Matrix of goal realization with the utopia vector.

## 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

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

where

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

At point

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

where

The vector function

## 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

The vector of evaluation

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

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

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

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

The transformation

Define by

The relation of equitable domination

The evaluation vector

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.

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

where

It is the problem of single-objective optimization with specially created scalaring function of two variables—the evaluation vector

Complete and sufficient parameterization of the set of equitably efficient decision

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

where

This function is called a function of achievement. Maximizing this function with respect to

A tool for searching the set of solutions is the function (11). Maximum of this function depends on the parameter

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:

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

where

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 criterion | Solution | ||
---|---|---|---|

Member's evaluation 1 | 900 | 900 | 300 |

Member's evaluation 2 | 750 | 1200 | 1100 |

Member's evaluation 3 | 220 | 880 | 1320 |

Utopia vector | 900 | 1200 | 1320 |

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.

where

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.

Iteration | Member 1 | Member 2 | Member 3 | |
---|---|---|---|---|

1. | Aspiration levels | 900 | 1200 | 1320 |

Solution | 750 | 1200 | 1100 | |

POF | 0.166 | 0 | 0.153 | |

2. | Aspiration levels | 850 | 1000 | 1200 |

Solution | 800 | 1192 | 1007 | |

POF | 0.111 | 0.006 | 0.224 | |

3. | Aspiration levels | 850 | 1000 | 1250 |

Solution | 775 | 1196 | 1053 | |

POF | 0.138 | 0.003 | 0.189 | |

4. | Aspiration levels | 850 | 1000 | 1300 |

Solution | 750 | 1200 | 1100 | |

POF | 0.166 | 0 | 0.153 | |

5. | Aspiration levels | 850 | 990 | 1300 |

Solution | 755 | 1199 | 1090 | |

POF | 0.161 | 0.0006 | 0.160 |

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:

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.