4. Technique of generating equitably efficient decisions

Equitably efficient decisions for a multiple criteria problem (1) are obtained by solving a special problem in multicriteria optimization—a problem with the vector function of the cumulative, evaluation vectors arranged in a nonincreasing order. This is the following problem.

$$\max\_{y} \left\{ \left( \overline{T}\_1(y), \overline{T}\_2(y), \dots, \overline{T}\_k(y) \right) : \quad y \in Y\_0 \right. \tag{9}$$

where

y ¼ y1; y2; …; yk � � 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, ordered evaluation vector, and Y<sup>0</sup> is the set of achievable evaluation vectors.

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

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

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

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 relation of equitable domination ≻<sup>e</sup> is a simple vector domination for evaluation vectors

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

The solution of choosing a group decision is to find the equitably efficient decision that best

Equitably efficient decisions for a multiple criteria problem (1) are obtained by solving a special problem in multicriteria optimization—a problem with the vector function of the cumulative,

<sup>T</sup>1ð Þ<sup>y</sup> ; <sup>T</sup>2ð Þ<sup>y</sup> ; …; Tkð Þ<sup>y</sup> � � : <sup>y</sup><sup>∈</sup> <sup>Y</sup><sup>0</sup>

evaluation vectors arranged in a nonincreasing order. This is the following problem.

Tið Þy for i ¼ 1, 2, …, k: (7)

y<sup>1</sup> ≻ ey<sup>2</sup> ⇔ T y<sup>1</sup> � � ≥ T y<sup>2</sup> � � (8)

� (9)

nondominated vector. The set of equitably efficient decisions is denoted by Xb0<sup>E</sup> [2, 6, 7].

the same sum of coordinates, but with their greater diversity.

50 Optimization Algorithms - Examples

cumulates nonincreasing coordinates of evaluation vector.

Tið Þ¼ <sup>y</sup> <sup>X</sup> i

with cumulated nonincreasing coordinates of evaluation vector [6, 7].

4. Technique of generating equitably efficient decisions

l¼1

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

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.

reflects the preferences of all members in the group.

max y

where

The efficient solution of multicriteria optimization problem (9) is an equitably efficient solution of the multicriteria problem (1).

To determine the solution of a multicriteria problem (9), the scalaring of this problem with the scalaring function <sup>s</sup> : <sup>Y</sup><sup>0</sup> � <sup>Ω</sup> ! <sup>R</sup><sup>1</sup> is solved:

$$\max\_{\mathbf{x}} \left\{ \mathbf{s}(\underline{y}, \overline{y}) : \mathbf{x} \in \mathbf{X}\_{\mathbf{o}} \right\} \tag{10}$$

where y ¼ y1; y2;…; yk � � is the evaluation vector and <sup>y</sup> <sup>¼</sup> <sup>y</sup>1; <sup>y</sup>2;…; yk � � is the control parameter for individual evaluations.

It is the problem of single-objective optimization with specially created scalaring function of two variables—the evaluation vector y∈ Y and control parameter y ∈ Ω ⊂Rk ; we have thus <sup>s</sup> : <sup>Y</sup><sup>0</sup> � <sup>Ω</sup> ! <sup>R</sup><sup>1</sup> . The parameter y ¼ y1; y2;…; yk � � is available to each member in the group that allows any member to review the set of equitably efficient solutions.

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 evaluation function that satisfies a given member.

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

$$s(\boldsymbol{y}, \overline{\boldsymbol{y}}) = \min\_{1 \le i \le k} \left( \overline{T}\_i(\boldsymbol{y}) - \overline{T}\_i(\overline{\boldsymbol{y}})\_i \right) + \boldsymbol{\varepsilon} \cdot \sum\_{i=1}^k \left( \overline{T}\_i(\boldsymbol{y}) - \overline{T}\_i(\overline{\boldsymbol{y}})\_i \right), \tag{11}$$

where y ¼ y1; y2;…; yk � � 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, ordered evaluation vector; y ¼ y1; y2;…; yk � � is the vector of aspiration levels; Tð Þ¼ y ð Þ T1ð Þy ; T2ð Þy ;…; Tkð Þy is the cumulative, ordered vector of aspiration levels; and ε is the arbitrary, small, positive adjustment parameter.

