Open access peer-reviewed chapter

# Multicriteria Support for Group Decision Making

Written By

Andrzej Łodziński

Submitted: 03 February 2018 Reviewed: 04 July 2018 Published: 05 September 2018

DOI: 10.5772/intechopen.79935

From the Edited Volume

## Optimization Algorithms - Examples

Edited by Jan Valdman

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## Abstract

This chapter presents the support method for group decision making. A group decision is when a group of people has to make one joint decision. Each member of the group has his own assessment of a joint decision. The decision making of a group decision is modeled as a multicriteria optimization problem where the respective evaluation functions are the assessment of a joint decision by each member. The interactive analysis that is based on the reference point method applied to the multicriteria problems allows to find effective solutions matching the group’s preferences. Each member of the group is able to verify results of every decision. The chapter presents an example of an application of the support method in the selection of the group decision.

### Keywords

• multicriteria optimization problem
• equitably efficient decision
• scalarizing function
• decision support systems

## 1. Introduction

The chapter presents the support method for group decision making—when a group of people who have different preferences want to make one joint decision.

The selection process of a group decision can be modeled with the use of game theory [1, 2, 3].

In this chapter, the choice of the group decision is modeled as a multicriteria problem. The individual coordinates of this optimization problem are functions to evaluate a joint decision by each person in the group. This allows one to take into account preferences of all members in the group. Decision support is an interactive process of proposals for subsequent decisions, by each member in the group and his evaluations. These proposals are parameters of the multicriteria optimization problem. The solution of this problem is assessed by members in the group. Each member can accept or refuse the solution. In the second case, a member gives his new proposal and the problem is resolved again.

## 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=1,2,,k, a decision evaluation function fi 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:

maxxf1xfkx:xX0,E1

where 1,2,,k are particular members, X0Rn is the feasible set, x=x1x2xnX0 is a group decision, f=f1f2fk is the vector function that maps the decision space X0=Rn into the criteria space Y0=Rk, and specific coordinates yi=fix,i=1,2,..,m represent the scalar evaluation functions—the result of a decision x ith member i=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 f1,,fk introduce a certain order in the set of decision variables—preference relations:

x1x2f1x1f2x2,,fkx1fkx2jfjx1>fjx2.E2

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:

maxx{y1yk:yY0},E3

where xX is a vector of decision variables, y=y1yk is the evaluation vector and particular coordinates yi represent the result of a decision x ith member i=1,2,,k, and Y0=fX0 is the set of evaluation vectors.

The vector function y=fx assigns to each vector of decision variables x an evaluation vector yY0 that measures the quality of decision x from the point of view of all members in the group. The set of results achieved Y0 is given in the implicit form—through a set of feasible decisions X0 and the mapping of a model f=f1f2fk. To determine the value y, the simulation of the model is necessary: y=fxforxX0.

## 3. Equitably efficient decision

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 y1,y2Rk is better? This is the following relation:

y1y2yi1yi2i=1,,mjy>j1yj2.E4

The vector of evaluation ŷY0 is called the nondominated vector; if there is no such vector yY0, that ŷ is dominated by y. Appropriate acceptable decisions are specified in the decision space. The decision x̂X0 is called efficient decision (Pareto efficient) if the corresponding vector of evaluations ŷ=fx̂ 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 y=y1y2ykRk and for any permutation P of the set 1k, the following property holds:

yP1yP2yPky1y2ykE5

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=y1y2ykRk:

yi'>yi"yεei'+εei"yfor0<yiyi<εE6

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 Ŷ0E. In the decision space, the equitably efficient decisions are specified. The decision x̂X0 is called an equitably efficient decision, if the corresponding evaluation vector ŷ=fx̂ is an equitably nondominated vector. The set of equitably efficient decisions is denoted by X̂0E [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 T¯:RkRk that cumulates nonincreasing coordinates of evaluation vector.

