A hybrid linguistic fuzzy multiple criteria group selection of a chief accounting officer.
Kersuliene, Violeta ; Turskis, Zenonas
Introduction
In most cases, modern real-world problems cannot be solved only by
considering precise and objective information. Existing work in data
mining from multiple data sources mainly falls into the following three
categories:
a) Data integration;
b) Model integration;
c) Relational learning.
The first objective of this study is to develop a decision making
approach to a problem of multiple information sources, which enables the
incorporation of both crisp data and fuzzy data represented as
linguistic variables or triangular fuzzy numbers into the analysis.
The second objective of this paper is to construct a Model for
Selection of a Chief Accounting Officer based on the study of ways used
by stakeholders selecting chief accounting officers. The presented model
reduces the time taken by stakeholders and managers to accumulate
experience in selection of a chief accounting officer, further
increasing the efficiency of the enterprises activities.
Human resources are one of the core competences for an organisation
to enhance its competitive advantage in the knowledge economy (Lin
2010). Personnel selection is the process of choosing among candidates,
who match the qualifications required to perform a defined job in the
best way (Dursun, Korsak 2010). The use of personality measures to
predict job performance has a long and storied history (Penney et al.
2010). However, methodological advances in meta-analytic techniques and
the advent of the now widely-accepted Big Five Model of
Personality--Conscientiousness, Extraversion, Agreeableness, Emotional
Stability and Openness to Experience--renewed the interest in
personality as a selection device among academics.
Kelemenis et al. (2011) presented an overview of recent studies on
the personnel selection problem (1992 to 2009). They indicated the use
of different techniques and conceptual models. The quality of human
capital is crucial for high-tech companies to maintain competitive
advantages in knowledge economy era (Chien, Chen 2008). The fuzzy set
appears as an essential tool to provide a decision framework that
incorporates imprecise judgements inherent to the personnel selection
process (Dursun, Korsak 2010).
A financial management department is special in the way it needs to
conform to standards, which is different from any other department.
Evaluation criteria for financial management are listed in Figure 1.
There are two fields of accounting: financial and managerial.
Management accounting provides customised, appropriate and timely
financial information to those internal managers entrusted with the
day-to-day operations of the organisation. Lambert and Pezet (2011)
analysed the practices through which a management accountant is
constructed as a knowing subject and becomes a producer of truthful
knowledge. The centrality of management accountant's role is
evidenced, among other aspects, by their participation in online reverse
auctions, wherein they commit themselves and their company to long-term
projects.
[FIGURE 1 OMITTED]
Chiapello and Medjad (2009) highlighted that accounting standards
concern a far greater audience than market actors (companies, auditors,
bankers and investors).
In the last two decades or so there has been a lively academic and
political debate about the continued gendering process of the
accountancy profession (Heidhues, Patel 2011). Accountancy is now well
established as an elite professional occupation in most parts of the
world and much of this status has been afforded through an association
with educational qualifications (Gammie, Kirkham 2008). Seifert et al.
(2010) applied the theory of organisational justice to the design of
whistle blowing policies and procedures. The emphasis in financial
accounting is on producing organisational summaries of financial
consequences of past activities and decisions. The prepared data is
objective, precise and verifiable, usually by an external auditor.
Tillmann and Goddard (2008) developed a substantive grounded theory
of strategic management accounting and sense-making. It is not enough to
'simply' know accounting or management accounting techniques,
but there is a need for a much broader know-how. Accounting is not a
'reality' in itself, but part of broader organisational
realities for whose understanding some non-accounting knowledge is
needed.
Jones and Lee (1998) stated that in recent years there have been
concerns for 'traditional' accounting approaches to investment
appraisal hinder companies' adoption of advanced manufacturing
technology. Primrose (1988) warned that traditional accounting methods,
when faced with engineering problems in trying to justify advanced
manufacturing technology have resulted in many companies investing in
wrong technologies or for wrong reasons. Some organisations (e.g.
governmental departments) distinguish between 'essential' and
'desirable' criteria. Essential criteria are those elements or
conditions of a job that the employer considers vital for successful
performance in a particular role. Desirable criteria are the ones that
are nice to have and may be of assistance in the role. In
highly-competitive recruitment situations, being able to address all the
desirable criteria may be necessary; however, do not be put off applying
for a role if you can address all the essential criteria.
Accurately defining performance criteria is a critical step in
empirical validation. However, defining performance criteria is also a
conceptual issue, as criteria should accurately represent all important
performance requirements of the target job (Penney, Borman 2005).
Applying the factor analysis research method, Lin (2008) has
empirically- developed 6 latent constructs about the desirable knowledge
and skill components that should be emphasized in accounting education
in order to meet the challenges stemming from the changing business
environment, i.e. business/management skills, business/management
knowledge, core accounting knowledge, personal characteristics, general
knowledge and basic techniques. The structural order of/the
interrelationship among these six dimensions of knowledge and skill
requirements in accounting education is also elaborated based on the
analysis of factor loading results.
1. Selection algorithm based on the fuzzy sets and MCDM methods
There are a lot of different MCDM methods. Selection of an
appropriate decision method depends on the aim of the problem, available
information, cost of the decision and qualification of actors
(decision-makers).
