Integrated fuzzy multiple criteria decision making model for architect selection/Integruotas neraiskusis daugiatikslis sprendimu priemimo modelis architektui atrinkti.

Kersuliene, Violeta ; Turskis, Zenonas

1. Introduction

The quality of human capital is crucial for high-tech companies to maintain competitive advantages in era of knowledge economy (Chien, Chen 2008). Human resources are one of the core competences for an organization 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 the best way (Dursun, Karsak 2010). It is one of the most important fields in human resources management, which directs company's present and future.

Among the functions of human resource management, personnel selection significantly affects the character of employees and quality of administration (Lin 2010). The personnel selection problem generally concerns with important and complex issues such as (Lin 2010):

--how to properly set the importance weights of criteria;

--how to use linguistic and numerical scales to evaluate the applicants under multiple criteria;

--and how to aggregate the evaluation results and then rank the applicants.

An effective personnel selection method should be able to assist the organization in selecting an appropriate person for a given work. The fuzzy set appears as an essential tool to provide a decision framework that incorporates imprecise judgements inherit in the personnel selection process (Dursun, Karsak 2010).

Many studies have been conducted to help organizations make effective selection decisions. Further applications of effective techniques in the personnel selection field are being developed. Kelemenis et al. (2011) presented an overview of recent studies on the personnel selection problem (from 1992 till 2009 year). They pointed that different techniques and conceptual models are used. For instance, fuzzy numbers, OWA operators, AHP, fuzzy analytic hierarchy process (AHP), analytic network process (ANP), fuzzy TOPSIS, fuzzy multiples objective programming, discriminant analysis, decision trees, analytic neural networks, total sum (TS) method, simple additive weighting (SAW), weighted product (WP) method, expert systems, neuro-fuzzy techniques, group TOPSIS, nominal group technique etc. are used.

Data mining involves various techniques including statistics, neural networks, decision tree, genetic algorithm, and visualization techniques that have been developed over the years. In the literature, there are a number of studies that have been conducted on resumes, interviews, assessment centers, job knowledge tests, work sample tests, cognitive tests, and personality tests in human resource management to help organizations make better personnel selection decisions, while only a few of them use MCDM techniques (Dursun, Karsak 2010).

Liang and Wang (1994) developed an algorithm for personnel selection. They presented a decision support tool for personnel selection using an integrated ANP and fuzzy DEA approach to effectively deal with the personnel selection problem. The method first aggregates decision-makers' linguistic assessments about subjective criteria weightings and ratings to obtain the fuzzy suitability index and its ranking value. Further, combining the subjective and objective ranking values, the final ranking values for personnel suitability evaluation are obtained.

Ling (2003) developed the model for the selection of architects from four theories: Theory of Job Performance, Theory of Contextual Performance, Network Theory of Embeddedness, and Theory of Firm. He described a problem by 40 attributes.

Chen and Cheng (2005) proposed an approach to rank fuzzy numbers by metric distance. The paper also developed a fuzzy computer-based group decision support system.

Chien and Chen (2008) developed a data mining framework based on decision tree and association rules to generate relationships between personnel profile data and their work behavior.

Huang et al. (2009) proposed a systematic approach with a feedback mechanism in which interrelations among positions and the differences among the selected employees are considered simultaneously. A fuzzy bi-objective binary integer programming model is formulated to solve a bi-objective personnel assignment problem.

Celik et al. (2009) proposed a fuzzy integrated multi-stage evaluation model under multiple criteria in order to manage the academic personnel selection and development processes. The model is based on Fuzzy AHP, Buckley's algorithm, fuzzy TOPSIS and SWOT.

Chen et al. (2010) presented a mechanism for partner selection that emphasizes the relation of criteria and motivation. AHP with fuzzy weighting and linguistic measurement is applied.

Dursun and Karsak (2010) applied fuzzy TOPSIS method with 2-tuple linguistic representation of criteria values.

Lin (2010) developed a decision support tool using an integrated analytic network process (ANP) and fuzzy data envelopment analysis (DEA) approach.

Kelemenis and Askounis (2010) presented a TOPSIS-based multi-criteria approach to personnel selection. This is based on the veto threshold, a critical characteristic of the main outranking methods. The ultimate decision criterion is not the similarity to the ideal solution but the distance of the alternatives from the veto set by the decision makers.

Lin et al. (2010) presented a hybrid particle swarm optimization model which utilizes random-key encoding and individual enhancement schemes.

Azadeh et al. (2011) applied an integrated Data Envelopment Analysis-Artificial Neural Network-Rough Set Algorithm for assessment of personnel efficiency.

Greco et al. (2011) introduced the concept of a representative value function in robust ordinal regression applied to multiple criteria sorting problems. The proposed approach can be seen as an extension of UTADISGMS, a new multiple criteria sorting method that aims at assigning actions to p pre-defined and ordered classes. This approach is applied to assess managers.

Shahhosseini and Sebt (2011) presented a fuzzy adaptive model to select the most competent construction personnel. The model is based on fuzzy AHP method.

Van Iddekinge et al. (2011) reconsidered some widely held beliefs concerning the (low) validity of interests for predicting criteria important to selection personnel, and reviewed theory and empirical evidence that challenge such beliefs. Then they described the development and validation of an interest-based selection measure.

Zhang and Liu (2011) proposed an intuitionistic fuzzy multi-criteria group decision making method with grey relational analysis. Intuitionistic fuzzy entropy is used to obtain the entropy weights of the criteria.

Each of aforementioned models does not present parity between each of considered alternatives with optimum alternative. The majority of the existing approaches require involved complex computations. The objective of this study is to develop a decision making approach to a multiple information sources problem, which enables to incorporate both crisp data and fuzzy data represented as linguistic variables or triangular fuzzy numbers into the analysis.

