## 3 Discrete Event Simulation Model
### Ranking Scenarios: VIKOR Method  
We use the VIKOR method to rank alternatives based on the criteria [4,5]. The working steps of are as follows:  
_Step 1:_ Normalize the decision matrix. Hence, the application of the VIKOR method starts with the development of the decision matrix, which shows the performance of the alternatives with respect to various criteria.  
A decision matrix \(F=(f_{ij})_{mn}\) with \(m\) alternatives and \(n\) criteria is constructed as follows:  
\[F=\left(\begin{array}{cccc}f_{11}&f_{12}&\cdots&f_{1n}\\ f_{21}&f_{22}&\cdots&f_{2n}\\ \vdots&\vdots&\cdots&\vdots\\ f_{m1}&f_{m2}&\cdots&f_{mn}\\ \end{array}\right);\]  
_Step 2:_ From the given normalized decision matrix, determine the best \(f_{(ij)_{mn}}\) and the worst \(f_{(ij)_{mn}}\) values of all the criteria;  
_Step 3:_ Calculate utility measure \(S_{i}\) and regret measure \(R_{i}\):  
\[S_{i}=\sum_{i=1}^{n}W_{i}\Big{[}\frac{(f_{ij})_{max}-f_{ij}}{(f _{ij})_{max}-(f_{ij})_{min}}\Big{]}\qquad\text{for beneficial attributes},\] \[S_{i}=\sum_{i=1}^{n}W_{i}\Big{[}\frac{f_{ij}-(f_{ij})_{min}}{(f _{ij})_{max}-(f_{ij})_{min}}\Big{]}\qquad\text{for non-beneficial attributes},\] \[R_{i}=\text{maximum of }\Big{\{}W_{i}\Big{[}\frac{(f_{ij})_{max}-f_{ij}}{(f _{ij})_{max}-(f_{ij})_{min}}\Big{]}\Big{\}}\qquad\text{for beneficial attributes},\] \[R_{i}=\text{maximum of }\Big{\{}W_{i}\Big{[}\frac{f_{ij}-(f_{ij})_{ max}}{(f_{ij})_{max}-(f_{ij})_{min}}\Big{]}\Big{\}}\qquad\text{for non-beneficial attributes},\]  
where \(W_{i}\) is the weight of the \(j^{th}\) criterion,  
_Step 4:_ Computing the VIKOR values (Q):  
\[Q_{i}=v\frac{S_{i}-S_{i-min}}{S_{i-max}-S_{i-min}}+(1-v)\frac{R_{i}-R_{i-min} }{R_{i-max}-R_{i-min}}\]  
where \(v\) is the weight of the strategy of 'the majority of attributes' (or 'the maximum group utility'). Its value lies between 0 and 1.The authors in [4,5] indicate that usually a value of \(v\) as 0.5 is preferred and that the compromise can be selected with 'voting by majority' (\(v>0.5\)), with 'consensus' (\(v=0.5\)) or with'veto' (\(v<0.5\)).  
Without loss of generality, it takes the value 0.5, where \(S_{i-max}=Max\ S_{i},S_{i-min}=Min\ S_{i},R_{i-max}=Max\ R_{i}\), and \(R_{i-min}=Min\ R_{i}\);  
_Step 5:_ Rank in ascending order the alternatives according to the values of \(S\), \(R\), and \(Q\). Once ranked, \(a^{{}^{\prime}}\) is considered to be the compromise solution (with minimum \(Q\) value) only if the following two conditions are satisfied:  
* Condition 1 : Acceptable advantage \(Q(a^{{}^{\prime\prime}})-Q(a^{{}^{\prime}})\geq DQ\) where \(a^{{}^{\prime\prime}}\) is the second best alternative in the ranking list by \(Q\) and \(DQ=\frac{1}{N-1}\);
* Condition 2 : Acceptable stability in decision making. Alternative \(a^{{}^{\prime}}\) must also be the best ranked by \(S\) or/and \(R\).  
If one of these conditions is not satisfied, a set of compromise solutions is then proposed as follows:  
* Alternatives \(a^{\prime}\) and \(a^{\prime\prime}\) if only Condition 2 is not satisfied, or
* Alternatives \(a^{\prime},a^{\prime\prime},...,a^{N}\) if Condition 1 is not satisfied and \(a^{N}\) is determined by the relation \(Q(a^{N})-Q(a^{\prime})<DQ\) for maximum \(N\).