The type of information collected can directly influence scale
construction. Different types of information could be measured in
different ways:
a) At the nominal level. That is, any numbers used are mere labels:
they express no mathematical properties.
b) At the ordinal level. Numbers indicate the relative position of
items, but not the magnitude of difference. An example is a preference
ranking.
c) At the interval level. Numbers indicate the magnitude of
difference between items, but there is no absolute zero point. Examples
are attitude scales and opinion scales.
d) At the ratio level. Numbers indicate magnitude of difference and
there is a fixed zero point. Examples include: age, income, price,
costs, sales revenue, sales volume, and market share.
A wider overview of MCDM methods, classification and applications
are presented by Zavadskas and Turskis (2011). In this research, two of
them are applied: ARAS-F and AHP. The multiple-criteria expert system
for problem solving can be described as shown in Figure 2.
[FIGURE 2 OMITTED]
1.1. Basic definitions
Fuzzy set theory, which was introduced by Zadeh (1975a, 1975b,
1975c). A fuzzy set can be defined mathematically by a membership
function, which assigns each element x in the universe of discourse X a
real number in the interval [0, 1].
A triangular fuzzy number can be defined by a triplet (a, y, b) as
illustrated in Figure 3.
A fuzzy set is a class of objects with a continuum of membership
grades. Such a set is characterized by a membership function which
assigns to each object a grade of membership ranging between zero and
one (Zadeh 1975a, 1975b, 1975c). A fuzzy set A defined in space X is a
set of pairs:
A = {(x, [[mu].sub.A] (x)), x [member of] X}, [for all]x [member
of] X, (1)
where the fuzzy set A is characterized by its membership function
[[mu].sub.A] : X [right arrow] [0;1] which associates with each element
x [member of] X, with a real number [[mu].sub.A] (x) [member of] [0;l].
The value [[mu].sub.A] (x) at x represents the grade of membership of x
in A and is interpreted as the membership degree to which x belongs to
A. So the closer the value [[mu].sub.A] (x) is to 1, the more x belongs
to A.
A crisp or ordinary subset A of X can also be viewed as a fuzzy set
in X with membership function as its characteristic function, i.e.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
The set X is called a universe of discourse. A fuzzy set A in X can
be represented as A = {(x, [[mu].sub.A] (x))}, where x [member of] X and
[[mu].sub.A] : X [right arrow] [0;l].
When the universe of discourse is discrete and finite with
cardinality n, that is X = {[x.sub.1],[x.sub.1],...,[x.sub.n]}, the
fuzzy set A can be represented as (Zadeh 1975d; Klir, Yuan 1995):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
[FIGURE 3 OMITTED]
When the universe of discourse X is an interval of real numbers,
the fuzzy set A can be expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
A fuzzy number is defined to be a fuzzy triangular number (a, b, y)
if its membership function is fully described by three parameters
([alpha] < [gamma] < [beta]):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
In order to obtain a crisp output, a defuzzification process is
needed to be applied. Various types of membership functions are used.
The most typical fuzzy set membership function is triangular membership
function (Fig. 3).
The basic operations of fuzzy triangular numbers [[??].sub.1] and
[[??].sub.2] (Van Laarhoven, Pedrycz 1983) are defined as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)
[k[??].sub.1] [congruent to] ([kn.sub.1[alpha]], [kn.sub.1[beta]],
[kn.sub.1[gamma]]) for multiplication by constant, (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
1.2. Additive Ratio Assessment method (ARAS) with fuzzy criteria
values (ARAS-F)
This section outlines the fuzzy MCDM approach, which is based on
ARAS with fuzzy criteria values method. ARAS method was developed by
Zavadskas and Turskis (2010). Later, modifications of ARAS
method--ARAS-G (grey relations are applied) and ARAS-F --were published
(Turskis, Zavadskas 2010a, 2010b; Turskis et al. 2012). There are only
few applications of ARAS method (Tupenaite et al. 2010; Zavadskas et al.
2010b, 2012a; Bakshi, Sarkar 2011; Susinskas et al. 2011; Kersuliene,
Turskis 2011).
According to the ARAS method, a utility function value determining
the complex relative efficiency of a reasonable alternative is directly
proportional to the relative effect of values and weights of the main
criteria considered in a project.
The first stage is dedicated to forming of the fuzzy
decision-making matrix (FDMM). Any problem which has to be solved is
represented by the following decision- making matrix (DMM) of
preferences for m reasonable alternatives (rows) rated on n criteria
(columns):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where m--number of alternatives, n--number of criteria describing
each alternative, [[??].sub.ij]--fuzzy value representing the
performance value of the i alternative in terms of the j criterion,
[[??].sub.0j]--optimal value of j criterion. A tilde "~" will
be placed above a symbol if the symbol represents a fuzzy set.
If optimal value of j criterion is unknown, then:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
Usually, the performance values [[??].sub.ij] and the criteria
weights [[??].sub.j] are viewed as the entries of a DMM. The system of
criteria as well as the values and initial weights of criteria are
determined by experts.
