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International Journal of Information Technology & Decision Making Vol. 16 (2017) c World Scienti¯c Publishing Company ° DOI: 10.1142/S0219622017500274

Entropy–KEMIRA Approach for MCDM Problem Solution in Human Resources Selection Task

Aleksandras Krylovas Department of Mathematical Modelling Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania [email protected]

Stanislavas Dadelo Department of Philosophy and Communication Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania [email protected]

Natalja Kosareva* Department of Mathematical Modelling Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania [email protected]

Edmundas Kazimieras Zavadskas Department of Construction Technology and Management Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania [email protected] Published Entropy–KEMIRA approach is proposed for criteria ranking and weights determining when solving Multiple Criteria Decision-Making (MCDM) problem in human resources selection task. For the ¯rst time the method is applied in the case of three groups of criteria. Weights are calculated by solving optimization problem of maximizing the number of elements, which are \best" according to all three criteria, and minimizing the number of \doubtful" elements. The algorithm of problem solution is presented in the paper. The numerical experiment with three groups of evaluation criteria describing 11 life goals was accomplished. Keywords: Entropy; MCDM methods; KEMIRA; criteria weights; human resources.

*Corresponding

author. 1


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1. Introduction Di®erent methods available for decision makers supply di®erent information. Therefore, by combining these methods we can make more subtle decisions. In the past decades, researches and development in the ¯eld of Multiple Criteria Decision-Making (MCDM) involving both quantitative and qualitative factors have accelerated and seem to continue growing. Studies1,2 systematically review the applications and methodologies of the MCDM techniques and approaches. The analysis of the complexity behind the individual choices, taking into account the relationship between economics and psychology under the notion of Bounded Rationality, is discussed in the article.3 The technical and social systems of the present day are large, complex and complicated objects. Several aspects concerning the utilization and technology of Decision Support Systems (DSS) in the context of Large-Scale Complex Systems (LSS) control are presented in Ref. 4. The research in Ref. 5 presents the evolution in time of the published DSS research materials and tests their usefulness and the relevance of the information provided by three scienti¯c databases. The task of determining criteria weights is rather di±cult. Di®erent criteria weights determination methods give sometimes very di®erent results.6 Having the goal of obtaining precise objective assessment of criteria weights and not losing information, we need to combine these methods. Reviewing the decision-making strategies, it can be said that situations generated by di®erent areas or di®erent circumstances form unalike challenges and their solutions require applying di®erent methods.7 The procedures of selecting the appropriate criteria are highly important for decision-making in the ¯eld of human resources management. The appropriate human resources management procedures become very important for creating speci¯c tasks, evaluating the person's life goals and their sets.8 In recent publications great attention is paid to the criteria weights assessment problem-solving. One of the most popular and cited method is Analytic Hierarchy Process (AHP) developed by Saaty,9 it is relevant to this day. Bayesian revision method for improving the individual pairwise comparison matrices when using AHP as the methodological support was proposed in Ref. 10. The paper11 analyzes a multiattribute decision-making method with generalized fuzzy numbers. The attribute weights are determined by a linear programming model. The Analytical Network Process (ANP) is widely used to assess the key factors of risks and analyze the impacts and preferences of decision alternatives. In Ref. 12 a maximum eigenvalue threshold is proposed as the consistency index for the ANP in risk assessment and decision analysis. Data consistency problem for a pairwise comparison matrix was analyzed in Refs. 13–16. Weighted least squares method17 and Delphi method18 are other methods belonging to the subjective methods category. There are not only subjective weight determining methods when the weights are established on the basis of an expert assessment, but also objective weight determining methods are being used in scienti¯c research. One of the most famous approach for determining the objective attribute weights is the entropy-based method


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Weight Balancing by Application of Entropy and KEMIRA

