## 2 Problem Modeling
### Calculating the Expected Utilities of the Owner Expected Utility  
The expectation value of the owner's final cost-utility could be written as below:  
\[EUV_{c}=E[u(c)]=\int_{-\infty}^{+\infty}u\ (c)\ f(c)dc \tag{6}\]  
\[ \begin{split} u_{1}(c)=0.0000001c^{2}+0.000184c+0.3109\ c\ \leq 1700\\ u_{2}(c)=0.000000c^{2}+0.000469c+0.2932\ c>1700\end{split}  \tag{7}\]  
\[EUV_{c}=\int_{-\infty}^{0}u_{1}(c)f(c)dc+\int_{0}^{1700}u_{1}(c)f(c)dc\]  
\[ \begin{split}+\int_{1700}^{+\infty}u_{2}(c)f(c)dc\\ =\int_{0}^{1700}u_{1}(c)f(c)dc+(1\\ -\int_{0}^{1700}u_{2}(c)\Big{)}f(c)dc\end{split}  \tag{8}\]  
\[ \begin{split}\Rightarrow EUV_{c}=\int_{0}^{1700}1\times& 10^{-7}c^{2}f(c)dc\\ +\int_{0}^{1700}1.84\times 10^{-4}cf(c)dc\\ +\int_{0}^{1700}3.109\times 10^{-1}f(c)dc+(1\\ -(\int_{0}^{1700}1\times 10^{-7}c^{2}f(c)dc\\ +\int_{0}^{1700}4.69\times 10^{-4}cf(c)dc\\ +2.932\times 10^{-1}f(c)dc\end{split}  \tag{9}\]  
Figure 11: Gamma dispensation and density function of completing the project (Source: researcher’s findings). It is necessary that the equation be clear by density function and \(f(c)\), and what was gained in the previous part was the density function of \(x\), which is the real cost of doing the project. So, \(C\) is the function of \(x\). \[ \begin{split} x\ \sim\ Gamma\ (k,\theta)\ \ \sim\ Gamma(24,103)\\ c=x+b+p_{c}(x_{0}-x)\end{split}  \tag{9}\]  
It also has the GAMA dispensation with \(\theta_{1},\ k_{1}\) parameters. To achieve the parameters of this new dispensation, we use the average and \(x\) dispensation variance and its relation to the average and c dispensation variance. We know that the average and variance of GAMA dispensation are received by the relations below:  
\[\mu=E(x)=k\theta=24^{*}103=2472 \tag{10}\]  
\[\delta^{2}=Var(x)=k\theta^{2}=24^{*}103^{2}=254616 \tag{11}\]  
The relation between \(c\) average and variance and \(x\) average and variance could be received by the way below:  
\[E[c]=E[x]+b=p_{c}x_{0}-p_{c}E[x]=b+p_{c}x_{0}+(1-p_{c})E[x]=k_{1}\theta_{1} \tag{12}\]  
\[Var[C]=(1-P_{C})^{2}\ Var(x)=k_{1}\theta_{1}{}^{2} \tag{13}\]  
By placing relations 10 and 11 in relations 12 and 13, the amounts of average and c dispensation variance based on \(p_{c}\) are gained as below:  
\[E[c]=k_{1}\theta_{1}=2772-472p_{c} \tag{14}\]  
\[Var[c]=k_{1}\theta_{1}{}^{2}=254616(1-p_{c})^{2} \tag{15}\]  
Table 6 shows the \(p_{c}\). Probability density function parameters. For example, for \(p_{c}=0.1\)  
\[\Rightarrow\theta_{1}=92.51,\ k_{1}=29.454\bigg{\{} \begin{matrix}k_{1}\theta_ {1}=2724.8\\ k_{1}\theta_{1}{}^{2}=252069.84\end{matrix} \ \Rightarrow \tag{16}\]  
Now we should be able to gain \(\int c^{2}f(c)dc,\ \int cf(C)dc\), in which \(C\) has the GAMA dispensation with the probability density function **f(c)** (Appendix A) and with the parameters of Table 6.  
\[\text{relation13}:c\ \sim\text{Gamma}(k_{1},\theta_{1})\Rightarrow f(c)= \begin{cases}\frac{c^{1}-1}{\epsilon^{\frac{mc}{p_{1}}}}{\Gamma(k_{1})\theta _{1}{}^{K_{1}}}&c\geq 0\\ 0&c<0\end{cases}  \tag{17}\]  
So, after the stages mentioned in the reference, Equation (18) is:  
\[c^{2}f(c)\ \sim(k_{1}+1)k_{1}\theta_{1}{}^{2}\text{Gamma}(k_{1}+2,\ \theta_{1}) \tag{18}\]  
Now, with respect to the relations and the given results, we can write the owner's final cost expected value in Equation (8) as below: \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \(\rho_{c}\) & **1** & **0.9** & **0.8** & **0.7** & **0.6** & **0.5** & **0.4** & **0.3** & **0.2** & **0.1** & **0** \\ \hline \(k_{1}\) & - & 2163.787 & 562.921 & 260.148 & 152.046 & 101.035 & 72.799 & 55.458 & 43.997 & 36 & 30.179 \\ \(\theta_{1}\) & 0 & 1.085 & 4.254 & 9.385 & 16.369 & 25.1 & 35.484 & 47.431 & 60.858 & 75.69 & 91.853 \\ \hline \end{tabular}
\end{table} 
Table 6: C probability density function parameters for \(p_{c}\). The proposed project's expected utility is obtained for the owner using Equation (7).