The purpose of the next stage is to calculate
dimensionless-normalized values. The initial values of all criteria are
normalized--defining values [[??].sub.ij] of normalised decision-making
matrix [??] :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
The criteria, whose preferable values are maxima, are normalized as
follows:
[[??].sub.ij] = [[??].sub.ij]/[m.summation over (i=0)]
[[??].sub.ij]. (17)
The criteria, whose preferable values are minima, are normalized by
applying two-stage procedure:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
The third stage is defining normalized-weighted matrix - [??]. It
is possible to evaluate the criteria with weights 0 < [[??].sub.j]
< 1. The values of weight [w.sub.j] are usually determined by the
expert evaluation method. The sum of weights [w.sub.j] would be limited
as follows:
[n.summation over (j=1)] [w.sub.j] = 1, (19)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
Normalized-weighted values of all the criteria are calculated as
follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
where [w.sub.j] is the weight (importance) of the j criterion and
[[bar.x.sub.ij] is the normalized rating of the j criterion.
The following step is determining values of effectiveness function:
[[??].sub.i] = [n.summation over (j=1)] [[??].sub.ij]; i = [bar.0,
m], (22)
where [[??].sub.i] is the value of effectiveness function of i-th
alternative.
The greater the value of the effectiveness function [[??].sub.i],
the more effective is the alternative.
The result of fuzzy decision making for each alternative is fuzzy
number [[??].sub.i]. The centre-of-area is the most practical and simple
to apply for defuzzification:
[S.sub.i] = 1/3([S.sub.i[alpha]] + [S.sub.i[beta]] +
[S.sub.i[gamma]]). (23)
The utility degree [K.sub.i] of an alternative [A.sub.i] is
determined by a comparison of the variant, which is analysed, with the
most ideal one
[K.sub.i] = [S.sub.i]/[S.sub.0]; i = [bar.0, m], (24)
where [S.sub.i] and [S.sub.0] are the optimal criterion values,
obtained from Eq. (23).
The complex relative efficiency of the reasonable alternative can
be determined according to the utility function values.
1.3. A fuzzy weighted-product model
Triantaphyllou and Lin (1995) presented the fuzzy weighted-product
model (WPM). The WPM uses multiplication to rank alternatives. Each
alternative is compared with others by multiplying a number of ratios,
one for each criterion. Each ratio is raised to the power of a
respective weight.
The two alternatives can be compared as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
where [[??].sub.Kj], [[??].sub.Lj], and [[??].sub.j] are fuzzy
triangular numbers. Alternative [A.sub.K] dominates alternative
[A.sub.L] if and only if the numerator in Eq. (25) is greater than the
denominator.
1.4. Determining criteria weights with the help of AHP
Methods of utility theory based on qualitative initial measurements
include two widely known groups of methods: AHP and fuzzy set theory
methods (Zimmermann 1985, 2000). Pioneering studies presented by Saaty
(Saaty 1977, 1980; Saaty, Zoffer 2011). Lootsma (1993) introduced
Multiplicative AHP, which is an exponential version of the simple
multi-attribute rating technique (SMART). Many AHP method applications
are suggested in recent researches: Ananda and Herath (2008) synthesised
stakeholder preferences related to regional forest planning and to
incorporate stakeholder preferences; Cebeci (2009) presented a fuzzy
approach to select a suitable enterprise resource planning system for
textile industry; Wu et al. (2009) adopted fuzzy AHP to rank the banking
performance and improve the gaps with three banks; Colombo et al. (2009)
proved that judicious use of AHP by experts can be used to represent
citizen views; Stemberger et al. (2009) applied it in business process
management; Maskeliunaite et al. (2009) and Sivilevicius and
Maskeliunaite (2010) solved the problem of improving the quality for
passenger transportation; Steuten et al. (2010) used AHP weights to fill
missing gaps in Markov decision models; Yan et al. (2011) presented new
developments and maintenances of the existing infrastructures under
limited government budget and time. Hadi-Vencheh and Niazi-Motlagh
(2011) applied improved voting analytic hierarchy process-data
envelopment analysis methodology for selection problem. Zavadskas et al.
(2012b) applied AHP method for determining managers skill weights.
There are various approaches for assessing weights (Zavadskas et
al. 2010a, 2010b), e.g., the eigenvector method, SWARA (Kersuliene et
al. 2010), expert method (Zavadskas, Vilutien? 2006), entropy method,
etc.
The decision is made by using the derived weights W of the
evaluative criteria (Saaty 1980). According to Saaty, his experiments
have shown that most individuals cannot compare more than seven objects
(plus/minus two). Based on this, Saaty established 9 objects as the
upper limit of his integer scale for multiple pairwise comparisons
(Table 1).
In AHP, the decision matrix is always a square matrix:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
Aggregate weight is determined as follows:
After obtaining the criteria weights from AHP, the synthesising of
ratio judgements is done.
Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is
aggregate weight for n criteria and [[??].sub.j] is fuzzy triangular
number:
[[??].sub.j] =([w.sub.j[alpha]], [w.sub.j[hamma]],
[w.sub.j[beta]]), (27)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is
minimum possible value,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the most
possible value and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the maximal
possible value of j-th criterion.
