Methodology and Multicriteria Algorithm for Group Decision Support

in Classification Problems

GEORGIOS RIGOPOULOS

Division of Mathematics-Informatics and Statistics-Econometrics, Department of Economics,

National and Kapodistrian University of Athens,

GREECE

Abstract: - In this work, a Group Decision methodology and algorithm for small collaborating teams is

introduced. It is based on a multicriteria algorithm for classification decisions, where aggregation of member

preferences is executed at the parameter level. The algorithm applies to relatively well-structured problems

guided by a process facilitator. Initially, a set of parameters is proposed by the facilitator to the group and next

group members evaluate the proposed parameter set and express their preferences in numeric or linguistic

format. Individual preferences are aggregated by appropriate operators, and a set of group parameter values is

generated, which is used as input for the classification algorithm. NeXClass multicriteria classification

algorithm is used for the classification of alternatives, initially at a training set of alternatives and later at the

entire set. Finally, group members evaluate results, and consensus, as well as satisfaction metrics, are

calculated. In case of a low acceptance level, problem parameters are reviewed by the facilitator, and the

aggregation phase is repeated. The methodology is a valid approach for group decision problems and can be

utilized in numerous business environments. The algorithm can be also utilized by software agents in

multiagent environments for automated decision-making, given the large volume of agent-based decision-

making in various settings today.

Key-Words: - group decision support, NexClass algorithm, WOWA, OWA, multicriteria classification

Received: July 5, 2022. Revised: August 21, 2023. Accepted: September 26, 2023. Published: November 20, 2023.

1 Introduction

Group Decision Support (GDS) is an active research

domain that has gained significant attention during

past decades due to its wide application in business

domains and automated agent-based decision-

making. Research in decision support systems aims

to equip decision-makers with tools and methods

and assist them in optimizing their decisions. Since

a decision support system must reflect decision

makers' preferences or their decision model,

building a Group Decision Support System (GDSS)

is not a trivial and straightforward process.

Moreover, several dimensions must be considered

as well, such as preference modeling, negotiation,

and coordination protocols, to name a few. Several

methodologies and tools have been developed to

support groups, ranging from collaborative

techniques to negotiation ones, depending on

whether group members share a common goal or

support individual goals. Technologies utilized for

GDSS development tend to follow Information

Technology advances, resulting in data-driven

support systems, that we can meet nowadays.

Incorporation of web and mobile technologies can

also support collaboration features in real-time, a

capability that could not be implemented in the early

days of GDSSs, [1]. A variety of methods have been

utilized for GDS ranging from algorithmic in well-

defined problems, to less structured ones for

problems requiring brainstorming and negotiation.

Multicriteria analysis methods have also been

utilized in various decision problems, however, due

to the inherent complex nature of group decision

settings there is no unique formulation and solution.

In GDS, the multicriteria analysis approach offers a

structured way for problem formulation and can

guide members to understand all requirements and

express their preferences effectively reflecting their

decision model, [2]. Despite its merits, multicriteria

analysis and relevant methodologies are rarely

utilized in group decision research. Reasons for this

can be partially attributed to the complexity of

aggregating mechanisms as well as negotiation and

consensus modeling requirements. Given the limited

number of works in this domain and considering the

need for automated agent-based decision-making,

this work aims to address the gap and introduce a

structured methodology that is based on

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multicriteria analysis and supports group

classification decisions.

In brief, the objective of the proposed method is

to assign a set of candidate alternatives to several

predefined non-ordered categories, according to

their ranking on a set of evaluation criteria, defined

by a group of decision-makers. Initially, a set of

parameters is defined by group members, and next,

each group member ranks the proposed parameter

set and expresses her preferences in numeric or

linguistic format. Individual preferences are

aggregated by aggregation operators, and a group

parameter set is produced and used as input for the

classification algorithm. NeXClass multicriteria

classification algorithm is used for the classification

of candidate alternatives, initially at a training set of

alternatives and later at the entire set. Finally, group

members evaluate results, and consensus, as well as

satisfaction metrics, are calculated. In case of a low

level of group consensus, problem parameters are

redefined by group members, and the aggregation

phase is repeated. The process can be administered

by a group facilitator role or can be automatically

run by group members.

In this work, we present the algorithm and the

way it can be used in a GDS problem. The structure

of the work is as follows. Initially, the introduction

sets the aims and highlights the approach. Next,

some brief background information is presented on

group decisions. Following this, we present the

group decision multicriteria methodology in detail.

