## 4 Methodology of Construction Schedule Robustness Measure
### Copula-CRITIC Method for Objective Weighting  
The Copula-CRITIC method is proposed to determine the objective weights. To overcome the shortcoming that the Pearson correlation coefficient function in the CRITIC method cannot effectively deal with, i.e., the difficulty in the intercriteria correlation among the three robustness criteria, the Copula function is introduced into the CRITIC method to replace the Pearson correlation coefficient function. In this way, the nonlinear correlation among \(r_{s}\), \(r_{p}\), and \(r_{c}\) can be obtained, and the objective weights incorporating both inherent importance and nonlinear intercriteria correlations can be obtained (as shown in Figure 3). The procedure of objective weighing is as follows:  
1. By using the Copula-CRITIC method, the objective weights are determined according to the inherent importance and nonlinear intercriteria correlation of three robustness criteria. Set \(C_{a}\) as the information contained in \(r_{a}\) (\(a=s,p,c\)), \(C_{a}\) is \[C_{a}=\sigma_{a}\sum_{i=s}^{c}(1-l_{i-a})\] (11) where \(C_{a}\) represents the criterion's information, \(\sigma_{a}\) is the standard deviation to represent its inherent importance, and \(l_{i-a}\) is the correlation coefficient and is computed using the Copula function. 2. Compute \(l_{i-a}\) using the Copula function: Five classical Copula functions [47] are applied to the distribution fitting of the robustness criteria. The unknown parameters in the five Copula functions are determined by semi-parameter estimations. Then, the best Copula function is chosen according to the squared Euclidean distance \(d^{2}\) between the fitting Copula function and the empirical Copula function. The smaller \(d^{2}\) is, the better the fitting degree is. The \(d^{2}\) is calculated as follows: \[d^{2}=\sum_{a=s}^{c}\left|\hat{C}\big{(}r_{s},r_{p},r_{c}\big{)}-C\big{(}r_{s },r_{p},r_{c}\big{)}\right|^{2}\] (12) where \(\hat{C}\big{(}r_{s},r_{p},r_{c}\big{)}\) represents the empirical Copula function, and \(C\big{(}r_{s},r_{p},r_{c}\big{)}\) represents the fitting Copula function. 3. Calculate the objective weights \(\rho_{\text{objective}}\): Considering the standard deviations \(\sigma_{a}\) and the correlation coefficients \(l_{i-a}\), the objective weights of each criterion can be calculated: \[\rho_{\text{objective-}a}=C_{a}/\sum_{a=s}^{a}(C_{a})\ (a=s,p,c).\] (13)  
Information-Entropy-Based Evidence Reasoning Method for the Combination of Subjective and Objective Weights  
In order to exclude invalid information from the weight combination process, the evidence reasoning method is improved by introducing the information-entropy-based effectiveness validation mechanism in the initial weight assignment stage in this paper, making the robustness measure a better  
Figure 3: The Copula-CRITIC method for objective weighting. reflection of bounded rational of subjective weights, inherent importance and interrelationships of objective weights [48]. The procedure of combination of subjective and objective weights is as follows:  
1. Define \(H\). \(H\) is the acceptance degree of \(\rho_{\text{subjective}}\) and \(\rho_{\text{objective}}\). It has five classifications {SN, N, A, Y, SY}, which correspond to {Strongly no, No, Accept, Yes, Strongly yes}. Their utility values are \(U\)(SN) = \(-1\), \(U\)(N) = \(-0.5\), \(U\)(A) = \(0\), \(U\)(Y) = \(0.5\), \(U\)(SY) = \(1\). The belief distribution of the robustness criterion \(r_{a}\) to weight \(\rho_{b}\) (\(b\) = subjective, objective) is: \[S(\rho_{b}(r_{a}))=\Big{\{}\Big{(}H_{x},\theta_{x,b}(r_{a})\Big{)},x\in H; \Big{(}H,\theta_{H,b}(r_{a})\Big{)}\Big{\}}.\] (14)
2. Compute BPA. Basic probability assignment (BPA) \(e\) represents the accepted probability of \(\rho_{b}\) with \(H\). The belief distribution \(S(\rho_{b}(r_{a}))\) is transformed to BPA by Equations (15)-(18).