2. Application of the developed model
For the assessment of a chief accounting officer, the set of
essential criteria consists of: education, academic level, long life
learning, working knowledge, working skills, work experience, culture,
competence, team player, leadership excellence, ability to work in
different business units, determination of a goal, problem solving
ability, decision making skills, strategic thinking, ability to sell
self and ideas, interpersonal skills, management experience, emotional
steadiness, communication skills, ability to maintain a good discussion,
personality assessment, computer skills, self-confidence, fluency in
foreign languages, responsibility, patience, effective time using and
age.
Bots et al. (2009) wrote "In 2002, the Accountants-in-Business
section of the International Federation of Accountants issued the
Competency Profiles for Management Accounting Practice and Practitioners
report". Birkett (2002) developed a framework for competencies
required of management accountants during their careers. The Birkett
Report distinguishes five experience levels of management accountants:
the novice practitioner level, the assistant practitioner level, the
competent practitioner level, the proficient practitioner level and the
expert practitioner level. Each level is characterized by its position
in the business hierarchy, activities and performance expectations.
Birkett determined a five-level system of competencies. Level one
consists of cognitive skills and behavioural skills. In level two,
cognitive skills are divided into technical skills, analytical/design
skills, and appreciative skills, while behavioural skills are divided
into personal skills, interpersonal skills, and organisational skills.
At level three, there are 38 essential skills, 80 skills at level four,
and 375 skills are at level five.
The problem's set of criteria was determined by three decision
makers (owners) of the company as follows:
[x.sub.1]--Education, academic level, long life learning;
[x.sub.2]--Working knowledge, working skills, work experience,
knowledge of legislation system;
[x.sub.3]--Responsibility;
[x.sub.4]--Strategic thinking;
[x.sub.5]--Leadership; ability to work in a team;
[x.sub.6]--Motivation to work in a particular position;
[x.sub.7]--Computer skills;
[x.sub.8]--Ability to work with clients, consultants and community.
At the first stage of problem solving, three decision makers
determined criteria ranks by applying AHP method. All of experts
prepared pairwise comparison matrixes. In Table 2 is shown of the
pairwise comparison matrix (Expert 1).
According to the calculations by using Eq. (27) aggregate weights
were established (Table 3).
In this study, the linguistic term set with associated semantics is
considered (Table 4).
The candidates were rated. Data related to the selection of a chief
accounting officer are given in Table 5.
According to Tables 4 and 5 the matrix with aggregate weights
(Table 6) and fuzzy decision making matrix with aggregate weights were
prepared (Table 7).
Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is
aggregate weight for n criteria and [[??].sub.j] is fuzzy triangular
number:
[[??].sub.j] =([w.sub.j[alpha]], [w.sub.j[gamma]],
[w.sub.j[beta]]), (28)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is
minimum possible value,
[w.sub.j[gamma]] = 1/p [[summation].sup.p.sub.i=1] [y.sub.jk], j =
[bar.1, n], k = [bar.1, p] is the most possible value and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the maximal
possible value of j-th criterion.
Solution results are presented in Table 8 (ARAS-F method) and Table
9 (WPM-F method).
According to the ARAS-F method, the second candidate is the best
alternative from those available.
According to the WPM-F and ARAS-F methods alternatives rank as
follows [A.sub.2] > [A.sub.1] > [A.sub.3] (Fig. 4):
The best candidate in both solution cases is the second candidate.
He/she was selected by decision-makers.
Conclusions
In the age of competitive markets, appropriate selection of
personnel determines success of organisations. A chief accounting
officer is one of the most important persons in each organisation. The
proposed model helps to overcome difficulties in the selection of a
chief accounting officer. The values of criteria set describing
candidates in most cases are lexical values. The fuzzy set theory is the
proper way to deal with uncertainty. It can be stated that the
effectiveness ratio with an optimal alternative may be used in cases
when it is sought to rank alternatives and find ways to improve them.
The presented case study showed that this model could successfully help
in cases when actors need to select from feasible candidates.
doi: 10.3846/16111699.2014.903201
Caption: Fig. 1. List of professional standards as the basis for
evaluation of financial management
Caption: Fig. 2. The multiple-criteria expert system for personnel
selection
Caption: Fig. 3. Triangular membership function
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Violeta Kersuliene [1], Zenonas Turskis [2]
[1] Department of Law, Faculty of Business Management, Vilnius
Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius,
Lithuania
[2] Department of Construction Technology and Management, Faculty
of Civil Engineering, Vilnius Gediminas Technical University, Sauletekio
al. 11, LT-10223 Vilnius, Lithuania E-mail: [1]
[email protected] (corresponding author); [2]
[email protected]
Received 16 January 2014; accepted 07 March 2014
Violeta KERSULIENE. Doctor, Assoc. Prof., Director of Legal Affairs
at the Department of Law of Vilnius Gediminas Technical University,
Lithuania. Author of 14 scientific papers. In 1991, she graduated from
the Civil Engineering Faculty of Vilnius Gediminas Technical University
and in 2008, she defended her doctoral thesis in the field of
technological sciences. In 1999, she acquired the lawyer's
qualification from the Faculty of Law of Vilnius University. Her
research interests include decision-making, dispute resolution and
decision support systems.