In the next section, we illustrate its usage and

applicability in the context of a GDSS and end with

a discussion and future research.

2 Background

Group decision-making is an essential component of

enterprise strategic planning and operations for

many organizations today. Complexity in a business

environment requires a decent level of knowledge

from a wide range of domains, so the contribution of

a domain experts’ team is the only way to achieve

efficiency in decisions. To support group needs,

researchers work towards developing tools and

methodologies, ranging from collaboration

technologies to decision support systems. Although

traditional decision support systems may look

outdated in the cloud and big data era today,

research is very active and evolves, as data-driven

models combined with machine learning

developments lead to novel approaches in the field,

[3], [4] [5].

Group decisions are inherently more complex

compared to single decision-making since several

contradicting factors are involved such as

individuals’ personal opinions, goals, and stakes,

resulting in a social procedure, where negotiation

and strategy play a critical role. Group decision-

making in real business environments also raises

some issues, such as conflicting individual goals,

not efficient knowledge, validity of information, and

individuals’ motivation, [6]. Despite the inherent

complexity, within a group decision-making setting

a member can express personal opinions and

suggest solutions from a personal perspective. In

addition, negotiation and voting advance decision

efficiency and increase consensus and adoption

since all participants have contributed to the result,

smoothening thus any disputes. In general, group

members can be motivated by individual

perceptions to work within the group either towards

collaboration or towards competition. While in the

first case, members express similar opinions and

goals, in the second one they state opposing

opinions. Although collaborative teams work

towards a common goal, contradiction may also

occur, [7]. Some key techniques that have been

acquired to facilitate group work and decisions

include brainstorming, nominal group technique,

Delphi method, voting, and multicriteria analysis.

In general, multicriteria analysis can be

incorporated as a method to model preferences and

facilitate decision-making within a group of

decision-makers. Modeling under a multicriteria

setting can be formulated under two major

approaches. Either as individual multicriteria

models, where separate solutions are generated and

aggregated into a group solution. Or, as one

multicriteria model, where group member

preferences are aggregated resulting in a group

parameter set that is the input for a multicriteria

method. Each approach has merits, and the selection

depends on the problem under study. A recent

systematic review can be found in the work of [8],

where we can see that most of the approaches

provide support to sorting and selection decisions.

Also, the Analytic Hierarchy Process methodology

is a popular method and web technologies are

relatively limited. Following the above and given

the limited number of works in the domain, we

argue that our approach provides a useful tool to

decision-makers, filling the gap in group

classification decision problems.

2.1 Fuzzy Majority

The majority notion, which is usually defined as a

threshold number of individuals, is a widely used

crisp criterion in group decisions and aggregation

operations. The fuzzy majority, on the other hand, is

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a soft majority concept expressed by a fuzzy

quantifier, which is manipulated via a fuzzy-logic-

based calculus of linguistically quantified

propositions and can be represented by fuzzy

quantifiers, [9]. One such approach is the Ordered

Weighted Averaging (OWA) operator, which

reflects fuzzy majority by means of fuzzy

quantifiers.

The concept of fuzzy quantifiers was introduced

by, [10]. This study suggested that the semantics of

a fuzzy quantifier can be captured by using fuzzy

subsets for its representation. He distinguished

between two types of fuzzy quantifiers, absolute and

proportional or relative. Absolute quantifiers are

used to represent amounts that are absolute such as

“about 2” or “more than 5”. These absolute

linguistic quantifiers are closely related to the

concept of the count or number of elements. He

defined these quantifiers as fuzzy subsets of the

non-negative real numbers. In this approach, an

absolute quantifier can be represented by a fuzzy

subset

, such that for any r the membership

degree of r in

)(rQ

, indicates the degree to

which the amount r is compatible with the quantifier

represented by

. Proportional quantifiers, such as

“most”, and “at least half”, can be represented by

fuzzy subsets of the unit interval, [0, 1]. For any

]1,0[r

)(rQ

indicates the degree to which the

proportion r is compatible with the meaning of the

quantifier it represents. Any quantifier of natural

language can be represented as a proportional

quantifier or given the cardinality of the elements

considered, as an absolute quantifier.