The transformation T¯:RkRk is defined as follows:

T¯iy=l=1iTiyfori=1,2,,k.E7

Define by Ty the vector with nonincreasing ordered coordinates of the vector y, i.e. Ty=T1yT2yTky, where T1yT2yTky and there is a permutation P of the set 1k, such that Tiy=yPi for i=1,..,k.

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

The evaluation vector y1 equitably dominates the vector y2 if the following condition is satisfied:

y1ey2T¯y1T¯y2E8

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.

## 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.

maxy{T¯1yT¯2yT¯ky:yY0}E9

where

y=y1y2yk is the evaluation vector, T¯y=T¯1yT¯2yT¯ky is the cumulative, ordered evaluation vector, and Y0 is the set of achievable evaluation vectors.

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 s:Y0×ΩR1 is solved:

maxx{syy¯:xXo},E10

where y=y1y2yk is the evaluation vector and y¯=y¯1y¯2y¯k 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 yY and control parameter y¯ΩRk; we have thus s:Y0×ΩR1. The parameter y¯=y¯1y¯2y¯k 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 X̂0E 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:

syy¯=min1ikT¯iyT¯iy¯i+εi=1kT¯iyT¯iy¯i,E11

where y=y1y2yk is the evaluation vector; T¯y=T¯1yT¯2yT¯ky is the cumulative, ordered evaluation vector; y¯=y¯1y¯2y¯k is the vector of aspiration levels; Ty¯=T1y¯T2y¯Tky¯ 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 y determines equitably nondominated vectors ŷ and the equitably efficient decision x̂. For any aspiration levels y¯, each maximal point ŷ of this function is an equitably nondominated solution. Note, the equitably efficient solution x̂ 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:

• Calculations—giving other equitably efficient decisions

• Interaction with the system—dialog with the members of the group, which is a source of additional information about the preferences of the group

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.

## 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.

X0={xR2:x1+5x275,3x1+5x295,x1+x225,5x1+2x2110,x10,x20} is the feasible set.

x=x1x2X0 is a group decision, belonging to the feasible set.

f1x=10x1+60x2 is the function of decision evaluation x by member 1.

f1x=40x1+60x2 is the function of decision evaluation x by member 2.

f1x=60x1+20x2 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:

maxx10x1+60x240x1+60x260x1+20x2xX0,E12

where X0 is the feasible set and x=x1x2X0 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.

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.

Optimization criterionSolution
ŷ1ŷ2ŷ3
Member's evaluation 1 y1900900300
Member's evaluation 2 y275012001100
Member's evaluation 3 y32208801320
Utopia vector90012001320

### Table 1.

Matrix of goal realization with the utopia vector.

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=yiuŷiyiu,i=1,2,3,E13

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

IterationMember 1Member 2Member 3
ŷ1ŷ2ŷ2
1.Aspiration levels y¯90012001320
Solution ŷ75012001100
POF0.16600.153
2.Aspiration levels y¯85010001200
Solution ŷ80011921007
POF0.1110.0060.224
3.Aspiration levels y¯85010001250
Solution ŷ77511961053
POF0.1380.0030.189
4.Aspiration levels y¯85010001300
Solution ŷ75012001100
POF0.16600.153
5.Aspiration levels y¯8509901300
Solution ŷ75511991090
POF0.1610.00060.160

### Table 2.

Interactive analysis of seeking a solution.

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 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: x5=14.8110.12.

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.

## 6. Summary

The chapter presents the method of supporting group decision making. The choice is made by solving the problem of multicriteria optimization.

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

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

• Then, each member determines the aspiration levels for particular results of decisions. These aspiration levels are determined adaptively in the learning process.

• The decision choice is not a single optimization act, but a dynamic process of searching for solutions in which each member may change his preferences.

• This process ends when the group finds a decision that makes it possible to achieve results meeting the member’s aspirations or closest to these aspirations in a sense.

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 decision-making process. The whole decision-making process is controlled by all the members in the group.

## References

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Written By

Andrzej Łodziński

Submitted: 03 February 2018 Reviewed: 04 July 2018 Published: 05 September 2018