3. Equitably efficient decision

the group. Each member can accept or refuse the solution. In the second case, a member gives

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

The problem of group decision making is modeled as multicriteria optimization problem: max<sup>x</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>x</sup> ;…; <sup>f</sup> <sup>k</sup>ð Þ<sup>x</sup> : <sup>x</sup> <sup>∈</sup> <sup>X</sup><sup>0</sup>

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

, and specific coordinates yi ¼ f <sup>i</sup>

The purpose of the problem (1) is to support the decision process to make a decision that will

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

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

The multicriteria optimization model (1) can be rewritten in the equivalent form in the space of

particular coordinates yi represent the result of a decision x i � th member i ¼ 1, 2, …, k, and

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

max<sup>x</sup> <sup>y</sup>1;…; yk

where x∈ X is a vector of decision variables, y ¼ y1;…; yk

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>

simulation of the model is necessary: y ¼ f xð Þ for x∈ X<sup>0</sup> .

scalar evaluation functions—the result of a decision x i � th member i ¼ 1, 2, …, k.

, (1)

: <sup>y</sup><sup>∈</sup> <sup>Y</sup>0<sup>g</sup> , (3)

is the evaluation vector and

. To determine the value y, the

ð Þx , i ¼ 1, 2, ::, m represent the

is the vector function that maps the decision space <sup>X</sup><sup>0</sup> <sup>¼</sup> Rn

<sup>x</sup><sup>1</sup> <sup>≻</sup>x<sup>2</sup> <sup>⇔</sup> <sup>f</sup> <sup>1</sup> <sup>x</sup><sup>1</sup> <sup>≥</sup> <sup>f</sup> <sup>2</sup> <sup>x</sup><sup>2</sup> , …, f <sup>k</sup> <sup>x</sup><sup>1</sup> <sup>≥</sup> <sup>f</sup> <sup>k</sup> <sup>x</sup><sup>2</sup> <sup>∧</sup> <sup>∃</sup>j f <sup>j</sup> <sup>x</sup><sup>1</sup> <sup>&</sup>gt; <sup>f</sup> <sup>j</sup> <sup>x</sup><sup>2</sup> : (2)

his new proposal and the problem is resolved again.

2. Modeling of group decision making

be the most satisfactory for all members in the group.

evaluations. Consider the following problem:

Y<sup>0</sup> ¼ f Xð Þ<sup>0</sup> is the set of evaluation vectors.

to be made by all members in the group.

group decision, f ¼ f <sup>1</sup>; f <sup>2</sup>;…; f <sup>k</sup>

into the criteria space <sup>Y</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup><sup>k</sup>

48 Optimization Algorithms - Examples

one is greater.

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 transfers.

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 better? This is the following relation:

$$y^1 \succ y^2 \Leftrightarrow y\_i^1 \succeq y\_i^2 \forall i = 1, \dots, m \quad \land \quad \exists j \ y\_j^1 \succ y\_j^2. \tag{4}$$

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 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].

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 preference relation.

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;…; � 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:

$$\left(y\_{P(1)}, y\_{P(2)}, \dots, y\_{P(k)}\right) \approx \left(y\_1, y\_2, \dots, y\_k\right) \tag{5}$$

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 nondominated vector.

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:

for the evaluation vector y ¼ y1; y2;…; yk � �∈R<sup>k</sup> :

$$y\_{j'} > y\_{i^\*} \Rightarrow y - \varepsilon \cdot \varepsilon\_{i'} + \varepsilon \cdot \varepsilon\_{i^\*} \succ y \text{ for } 0 < y\_{j'} - y\_{j''} < \varepsilon \tag{6}$$

Equalizing transfer is a slight deterioration of a better coordinate of evaluation vector and, simultaneously, improvement of a poorer coordinate. The resulting evaluation vector is strictly preferred in comparison to the initial evaluation vector. This is a structure of equalizing—the evaluation vector with less diversity of coordinates is preferred in relation to the vector with the same sum of coordinates, but with their greater diversity.

A nondominated vector satisfying the anonymity property and the principle of transfers is called equitably nondominated vector. The set of equitably nondominated vectors is denoted by Yb0E. In the decision space, the equitably efficient decisions are specified. The decision <sup>b</sup><sup>x</sup> <sup>∈</sup> <sup>X</sup><sup>0</sup> is called an equitably efficient decision, if the corresponding evaluation vector <sup>b</sup><sup>y</sup> <sup>¼</sup> <sup>f</sup>ð Þ <sup>b</sup><sup>x</sup> is an equitably nondominated vector. The set of equitably efficient decisions is denoted by Xb0<sup>E</sup> [2, 6, 7].

Equitable dominance can be expressed as the relation of inequality for cumulative, ordered evaluation vectors. This relation can be determined with the use of mapping <sup>T</sup> : <sup>R</sup><sup>k</sup> ! <sup>R</sup><sup>k</sup> that cumulates nonincreasing coordinates of evaluation vector.