ARAS, which is a newly developed multi-attribute decision making technique, is based on the intuitive principle that the preferred alternative should have the biggest ratio to the optimal solution (Zavadskas, Turskis 2010).

The significance of the model is that it reduces the time taken by project managers to accumulate experience in architect selection, further increasing the efficiency of the construction industry.

2. Selection algorithm based on the fuzzy sets and multiple criteria decision making methods

[FIGURE 1 OMITTED]

There are a lot of different multiple criteria decision making methods. The selection of appropriate decision method depends on the aim of the problem, available information, costs of decision and actors' (persons which are making decisions) qualification. A wider overview of multiple criteria decision making methods, classification and applications are presented by Zavadskas and Turskis (2011). In this research two of them are applied: ARAS-F and SWARA. The multiple criteria expert system for problem solution can be described as shown in Fig. 1.

In marketing research and particularly in the context of customer satisfaction measurement it is often attempted to measure attitudes and human perceptions. This raises a number of questions regarding appropriate scales to use, such as the number of response alternatives.

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.

2.1. Basic definitions

Fuzzy multiple criteria analysis concerns about selecting or prioritizing alternatives with respect to multiple, usually conflicting criteria in a fuzzy environment (Deng 2009). There are many misconceptions about fuzzy logic. To begin with, fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision and approximate reasoning (Zadeh 1975d, 1979). The real-world is pervaded with fuzziness. Fuzzy logic is needed to deal effectively with fuzzy reality. More specifically, fuzzy logic may be viewed as an attempt at formalization/mechanization of two remarkable human capabilities (Zadeh 2008). By decision-making in a fuzzy environment a decision process is meant, in which the goals and/or the constraints, but not necessarily the system under control, are fuzzy in nature. This means that the goals and/or the constraints constitute classes of alternatives whose boundaries are not sharply defined. The task of developing a general theory of decision making in a fuzzy environment is one of very considerable magnitude and complexity (Bellman, Zadeh 1970). Fuzzy goals and fuzzy constraints can be defined precisely as fuzzy sets in the space of alternatives. Fuzzy set theory, which was introduced by Zadeh (1975a, b, c) to deal with problems in which a source of vagueness is involved, has been utilized for incorporating imprecise data into the decision framework. Deng (2009) presented an overview of the development in fuzzy multiple criteria analysis. Zavadskas and Turskis (2011) presented an overview of multiple criteria decision making methods in economics.

Classification of the most of fuzzy multiple criteria decision making methods in the literature is presented by Olcer and Odabasi (2005).

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 ([alpha], [gamma], [beta]) as illustrated in Fig. 2.

In most cases, the classes of objects encountered in the real physical world do not have precisely defined criteria of membership. A fuzzy set is a class of objects with a continuum of membership grades. Such set is characterized by a membership function which assigns to each object a grade of membership ranging between zero and one (Zadeh 1975a, b, c). A fuzzy set A defined in space X is a set of pairs:

A = j{(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; l] which associates with each element x [member of] X, a real number [[mu].sub.A](x) [member of] [0; 1]. 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 and can be written [subset or equal to] X. Sometimes a fuzzy set A in X is denoted by the list of ordered pairs (x, [[mu].sub.a](x)), where the elements with zero degree are usually not listed. Thus 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;1].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

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, [beta], [gamma]) if its membership function is fully described by three parameters (a < [beta] < [gamma]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

The most typical fuzzy set membership function is triangular membership function (Fig. 2).

The basic operations of fuzzy triangular numbers [[??].sub.1] and [[??].sub.2] (van Laarhoven, Pedrycz 1983) are defined as follows:

[[??].sub.1] [cross product] [[??].sub.2] =([n.sub.1[alpha]] + [n.sub.2[alpha]], [n.sub.1[gamma]] + [n.sub.2[gamma]], [n.sub.1[beta]] + [n.sub.2[beta])] addition, (6)

[[??].sub.1] (-) [[??].sub.2] = ([n.sub.1[alpha]] - [n.sub.2[beta]], [n.sub.1[gamma]] - [n.sub.2[gamma]], [n.sub.1[beta]] - [n.sub.2[alpha]]) substraction, (7)

[[??].sub.1] [cross product] [[??].sub.2] = ([n.sub.1[alpha]] [cross product] [n.sub.2[alpha]], [n.sub.1[gamma]] [cross product] n2[gamma], [n.sub.1[beta]] [cross product] [n.sub.2[beta]]) multiplication, (8)

[[??].sub.1] (/) ([[??].sub.2] = [[n.sub.1[alpha]]/[n.sub.2[beta]], [[n.sub.1[gamma]]/[n.sub.2[gamma]]], [[n.sub.1[beta]]/[n.sub.2[alpha]]]) division, (9)

[k[??].sub.1] = ([kn.sub.1[alpha]], [kn.sub.1[beta]], [kn.sub.1[gamma]]) multiplication of any real number k and a fuzzy number, (10)

[([[??].sup.1]).sup.-1] = (1/[n.sub.1[beta]], 1/[n.sub.1[gamma]], 1/[n.sub.1[alpha]]) inverse of triangular fuzzy number. (11)

Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is fuzzy group 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]]). (12)

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.

[FIGURE 2 OMITTED]

In order to obtain a crisp output, a defuzzification process is needed to be applied. Defuzzification is the process of producing a quantifiable result in fuzzy logic, given fuzzy sets and corresponding membership degrees. The output of the defuzzification process is a single number. Many defuzzification techniques have been proposed in the literature.

2.2. 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 (Zavadskas, Turskis 2010).