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 reachable or not [4, 8].

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 for supporting selection of group decisions is as follows:


A solution which is as satisfying as possible for all members in the group is searched for. All members in the problem of decision making in a group should be treated in the same way, no member should be favored. The decision-making model should have the anonymity properties of preference relation and satisfy the principle of transfers. The solution of the problem should

At the beginning of the analysis, a separate single-criterion optimization is carried out for each member in the group. In this way, the best results for each member are obtained separately. This is a utopia point of the multicriteria optimization problem. This also gives information about the conflict of evaluations of group members in the decision-making problem [9, 10].

When analyzing Table 1, it might be observed that the big selection possibilities have members

For each iteration, the price of fairness (POF) for each member is calculated [4]. It is the quotient of the difference between the utopia value of a solution and the value from the solution of the

where yiu is the utopia value of a member i, i ¼ 1, 2, 3, and yiu is the value from the solution of

The value of the POFs is a number between 0 and 1. POF values closer to zero are preferred by the members, as the solution is closer to a utopia solution. The more the values of the POFs of

People in the group do control the process by means of aspiration levels. The multicriteria analysis

At the beginning of the analysis (Iteration 1), members in the group define their preferences as aspiration levels equal to the values of utopia. The obtained effective leveling solution is ideal for member 2, while member 1 and member 3 would like to correct their solutions. In the next iteration, all members reduce their levels of aspiration. As a result (Iteration 2), the solution for

Member's evaluation 1 y1 900 900 300 Member's evaluation 2 y2 750 1200 1100 Member's evaluation 3 y3 220 880 1320 Utopia vector 900 1200 1320

yiu , i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, (13)

Multicriteria Support for Group Decision Making http://dx.doi.org/10.5772/intechopen.79935 53

<sup>b</sup>y<sup>1</sup> <sup>b</sup>y<sup>2</sup> <sup>b</sup>y<sup>3</sup>

POF <sup>¼</sup> yiu � <sup>b</sup>yi

be an equitably efficient decision of the problem (12).

multicriteria problem, in relation to the utopia value.

the multicriteria problems of a member, i i ¼ 1, 2, 3.

Optimization criterion Solution

Table 1. Matrix of goal realization with the utopia vector.

the members get closer to each other, the better the solution.

2 and 3 and lower member 1.

is presented in Table 2.

For solving the problem (12) the method of reference point is used.

The method of selecting group decision is presented in Figure 1.

The computer will not replace members of the group in the decision-making process; the whole process of selecting a decision is guided by all members in the group.

Figure 1. The method of selecting group decision.

## 5. Example

To illustrate the process of supporting group decision making, the following example is presented—selection of group decision by three members [8].

The problem of selecting the decision is the following:

1, 2, 3 are the members in the group.

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>∈</sup> <sup>R</sup><sup>2</sup> : <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>5</sup> � <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>75</sup>, <sup>3</sup> � <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>5</sup> � <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>95</sup>, x<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>25</sup>, <sup>5</sup> � <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup> � <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>110</sup>, x<sup>1</sup> <sup>≥</sup> <sup>0</sup>, x<sup>2</sup> <sup>≥</sup> <sup>0</sup> is the feasible set.

x ¼ ð Þ x1; x2 ∈ X<sup>0</sup> is a group decision, belonging to the feasible set.

f <sup>1</sup>ð Þ¼ x 10 � x<sup>1</sup> þ 60 � x2 is the function of decision evaluation x by member 1.

f <sup>1</sup>ð Þ¼ x 40 � x<sup>1</sup> þ 60 � x2 is the function of decision evaluation x by member 2.

f <sup>1</sup>ð Þ¼ x 60 � x<sup>1</sup> þ 20 � x2 is the function of decision evaluation x by member 3.

