## 4 Methodology of Construction Schedule Robustness Measure
### Improved Prospect Theory for Subjective Weighting  
Improved prospect theory is proposed to determine the subjective weights. To overcome the shortcoming that the subjective weights given by experts are rationally bounded because of the experts' individual preferences, the experts' individual preferences are calculated and eliminated by prospect theory. In this way, the subjective weights of the bounded rationality can be transformed into subjective weights of rationality. Meanwhile, the interval distance formula is introduced into the prospect theory to overcome the shortcoming that prospect theory cannot deal with complete expert evaluation information. The procedure of subjective weighing is as follows:  
1.  
2.  
where, \(vp_{s}\), \(vp_{p}\) and \(vp_{c}\) represent the reference intervals. \(f(\cdot)\) is the arithmetic mean function.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\begin{array}{c}\hline\hline\text{Expert}\text{C}\text{C}\text{C}\text{C} \text{C}\text{C}\text{C}\text{C}\text{C}\text{C}\text{C}\text{C}\text{C}\  
3. Establish the matrix of reference intervals \(VP\): \[VP=[vp_{s},\,vp_{p},\,vp_{c}]\] (6)
4. Obtain the gain/loss matrix \(M\), which is the matrix reflecting experts' individual preference. Gain implies that the subjective weight given by the expert is larger than \(RP\). Loss implies that the subjective weight given by the expert is smaller than \(RP\). \(M\) is computed using interval distance formula, i.e., \[\begin{array}{ll}M&=m\Big{(}vp_{a},\,\rho_{an}^{\text{sub}}\Big{)}\\ &=\int_{0}^{1}\Big{|}\Big{(}vp_{a}^{-}+x\Big{(}vp_{a}^{+}-vp_{a}^{-}\Big{)} \Big{)}-\Big{[}\rho_{an}^{\text{sub-}}+x\Big{(}\rho_{an}^{\text{sub+}}-\rho_{ an}^{\text{sub-}}\Big{)}\Big{]}\Big{|}dx\\ &=\int_{0}^{1}\Big{|}\Big{(}vp_{a}^{-}-\rho_{an}^{\text{sub-}}\Big{)}+\Big{(} -vp_{a}^{-}+vp_{a}^{+}+\rho_{an}^{\text{sub-}}-\rho_{an}^{\text{sub+}}\Big{)}x \Big{|}dx\end{array}\] (7) where \(a=s,p,c\) represents the three robustness criteria, \(\rho_{an}^{\text{sub}}\) is the subjective weights given by experts, \(m\big{(}vp_{a},\,\rho_{an}^{\text{sub}}\Big{)}\) is the distance between \(vp_{a}\) and \(\rho_{an}^{\text{sub}}\). \(m\big{(}vp_{a},\,\rho_{an}^{\text{sub}}\big{)}\) is calculated as follows: take the weighted absolute value of the left and right endpoint values of two intervals, then the value is traversed in \([0,1]\) and the combined result is the distance between the intervals. There are six kinds of positional relationships between \(vp_{a}\) and \(\rho_{an}^{\text{sub}}\). The positional relationships and the corresponding distance formulas derived from Equation (7) are shown in Figure 2.
5. Calculate the value function through \(M=m(vp_{a},\,\rho_{an}^{\text{sub}})\). The value function is the function reflecting the value of weights given by experts, with the gain/loss as an independent variable and the value of weights given by experts as a dependent variable. It is calculated as \[v(m)=\left\{\begin{array}{ll}m^{\alpha}&m\geq 0\\ -\lambda(-m)^{\beta}&m<0\end{array}\right.\] (8) where \(v(\cdot)\) is the value function, \(\alpha\) is the positive risk attitude coefficient, while \(\beta\) is the negative risk attitude coefficient, and \(0\leq\alpha,\,\,\,\beta\leq 1\). \(\lambda\) is the loss avoidance coefficient. It is suggested that \(\alpha=0.89\), \(\beta=0.92\) and \(\lambda=2.25\)[46].
6. Calculate the comprehensive prospect value: \[V=\sum\pi(VP)\cdot v(m)\] (9) where \(\pi(VP)\) is the weight function and weights of different experts are assigned identically.
7. Finally, the subjective weights, considering bounded rationality can be calculated: \[\rho_{\text{subjective-e}}=\frac{v(m)}{V}.\] (10)  
Figure 2: Positional relationships and distance formulas between \(vp_{a}\) and \(\rho_{an}^{\text{sub}}\).