Aras method was developed by Zavadskas and Turskis (2010). Later, modifications of ARAS method, such as ARAS-G (grey relations are applied) and ARAS-F, were published (Turskis, Zavadskas 2010a, b). There are only a few applications of ARAS method (Tupenaite et al. 2010; Zavadskas et al. 2010b; Bakshi, Sarkar 2011).

ARAS method (Zavadskas, Turskis 2010) is based on the argument that the phenomena of complicated world could be understood by using simple relative comparisons. It is argued that the ratio of the sum of normalized and weighted criteria scores, which describe alternative under consideration, to the sum of the values of normalized and weighted criteria, which describes the optimal alternative, is degree of optimality, which is reached by the alternative under comparison.

According to the ARAS method (Zavadskas, Turskis 2010), 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 fuzzy decision-making matrix (FDMM) forming. In the FMCDM of the discrete optimization problem any problem which has to be solved is represented by the following DMM of preferences for m reasonable alternatives (rows) rated on n criteria (columns):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

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]. (14)

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 information can be corrected by the interested parties, taking into account their goals and opportunities.

Then the determination of the priorities of alternatives is carried out in several stages.

Usually, the criteria have different dimensions. The purpose of the next stage is to receive dimensionless weighted values from the comparative criteria. In order to avoid the difficulties caused by different dimensions of the criteria, the ratio to the optimal value is used. There are various theories describing the ratio to the optimal value. However, the values are mapped either on the interval [0;1] or the interval [0; [infinity]) by applying the normalization of a DMM.

In the second stage the initial values of all the criteria are normalized--defining values [[??].sub.ij] of normalised decision-making matrix [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

The criteria, whose preferable values are maxima, are normalized as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

The criteria, whose preferable values are minima, are normalized by applying two-stage procedure:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

When the dimensionless values of the criteria are known, all the criteria, originally having different dimensions, can be compared.

The third stage is defining normalized-weighted matrix--[??]. It is possible to evaluate the criteria with weights 0 < [[??].sub.j] < 1. Only well-founded weights should be used because weights are always subjective and influence the solution. 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. (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Normalized-weighted values of all the criteria are calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

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 task is determining values of optimality function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

where [[??].sub.i] is the value of optimality function of i-th alternative.

The biggest value is the best, and the smallest one is the worst. Taking into account the calculation process, the optimality function [[??].sub.i] has a direct and proportional relationship with the values [[??].sub.ij] and weights [[??].sub.j] of the investigated criteria and their relative influence on the final result. Therefore, the greater the value of the optimality function [[??].sub.i], the more effective the alternative. The priorities of alternatives can be determined according to the value [[??].sub.i]. Consequently, it is convenient to evaluate and rank decision alternatives when this method is used.

The result of fuzzy decision making for each alternative is fuzzy number [[??].sub.i]. There are several methods for defuzzification. The centre-of-area is the most practical and simple to apply to:

[S.sub.i] = 1/3([S.sub.i[alpha]] + [S.sub.i[beta]] + [S.sub.i[gamma]]). (22)

The degree of the alternative utility is determined by a comparison of the variant, which is analysed, with the most ideal [S.sub.0]. The equation used for the calculation of the utility degree [K.sub.i] of an alternative [A.sub.i]. is given below:

[K.sub.i] = [S.sub.i]/[S.sub.0]; i = [bar.0, m], (23)

where [S.sub.i] and [S.sub.0] are the optimal criterion values, obtained from Eq. (22).

It is clear, that the calculated values [K.sub.i] are in the interval [0;1] and can be ordered in an increasing sequence, which is the wanted order of precedence. The complex relative efficiency of the reasonable alternative can be determined according to the utility function values.

2.3. Criteria weights determination

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).

There are various approaches for assessing weights (Zavadskas et al. 2010a, b), e.g. the eigenvector method, SWARA (Kersuliene et al. 2010), expert method (Zavadskas, Vilutiene 2006), analytic hierarchy process (AHP) (Saaty 1977, 1980), Entropy method, etc.

Each of experts first of all ranks criteria. The most significant criterion is given rank 1, and the least significant criterion is given rank 8. The overall ranks to the group of experts are determined according to the mediocre value of ranks.

Later, SWARA method is applied to determine fuzzy group weights of criteria.

The step-wise weight assessment ratio analysis (SWARA) (Kersuliene et al. 2010) methodology is developed in 2010 and applied for the selection of rational dispute resolution method. The new procedure for the criteria weights determination can be described as is presented in Fig. 2.

However, according to the above mentioned methods the attribute weights cannot be valued as one weight of attribute is higher/lower significant than the other attribute, because attributes are ranked according to preferences of expert decision-making. The SWARA procedure for the attributes weights determination which provides the opportunity to estimate the differences of their significances can be described as presented in Fig. 3.

This method allows including experts opinion about significance ratio of the criteria in the process of rational decision determination.

[FIGURE 3 OMITTED]

3. Architect selection using fuzzy MCDM approach

Each personnel selection problem is individual and needs own criteria set. For architect's assessment 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 goal, problem solving ability, decision making skills, strategic thinking, ability to sell oneself and ideas, interpersonal skills, management experience, emotional steadiness, communication skills, ability of good discussion, personality assessment, computer skills, self-confidence, fluency in foreign languages, responsibility, patience, effective time using, and age.

Ling (2003) presented a conceptual model for selection of architects by project managers. He determined four main criteria groups to select an architect:

1. Task performance criterion, which includes: creativity; innovative decisions; strategic decision making; knowledge of economics; knowledge of construction and construction technologies; knowledge of design legislation system, regulations and requirements; accuracy of work and good design skills in detailing; good knowledge of contracting and job experience.