Zenonas TURSKIS. Professor, senior research fellow at the
Construction Technology and Management Laboratory of Vilnius Gediminas
Technical University, Lithuania. His research interests include building
technology and management, decision-making theory, computer-aided design
and expert systems. Author of more than 90 research papers.
Table 1. The nine-point scale of pair wise comparison
(according to Saaty 1980)
Intensity of 1 3 5
importance
Definition Criteria i and j Criterion z is Criterion z
have equal weakly more is
importance important than essentially
criterion j (strongly)
more
important
than
criterion j
Intensity of 7 9 2, 4, 6,
importance
Definition Criterion i is (Criterion Intermediate
very strongly z is values between
absolutely the two adjacent
more important judgements
than criterion
j
Intensity of Reciprocals
importance nonzero
Definition If activity z has one of the
nonzero numbers assigned to
it when compared with activity
j then, then j has reciprocal
value when compared with i
Table 2. Pairwise comparisons among criteria
Expert 1
[x.sub.1] [x.sub.2] [x.sub.3] [x.sub.4] [x.sub.5]
[x.sub.1] 1.00 2.00 3.00 4.00 5.00
[x.sub.2] 0.50 1.00 2.00 3.00 4.00
[x.sub.3] 0.33 0.50 1.00 2.00 3.00
[x.sub.4] 0.25 0.33 0.50 1.00 2.00
[x.sub.5] 0.20 0.25 0.33 0.50 1.00
[x.sub.6] 0.17 0.20 0.25 0.33 0.50
[x.sub.7] 0.14 0.17 0.20 0.25 0.33
[x.sub.8] 0.13 0.14 0.17 0.20 0.25
[x.sub.6] [x.sub.7] [x.sub.8] Weights Products
[x.sub.1] 6.00 7.00 8.00 0.33 2.78
[x.sub.2] 5.00 6.00 7.00 0.23 1.94
[x.sub.3] 4.00 5.00 6.00 0.16 1.33
[x.sub.4] 3.00 4.00 5.00 0.11 0.90
[x.sub.5] 2.00 3.00 4.00 0.07 0.60
[x.sub.6] 1.00 2.00 3.00 0.05 0.40
[x.sub.7] 0.50 1.00 2.00 0.03 0.27
[x.sub.8] 0.33 0.50 1.00 0.02 0.20
CI= 0.04 CI/RI=
Ratio
[x.sub.1] 8.51
[x.sub.2] 8.55
[x.sub.3] 8.47
[x.sub.4] 8.33
[x.sub.5] 8.17
[x.sub.6] 8.07
[x.sub.7] 8.07
[x.sub.8] 8.16
0.03
Table 3. Aggregate weights
Criteria weights
Expert 1 Expert 2 Expert 3 Expert 4 Expert 5
[x.sub.1] 0.33 0.16 0.16 0.23 0.23
[x.sub.2] 0.23 0.33 0.11 0.33 0.16
[x.sub.3] 0.16 0.23 0.07 0.16 0.07
[x.sub.4] 0.11 0.07 0.02 0.07 0.02
[x.sub.5] 0.07 0.11 0.33 0.11 0.33
[x.sub.6] 0.05 0.03 0.23 0.03 0.11
[x.sub.7] 0.03 0.05 0.05 0.02 0.03
[x.sub.8] 0.02 0.02 0.03 0.05 0.05
Aggregate weights
[w.sub.j[alpha]] [w.sub.j[gamma] [w.sub.j[beta]]
[x.sub.1] 0.16 0.21 0.33
[x.sub.2] 0.11 0.21 0.33
[x.sub.3] 0.07 0.13 0.23
[x.sub.4] 0.02 0.05 0.11
[x.sub.5] 0.07 0.17 0.33
[x.sub.6] 0.03 0.08 0.23
[x.sub.7] 0.02 0.03 0.05
[x.sub.8] 0.02 0.03 0.05
Table 4. Label set
Label set Linguistic term Fuzzy number
[alpha] [gamma] [beta]
[s.sub.1] Nothing answered, task 0 0 0.2
was not completed
[s.sub.2] Bad 0 0.2 0.4
[s.sub.3] Weak 0.2 0.4 0.6
[s.sub.4] Satisfactory 0.4 0.6 0.8
[s.sub.5] Good 0.6 0.8 1.0
[s.sub.6] Excellent 0.8 1.0 1.0
Table 5. Rating of candidates with respect to subjective criteria
Criteria Candidates Decision makers
[D.sub.1] [D.sub.2] [D.sub.3]
[x.sub.1] [A.sub.1] [s.sub.6] [s.sub.4] [s.sub.4]
[A.sub.2] [s.sub.5] [s.sub.6] [s.sub.4]
[A.sub.3] [s.sub.4] [s.sub.6] [s.sub.4]
[x.sub.2] [A.sub.1] [s.sub.4] [s.sub.4] [s.sub.4]
[A.sub.2] [s.sub.5] [s.sub.5] [s.sub.5]
[A.sub.3] [s.sub.4] [s.sub.5] [s.sub.5]
[x.sub.3] [A.sub.1] [s.sub.5] [s.sub.4] [s.sub.5]
[A.sub.2] [s.sub.6] [s.sub.6] [s.sub.4]