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and its variations.19 In MCDM, the greater value of the entropy, which is a general measure of uncertainty,20 corresponds to the smaller attributes weight, the less discriminate power of that attribute in decision-making process. In Ref. 21 the best features of the entropy method and the Criterion Impact Loss (CILOS) approach have been combined to obtain a new method namely Integrated Determination of Objective CRIteria Weights (IDOCRIW). Multiple objective programming22 is the alternative weight determining method. A new analytical model for determining the relative weights of evaluation criteria using trapezium fuzzy numbers in decision-making problems is proposed in Ref. 23. A two-phase heuristic simulated annealing is presented as solution methodology. A practical common weight maximin approach with an improved discriminating power for technology selection is introduced.24 The proposed maximin approach enables the evaluation of the relative e±ciency of decision-making units (DMUs) with respect to multiple outputs and a single exact input with common weights. An improved weighted correlation coe±cient based on integrated weight for Interval Neutrosophic Sets (INSs) is developed.25 Using the proposed measure of INSs, a decision-making method is developed, which takes into account the in°uence of the evaluations' uncertainty and both the objective and subjective weights. An integrated fuzzy entropy-weight multiple criteria decision-making method was proposed and applied to risk assessment of hydropower stations in the Xiangxi River.26 It integrates the fuzzy set theory, the entropy-weight method and the multiple criteria decision-making method within a risk assessment framework. The other well-known criteria weights assessment method is Measuring Attractiveness through a Category-Based Evaluation Technique (MACBETH) designed by Bana-e-Costa et al.27 The method requires only qualitative judgments about di®erences of attractiveness to help a decision maker, or a decision-advisory group, to quantify the relative value of options. A case study based on a real-world application of MACBETH for multicriteria value measurement of IT solutions is presented in this paper. A multicriteria decision-making context in which the decision makers' preferences are represented by a multiattribute additive value function is considered.28 Two new methods based on dominance intensity measures aimed at ranking alternatives are proposed. The ranking performance of six most popular and easily comprehensive MCDM methods is investigated using two real-time industrial robot selection problems.29 Both single-dimensional and high-dimensional weight sensitivity analyses are performed to study the e®ects of weight variations on the ranking stability of all the six considered MCDM methods. It was determined that MULTIMOORA method is the most robust method being least a®ected by the changing weights. The KEmeny Median Indicator Ranks Accordance (KEMIRA) method accomplishes the procedure of criteria ranking and allows simultaneous identi¯cation of criteria weights. This method is especially e±cient when there are few separate groups of criteria and one must determine criteria priority and weights in each group. Such situation arises, for example, when there is a combination of certain external and internal objects or subjective and objective evaluation criteria. This method


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presents a new KEMIRA approach for determining criteria priority and selection criteria weights.30 It allows reducing the amount of expert work and increases the accuracy of calculations considerably. Possibilities of KEMIRA method can be illustrated with examples. Combined KEMIRA method was applied for solving personnel ranking and selection problem in the case of two subgroups of evaluating criteria.31 The example of KEMIRA-M decision method application in solving the task of construction site choice for nonhazardous waste incineration plant is presented in Ref. 32. The current paper exposes that KEMIRA method can be applicable as well in the case of three or more subgroups of evaluating criteria. Usually, in the ¯rst stage, Kemeny median33 is applied to set criteria priorities. In this study entropy method is applied for criteria prioritization. In the second stage criteria weights are calculated by Indicator Rank Accordance method. Alternatives ranking could be accomplished by any of the well-known methods TOPSIS, SAW, VIKOR, WASPAS, etc. Modi¯ed KEMIRA method can be applied for solving problem of choosing the best alternative among several alternatives relevant in various ¯elds of management process, economic activity, marketing, capital investments, logistics, construction, cultural heritage management, hospitality management, location of facilities, etc. Optimal solution in a variety of di®erent problems can help making strategic decision not only for saving money and resources, but also for improving social and environmental situation.32 New KEMIRA approach is designed to optimize the decisionmaking process. This method provides the opportunity to combine factors of different origin and nature and can be integrated into many ¯elds of human activity. One of such ¯elds is human resources. The quality of university education is an important factor in the progress of society, a®ecting the cultural, scienti¯c and technological progress. Optimization of university studies is increasingly global process. One of the main purposes of university studies is formation of a certain stratum of professionals, manifesting themselves in a variety of cultural and technological ¯elds as initiators and guides to innovation, a kind of generators of progress. This is possible if the pursuit of quality of the educational system is based on modern technological advancements, applying the design methodology of decision-making principles.34 First of all, experts must provide visions of future development in various ¯elds of human activity, policy and practice of innovation application. It forms the basis for understanding the importance of innovation and knowledge penetration and breakthrough. University education process provides combining needs of di®erent social groups students, research sta®, administrative sta® and employers of graduates. Education process optimization requires combining of stakeholder needs and interests.35 Persons' assessment should be carried out taking into account a lot of subjective and objective criteria.36–39 A description of the phenomenon considered is based on the relationships between all the criteria. Often the educational process faces the challenge of how to objectively select from distinguished students with optimal personal values by distancing themselves from subjective assessments. These results provide new understanding of the objective selection.


