# 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
\begin{table}
\begin{tabular}{c c} \hline \hline
**Terminology** & **Notation** \\ \hline Set of decision-makers & \(DM\) \\ Decision-maker & \(dm_{k}\) \\ Set of alternatives & \(A\) \\ Set of criteria & C \\ Alternative & \(a_{i}\) \\ Criterion & \(c_{j}\) \\ Decision matrix & \(D\) \\ Weight or preference given to a certain alternative \(a_{i}\) by a decision-maker \(dm_{k}\) & \(wu_{dm_{k_{j_{i}}}}\) \\ Set of alternatives weights of a decision-maker \(dm_{k}\) & \(Wu_{dm_{k_{j}}}\) \\ Weight or preference given to a certain criterion \(c_{j}\) by a decision-maker \(dm_{k}\) & \(wu_{dm_{k_{j}}}\) \\ Set of criteria weights of a decision-maker \(dm_{k}\) & \(W_{cd_{m_{j}}}\) \\ Set of alternatives weights of a set of decision-makers & \(W_{dm}\) \\ Set of criteria weights of a set of decision-makers & \(W_{CDM}\) \\ Alternatives preference matrix that relates each alternative with the correspondent evaluation made & \(AP_{DM}\) \\ \multirow{2}{*}{Criteria preference matrix that relates each criterion with the correspondent evaluation made by} & \multirow{2}{*}{\(CP_{DM}\)} \\  & each decision-maker & \\ Predicted importance of alternative \(a_{i}\) to decision-maker \(dm_{k}\) & \(imp_{dm_{k_{j}}}\) \\ Set of importance values for all the alternatives to decision-maker \(dm_{k}\) & \(Amp_{dm_{k_{j}}}\) \\ Predicted importance of criterion \(c_{j}\) to decision-maker \(dm_{k}\) & \(imp_{dm_{k_{j}}}\) \\ Set of importance values for all the criteria to decision-maker \(dm_{k}\) & \(Clmp_{dm_{k}}\) \\ Set of importance values for all the alternatives of a set of decision-makers & \(Almp_{DM}\) \\ Set of importance values for all the criteria of a set of decision-makers & \(Clmp_{DM}\) \\ \multirow{2}{*}{Alternatives evaluation matrix} & \multirow{2}{*}{\(AP_{DM}\)} \\  & Normalized decision matrix & \(D\) \\ Consistency matrix of the decision-maker \(dm_{k}\) & & \(CM_{dm_{k}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation correspondence. **Definition 3**. : _An alternative \(a_{i}=\left\{id_{a_{i}}\right\}\) consists of:_  
* \(\forall a_{i}\in A,i\in\{1,2,\ldots,m\}\)_;_
* \(id_{a_{i}}\) _is the identification of a particular alternative._  
We can now define the desirable set of alternatives:  
* \(zingara=\{a_{1}\}\)_;_
* _thaiPalace_ \(=\{a_{2}\}\)_;_
* \(nosh=\{a_{3}\}\)_._  
In our example, the decision matrix is as follows:  
\[D=A\times C=\left[ \begin{array}{ccccccccc}5&0&90&3&25&1&1&1&1\\ 10&10&60&3&15&1&1&1&1\\ 1&0&15&1&7&0&0&1&0\end{array} \right]\]  
As can be seen in the example, although the criteria value can be Numeric, Boolean, or Classificatory, internally they are all interpreted as Numeric. When the criterion type is Boolean, we assume \(false=0\) and \(true=1\), when the criterion type is Classificatory, the scale is converted into the corresponding values (in our example \(Good=3\), \(Ok=2\) and \(Poor=1\)), and when the criterion greatness is Subjective, the instantiation of each criterion in each alternative is always 1 (as is the case of the criterion cuisineStyle). Usually, most MCDA methods only consider the decision-makers' preferences on alternatives and criteria, i.e., the decision process is seen as a fully rational process. However, there are many human particularities that affect and are important in decision processes. According to the winner of 2017 Economics Nobel Prize ([PERSON]), much more should be considered, and a decision process should not be seen only from a rational perspective. In this work, we follow that vision. We consider the existence of a heterogeneous group of decision-makers and consequently, we contemplate a set of attributes to model each decision-maker according to his/her own particularities, interests, and intentions. Next, we introduce the decision-maker definition and all the definitions associated with the decision-makers representation. **Definition 4**. : _Let \(DM\) be a set of decision-makers where:_  
* \(DM=\{dm_{1},dm_{2},\ldots,dm_{n}\},n>1\)_._  
**Definition 5**.