The problem of selection of group decision is expressed in the form of multicriteria optimization problem with three evaluation functions:

$$\max\_{\mathbf{x}} \left\{ (10 \cdot \mathbf{x}\_1 + 60 \cdot \mathbf{x}\_2, 40 \cdot \mathbf{x}\_1 + 60 \cdot \mathbf{x}\_2, 60 \cdot \mathbf{x}\_1 + 20 \cdot \mathbf{x}\_2) \mid \mathbf{x} \in X\_0 \right\},\tag{12}$$

where X<sup>0</sup> is the feasible set and x ¼ ð Þ x1; x2 ∈ X<sup>0</sup> is a group decision.

A solution which is as satisfying as possible for all members in the group is searched for. All members in the problem of decision making in a group should be treated in the same way, no member should be favored. The decision-making model should have the anonymity properties of preference relation and satisfy the principle of transfers. The solution of the problem should be an equitably efficient decision of the problem (12).

For solving the problem (12) the method of reference point is used.

• Calculations—giving other equitably efficient decisions

52 Optimization Algorithms - Examples

additional information about the preferences of the group

whole process of selecting a decision is guided by all members in the group.

The method of selecting group decision is presented in Figure 1.

presented—selection of group decision by three members [8].

x ¼ ð Þ x1; x2 ∈ X<sup>0</sup> is a group decision, belonging to the feasible set.

where X<sup>0</sup> is the feasible set and x ¼ ð Þ x1; x2 ∈ X<sup>0</sup> is a group decision.

f <sup>1</sup>ð Þ¼ x 10 � x<sup>1</sup> þ 60 � x2 is the function of decision evaluation x by member 1. f <sup>1</sup>ð Þ¼ x 40 � x<sup>1</sup> þ 60 � x2 is the function of decision evaluation x by member 2. f <sup>1</sup>ð Þ¼ x 60 � x<sup>1</sup> þ 20 � x2 is the function of decision evaluation x by member 3.

The problem of selecting the decision is the following:

1, 2, 3 are the members in the group.

Figure 1. The method of selecting group decision.

tion problem with three evaluation functions:

5. Example

is the feasible set.

• Interaction with the system—dialog with the members of the group, which is a source of

The computer will not replace members of the group in the decision-making process; the

To illustrate the process of supporting group decision making, the following example is

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>∈</sup> <sup>R</sup><sup>2</sup> : <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>5</sup> � <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>75</sup>, <sup>3</sup> � <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>5</sup> � <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>95</sup>, x<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>25</sup>, <sup>5</sup> � <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup> � <sup>x</sup><sup>2</sup> <sup>≤</sup> <sup>110</sup>, x<sup>1</sup> <sup>≥</sup> <sup>0</sup>, x<sup>2</sup> <sup>≥</sup> <sup>0</sup>

The problem of selection of group decision is expressed in the form of multicriteria optimiza-

max<sup>x</sup> f g ð Þ <sup>10</sup> � <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>60</sup> � <sup>x</sup>2; <sup>40</sup> � <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>60</sup> � <sup>x</sup>2; <sup>60</sup> � <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>20</sup> � <sup>x</sup><sup>2</sup> <sup>x</sup> <sup>∈</sup> <sup>X</sup><sup>0</sup> , (12)

At the beginning of the analysis, a separate single-criterion optimization is carried out for each member in the group. In this way, the best results for each member are obtained separately. This is a utopia point of the multicriteria optimization problem. This also gives information about the conflict of evaluations of group members in the decision-making problem [9, 10].

When analyzing Table 1, it might be observed that the big selection possibilities have members 2 and 3 and lower member 1.

For each iteration, the price of fairness (POF) for each member is calculated [4]. It is the quotient of the difference between the utopia value of a solution and the value from the solution of the multicriteria problem, in relation to the utopia value.

$$POF = \frac{y\dot{u} - \widehat{y}\dot{u}}{y\dot{u}},\\ i = 1, 2, 3,\tag{13}$$

where yiu is the utopia value of a member i, i ¼ 1, 2, 3, and yiu is the value from the solution of the multicriteria problems of a member, i i ¼ 1, 2, 3.

