# A Multiple Criteria Decision Analysis Framework for Dispersed Group Decision-Making Contexts
## 3 The Multiple Criteria Decision Analysis Framework
### Recommending Alternatives to Reject and as Solution
#### 3.3.2 Recommending Based on the Prediction of Decision Satisfaction  
So far we have presented all the necessary entities and definitions to implement a MCDA method capable of proposing solutions, recommending alternatives to reject and fundamentally, of executing those tasks using a method that deduces the reasons behind those proposals/recommendations. However, it is also essential to predict how the decision quality is perceived from the perspective of each decision-maker and the group. For instance, decision-makers who consider unsatisfactory a certain solution may be advised to review inconsistent assessments or to share with the group the reasons why certain criteria are more important. The discussion about the importance of the criteria may lead decision-makers to consider other perspectives, influencing them to redefine their preferences, promoting consensus. Next, we will introduce a set of definitions that intends to predict the decision quality through the analysis of the decision-makers' satisfaction. As is evidenced in several studies, the study of the decision-makers' satisfaction can be a strong indicator of the decision quality [37; 38]. The satisfaction analysis allows to understand the impact of a certain choice in a certain time instant according to the context of that time instant. The context is not only the social context or the current dynamics but also a reflection of the decision-makers' sentiments regarding a possible solution, according to the knowledge they have in that time instant. Therefore, we adapted to this framework a simpler version of the model proposed in [38] that intends to predict the decision-makers' satisfaction (perception of the decision quality). The used satisfaction scale is presented in Table 9, with the predicted satisfaction belonging to the interval \([-1,\ 1]\). **Definition 24**. : _Let \(F_{\mathit{SatisfactionWSB}}(\mathit{dm}_{k_{x}},a_{i})\) be a function which returns the predicted satisfaction of decision-maker \(\mathit{dm}_{k}\) regarding alternative \(a_{i}\) without the inclusion of the decision-maker's style of behavior (\(a_{y}\) is the decision-maker's \(\mathit{dm}_{k}\) most preferred alternative with the initial weights, i.e., before the readjustment performed in definition 12):_  
\[F_{\mathit{SatisfactionWSB}}(\mathit{dm}_{k},a_{i})=\left(1-\left|2*\mathit{ wa}_{\mathit{dm}_{k_{x}}}-1\right|\right)*\left(\mathit{wa}_{\mathit{dm}_{k_{x}}}- \mathit{wa}_{\mathit{dm}_{k_{y}}}\right)+2*\mathit{wa}_{\mathit{dm}_{k_{x_{i} }}}-1 \tag{8}\]  
In our example, \(F_{\mathit{SatisfactionWSB}}(\mathit{harry},\mathrm{zingara})=0.425\), \(F_{\mathit{SatisfactionWSB}}(\mathit{george},\mathrm{zingara})=-0.25\) and \(F_{\mathit{SatisfactionWSB}}(\mathit{jane},\mathrm{zigara})=0.8\). **Definition 25**. : _Let \(F_{\mathit{SatisfactionCred}}(\mathit{dm}_{k_{x}},a_{i})\) be a function which returns the average satisfaction of the \(\mathit{DMcredible}_{\mathit{dm}_{k}}\):_  
\[F_{\mathit{SatisfactionCred}}(\mathit{dm}_{k_{x}},a_{i})=\frac{\sum_{w=1}^{( \mathit{DMcredible}_{\mathit{dm}_{k}})}F_{\mathit{SatisfactionWSB}}( \mathit{dm}_{w},a_{i})}{\left\langle\mathit{DMcredible}_{\mathit{dm}_{k}} \right\rangle} \tag{9}\]  
In our example, \(F_{\mathit{SatisfactionCred}}(\mathit{harry},\mathrm{zigara})=0.275\), \(F_{\mathit{SatisfactionCred}}(\mathit{george},\mathrm{zigara})=0.6125\) and \(F_{\mathit{SatisfactionCred}}(\mathit{jane},\mathrm{zigara})=0.0875\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Designation** & **Interval** \\ \hline Extremely satisfied & \([0.75,\ 1]\) \\ Much satisfaction & \([0.5,\ 0.75]\) \\ Satisfaction & \([0.25,\ 0.5]\) \\ Some satisfaction & \([0,\ 0.25]\) \\ Some dissatisfaction & \([-0.25,\ 0]\) \\ Dissatisfied & \([-0.5,-0.25]\) \\ Very dissatisfied & \([-0.75,-0.5]\) \\ Extremely dissatisfied & \([-1,-0.75]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Scale of satisfaction (adapted from [38]). **Definition 26**.