The transformation <sup>T</sup> : <sup>R</sup><sup>k</sup> ! <sup>R</sup><sup>k</sup> is defined as follows:

$$\overline{T}\_i(\underline{y}) = \sum\_{l=1}^i T\_i(\underline{y}) \quad \text{for } i = 1, 2, \dots, k. \tag{7}$$

y ¼ y1; y2; …; yk

of the multicriteria problem (1).

where y ¼ y1; y2;…; yk

where y ¼ y1; y2;…; yk

hable or not [4, 8].

<sup>s</sup> : <sup>Y</sup><sup>0</sup> � <sup>Ω</sup> ! <sup>R</sup><sup>1</sup>

ter for individual evaluations.

scalaring function <sup>s</sup> : <sup>Y</sup><sup>0</sup> � <sup>Ω</sup> ! <sup>R</sup><sup>1</sup> is solved:

tion function that satisfies a given member.

s yð Þ¼ ; y min

lative, ordered evaluation vector; y ¼ y1; y2;…; yk

for supporting selection of group decisions is as follows:

the arbitrary, small, positive adjustment parameter.

1 ≤ i ≤ k

� � is the evaluation vector, T yð Þ¼ <sup>T</sup>1ð Þ<sup>y</sup> ; <sup>T</sup>2ð Þ<sup>y</sup> ;…; Tkð Þ<sup>y</sup> � � is the cumulative,

The efficient solution of multicriteria optimization problem (9) is an equitably efficient solution

To determine the solution of a multicriteria problem (9), the scalaring of this problem with the

It is the problem of single-objective optimization with specially created scalaring function of

Complete and sufficient parameterization of the set of equitably efficient decision Xb <sup>0</sup><sup>E</sup> can be achieved, using the method of the reference point for the problem (9). In this method the aspiration levels are applied as control parameters. Aspiration level is the value of the evalua-

> X k

> > i¼1

� � is the evaluation vector; T yð Þ¼ <sup>T</sup>1ð Þ<sup>y</sup> ; <sup>T</sup>2ð Þ<sup>y</sup> ;…; Tkð Þ<sup>y</sup> � � is the cumu-

Tð Þ¼ y ð Þ T1ð Þy ; T2ð Þy ;…; Tkð Þy is the cumulative, ordered vector of aspiration levels; and ε is

This function is called a function of achievement. Maximizing this function with respect to <sup>y</sup> determines equitably nondominated vectors <sup>b</sup><sup>y</sup> and the equitably efficient decision <sup>b</sup>x. For any aspiration levels <sup>y</sup>, each maximal point <sup>b</sup><sup>y</sup> of this function is an equitably nondominated solution. Note, the equitably efficient solution <sup>b</sup><sup>x</sup> depends on the aspiration levels y. If the aspiration levels y are too high, then the maximum of this function is smaller than zero. If the aspiration levels y are too low, then the maximum of this function is larger than zero. This is the information for the group, whether a given aspiration level is reac-

A tool for searching the set of solutions is the function (11). Maximum of this function depends on the parameter y, which is used by the members of the group to select a solution. The method

max<sup>x</sup> s yð Þ ; <sup>y</sup> : <sup>x</sup> <sup>∈</sup> Xo <sup>f</sup> , (10)

� � is available to each member in the group

Tið Þ� y Tið Þy <sup>i</sup>

� � is the vector of aspiration levels;

� � is the control parame-

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

� �, (11)

; we have thus

51

ordered evaluation vector, and Y<sup>0</sup> is the set of achievable evaluation vectors.

� � is the evaluation vector and <sup>y</sup> <sup>¼</sup> <sup>y</sup>1; <sup>y</sup>2;…; yk

two variables—the evaluation vector y∈ Y and control parameter y ∈ Ω ⊂Rk

. The parameter y ¼ y1; y2;…; yk

that allows any member to review the set of equitably efficient solutions.

The scalaring function defined in the method of reference point is as follows:

Tið Þ� y Tið Þy <sup>i</sup> � � <sup>þ</sup> <sup>ε</sup> �

Define by T yð Þ the vector with nonincreasing ordered coordinates of the vector y, i.e. T yð Þ¼ ð Þ T1ð Þy ; T2ð Þy ;…; Tkð Þy , where T1ð Þy ≤ T2ð Þy ≤ …≤ Tkð Þy and there is a permutation P of the set 1f g ; …; <sup>k</sup> , such that Tið Þ¼ <sup>y</sup> yP ið Þ for <sup>i</sup> <sup>¼</sup> <sup>1</sup>, ::, k.

The relation of equitable domination ≻<sup>e</sup> is a simple vector domination for evaluation vectors with cumulated nonincreasing coordinates of evaluation vector [6, 7].

The evaluation vector y<sup>1</sup> equitably dominates the vector y<sup>2</sup> if the following condition is satisfied:

$$y^1 \succ\_\epsilon y^2 \Leftrightarrow \overline{T}(y^1) \ge \overline{T}(y^2) \tag{8}$$

The solution of choosing a group decision is to find the equitably efficient decision that best reflects the preferences of all members in the group.
