# 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
: _Let \(wa_{dm_{k_{i}}}\) be the weight or preference given to a certain alternative \(a_{i}\in A\) by a decision-maker \(dm_{k}\in DM\)._  
**Rule 2**. : A decision-maker \(dm_{k}\) can define a set of alternatives weights where:  
* \(Wa_{dm_{k_{k}}}=\left\{wa_{dm_{k_{1}}},wa_{dm_{k_{2}}},\ldots,\ wa_{dm_{k_{m}}} \right\},m>1\);
* \(0\leq wa_{dm_{k_{i}}}\leq 1,\forall i\in\{1,2,\ldots,m\}\);
* \(\left\langle Wa_{dm_{k_{i}}}\right\rangle=\langle A\rangle\). **Definition 6**. : _Let \(wc_{dm_{k_{i}}}\) be the weight or preference given to a certain criterion\(c_{j}\in C\) by a decision-maker \(dm_{k}\in DM\)._  
**Rule 3**. : A decision-maker \(dm_{k}\) can define a set of criteria weights where:  
* \(Wc_{dm_{k}}=\left\{wc_{dm_{k_{1}}},wc_{dm_{k_{2}}},\ldots,\ wc_{dm_{k_{m}}} \right\}\)_,_ \(p>1\)* \(0\leq wc_{dm_{k_{j}}}\leq 1,\forall j\in\{1,2,\ldots,p\}\);
* \(\left\langle Wc_{dm_{k}}\right\rangle=\langle C\rangle\). In order to help decision-makers express their preferences regarding alternatives and criteria [25; 31; 32], several strategies can be applied. One of the preference structures most used in GDM problems defined under uncertainty, is the so-called fuzzy preference relation [33], but there are others, such as multiplicative preference relation [34] and linguistic preference relation [35]. In this work, we follow the one presented in [36], which consists of classifying the preferences individually according to an adaptation of the Visual Analog Scale, although, due to the alternatives and criteria alignment on the screen, they are unconsciously being visually compared to each other. As we will see later, the decision-makers' preferences regarding alternatives and criteria are used to define the alternatives and criteria importance according to a specific scale. Decision-makers are not obliged to define a preference for all alternatives and criteria, however, the alternatives and criteria without a defined preference will be (internally) considered as 0. **Definition 7**. : _A decision-maker \(dm_{k}=\left\{id_{dm_{k}},Wa_{dm_{k}},Wc_{dm_{k}},DMcredible_{dm_{k}},CS_{dm_{k }},CO_{dm_{k}},e_{dm_{k}}\right\}\) consists of:_  
* \(id_{dm_{k}}\) is the identification of a particular decision-maker;
* \(DMcredible_{dm_{k}}\) is the set of decision-makers that decision-maker \(dm_{k}\) considers as credible, \(DMcredible_{dm_{k}}\in DM\); \(CS_{dm_{k}}\) is the value of the Concern for Self of decision-maker \(dm_{k}\) chosen style of behavior;
* \(CO_{dm_{k}}\) is the value of the Concern for Others of decision-maker \(dm_{k}\) chosen style of behavior;
* \(e_{dm_{k}}\) is the expertise level of decision-maker \(dm_{k}\). Besides the unique identification and the preferences for alternatives and criteria, it is possible to define for each decision-maker \(dm_{k}\) the set of decision-makers (from the set of decision-makers involved in the decision process) that he/she considers credible in a specific decision problem. The credibility is multifold, i.e., a decision-maker can identify another decision-maker as credible due to different reasons, such as reputation, expertise level and/or hierarchy. Following the behavior styles proposed by the authors in [15], which can be used to model decision-makers according to their intentions, it is possible to represent the decision-maker's concerns in attaining his/her own objectives (Concern for Self) and the other decision-makers' objectives (Concern for Others). This is possible due to the solid \"operating values\" the authors found for the five behavior styles (see Table 2). The proposed behavior styles vary in four dimensions being two of them Concern for Self and Concern for Others. The \"operating values\" resulted from a questionnaire, where the authors asked the participants to classify the five behavior styles in the four dimensions. The objective was to understand if the participants had a similar perception of the meaning of each behavior style. The results demonstrated high levels of agreement, which allowed one to define standards for each behavior style.