Fuzzy quantifiers are usually of one of three

types, increasing, decreasing, and unimodal. A non-

decreasing quantifier

satisfies the expression

)()( ,bQaQthenbaifba 

and its

membership function is given by the following

expression





















brif

braif

arif

)(

with

]1,0[,, rba

. For our algorithm, we select the

following values which represent the concept of

fuzzy majority

)8.0,3.0(),( ba

2.2 Social Judgement Scheme

The aggregation of individual preferences in a group

decision setting has been studied extensively. Davis

has introduced the Social Decision Scheme (SDS)

theory providing a formal way to analyse different

aggregation processes by representing them as

stochastic matrices called decision schemes. SDS

theory suggests a systematic way to investigate

which decision aggregation model best defines the

actual consensus process in a given context, [11],

[12]. In addition to the SDS approach Davis

proposed the Social Judgment Scheme (SJS) theory,

which applies to continuous judgment cases. This

model assumes a dominant role of members whose

opinions are relatively central in the group. Thus,

each decision-maker is given a weight depending on

the centrality of his/her position relative to the other

members of the group and the group decision is a

weighted sum of the members’ preferences. This

model has been tested empirically with sufficient

results, [12]. In our model, we implement the SJS

model for aggregating numeric values assigned by

decision-makers to problem parameters.

For example, we consider the case where a

decision maker expresses her individual opinion on

the weight of a criterion in numerical format. If

is the weight of

ith

criterion as defined by

jth

decision maker, then the group weight

ith

criterion is defined as





n

ijijiwvc

where

is the

consensus weight of

jth

decision maker relative to

ith

criterion. Consensus weight depends on how

close the position of a decision maker’s opinion

with respect to the rest of the members’ opinions is.

The closer the opinion of the decision maker to the

team’s opinion is, the greater weight is calculated

for this decision maker for the specific criterion.

Consensus weights are calculated according to the

formula











n

jll

ilij

jll

ilij

,11

|)|exp(

3 Proposed Group Decision

Methodology

The main objective of this work is to introduce a

method to support a group of decision-makers in

classification problems. The problem refers to the

assignment of a set of alternatives to several

predefined non-ordered categories, according to

their ranking on a set of evaluation criteria. For this

reason, we have developed a structured group

decision methodology, based on the following

principles:

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 The decision group is a small homogeneous

team of collaborating decision makers.

Although the methodology can be extended to

large decision teams, our approach is based on

collaborative teams, which target maximizing

consensus. Non-collaborative teams require a

negotiation-based approach, which is out of the

scope of the present methodology.

 A facilitator coordinates the entire decision

process. The entire group decision process is

coordinated by a facilitator. Usually, in group

decision making a negotiation phase takes place

at the preliminary steps of the decision problem

formation. During this negotiation, which can

be either structured or not, basic parameters are

defined. Since our methodology does not focus

on group formation procedure and initial

negotiations, we consider that a preliminary

negotiation step has already taken place,

possibly by utilizing a brainstorming technique,

between stakeholders, and the outcome of this

process is an initial set of proposed parameters.

This set is expressed by the facilitator as the

initial proposal upon which group members will

express their preferences. The facilitator drives

the entire process to generate efficient and

timely results.

 A decision problem is structured or semi-

structured. The team solves a structured

classification problem based on their personal

preferences. Non-structured problems are out of

scope.

 A multicriteria analysis is utilized for the

classification. For the classification problem, we

utilize multicriteria analysis which provides

appropriate support to this type of problem.

Following the above principles, we developed a

group decision methodology comprising the

following phases:

 Problem initiation. In this phase, the

facilitator defines the basic parameters of

the problem. The parameters are related to

the specific multicriteria methodology and

refer to criteria, alternatives, and categories.

 Aggregation of individual parameters.

During this phase, each member evaluates

the proposed parameter set and expresses

her preferences in numeric and linguistic

format. Next, individual preferences are

aggregated, and a group parameter set is

produced which is used as input for the

classification algorithm.

 Application of NexClass multicriteria

classification algorithm. In this phase, using

the group parameter set, the NexClass

multicriteria algorithm is applied initially to

a training set of alternatives, [13]. Group

members evaluate results and if accepted,

the same parameter set is used for the

classification of the entire set of

alternatives.

 Results evaluation. At this phase, group

members evaluate the classification results

of the entire set expressing their opinions.