2. Contextual performance criterion includes: speed of generating and preparing drawings of a design; close attention to the essential details of the project; fair done specifications of the project; innovative ideas how to improve design; ability to satisfy clients requirements in a proper way both to the company and client; ability to follow the client's and project manager's instructions and orders; ability to work independently; ability to revise project's quality and achieving determined goals; leadership; ability to control subcontractors and staff.

3. Network criterion: reputation; ability to work in the team; prior work with consultants and clients.

4. Price criterion: low fee; architect allows the client to delay payments of professional fees.

The problem's set of criteria was determined by three decision makers (owners) of the designing firm as follows:

[x.sub.1]--Working knowledge, working skills, work experience, knowledge of design process and legislation system;

[x.sub.2]--Education, academic level, long life learning;

[x.sub.3]--Ability to revise project's quality and achieving determined goals; leadership; ability to work in team; ability to control subcontractors and staff;

[x.sub.4]--Creativity and strategic decision making;

[x.sub.5]--Ability to satisfy client's and project manager's requirements;

[x.sub.6]--Ability to work with clients, consultants and community;

[x.sub.7]--Culture and communication skills;

[x.sub.8]--Responsibility and ability in detailing of the project.

At the first stage of problem solution three decision makers determined criteria ranks by simple ranking.

The criteria ranks are determined according to the ranks as is shown in Table 1.

At the second stage SWARA method was applied. The decision makers prepared Table 2, Table 3 and Table 4.

Calculation results are shown in tables. The experts were allowed to determine criteria weights according to the group ranks which are established in Table 1. For instance, criterion [x.sub.6] must be evaluated as the least significant, or, at least, to be equally significant as criterion [x.sub.7], criterion [x.sub.3] must be evaluated as the most significant, or at least, to be equally significant as criterion [x.sub.2].

According to the calculations by applying SWARA method, fuzzy group criteria weights were established as is shown in Table 5.

As mentioned, the main feature of SWARA method is the possibility to estimate experts or interest groups opinion about significance ratio of the criteria in the process of their weights determination.

In this study, the eleven linguistic term set with associated semantic is considered (Table 6 and Fig. 4).

[FIGURE 4 OMITTED]

The candidates were rated. Data related to architect selection problem are given in Table 8.

According to the Table 6, Table 7 and Fig. 4, it is a prepared matrix with fuzzy group criteria values (Table 8) and fuzzy decision making matrix with fuzzy group weights (Table 9).

Normalized fuzzy decision making matrix is presented in Table 10. Solution results are presented in Table 11.

The best candidate from available and feasible is the second architect. He was selected by decision makers.

4. Conclusions

In the era of competitive markets, appropriate selection of personnel determines success of organizations. In this paper a sequential decision making process in group, where preferences of actors are presented by linguistic preference relations is given. The proposed model helps to overcome difficulties in personnel selection process. This allows to find consensus under a linguistic assessment approach and to cooperate in the solution finding of the group decision problem. The values of criteria set describing candidates in most cases are lexical values. The fuzzy set theory is a proper way to deal with uncertainty. It can be stated that the ratio with an optimal alternative may be used in cases when it is seeking to rank alternatives and find ways to improve them. The presented case study shows that this model successfully could help in cases when actors need to select among feasible candidates.

http://dx.doi.org/10.3846/20294913.2011.635718

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Violeta Kersuliene (1), Zenonas Turskis (2)

Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

(1) Office of Legal Affairs, (2) Faculty of Civil Engineering

E-mails: (1) [email protected] (corresponding author); (2) [email protected]

Received 9 May 2011; accepted 7 October 2011

Violeta KERSULIENE has a PhD and is a Director of Legal Affairs Dept. at Vilnius Gediminas Technical University, Lithuania. Her research interests include building technology and management, decision-making theory, computer-aided automation in design and expert systems. She is the author of more than 10 research papers.

Zenonas TURSKIS has a PhD and is a chief research worker at Laboratory of Construction Technology and Management at Vilnius Gediminas Technical University, Lithuania. His research interests include building technology and management, decision-making theory, computer-aided automation in design and expert systems. He is the author of more than 80 research papers.

Table 1. Average criteria ranks

                    Ranks of criteria

Criteria     Expert 1   Expert 2   Expert 3   [MATHEMATICAL    Group
                                              EXPRESSION NOT   rank
                                               REPRODUCIBLE
                                                IN ASCII]

[x.sub.1]       1          1          4            4.04          3
[x.sub.2]       2          3          3            3.96          2
[x.sub.3]       3          2          1            3.30          1
[x.sub.4]       6          6          5            8.23          6
[x.sub.5]       5          4          2            5.82          4
[x.sub.6]       7          8          8           11.10          8
[x.sub.7]       8          7          7           10.62          7
[x.sub.8]       4          5          6            7.40          5

n--number of criteria; k--number of experts

Table 2. Criteria describing candidates and their weights determined
by applying SWARA method (Expert 1)

             Criteria                  Comparative      Coefficient
                                      importance of     [k.sub.j] =
                                      average value    [s.sub.j] + 1
                                        [s.sub.j]

Expert 1

Ability to revise project's           0.00             1
quality and achieving determined
goals; leadership; ability to
work in team; ability to control
sub-contractors and staff--
[x.sub.3]

Education, academic level, long       0.33             1
life learning--[x.sub.2]

Working knowledge, working            0.00             1.33
skills, work experience, knowledge
of design process and legislation
system--[x.sub.1]

Creativity and strategic decision     0.33             1
making--[x.sub.4]

Ability to satisfy client's and       0.40             1.33
project manager's requirements--
[x.sub.5]

Ability to work with clients,         0.40             1.40
consultants and community--
[x.sub.6]

Culture and communication             0.40             1.40
skills--[x.sub.7]

Responsibility and ability in                          1.40
detailing of the project--
[x.sub.8]