The value of the POFs is a number between 0 and 1. POF values closer to zero are preferred by the members, as the solution is closer to a utopia solution. The more the values of the POFs of the members get closer to each other, the better the solution.

People in the group do control the process by means of aspiration levels. The multicriteria analysis is presented in Table 2.

At the beginning of the analysis (Iteration 1), members in the group define their preferences as aspiration levels equal to the values of utopia. The obtained effective leveling solution is ideal for member 2, while member 1 and member 3 would like to correct their solutions. In the next iteration, all members reduce their levels of aspiration. As a result (Iteration 2), the solution for


Table 1. Matrix of goal realization with the utopia vector.


6. Summary

follows:

Author details

Andrzej Łodziński

Poland

References

Verlag; 1989

ling. 1982;3:391-405

solving the problem of multicriteria optimization.

The chapter presents the method of supporting group decision making. The choice is made by

Multicriteria Support for Group Decision Making http://dx.doi.org/10.5772/intechopen.79935 55

The decision support process is not a one-step act, but an iterative process, and it proceeds as

• Then, each member determines the aspiration levels for particular results of decisions.

• The decision choice is not a single optimization act, but a dynamic process of searching for

• This process ends when the group finds a decision that makes it possible to achieve results

This method allows the group to verify the effects of each decision and helps find the decision which is the best for their aspiration levels. This procedure does not replace the group in decisionmaking process. The whole decision-making process is controlled by all the members in the group.

Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences, Warsaw,

[2] Malawski M, Wieczorek A, Sosnowska H. Competition and cooperation. Game Theory in

[4] Lewandowski A, Wierzbicki A, editors. Aspiration Based Decision Support Systems. Lecture Notes in Economics and Mathematical Systems. Vol. 331; Berlin-Heidelberg: Springer-

[5] Wierzbicki AP. A mathematical basis for satisficing decision making. Mathematical Model-

[1] Luce D, Raiffa H. Games and Decisions. Warsaw: PWN; 1966. (in Polish)

Economics and the Social Sciences. Warsaw: PWN; 1997. (in Polish)

[3] Straffin PhD. Game Theory. Warsaw: Scolar; 2004. (in Polish)

• Each member of the group participates in the decision-making process.

solutions in which each member may change his preferences.

Address all correspondence to: andrzej\_lodzinski@sggw.pl

These aspiration levels are determined adaptively in the learning process.

meeting the member's aspirations or closest to these aspirations in a sense.

Table 2. Interactive analysis of seeking a solution.

member 1 has improved, while the solution for member 2 and member 3 has deteriorated. The group now wishes to correct the solution for member 3 and increases the aspiration level for member 3, but does not change the aspiration levels for members 1 and 2. As a result (Iteration 3), the solution for member 2 and member 3 has improved, while the solution for member 1 has deteriorated. The group still wishes to correct the solution for member 3 and provides a higher value of the aspiration level for member 3, but does not change the aspiration levels for members 1 and 2. As a result (Iteration 4), the solution for member 2 and member 3 has improved, but the solution for member 1 has deteriorated. The group now wishes to correct the solution for member 1 and member 3 and reduces the aspiration level for member 2, but does not reduce the aspiration levels for members 1 and 3. As a result (Iteration 5), the solution for member 1 has improved, while the solution for members 2 and 3 has deteriorated. A further change to the value of the aspiration levels causes either an improvement in the solution for member 1 and at the same time a deterioration in the solution for member 3 or vice versa, as well as slight changes in the solution for member 2. Such a solution results from the specific nature of the examined problem—the solution for member 2 lies between solutions for members 1 and 3. The group decision for Iteration 5 is as follows: <sup>x</sup><sup>5</sup> <sup>¼</sup> ð Þ <sup>14</sup>:81; <sup>10</sup>:<sup>12</sup> .

The final choice of a specific solution depends on the preferences of the members in the group. This example shows that the presented method allows the members to get to know their decision-making possibilities within interactive analysis and to search for a solution that would be satisfactory for the group.
