## 4 Case Example: Australian Public-Private Partnership (PPP) Schools
### Data Description  
Multiple-project analysis was conducted to simulate the uncertain environment and to test the framework's feasibility. Many cultural and jurisdictional differences were avoided in the sample by selecting three schools that form part of one PPP contract from one jurisdiction in Australia, thus, their difference in data can be viewed as random variables under uncertainty. Abatement costs and the end-user's satisfaction levels are considered as vital variables as, on the one hand, they can show how the project service is delivered by the private sector, and on the other hand, how the service is  
Figure 4: Time-based performance curve for social PPPs. perceived by the end-user. In this study, abatement data and actual service payments from 2010 to 2018 for school A, B and C are listed in the way of the ratio of abatement cost in Table 1.  
Notably, the performance-related service payment for these PPP schools was 6% of the contract sum, based on the historical data collected in this project. The service payment is much smaller in t = 1, i.e., the year 2010, due to only one quarter service payment in the first open year. Also, the criterion for good and acceptable is set as 10% (\(l_{2}\)) and 20% (\(l_{2}\)) respectively in this study, i.e., when the probability of failure \(P_{f,t}\leq 10\%\), it means the service performance is good so as to meet the principals and teachers' need very well. When the \(P_{f,t}>20\%\), it indicates the government should consider major changes in the original project scope to satisfy the principal's and teachers' expectation. When the \(P_{f,t}\) falls on \([10\%,20\%]\), it is considered as acceptable. These criteria could be referred to from MMG Education [49], which has the lowest satisfaction score from 4545 benchmarked schools of 79%, and thus the acceptable bottom line is 21%. The criterion could be changed to adapt to different cases. The perceived service quality by the end-user was inferred on the basis of data collected using in-depth interviews with school principals and senior teachers. The interviewee profile is provided in Table 2. It shows the average working length for all interviewees is 17.3 years, with a minimum working length of 7.5 years and the average working school number is about 7, from which it could be concluded that the interviewees are sufficiently experienced to give the objective opinions on service delivered in PPP schools. The in-depth interview was supported by the questionnaire with a seven-point Likert scale beginning at 1 representing extremely dissatisfied, very dissatisfied, dissatisfied, unsure, satisfied, very satisfied and extremely satisfied in turn. To be consistent to the interview scale, a liner relationship between abatement cost and performance score (assumed no repeated service failure, as it would cost more deductions) is established, where no abatement cost means 95% and above performance [15]. Hereby, the zone of 4.95-5 represents satisfied, where payments are paid in full according to payment regime, 4 represents full abatement cost, which corresponds to the meaning of deduction, and thus might not be fit for purpose, 3 is too terrible that it means no availability. The scale description can be seen in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Year** & \(A(t)_{A}\) & \(A(t)_{B}\) & \(A(t)_{C}\) & _Mean_ & _S.D._ \\ \hline t = 1 & 0 & 0 & 0 & 0 & 0 \\ t = 2 & 0.1532 & 0.0512 & 0.0512 & 0.0852 & 0.0588 \\ t = 3 & 0 & 0 & 0 & 0 & 0 \\ t = 4 & 0.0142 & 0.0235 & 0.0217 & 0.0198 & 0.0049 \\ t = 5 & 0.0366 & 0.0604 & 0.0535 & 0.0503 & 0.0122 \\ t = 6 & 0.0054 & 0.0090 & 0.0064 & 0.0069 & 0.0018 \\ t = 7 & 0.0097 & 0.0160 & 0.0171 & 0.0143 & 0.0039 \\ t = 8 & 0.0065 & 0.0108 & 0.0109 & 0.0094 & 0.0025 \\ t = 9 & 0.0376 & 0.0622 & 0.0625 & 0.0541 & 0.0142 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cost information for PPP schools (A$k).