                                                        Weight
             Criteria                  Recalculated     [w.sub.j] =
                                          weight         [g.sub.j]/
                                       [q.sub.j] =     [[summation].
                                       [x.sub.j-1]/    sup.n.sub.j=1]
                                        [k.sub.j]         [q.sub.j

Expert 1

Ability to revise project's           1                0.201
quality and achieving determined
goals; leadership; ability to
work in team; ability to control
sub-contractors and staff--
[x.sub.3]

Education, academic level, long       1                0.201
life learning--[x.sub.2]

Working knowledge, working            0.752            0.151
skills, work experience, knowledge
of design process and legislation
system--[x.sub.1]

Creativity and strategic decision     0.752            0.151
making--[x.sub.4]

Ability to satisfy client's and       0.565            0.114
project manager's requirements--
[x.sub.5]

Ability to work with clients,         0.404            0.081
consultants and community--
[x.sub.6]

Culture and communication             0.288            0.058
skills--[x.sub.7]

Responsibility and ability in         0.206            0.041
detailing of the project--
[x.sub.8]

Table 3. Criteria describing candidates and their weights determined
by applying SWARA method (Expert 2)

              Criteria                  Comparative      Coefficient
                                       importance of     [k.sub.j] =
                                       average value    [s.sub.j] + 1
                                         [s.sub.j]

Expert 2

Ability to revise project's            0.00             1
quality and achieving determined
goals; leadership; ability to
work in team; ability to control
sub-contractors and staff--
[x.sub.3]

Education, academic level, long        0.50             1
life learning--[x.sub.2]

Working knowledge, working             0.70             1.50
skills, work experience, knowledge
of design process and legislation
system--[x.sub.1]

Creativity and strategic decision      0.00             1.70
making--[x.sub.4]

Ability to satisfy client's and
project manager's requirements--       0.70             1.00
[x.sub.5]

Ability to work with clients,
consultants and community--            0.00             1.70
[x.sub.6]

Culture and communication              0.50             1.00
skills--[x.sub.7]

Responsibility and ability in                           1.50
detailing of the project--
[x.sub.8]

                                                            Weight
              Criteria                  Recalculated     [w.sub.j] =
                                           weight         [q.sub.j]
                                        [q.sub.j] =     [[summation].
                                        [x.sub.j-1]/    sup.n.sub.j=1]
                                         [k.sub.j]         [q.sub.j

Expert 2

Ability to revise project's            1                0.246
quality and achieving determined
goals; leadership; ability to
work in team; ability to control
sub-contractors and staff--
[x.sub.3]

Education, academic level, long        1                0.246
life learning--[x.sub.2]

Working knowledge, working             0.667            0.164
skills, work experience, knowledge
of design process and legislation
system--[x.sub.1]

Creativity and strategic decision      0.392            0.096
making--[x.sub.4]

Ability to satisfy client's and
project manager's requirements--       0.392            0.096
[x.sub.5]

Ability to work with clients,
consultants and community--            0.231            0.057
[x.sub.6]

Culture and communication              0.231            0.057
skills--[x.sub.7]

Responsibility and ability in          0.154            0.038
detailing of the project--
[x.sub.8]

Table 4. Criteria describing candidates and their weights determined
by applying SWARA method (Expert 3)

              Criteria                  Comparative      Coefficient
                                       importance of     [k.sub.j] =
                                       average value    [s.sub.j] + 1
                                         [s.sub.j]

Expert 3

Ability to revise project's
quality and achieving determined       0.00             1
goals; leadership; ability to
work in team; ability to control
sub-contractors and staff--
[x.sub.3]

Education, academic level, long        0.00             1
life learning--[x.sub.2]

Working knowledge, working
skills, work experience, knowledge     0.70             1
of design process and legislation
system--[x.sub.1]

Creativity and strategic decision      0.00             1.7
making--[x.sub.4]

Ability to satisfy client's and        0.70             1
project manager's requirements--
[x.sub.5]

Ability to work with clients,          0.00             1.70
consultants and community--
[x.sub.6]

Culture and communication              0.00             1
skills--[x.sub.7]

Responsibility and ability in                           1
detailing of the project--
[x.sub.8]

                                                          Weight
              Criteria                  Recalculated     [w.sub.j] =
                                           weight         [q.sub.j]
                                        [q.sub.j] =     [[summation].
                                        [x.sub.j-1]/    sup.n.sub.j=1]
                                         [k.sub.j]         [q.sub.j

Expert 3

Ability to revise project's
quality and achieving determined       1                0.192
goals; leadership; ability to
work in team; ability to control
sub-contractors and staff--
[x.sub.3]

Education, academic level, long        1                0.192
life learning--[x.sub.2]

Working knowledge, working
skills, work experience, knowledge     1                0.192
of design process and legislation
system--[x.sub.1]

Creativity and strategic decision      0.588            0.113
making--[x.sub.4]

Ability to satisfy client's and        0.588            0.113
project manager's requirements--
[x.sub.5]

Ability to work with clients,          0.346            0.066
consultants and community--
[x.sub.6]

Culture and communication              0.346            0.066
skills--[x.sub.7]

Responsibility and ability in          0.346            0.066
detailing of the project--
[x.sub.8]

Table 5. Fuzzy group criteria weights

                             Criteria weights

                Expert 1         Expert 2         Expert 3

[x.sub.1]        0.151            0.192            0.164
[x.sub.2]        0.201            0.192            0.246
[x.sub.3]        0.201            0.192            0.246
[x.sub.4]        0.151            0.113            0.096
[x.sub.5]        0.114            0.113            0.096
[x.sub.6]        0.081            0.066            0.057
[x.sub.7]        0.058            0.066            0.057
[x.sub.8]        0.041            0.346            0.038