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The aim of the current research is to use KEMIRA integrated method to make students ranking according to the di®erent life goals criteria. Previous research studies on overall social motivation in support professions have established that life motivation plays a central role in the decisions made by individuals to enter their profession. Information about the person's life goals and values can help employers to form employment policies, to identify personal professional features, to help educational institutions manage the educational process and to shape education policy for motivating the personal development of students. 2. The Mathematical Task Formulation Suppose we have N respondents' survey results X ij , Y kj , Z lj in the assessment of certain phenomena by three groups of criteria X, Y , Z . Here i ¼ 1; 2; . . . ; n x , k ¼ 1; 2; . . . ; n y , l ¼ 1; 2; . . . ; n z , j ¼ 1; 2; . . . ; N . We consider these criteria independently measuring the same phenomena. Our task is to calculate weights of all criteria and choose the \best" respondents according to three criteria groups by formulating and solving the optimization problem. The initial data are modi¯ed by below normalization: x ij ¼

X ij min j X ij max j X ij min j X ij

;

y kj ¼

Y kj min j Y kj max j Y kj min j Y kj

;

z lj ¼

Z lj min j Z lj max j Z lj min j Z lj

:

ð1Þ Calculate entropy e xi of each data column

x ij .

Form a table as shown below:

x~

x~ 1

x~ 2

x~ s

p~

p~1

p~2

p~s

Here x~ t < x~ tþ1 values x ij ; j ¼ 1; 2; . . . ; N , listed in the increasing order; p~t are their Ps ~t ¼ N . Thus, we obtain the analog of discrete random variable frequencies: t¼1 p empirical probability distribution. The entropy is calculated in the usual way (see, for example, Ref. 20) and is used as a characterization of the information content of a data source: e xi ¼

s X p~t t¼1

N

ln

p~t : N

Entropies e yk and e zl are calculated by analogy. 2.1. Determining criteria priorities Criteria priorities are established according to the descending values of the entropy. Suppose that the following priorities of criteria are established: i1 i2 inx ;

k1 k2 kny ;

l1 l2 lnz :

ð2Þ


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It means that for construction of the criteria weighted averages (4) we must require corresponding restrictions on the criteria weights: w xi 1 w xi 2 w xi n x ; w yk 1 w yk 2 w yk n y ; w zl 1 w zl 2 w zl n z :

ð3Þ

Notice that one task of the study is ¯nding the weights satisfying requirements (3). The weights show the relative importance of the evaluation criteria. In the previous works of authors,30,31,40 priorities (2) were obtained from expert estimates. This paper examines the case when only the measurement results are present and priorities (2) are determined by the entropy of measurement results. It is known20,41 that entropy shows informativeness of data. The bigger the entropy, the more information the corresponding criterion has and the more important this criterion is. So, having a respondent's data vector ðX; Y ; Z Þ, we calculate the values of weighted averages: W x ðXÞ ¼ W y ðY Þ ¼ W z ðZ Þ ¼

nx X

w xi s x i s ¼ w xi 1 x i 1 þ w xi 2 x i 2 þ þ w xi n x x i n x ;

s¼1 ny X s¼1 nz X

w yi s y i s ¼ w yk 1 y k 1 þ w yk 2 y k 2 þ þ w yk n y y k n y ;