[A.sub.3] [s.sub.5] [s.sub.5] [s.sub.5]
[x.sub.4] [A.sub.1] [s.sub.5] [s.sub.6] [s.sub.4]
[A.sub.2] [s.sub.5] [s.sub.5] [s.sub.5]
[A.sub.3] [s.sub.4] [s.sub.4] [s.sub.5]
[x.sub.5] [A.sub.1] [s.sub.4] [s.sub.5] [s.sub.4]
[A.sub.2] [s.sub.5] [s.sub.4] [s.sub.4]
[A.sub.3] [s.sub.4] [s.sub.6] [s.sub.5]
[x.sub.6] [A.sub.1] [s.sub.5] [s.sub.6] [s.sub.5]
[A.sub.2] [s.sub.6] [s.sub.5] [s.sub.4]
[A.sub.3] [s.sub.5] [s.sub.6] [s.sub.5]
[x.sub.7] [A.sub.1] [s.sub.5] [s.sub.6] [s.sub.5]
[A.sub.2] [s.sub.4] [s.sub.5] [s.sub.4]
[A.sub.3] [s.sub.4] [s.sub.4] [s.sub.4]
[x.sub.8] [A.sub.1] [s.sub.5] [s.sub.5] [s.sub.4]
[A.sub.2] [s.sub.4] [s.sub.4] [s.sub.5]
[A.sub.3] [s.sub.5] [s.sub.4] [s.sub.4]
Criteria Candidates Decision makers
[D.sub.4] [D.sub.5]
[x.sub.1] [A.sub.1] [s.sub.5] [s.sub.4]
[A.sub.2] [s.sub.6] [s.sub.4]
[A.sub.3] [s.sub.5] [s.sub.5]
[x.sub.2] [A.sub.1] [s.sub.6] [s.sub.5]
[A.sub.2] [s.sub.5] [s.sub.4]
[A.sub.3] [s.sub.4] [s.sub.5]
[x.sub.3] [A.sub.1] [s.sub.5] [s.sub.5]
[A.sub.2] [s.sub.4] [s.sub.4]
[A.sub.3] [s.sub.4] [s.sub.4]
[x.sub.4] [A.sub.1] [s.sub.6] [s.sub.4]
[A.sub.2] [s.sub.6] [s.sub.4]
[A.sub.3] [s.sub.5] [s.sub.5]
[x.sub.5] [A.sub.1] [s.sub.6] [s.sub.5]
[A.sub.2] [s.sub.5] [s.sub.4]
[A.sub.3] [s.sub.5] [s.sub.5]
[x.sub.6] [A.sub.1] [s.sub.4] [s.sub.4]
[A.sub.2] [s.sub.6] [s.sub.5]
[A.sub.3] [s.sub.5] [s.sub.5]
[x.sub.7] [A.sub.1] [s.sub.6] [s.sub.4]
[A.sub.2] [s.sub.6] [s.sub.4]
[A.sub.3] [s.sub.4] [s.sub.4]
[x.sub.8] [A.sub.1] [s.sub.5] [s.sub.5]
[A.sub.2] [s.sub.4] [s.sub.4]
[A.sub.3] [s.sub.4] [s.sub.4]
Table 6. The aggregate weight criteria values
Ratings
Criteria Candidates [D.sub.1]
[alpha] [gamma] [beta]
[x.sub.1] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.2] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.3] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.8 1.0 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.4] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.5] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.6] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.8 1.0 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.7] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.4 0.6 0.8
[x.sub.8] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.6 0.8 1.0
Ratings
Criteria Candidates [D.sub.2]
[alpha] [gamma] [beta]
[x.sub.1] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.8 1.0 1.0
[A.sub.3] 0.8 1.0 1.0
[x.sub.2] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.3] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.4] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.5] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.8 1.0 1.0
[x.sub.6] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.8 1.0 1.0
[x.sub.7] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.8] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.4 0.6 0.8
Ratings
Criteria Candidates [D.sub.3]
[alpha] [gamma] [beta]
[x.sub.1] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.4 0.6 0.8
[x.sub.2] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.3] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.6 0.8 1.0
[x.sub.4] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.5] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.6 0.8 1.0