                           Fuzzy group criteria weights

             [w.sub.j[alpha]]   [w.sub.j[gamma]]   [w.sub.j[beta]]

[x.sub.1]         0.151              0.246              0.192
[x.sub.2]         0.192              0.311              0.246
[x.sub.3]         0.192              0.311              0.246
[x.sub.4]         0.096              0.179              0.151
[x.sub.5]         0.096              0.156              0.114
[x.sub.6]         0.057              0.100              0.081
[x.sub.7]         0.057              0.087              0.066
[x.sub.8]         0.038              0.346              0.346

Table 6. Label set

Label        Linguistic term                Fuzzy number
set
                                      [alpha]   [gamma]   [beta]

[s.sub.0]    Absent                   0         0         0.1

[s.sub.1]    Nothing answered, task   0         0.1       0.2
             was not completed

[s.sub.2]    Very bad                 0.1       0.2       0.3

[s.sub.3]    Bad                      0.2       0.3       0.4

[s.sub.4]    Weak                     0.3       0.4       0.5

[s.sub.5]    Satisfactory enough      0.4       0.5       0.6

[s.sub.6]    Satisfactory             0.5       0.6       0.7

[s.sub.7]    Good enough              0.6       0.7       0.8

[s.sub.8]    Good                     0.7       0.8       0.9

[s.sub.9]    Very good                0.8       0.9       1.0

[s.sub.10]   Excellent                0.9       1.0       1.0

Table 7. Rating of the 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.9]    [s.sub.8]    [s.sub.5]
             [A.sub.2]    [s.sub.8]    [s.sub.6]    [s.sub.8]
             [A.sub.3]    [s.sub.8]    [s.sub.9]    [s.sub.5]

[x.sub.2]    [A.sub.1]    [s.sub.6]    [s.sub.7]    [s.sub.7]
             [A.sub.2]    [s.sub.5]    [s.sub.9]    [s.sub.8]
             [A.sub.3]    [s.sub.8]    [s.sub.9]    [s.sub.5]

[x.sub.3]    [A.sub.1]    [s.sub.5]    [s.sub.8]    [s.sub.6]
             [A.sub.2]    [s.sub.8]    [s.sub.9]    [s.sub.5]
             [A.sub.3]    [s.sub.9]    [s.sub.8]    [s.sub.8]

[x.sub.4]    [A.sub.1]    [s.sub.8]    [s.sub.9]    [s.sub.6]
             [A.sub.2]    [s.sub.5]    [s.sub.8]    [s.sub.5]
             [A.sub.3]    [s.sub.5]    [s.sub.5]    [s.sub.5]

[x.sub.5]    [A.sub.1]    [s.sub.8]    [s.sub.8]    [s.sub.5]
             [A.sub.2]    [s.sub.6]    [s.sub.5]    [s.sub.8]
             [A.sub.3]    [s.sub.8]    [s.sub.6]    [s.sub.8]

[x.sub.6]    [A.sub.1]    [s.sub.9]    [s.sub.9]    [s.sub.5]
             [A.sub.2]    [s.sub.8]    [s.sub.9]    [s.sub.8]
             [A.sub.3]    [s.sub.8]    [s.sub.8]    [s.sub.8]

[x.sub.7]    [A.sub.1]    [s.sub.8]    [s.sub.9]    [s.sub.6]
             [A.sub.2]    [s.sub.5]    [s.sub.8]    [s.sub.5]
             [A.sub.3]    [s.sub.5]    [s.sub.5]    [s.sub.5]

[x.sub.7]    [A.sub.1]    [s.sub.8]    [s.sub.8]    [s.sub.5]
             [A.sub.2]    [s.sub.6]    [s.sub.5]    [s.sub.8]
             [A.sub.3]    [s.sub.8]    [s.sub.6]    [s.sub.8]

Table 8. The fuzzy group criteria values

 Criterion    Candidates                   [D.sub.1]

                              [alpha]       [gamma]       [beta]

[x.sub.1]      [A.sub.1]        0.8           0.9           1.0
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.7           0.8           0.9

[x.sub.2]      [A.sub.1]        0.5           0.6           0.7
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.7           0.8           0.9

[x.sub.3]      [A.sub.1]        0.4           0.5           0.6
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.8           0.9           1.0

[x.sub.4]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.4           0.5           0.6

[x.sub.5]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.5           0.6           0.7
               [A.sub.3]        0.7           0.8           0.9

[x.sub.6]      [A.sub.1]        0.8           0.9           1.0
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.7           0.8           0.9

[x.sub.7]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.4           0.5           0.6

[x.sub.8]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.5           0.6           0.7
               [A.sub.3]        0.7           0.8           0.9

 Criterion    Candidates                   [D.sub.2]

                              [alpha]       [gamma]       [beta]

[x.sub.1]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.5           0.6           0.7
               [A.sub.3]        0.8           0.9           1.0

[x.sub.2]      [A.sub.1]        0.6           0.7           0.8
               [A.sub.2]        0.8           0.9           1.0
               [A.sub.3]        0.8           0.9           1.0

[x.sub.3]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.8           0.9           1.0
               [A.sub.3]        0.7           0.8           0.9

[x.sub.4]      [A.sub.1]        0.8           0.9           1.0
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.4           0.5           0.6

[x.sub.5]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.5           0.6           0.7

[x.sub.6]      [A.sub.1]        0.8           0.9           1.0
               [A.sub.2]        0.8           0.9           1.0
               [A.sub.3]        0.7           0.8           0.9

[x.sub.7]      [A.sub.1]        0.8           0.9           1.0
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.4           0.5           0.6

[x.sub.8]      [A.sub.1]        0.7           0.8           0.9
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.5           0.6           0.7

 Criterion    Candidates                   [D.sub.3]

