2. Modeling of group decision making

The problem of choosing a group decision is as follows. There is a group of k members. There is given a set X0—the feasible set. For each member i, i ¼ 1, 2, …, k, a decision evaluation function f <sup>i</sup> is defined, which is an assessment of a joint decision. The assessment of the joint decision is to be made by all members in the group.

The problem of group decision making is modeled as multicriteria optimization problem:

$$\max\_{\mathbf{x}} \left\{ \left( f\_1(\mathbf{x}), \dots, f\_k(\mathbf{x}) \right) : \ \mathbf{x} \in X\_0 \right\}, \tag{1}$$

3. Equitably efficient decision

better? This is the following relation:

y<sup>1</sup> ≻y<sup>2</sup> ⇔ y<sup>1</sup>

vector of evaluations <sup>b</sup><sup>y</sup> <sup>¼</sup> <sup>f</sup>ð Þ <sup>b</sup><sup>x</sup> is a nondominated vector [4, 5].

<sup>i</sup> <sup>≥</sup> <sup>y</sup><sup>2</sup>

transfers.

preference relation.

nondominated vector.

for the evaluation vector y ¼ y1; y2;…; yk

yi

Group decision making is modeled as a special multicriteria optimization problem—the solution should have the feature of anonymity—no distinction is made between the results that differ in the orientation coordinates and the principle of transfers. This solution of the problem is named an equitably efficient decision. It is an efficient decision that satisfies the additional property‑the property of preference relation anonymity and the principle of

Nondominated solutions (optimum Pareto) are defined with the use of preference relations which answer the question: which one of the given pair of evaluation vectors y<sup>1</sup>, y<sup>2</sup> ∈ Rk is

The vector of evaluation <sup>b</sup><sup>y</sup> <sup>∈</sup>Y<sup>0</sup> is called the nondominated vector; if there is no such vector <sup>y</sup><sup>∈</sup> <sup>Y</sup>0, that <sup>b</sup><sup>y</sup> is dominated by <sup>y</sup>. Appropriate acceptable decisions are specified in the decision space. The decision <sup>b</sup><sup>x</sup> <sup>∈</sup> <sup>X</sup><sup>0</sup> is called efficient decision (Pareto efficient) if the corresponding

In the multicriteria problem (1), which is used to make a group decision for a given set of the evaluation functions, only the set of the evaluation functions is important without taking into account which function is taking a specific value. No distinction is made between the results that differ in the arrangement. This requirement is formulated as the property of anonymity of

The relation is called an anonymous (symmetric) relation if, for every vector <sup>y</sup> <sup>¼</sup> <sup>y</sup>1; <sup>y</sup>2;…; �

yPð Þ<sup>1</sup> ; yPð Þ<sup>2</sup> ;…; yP kð Þ � � <sup>≈</sup> <sup>y</sup>1; <sup>y</sup>2;…; yk

The relation of preferences that would satisfy the anonymity property is called symmetrical relation. Evaluation vectors having the same coordinates, but in a different order, are identified. A nondominated vector satisfying the anonymity property is called symmetrically

Moreover, the preference model in group decision making should satisfy the principle of transfers. This principle states that the transfer of small amount from an evaluation vector to any relatively worse evaluation vector results in a more preferred evaluation vector. The relation of preferences satisfies the principle of transfers, if the following condition is satisfied:

:

<sup>0</sup> þ ε � ei} ≻ y for 0 < yi

<sup>0</sup> � yi

<sup>00</sup> < ε (6)

yk<sup>Þ</sup> <sup>∈</sup>Rk and for any permutation <sup>P</sup> of the set 1f g ;…; <sup>k</sup> , the following property holds:

� �∈R<sup>k</sup>

<sup>0</sup> > yi} ) y � ε � ei

<sup>i</sup> <sup>∀</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, m <sup>∧</sup> <sup>∃</sup> <sup>j</sup> <sup>y</sup><sup>1</sup>

<sup>j</sup> > y<sup>2</sup>

<sup>j</sup> : (4)

Multicriteria Support for Group Decision Making http://dx.doi.org/10.5772/intechopen.79935 49

� � (5)

where 1, <sup>2</sup>,…, k are particular members, <sup>X</sup><sup>0</sup> <sup>⊂</sup> Rn is the feasible set, <sup>x</sup> <sup>¼</sup> ð Þ <sup>x</sup>1; <sup>x</sup>2;…; xn <sup>∈</sup> <sup>X</sup><sup>0</sup> is a group decision, f ¼ f <sup>1</sup>; f <sup>2</sup>;…; f <sup>k</sup> is the vector function that maps the decision space <sup>X</sup><sup>0</sup> <sup>¼</sup> Rn into the criteria space <sup>Y</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>k</sup> , and specific coordinates yi ¼ f <sup>i</sup> ð Þx , i ¼ 1, 2, ::, m represent the scalar evaluation functions—the result of a decision x i � th member i ¼ 1, 2, …, k.

The purpose of the problem (1) is to support the decision process to make a decision that will be the most satisfactory for all members in the group.

Functions f <sup>1</sup>, …, f <sup>k</sup> introduce a certain order in the set of decision variables—preference relations:

$$\mathbf{x}^1 \succ \mathbf{x}^2 \Leftrightarrow f\_1(\mathbf{x}^1) \succeq f\_2(\mathbf{x}^2), \dots, f\_k(\mathbf{x}^1) \succeq f\_k(\mathbf{x}^2) \quad \land \quad \exists j \\ f\_j(\mathbf{x}^1) > f\_j(\mathbf{x}^2). \tag{2}$$

At point x1, all functions have values greater than or equal to the value at point x2, and at least one is greater.

The multicriteria optimization model (1) can be rewritten in the equivalent form in the space of evaluations. Consider the following problem:

$$\max\_{\mathbf{x}} \left\{ (y\_1, \dots, y\_k) : \quad y \in Y\_0 \right\}, \tag{3}$$

where x∈ X is a vector of decision variables, y ¼ y1;…; yk is the evaluation vector and particular coordinates yi represent the result of a decision x i � th member i ¼ 1, 2, …, k, and Y<sup>0</sup> ¼ f Xð Þ<sup>0</sup> is the set of evaluation vectors.

The vector function y ¼ f xð Þ assigns to each vector of decision variables x an evaluation vector y∈ Y<sup>0</sup> that measures the quality of decision x from the point of view of all members in the group. The set of results achieved Y<sup>0</sup> is given in the implicit form—through a set of feasible decisions X<sup>0</sup> and the mapping of a model f ¼ f <sup>1</sup>; f <sup>2</sup>; …; f <sup>k</sup> . To determine the value y, the simulation of the model is necessary: y ¼ f xð Þ for x∈ X<sup>0</sup> .