3.1 Phases

Notations used:



},...,,{ 21 m

aaaA 

: a set of alternatives

for classification in a number of categories,

 : a set of evaluation

criteria,

 : a set of categories,



},...,,{ 21

hhh bbbB 

: a set of prototypes

for category h, where

},...1,,..1|{ h

hLhkibB 

and

the ith prototype of hth category. These

prototypes define the category as thresholds

of entrance to the category.

 Alternatives’ performance on criteria is

calculated in a way such that

))(),...,(),(()(, 21 agagagaga n



and

))(),...,(),(()(, 21

ibgbgbgbgb 

Phase 1. Problem initiation. In this phase the

facilitator initiates the decision problem, defining all

appropriate parameters. In details:

1. Basic parameters. Initially, the facilitator

defines several parameters, related to the

classification problem such as the number of

group members, the number of categories, the

number of criteria, and to results assessment

such as the consensus, satisfaction, and

acceptance levels. These levels define the

minimum required levels for the group decision.

In case they are not satisfied, a second round is

executed with modification of individual

preferences.

2. Members. The facilitator defines group

members

},...mm,{mMn21



assigning all

necessary contact details.

3. Categories. The facilitator defines the set of

categories for the

classification of alternatives.

{ , ,..., }

G g g g

{ , ,..., }

C C C C

{ , ,..., }

C C C C

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4. Evaluation criteria. The facilitator defines the

set of evaluation criteria

according to problem requirements.

5. Criteria weights. The facilitator defines the

criteria weights.

6. Alternatives. The facilitator defines the set of

alternatives

},...,,{ 21 m

aaaA 

for

classification and defines their performance on

the evaluation criteria

))(),...,(),(()(, 21 agagagaga n



7. Entrance thresholds. The facilitator defines

appropriate entrance thresholds

},...,,{ 21

hhh bbbB 

for each category

For each threshold the

facilitator defines preference, indifference, and

veto thresholds.

8. Training set. The facilitator defines a subset of

alternatives as a training set, to evaluate the

parameters’ accuracy. After the initiation of the

parameters, the facilitator communicates

through the GDSS with group members

informing them about the problem and asking

them to submit their preferences.

Phase 2. Aggregation of individual parameters. In

this phase group members express their preferences

on the proposed parameter set. Member preferences

are expressed in numeric values and linguistic

preferences. For the aggregation of numeric values,

we utilize the Social Judgment Scheme (SJS), while

linguistic terms are aggregated in terms of an

Ordered Weighted Averaging Operator (OWA),

[13].

1. Numeric value aggregation. For numeric values,

we follow the SJS approach as presented in the

previous section.

2. Linguistic value aggregation. For non-numeric

values, we follow the Ordered Weighted

Averaging Operator (OWA) approach

introduced in, [13].

Aggregation of member preferences is executed for

the following parameters.

1. Criteria. Group members express their

acceptance of each proposed criterion on a five-

point linguistic scale and their preferred weight

in numeric value.

2. Alternatives. Group members express their

acceptance of alternatives’ performance or

submit their preference in numeric value.

3. Categories. Group members express their

acceptance of each category definition and

submit their preferences on category thresholds

in numeric value.

The facilitator proceeds with the validation of

members’ input and aggregates the values.

Parameters with low acceptance levels are marked

and are subject to review if the final results are not

acceptable to group members.

Phase 3. Application of multicriteria classification

algorithm. After the aggregation of individual

members’ parameters, a group parameter set is

created and the NeXClass algorithm for multicriteria

classification is applied to this group parameter set,

[13].

NeXClass algorithm classifies an alternative to a

specific category with respect to the alternative’s

performance to the evaluation criteria, considering a

set of alternatives, a set of predefined non-ordered

categories, and a set of evaluation criteria. In more

detail, the algorithm works as follows:

1. For each category , the

decision maker defines an entrance threshold

},...,,{ 21

hhh bbbB 

using available

information. This threshold represents the

minimum requirements for an alternative in

terms of performance on the evaluation criteria

to be included in this category.

2. The decision maker defines the alternatives’

performance

))(),...,(),(()(, 21 agagagaga n



on the

evaluation criteria

},...,,{ 21 n

gggF 

3. For each alternative, an excluding degree

),(1

),(

tot

iba







is calculated for every

category threshold, based on outranking

relations, following a similar approach to the

ELECTRE TRI method.

4. Next, the fuzzy excluding degree

tothh baPCa



 ),(),(

of an alternative

over a category

C

is calculated.