                              [alpha]       [gamma]       [beta]

[x.sub.1]      [A.sub.1]        0.4           0.5           0.6
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.4           0.5           0.6

[x.sub.2]      [A.sub.1]        0.6           0.7           0.8
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.4           0.5           0.6

[x.sub.3]      [A.sub.1]        0.5           0.6           0.7
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.7           0.8           0.9

[x.sub.4]      [A.sub.1]        0.5           0.6           0.7
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.4           0.5           0.6

[x.sub.5]      [A.sub.1]        0.4           0.5           0.6
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.7           0.8           0.9

[x.sub.6]      [A.sub.1]        0.4           0.5           0.6
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.7           0.8           0.9

[x.sub.7]      [A.sub.1]        0.5           0.6           0.7
               [A.sub.2]        0.4           0.5           0.6
               [A.sub.3]        0.4           0.5           0.6

[x.sub.8]      [A.sub.1]        0.4           0.5           0.6
               [A.sub.2]        0.7           0.8           0.9
               [A.sub.3]        0.7           0.8           0.9

 Criterion    Candidates               Fuzzy group value

                              [alpha]       [gamma]       [beta]

[x.sub.1]      [A.sub.1]        0.4          0.73           1
               [A.sub.2]        0.5          0.73           0.9
               [A.sub.3]        0.4          0.73           1

[x.sub.2]      [A.sub.1]        0.5          0.67           0.8
               [A.sub.2]        0.4          0.73           1
               [A.sub.3]        0.4          0.73           1

[x.sub.3]      [A.sub.1]        0.4          0.63           0.9
               [A.sub.2]        0.4          0.73           1
               [A.sub.3]        0.7          0.83           1

[x.sub.4]      [A.sub.1]        0.5          0.77           1
               [A.sub.2]        0.4          0.60           0.9
               [A.sub.3]        0.4          0.50           0.6

[x.sub.5]      [A.sub.1]        0.4          0.70           0.9
               [A.sub.2]        0.4          0.63           0.9
               [A.sub.3]        0.5          0.73           0.9

[x.sub.6]      [A.sub.1]        0.4          0.77           1
               [A.sub.2]        0.7          0.83           1
               [A.sub.3]        0.7          0.80           0.9

[x.sub.7]      [A.sub.1]        0.5          0.77           1
               [A.sub.2]        0.4          0.60           0.9
               [A.sub.3]        0.4          0.50           0.6

[x.sub.8]      [A.sub.1]        0.4          0.70           0.9
               [A.sub.2]        0.4          0.63           0.9
               [A.sub.3]        0.5          0.73           0.9

Table 9. The fuzzy decision making matrix with fuzzy group weights
(all criteria should to be maximized and optimal value equals to
1.0)

                                Alternatives

Criterion      [A.sub.0]               [A.sub.1]

                Ratings                 Ratings

               [alpha];     [alpha]     [gamma]     [beta]
               [gamma];
                [beta]

[x.sub.l]         1.0         0.7         0.8         0.9
[x.sub.2]         1.0         0.7         0.8         0.9
[x.sub.3]         1.0         0.5         0.6         0.7
[x.sub.4]         1.0         0.4         0.5         0.6
[x.sub.5]         1.0         0.7         0.8         0.9
[x.sub.6]         1.0         0.4         0.5         0.6
[x.sub.7]         1.0         0.7         0.8         0.9
[x.sub.8]         1.0         0.8         0.9         1.0

                           Alternatives

Criterion                  [A.sub.2]

                            Ratings

                [alpha]     [gamma]     [beta]

[x.sub.l]         0.5         0.6         0.7
[x.sub.2]         0.8         0.9         1.0
[x.sub.3]         0.6         0.7         0.8
[x.sub.4]         0.8         0.9         1.0
[x.sub.5]         0.8         0.9         1.0
[x.sub.6]         0.7         0.8         0.9
[x.sub.7]         0.8         0.9         1.0
[x.sub.8]         0.7         0.8         0.9

                           Alternatives

Criterion                  [A.sub.3]

                            Ratings

                [alpha]     [gamma]     [beta]

[x.sub.l]         0.7         0.8         0.9
[x.sub.2]         0.4         0.5         0.6
[x.sub.3]         0.6         0.7         0.8
[x.sub.4]         0.7         0.8         0.9
[x.sub.5]         0.4         0.5         0.6
[x.sub.6]         0.5         0.6         0.7
[x.sub.7]         0.4         0.5         0.6
[x.sub.8]         0.7         0.8         0.9

                           Alternatives

Criterion                    Total

                [alpha]     [gamma]     [beta]

[x.sub.l]         2.9         3.2         3.5
[x.sub.2]         2.9         3.2         3.5
[x.sub.3]         2.7         3           3.3
[x.sub.4]         2.9         3.2         3.5
[x.sub.5]         2.9         3.2         3.5
[x.sub.6]         2.6         2.9         3.2
[x.sub.7]         2.9         3.2         3.5
[x.sub.8]         3.2         3.5         3.8

                                 Fuzzy group weight

Criterion                           [[??].sub.j]

               [w.sub.j[alpha]]   [w.sub.j[gamma]]   [w.sub.j[beta]]

[x.sub.l]           0.151              0.246              0.192
[x.sub.2]           0.192              0.311              0.246
[x.sub.3]           0.192              0.311              0.246
[x.sub.4]           0.096              0.179              0.151
[x.sub.5]           0.096              0.156              0.114
[x.sub.6]           0.057              0.100              0.081
[x.sub.7]           0.057              0.087              0.066
[x.sub.8]           0.038              0.346              0.346

Table 10. The fuzzy normalized decision making matrix

                                  Alternatives

Criterion                          [A.sub.0]

