# A Multiple Criteria Decision Analysis Framework for Dispersed Group Decision-Making Contexts
## 3 The Multiple Criteria Decision Analysis Framework
### Definition of the Entities and How They Relate
: _The correlation between the decision-maker's style of behavior, expertise level and decision-makers he/she considers credible is defined as follows:_  
\[\forall dm_{k}\in DM,\forall a_{i}\in A,wa_{dm_{k_{i}}}=\frac{wa_{dm_{k_{i}}} \times CS_{dm_{k}}\times e_{dm_{k}}+\left(\frac{TP}{ND}\right)\times CO_{dm_{k }}\times e_{dm_{k}}}{CS_{dm_{k}}\times e_{dm_{k}}+CO_{dm_{k}}\times e_{dm_{k}}} \tag{1}\]where:  
* _TP_ is the sum of the weights given to alternative \(a_{i}\) by each of the credible decision-makers in \(\textit{DMcredible}_{dm_{k}}\), \(\textit{TP}=\sum\limits_{x=1}^{ND}wa_{dm_{x_{i}}}\);
* _ND_ is the number of credible decision-makers to \(dm_{k}\) such that \(ND=\textit{DMcredible}_{dm_{k}}\);
* \(e_{dm_{k}}\) is the inverse of expertise level of decision-maker \(dm_{k}\) (calculated as \(1-e_{dm_{k}}\)). Each \(wa_{dm_{x_{i}}}\) is associated with the \(dm_{k}\)'s Concern for Self and expertise level, while the average preference for \(a_{i}\) to the decision-makers marked as credible by \(dm_{k}\) is associated with the \(dm_{k}\)'s Concern for Others and \(dm_{k}\)'s inverse of expertise level. In this way, after the readjustment, the new \(wa_{dm_{x_{i}}}\) value will reflect not only the decision-makers' preferences but also the context, their motivations and more. After readjusting the decision-makers' preferences, the difference between the alternative/criterion with the highest evaluation and the alternative/criterion with the lowest evaluation is compared, since we believe the evaluation range used by the decision-makers can be an important indicator. **Definition 13**. : _Let \(F_{Dif}(\text{arg})\) be a function that returns the difference (according to the preferences defined by \(dm_{k}\)) between the maximum and minimum weights given to alternatives or criteria that belong to a set of weights (arg is a list of alternatives or criteria weights):_  
\[F_{Dif}\Big{(}Wa_{dm_{k}}\Big{)}=\left\{ \begin{array}{c}\max\Big{(}Wa_{dm_{ k}}\Big{)}-\min\Big{(}Wa_{dm_{k}}\Big{)},\ if\max\Big{(}Wa_{dm_{k}}\Big{)}\
eq\min\Big{(}Wa_{dm_{k}}\Big{)}\\ \max\Big{(}Wa_{dm_{k}}\Big{)}\end{array} \right. \tag{2}\]  
\[F_{Dif}\Big{(}Wc_{dm_{k}}\Big{)}=\left\{ \begin{array}{c}\max\Big{(}Wc_{dm_{ k}}\Big{)}-\min\Big{(}Wc_{dm_{k}}\Big{)},\ if\max\Big{(}Wc_{dm_{k}}\Big{)}\
eq\min\Big{(}Wc_{dm_{k}}\Big{)}\\ \max\Big{(}Wc_{dm_{k}}\Big{)}\end{array} \right. \tag{3}\]  
\(F_{Dif}\Big{(}Wa_{dm_{k}}\Big{)}\) returns \(dif\), the difference (according to the preferences defined by \(dm_{k}\)) between the alternative with the highest weight and the alternative with the lowest weight, while \(F_{Dif}\Big{(}Wc_{dm_{k}}\Big{)}\) returns the difference between the criterion with the highest weight and the criterion with the lowest weight. The result of \(F_{Dif}\) can be classified into one of the five possible levels presented in Table 3.  
The difference (using function \(F_{Dif}\)) between the criterion or alternative with more weight and the criterion or alternative with less weight allows to obtain (according to Table 3) the \(l\) value. **Definition 14**. : _Let \(F_{l}(dif)\) be a function that returns the \(l\) value for \(dif\) according to Table 3, \(0\leq dif\leq 1\)._  
The \(l\) value allows to have another look over the preferences defined by the decision-makers. Let us suppose \(dm_{y}\) defined the preferences [0.4, 0.4, 0.8] and \(dm_{z}\) defined the preferences [0.5, 0.5, 0.9], both for alternatives \(a_{1}\), \(a_{2}\) and \(a_{3}\), respectively. Can we say that \(dm_{y}\) prefers \(a_{3}\) more than \(dm_{z}\) prefers \(a_{3}\)? We believe not, because people have different perceptions of the world around them. That is why we try to understand the decision-makers' configurations in terms of meaning. \begin{table}
\begin{tabular}{c l} \hline \hline
**Level (\(l\))** & \(F_{Dif}\) \\ \hline
5 & \(dif\geq 0.80\) \\
4 & \(0.60\leq dif<0.80\) \\
3 & \(0.40\leq dif<0.60\) \\
2 & \(0.20\leq dif<0.40\) \\
1 & \(dif<0.20\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(F_{Dif}\) levels.