## 3 Characterization of GDM-MOPs Under-Study  
In this paper, we address group multi-objective optimization problems described by the following characteristics:  
* There is a group moderator who is in charge to control and guide the consensus reaching process;
* The individual DMs participate in the decision process providing information about their preferences, beliefs, and level of conservatism, and modifying this information during consecutive steps of the CRP;
* Some (even all) objective values may be not the same for different DMs;* Each DM may handle a different set of objective functions;
* The objective values may be imperfectly known (subject to imprecision or uncertainty);
* The availability of resources may be imperfectly known;
* Resource requirements per activity (project, in case of portfolio optimization) may be imperfectly known;
* Each group member has their own opinion about the availability of resources and requirements per activity (project, in case of portfolio optimization);
* All the DMs consider a common point in the decision variable space. Points c, d, e, f, g, and h imply that each group member faces their own MOO problem under imperfect knowledge. The model of such imperfect knowledge is based on the following assumption:  
**Assumption 1**. : _Each DM is willing to represent as interval numbers the imperfect information on objective values, availability, and requirement of resources._  
In the framework of project portfolio optimization, the above statement has been assumed by [PERSON] et al. (2007, 2008), [PERSON] and [PERSON] (2016), [PERSON] and [PERSON] (2017), and [PERSON] et al. (2019). This assumption is a bit more restrictive than it appears. As stated by [41], if we assume that a magnitude is an interval number, one should admit that (i) this magnitude can take any value within this interval, and (ii) the DM does not privilege the credibility of any particular value within the range. Let \(\mathbf{AR}^{j}_{i}\) be the interval number used to represent the estimated availability of the \(j\)-th resource from the point of view of the \(i\)-th DM; \(\mathbf{NR}^{j}_{i}\) is the interval number used to denote the aggregation of required resources, which depends on the decision variables vector \(z\). In order to find a feasibility condition for the \(i\)-th DM, a solution \(z\) must satisfy \(\mathbf{AR}^{j}_{i}>\xi^{i}\mathbf{NR}^{j}_{i}\) (\(z\)) according to the order relation given by Equation (10). The \(i\)-th DM faces the MOP in Equation (11). \[\underset{z\in R_{F_{i}}}{\text{Maximize}}F_{i}=(f_{1i}(z),f_{2i}(z),\ \ldots,f_{Ni}(z)) \tag{11}\]  
where \(N_{i}\) is the number of the evaluation criterion for the \(i\)-th member of a group; \(F_{i}\) is the vector function being considered by the \(i\)-th DM; \(z\) is the vector of decision variables; in project portfolio optimization, \(z\) is a vector composed of binary variables, in which \"1\" means \"the related project belongs to the portfolio\", and \"0\" otherwise. \(R_{F_{i}}\) is the feasible region from the point of view of the \(i\)-th DM. It is defined by the interval-based resource constraints \(\mathbf{AR}^{j}_{i}>\xi^{i}\mathbf{NR}^{j}_{i}\) (\(z\)), and perhaps other non-resource constraints. In Problem 11, \(\xi^{i}\) is the credibility threshold for the expression \"the available resources are sufficient to satisfy the requirements\". The more conservative the \(i\)-th DM, the higher the value of \(\xi^{i}\). **Definition 5** (\(\alpha\)-dominance). : _Let \(O_{i}\) be the image of the feasible region of Problem 11. Consider an element (x,y) belonging to \(O_{i}\times O_{i}\), where \(x\
eq y\), and a a real number in [0.5, 1]. The solution \(y\) is \(\alpha\)-dominated by \(x\) (denoted by xD(\(\alpha\))y) iff \(min[P(f_{yi}(x)\geq f_{yi}(y))\), \(j=1\),..., \(N_{i}]=\alpha\) and \(P(f_{ki}(x)\geq f_{ki}(y))>0.5\) for some \(k\)._  
Identifying satisfactory solutions is a preference judgment.