                                    Ratings

                 [alpha]             [beta]            [gamma]

[x.sub.1]         0.2857             0.3125             0.3448
[x.sub.2]         0.2857             0.3125             0.3448
[x.sub.3]         0.3030             0.3333             0.3704
[x.sub.4]         0.2857             0.3125             0.3448
[x.sub.5]         0.2857             0.3125             0.3448
[x.sub.6]         0.3125             0.3448             0.3846
[x.sub.7]         0.2857             0.3125             0.3448
[x.sub.8]         0.2632             0.2857             0.3125

                                  Alternatives

Criterion                          [A.sub.1]

                                    Ratings

                 [alpha]             [beta]            [gamma]

[x.sub.1]         0.2000             0.2500             0.3103
[x.sub.2]         0.2000             0.2500             0.3103
[x.sub.3]         0.1515             0.2000             0.2593
[x.sub.4]         0.1143             0.1563             0.2069
[x.sub.5]         0.2000             0.2500             0.3103
[x.sub.6]         0.1250             0.1724             0.2308
[x.sub.7]         0.2000             0.2500             0.3103
[x.sub.8]         0.2105             0.2571             0.3125

                                  Alternatives

Criterion                          [A.sub.2]

                                    Ratings

                 [alpha]             [beta]            [gamma]

[x.sub.1]         0.1429             0.1875             0.2414
[x.sub.2]         0.2286             0.2813             0.3448
[x.sub.3]         0.1818             0.2333             0.2963
[x.sub.4]         0.2286             0.2813             0.3448
[x.sub.5]         0.2286             0.2813             0.3448
[x.sub.6]         0.2188             0.2759             0.3462
[x.sub.7]         0.2286             0.2813             0.3448
[x.sub.8]         0.1842             0.2286             0.2813

                                  Alternatives

Criterion                          [A.sub.3]

                                    Ratings

                 [alpha]             [beta]            [gamma]

[x.sub.1]         0.2000             0.2500             0.3103
[x.sub.2]         0.1143             0.1563             0.2069
[x.sub.3]         0.1818             0.2333             0.2963
[x.sub.4]         0.2000             0.2500             0.3103
[x.sub.5]         0.1143             0.1563             0.2069
[x.sub.6]         0.1563             0.2069             0.2692
[x.sub.7]         0.1143             0.1563             0.2069
[x.sub.8]         0.1842             0.2286             0.2813

Criterion                      Fuzzy group weight

                                  [[??].sub.j]

             [w.sub.j[alpha]]   [w.sub.j[gamma]]   [w.sub.j[beta]]

[x.sub.1]         0.151              0.246              0.192
[x.sub.2]         0.192              0.311              0.246
[x.sub.3]         0.192              0.311              0.246
[x.sub.4]         0.096              0.179              0.151
[x.sub.5]         0.096              0.156              0.114
[x.sub.6]         0.057              0.100              0.081
[x.sub.7]         0.057              0.087              0.066
[x.sub.8]         0.038              0.346              0.346

Table 11. The normalized-weighted fuzzy decision making matrix and
solution results

                                Alternatives

Criterion                        [A.sub.0]

                                  Ratings

                   [alpha]         [beta]        [gamma]

[x.sub.1]           0.043          0.077          0.066
[x.sub.2]           0.055          0.097          0.085
[x.sub.3]           0.058          0.104          0.091
[x.sub.4]           0.027          0.056          0.052
[x.sub.5]           0.027          0.049          0.039
[x.sub.6]           0.018          0.034          0.031
[x.sub.7]           0.016          0.027          0.023
[x.sub.8]           0.010          0.099          0.108
[[??].sub.i]        0.255          0.543          0.496
[S.sub.i]                          0.431
[K.sub.i]                          1.000

                                Alternatives

Criterion                        [A.sub.1]

                                  Ratings

                   [alpha]         [beta]        [gamma]

[x.sub.1]           0.030          0.062          0.060
[x.sub.2]           0.038          0.078          0.076
[x.sub.3]           0.029          0.062          0.064
[x.sub.4]           0.011          0.028          0.031
[x.sub.5]           0.019          0.039          0.035
[x.sub.6]           0.007          0.017          0.019
[x.sub.7]           0.011          0.022          0.020
[x.sub.8]           0.008          0.089          0.108
[[??].sub.i]        0.154          0.396          0.414
[S.sub.i]                          0.321
[K.sub.i]                          0.745

                                Alternatives

Criterion                        [A.sub.2]

                                  Ratings

                   [alpha]         [beta]        [gamma]

[x.sub.1]           0.022          0.046          0.046
[x.sub.2]           0.044          0.087          0.085
[x.sub.3]           0.035          0.073          0.073
[x.sub.4]           0.022          0.050          0.052
[x.sub.5]           0.022          0.044          0.039
[x.sub.6]           0.012          0.028          0.028
[x.sub.7]           0.013          0.024          0.023
[x.sub.8]           0.007          0.079          0.097
[[??].sub.i]        0.177          0.432          0.444
[S.sub.i]                          0.351
[K.sub.i]                          0.813

                                Alternatives

Criterion                        [A.sub.3]

                                  Ratings

                   [alpha]         [beta]        [gamma]

[x.sub.1]           0.030          0.062          0.060
[x.sub.2]           0.022          0.049          0.051
[x.sub.3]           0.035          0.073          0.073
[x.sub.4]           0.019          0.045          0.047
[x.sub.5]           0.011          0.024          0.024
[x.sub.6]           0.009          0.021          0.022
[x.sub.7]           0.007          0.014          0.014
[x.sub.8]           0.007          0.079          0.097
[[??].sub.i]        0.140          0.365          0.387
[S.sub.i]                          0.297
[K.sub.i]                          0.689