5. Assignment to a category is based on the rule

}},...,1{/),(min{),( kiCaCaCa ihh 



which states that alternative is assigned

to the category

C

for which the excluding

degree over the entrance threshold is minimum.

Application of the NeXClass classification

algorithm is executed through the following steps.

{ , ,..., }

G g g g

{ , ,..., }

C C C C

{ , ,..., }

C C C C

Aa 

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1. Training set classification. The classification

algorithm is initially applied to the training set,

as it has been defined by group members.

Classification is executed by the Facilitator, and

group members are informed to assess the

results.

2. Evaluation of results. Each member expresses

her preference for the results on a five-point

linguistic scale, and in case of a low acceptance

level, the Facilitator executes a second round of

parameter definition from members to calibrate

the model. When training set classification is

acceptable, the Facilitator proceeds with the

classification of the entire set of alternatives. In

case of a low acceptance level after the second

round, the Facilitator terminates the process to

revise the problem with stakeholders.

3. Training set classification. The classification

algorithm is finally applied to the entire set by

the Facilitator, and group members are informed

to assess the results.

Phase 4. Results assessment. Group members assess

the results expressing their preference in a five-point

linguistic scale. In case of a low acceptance level,

the Facilitator reruns the model, requesting

modifications from members.

3 Discussion

In the previous sections, we introduced a novel

methodology for group classification decisions in

nominal categories, based on the multicriteria

algorithm NeXClass for the aggregation of

individual preferences. The approach aggregates the

individual preferences under the fuzzy majority

approach and the resulting set is used as input for

the classification algorithm. At the end of the

process, consensus is measured and if it does not

reach the baseline the process is repeated. An

alternative approach would be to apply the

classification algorithm at the member level and

then aggregate the classification results. This

approach is not suitable for nominal categories, as

there is not no way to aggregate results on

categories, while numeric preferences are easier to

aggregate by applying OWA family operators.

Examining the scenario of aggregating class

preferences will be part of future research on this

domain.

As a general comment, the methodology

introduced contributes to existing GDS research, as

it presents an integrated methodology for group

classification problems in small-group settings. The

methodology is based on a solid foundation for

aggregation of preferences and its structured

approach can be easily implemented in a web GDSS

or a mobile application. In addition, it can be easily

utilized in multiagent-based decision-making and

automated decisions in collaborative environments

where agents interact and try to reach a consensus.

As mentioned earlier, due to the complex nature

of decision problems, it is not feasible to provide a

generic methodology that fits all problems, and this

is the reason for the diversity of methods in the

literature. The methodology presented here is not

very specific and can be extended to various

applications and generalized as a model to fit more

complex scenarios. However, some limitations can

be identified in the present form. The following

restrictions exist regarding the problems that can be

solved by the methodology.

• Since the methodology requires a relatively

substantial number of parameters, it is possible

that group members who are not familiar

enough with the methodology will be confused.

Thus, the number of criteria and parameters

should be kept to an optimum number to

minimize complexity without losing critical

problem parameters.

• Another limitation is that the number of

members should be kept within the limits of a

small collaborating team. If members are quite a

few, anonymity is not so well established since

preferences can be easily identified. On the

other hand, a large number of members will

increase the complexity and extra facilitation

will be necessary. A large number of members

require alternative aggregation approaches,

while very large numbers require a statistical

approach or even sampling.

4 Conclusion

In this work, we presented a Group Decision

Support System methodology for small

collaborating teams based on multicriteria analysis

and aggregation operators. It implements a group

multicriteria decision methodology for classification

decisions where aggregation of members’

preferences is executed at the parameter level. We

presented the methodology and the steps in detail so

it can be easily implemented in software

applications, like GDSS based on web or mobile

technology, and can be easily integrated within

existing business infrastructure or business

intelligence context. Future work will focus on

empirical findings from the application of the

methodology and analysis of user adoption in

business environments. We believe that this

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methodology and a relevant GDSS can be easily

deployed to support group decisions in

contemporary business environments, either in

physical decision-making or in artificial

environments with multiagent settings.

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WSEAS Transactions on International Journal

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Georgios Rigopoulos is the sole author of the work.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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DOI: 10.37394/23205.2023.22.25

Georgios Rigopoulos

E-ISSN: 2224-2872

224

Volume 22, 2023