ \hline \(r_{u},r_{c}\) & **Function** & **[PERSON]** & **t-Copula** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\  & **Estimated value** & 0.9969 & 0.9977 & 20.4307 & 24.5123 & 121.3891 \\ \hline \(r_{p}\,r_{c}\) & **Function** & **[PERSON]** & **t-Copula** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\  & **Estimated value** & 0.9982 & 0.9986 & 27.0803 & 32.3490 & 121.1006 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameter estimation of the Copula function.  
Figure 7: Distributions of the [PERSON] function.  
The parameter estimation results in Table 3 are applied to the [PERSON] function, and the rank correlation coefficients can be obtained: \(l_{s\_p}\) = 0.9697, \(l_{s\_c}\) = 0.9675, \(l_{p\_c}\) = 0.9674. In addition, the standard variance reflecting the self-strength of the different criteria are \(\sigma_{s}=28.61\), \(\sigma_{p}=2.02\), \(\sigma_{c}=17.80\).  
Here the objective weights of each criterion, which could incorporate both inherent importance and nonlinear intercriteria correlations, can be gained:  
\[\rho_{\text{objective}}=\left[\begin{array}{ccc}0.58&0.05&0.37\end{array}\right]\]