[x.sub.6] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.6 0.8 1.0
[x.sub.7] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.4 0.6 0.8
[x.sub.8] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
Ratings
Criteria Candidates [D.sub.4]
[alpha] [gamma] [beta]
[x.sub.1] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.8 1.0 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.2] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.3] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.4 0.6 0.8
[x.sub.4] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.8 1.0 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.5] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.6 0.8 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.6] [A.sub.1] 0.4 0.6 0.8
[A.sub.2] 0.8 1.0 1.0
[A.sub.3] 0.6 0.8 1.0
[x.sub.7] [A.sub.1] 0.8 1.0 1.0
[A.sub.2] 0.8 1.0 1.0
[A.sub.3] 0.4 0.6 0.8
[x.sub.8] [A.sub.1] 0.6 0.8 1.0
[A.sub.2] 0.4 0.6 0.8
[A.sub.3] 0.4 0.6 0.8
Ratings
Criteria Candidates Group fuzzy
[alpha] [gamma] [beta]
[x.sub.1] 0.4 0.66 1
0.4 0.73 1
0.4 0.66 1
[x.sub.2] 0.4 0.62 1
0.6 0.71 1
0.4 0.63 1
[x.sub.3] 0.4 0.67 1
0.4 0.66 1
0.4 0.67 1
[x.sub.4] 0.4 0.73 1
0.6 0.73 1
0.4 0.63 1
[x.sub.5] 0.4 0.66 1
0.4 0.63 1
0.4 0.70 1
[x.sub.6] 0.4 0.70 1
0.4 0.73 1
0.6 0.73 1
[x.sub.7] 0.6 0.76 1
0.4 0.66 1
0.4 0.54 0.8
[x.sub.8] 0.4 0.67 1
0.4 0.59 1
0.4 0.59 1
Table 7. The fuzzy decision making matrix with aggregate weights
(all criteria should to be maximized and optima value equal to 1.0)
Alternatives
Criterion [A.sub.0] [A.sub.1]
Ratings
[alpha]; [gamma]; [beta] [alpha] [gamma] [beta]
[x.sub.1] 1.0 0.4 0.66 1
[x.sub.2] 1.0 0.4 0.62 1
[x.sub.3] 1.0 0.4 0.67 1
[x.sub.4] 1.0 0.4 0.73 1
[x.sub.5] 1.0 0.4 0.66 1
[x.sub.6] 1.0 0.4 0.70 1
[x.sub.7] 1.0 0.6 0.76 1
[x.sub.8] 1.0 0.4 0.67 1
Alternatives
Criterion [A.sub.2] [A.sub.3]
Ratings
[alpha] [gamma] [beta] [alpha] [gamma] [beta]
[x.sub.1] 0.4 0.73 1 0.4 0.66 1
[x.sub.2] 0.6 0.71 1 0.4 0.63 1
[x.sub.3] 0.4 0.66 1 0.4 0.67 1
[x.sub.4] 0.6 0.73 1 0.4 0.63 1
[x.sub.5] 0.4 0.63 1 0.4 0.70 1
[x.sub.6] 0.4 0.73 1 0.6 0.73 1
[x.sub.7] 0.4 0.66 1 0.4 0.54 0.8
[x.sub.8] 0.4 0.59 1 0.4 0.59 1
Alternatives
Criterion Total
Ratings
[alpha] [gamma] [beta]
[x.sub.1] 2.2 3.05 4
[x.sub.2] 2.4 2.96 4
[x.sub.3] 2.2 3 4
[x.sub.4] 2.4 3.09 4
[x.sub.5] 2.2 2.99 4
[x.sub.6] 2.4 3.16 4
[x.sub.7] 2.4 2.96 3.8
[x.sub.8] 2.2 2.85 4
Table 8. The normalized-weighted fuzzy decision making matrix and
solution results (ARAS-F method)
Alternatives
Criterion [A.sub.0]
Ratings
[alpha] [gamma] [beta]
[x.sub.1] 0.0400 0.0689 0.1500
[x.sub.2] 0.0275 0.0709 0.1375
[x.sub.3] 0.0175 0.0433 0.1045
[x.sub.4] 0.0050 0.0162 0.0458
[x.sub.5] 0.0175 0.0569 0.1500
[x.sub.6] 0.0075 0.0253 0.0958
[x.sub.7] 0.0053 0.0101 0.0208
[x.sub.8] 0.0050 0.0105 0.0227
[[??].sub.i] 0.1253 0.3021 0.7273
[S.sub.i] 0.770
[K.sub.i] 1.000
Alternatives
Criterion [A.sub.1]
Ratings
[alpha] [gamma] [beta]
[x.sub.1] 0.0160 0.0454 0.1500
[x.sub.2] 0.0110 0.0440 0.1375
[x.sub.3] 0.0070 0.0290 0.1045
[x.sub.4] 0.0020 0.0118 0.0458
[x.sub.5] 0.0070 0.0375 0.1500
[x.sub.6] 0.0030 0.0177 0.0958
[x.sub.7] 0.0032 0.0077 0.0208
[x.sub.8] 0.0020 0.0071 0.0227
[[??].sub.i] 0.0512 0.2003 0.7273
[S.sub.i] 0.652
[K.sub.i] 0.848
Alternatives
