## 3 Research Methodology  
Qualitative and quantitative methods are necessary to inform specific values for \(P\) and \(A\) adopted in the RDSF. Specifically, the data for \(A\) are derived from the contract documents and payment records for abatement costs, and the data on \(P\) are collected through in-depth interviews to end-user. Reliability analysis is utilized to measure the above probability of relationships, see the definitions in Equation (1), and of particular interest is the intersection of the two variables, i.e., \(P\) and \(A\). It is a risk-based design concept to incorporate uncertainties into the framework that has widely been used in the area of system engineering lifecycle management, dealing with the estimation, prevention, management of lifetime engineering uncertainty and risks of failure [47]. In this context, the uncertainties could be captured by analyzing the input variables relating to different scenarios. Hence, this approach presents government with an early indicator of when genuine change may be warranted and thus better satisfy the end-user's expectation through the whole lifetime of social PPP operations. Taking the failure as defined in Equation (1), \(P_{f}\), i.e., the key indicator for social PPPs service performance improvement, could be illustrated as the following equation adapted from  
the result variable \((P-A)\) corresponds to N\(\left(\mu_{P}-\mu_{A},\ \sqrt{\sigma_{P}^{2}+\sigma_{A}^{2}}\right)\). The probability of failure \((P_{f})\) can be expressed as the Equation (3) from [48]. \[ \begin{array}{l}P-A\ =\ h(X)\ =\ h(X_{1},X_{2},\ldots,X_{n})\\ =\ \int\cdots\int_{(1)<0}\ f_{X}(X_{1},X_{2},\ldots,X_{n})dX_{1}dX_{2}\ldots dX_ {n},\end{array}  \tag{3}\]  
where \(f_{X}(X_{1},X_{2},\ldots,X_{n})\) is the joint probability density function for the random variables of \(X_{1},X_{2},\ldots,X_{n}\); However, the joint probability density function and the data on these random variables are difficult to be available. The first-order second moment reliability methods, thereby, is used to get the approximations of the result using the means and standard deviations. For this study, the probability of failure \((P_{f})\) can be expressed as Equation (4),  
\[P_{f}\ =\ P(P-\ A<0)\ =\ \varnothing\left[\frac{0-(\mu_{P}-\mu_{A})}{\sqrt{ \sigma_{P}^{2}+\sigma_{A}^{2}}}\right], \tag{4}\]  
where \(\varnothing\) is the cumulative distribution function of the standard normal variable; \(P_{f}\) is dependent on the ratio between the mean and standard deviation of variable \(P-A\), which is called reliability index, and expressed as,  
\[\text{Reliability index}\ (\theta)\ =\ \frac{\mu_{P}-\mu_{A}}{\sqrt{\sigma_{P}^{2}+ \sigma_{A}^{2}}}. \tag{5}\]  
Hence, the probability of failure \((P_{f})\), for social PPPs service performance can be illustrated as Equation (6). \[P_{f}\ =\ P(P-\ A<0)\ =\ \varnothing[-\theta]\ =\ 1-\varnothing(\theta). \tag{6}\]  
Additionally, In the case of multicriteria weight analysis, the mean value of perceived service quality \((P)\) equals the weighted mean value for each category, i.e., space \((s)\), operation \((o)\) and maintenance \((m)\) activities. The standard deviation of perceived service quality \((P)\) equals the square root of the sum square of weighted standard deviation of each category. Thus, the probability of failure \((P_{f})\) could be interpreted as,  
\[P_{f}\ =\ 1-\varnothing\left(\frac{w_{s}\mu_{s}+w_{2}\mu_{a}+w_{m}\mu_{m}-\mu_{A }}{\sqrt{w_{s}\sigma_{s}^{2}+w_{2}\sigma_{a}^{2}+w_{m}^{2}\sigma_{m}^{2}+ \sigma_{A}^{2}}}\right). \tag{7}\]  
In this study, Equation (2), which is commonly used in structural reliability analysis and prediction [18], has already been used in the appraisal of a desalination plant [19], residential buildings [20] and port infrastructure [21].