Criterion [A.sub.2]
Ratings
[alpha] [gamma] [beta]
[x.sub.1] 0.0160 0.0503 0.1500
[x.sub.2] 0.0165 0.0504 0.1375
[x.sub.3] 0.0070 0.0286 0.1045
[x.sub.4] 0.0030 0.0118 0.0458
[x.sub.5] 0.0070 0.0358 0.1500
[x.sub.6] 0.0030 0.0185 0.0958
[x.sub.7] 0.0021 0.0067 0.0208
[x.sub.8] 0.0020 0.0062 0.0227
[[??].sub.i] 0.0566 0.2082 0.7273
[S.sub.i] 0.661
[K.sub.i] 0.859
Alternatives
Criterion [A.sub.3]
Ratings
[alpha] [gamma] [beta]
[x.sub.1] 0.0160 0.0454 0.1500
[x.sub.2] 0.0110 0.0447 0.1375
[x.sub.3] 0.0070 0.0290 0.1045
[x.sub.4] 0.0020 0.0102 0.0458
[x.sub.5] 0.0070 0.0398 0.1500
[x.sub.6] 0.0045 0.0185 0.0958
[x.sub.7] 0.0021 0.0055 0.0167
[x.sub.8] 0.0020 0.0062 0.0227
[[??].sub.i] 0.0516 0.1993 0.7231
[S.sub.i] 0.649
[K.sub.i] 0.844
Table 9. The solution results (WPM-F method)
Alternatives
Criterion [A.sub.0] [A.sub.1]
Ratings Ratings
[alpha] [gamma] [beta] [alpha] [gamma] [beta]
[x.sub.1] 1 1 1 0.4 0.66 1
[x.sub.2] 1 1 1 0.4 0.62 1
[x.sub.3] 1 1 1 0.4 0.67 1
[x.sub.4] 1 1 1 0.4 0.73 1
[x.sub.5] 1 1 1 0.4 0.66 1
[x.sub.6] 1 1 1 0.4 0.7 1
[x.sub.7] 1 1 1 0.6 0.76 1
[x.sub.8] 1 1 1 0.4 0.67 1
Alternatives
Criterion [A.sub.2] [A.sub.3]
Ratings Ratings
[alpha] [gamma] [beta] [alpha] [gamma] [beta]
[x.sub.1] 0.4 0.73 1 0.4 0.66 1
[x.sub.2] 0.6 0.71 1 0.4 0.63 1
[x.sub.3] 0.4 0.66 1 0.4 0.67 1
[x.sub.4] 0.6 0.73 1 0.4 0.63 1
[x.sub.5] 0.4 0.63 1 0.4 0.7 1
[x.sub.6] 0.4 0.73 1 0.6 0.73 1
[x.sub.7] 0.4 0.66 1 0.4 0.54 0.8
[x.sub.8] 0.4 0.59 1 0.4 0.59 1
Aggregate weight
[[??].sub.j]
[w.sub.j[alpha]] [w.sub.j[gamma]] [w.sub.j[beta]]
[x.sub.1] 0.16 0.21 0.33
[x.sub.2] 0.11 0.21 0.33
[x.sub.3] 0.07 0.13 0.23
[x.sub.4] 0.02 0.05 0.11
[x.sub.5] 0.07 0.17 0.33
[x.sub.6] 0.03 0.08 0.23
[x.sub.7] 0.02 0.03 0.05
[x.sub.8] 0.02 0.03 0.05
Exponential decision making matrix
[alpha] [gamma] [beta] [alpha] [gamma] [beta]
[x.sub.1] 1.000 1.000 1.000 0.864 0.916 1.000
[x.sub.2] 1.000 1.000 1.000 0.904 0.904 1.000
[x.sub.3] 1.000 1.000 1.000 0.938 0.949 1.000
[x.sub.4] 1.000 1.000 1.000 0.982 0.984 1.000
[x.sub.5] 1.000 1.000 1.000 0.938 0.932 1.000
[x.sub.6] 1.000 1.000 1.000 0.973 0.972 1.000
[x.sub.7] 1.000 1.000 1.000 0.990 0.992 1.000
[x.sub.8] 1.000 1.000 1.000 0.982 0.988 1.000
[PI] 1.000 1.000 1.000 0.638 0.687 1.000
[PI] 1.000 1.000 1.000 1.000 1.455 1.568
([A.sub.0])
/[PI]
([A.sub.i])
Defuzzified 1.000 1.341
values
[PI] 1.000 0.746
([A.sub.0])
/[PI]
([A.sub.i])
Ranks of 0 2
alternatives
Exponential decision making matrix
[alpha] [gamma] [beta] [alpha] [gamma] [beta]
[x.sub.1] 0.864 0.936 1.000 0.864 0.916 1.000
[x.sub.2] 0.945 0.931 1.000 0.904 0.908 1.000
[x.sub.3] 0.938 0.947 1.000 0.938 0.949 1.000
[x.sub.4] 0.990 0.984 1.000 0.982 0.977 1.000
[x.sub.5] 0.938 0.924 1.000 0.938 0.941 1.000
[x.sub.6] 0.973 0.975 1.000 0.985 0.975 1.000
[x.sub.7] 0.982 0.988 1.000 0.982 0.982 0.989
[x.sub.8] 0.982 0.984 1.000 0.982 0.984 1.000
[PI] 0.667 0.712 1.000 0.640 0.684 0.989
[PI] 1.000 1.405 1.500 1.011 1.462 1.562
([A.sub.0])
/[PI]
([A.sub.i])
Defuzzified 1.302 1.345
values
[PI] 0.768 0.744
([A.sub.0])
/[PI]
([A.sub.i])
Ranks of 1 3
alternatives
Fig. 4. Ranks of candidates
Alternatives
[A.sub.0] [A.sub.1] [A.sub.2] [A.sub.3]
[PI}([A.sub,i]) 1 0.746 0.768 0.744
/[PI}(A.sub.0]
K 1 0.848 0.859 0.844
Note: Table made from bar graph.