ð4Þ

w zi s z i s ¼ w zl 1 z l 1 þ w zl 2 z l 2 þ þ w zl n z z l n z ;

s¼1

where weights w x ; w y ; w z satisfy requirements (3). 2.2. The construction of objective function Methodological assumption of this research: all three features X; Y ; Z are equivalent, and the bigger values of variables x i ; y k ; z l represent better satisfaction of certain properties. Thus, the better given respondent meets the relevant requirements, the greater values acquire functions W x ðXÞ; W y ðY Þ; W z ðZ Þ. Choose numbers 0 < w x ; w y ; w z < 1 and de¯ne sets A xw x ; A yw y ; A zw z by formulas (5) including respondents who meet the requirements not less than certain ¯xed levels wx ; wy ; wz : A xw x ¼ fj 2 f1; 2; . . . ; N g : W x ðX j Þ w x g; A yw y ¼ fj 2 f1; 2; . . . ; N g : W y ðY j Þ w y g; A zw z ¼ fj 2 f1; 2; . . . ; N g : W z ðZ j Þ w z g:

ð5Þ

The sets A xwx ; A ywy ; A zw z consist from the test takers j 2 J f1; 2; . . . ; N g, whose values of criteria (4) are not lower than selected levels w x ; w y ; w z , in other words, they are sets of the \best" respondents. Notice that A x0 ¼ A y0 ¼ A z0 ¼ J and A x1þ ¼ A y1þ ¼ A z1þ ¼ ;, for each > 0 [since the values of sums (4) are always between 0 and 1].


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The weights w xi ; w yk ; w zl must be selected in such a way that criteria (4) are best aligned to each other. Mathematically, this correspondence of three criteria can be measured in this way. Suppose that under certain considerations the numbers w x ; w y ; w z 2 ð0; 1Þ, indicating the minimum criteria (4) values have been chosen. Create two more sets A ¼ A xw x \ A yw y \ A zw z ;

B ¼ ðA xw x [ A yw y [ A zw z ÞnA:

ð6Þ

The set A consists from the test takers j 2 J ful¯lling all three criteria W x;y;z ðX j ; Y j ; Z j Þ w x ; w y ; w z and the set B includes test takers ful¯lling at least one of them, but not all. Graphical representation of the sets A and B is depicted in Fig. 1. Denote nA the number of set A elements. Thus, the proposed mathematical de¯nition of the \best" correspondence of criteria is max nA:

w xi ;w yk ;w zl

ð7Þ

The best choice of weights w xi ; w yk ; w zl may guarantee the largest size of set A. The weights w xi ; w yk ; w zl must satisfy restrictions (3) and thresholds w x ; w y ; w z 2 ð0; 1Þ chosen in such way that only test takers with high criteria (4) values would be analyzed, for example 20–30% of the \best". When there is a choice of several alternatives satisfying the condition (7), together with it we analyze another objective function: min nB:

w xi ;w yk ;w zl

ð8Þ

We minimize the number of doubtful respondents, those who are the best according to one or two, but not all three criteria. In Fig. 2 maximization of the set A and minimization of the set B is represented.

Fig. 1. Graphical representation of the sets A and B.


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Fig. 2.

Maximization of the set A (on the left) and minimization of the set B (on the right).

3. Algorithm of Problem Solution For solving the optimization problem (7) in the paper,40 the authors chose the alternative objective function: W2 ¼

N X

½ðW x ðX j Þ W y ðY j ÞÞ2

j¼1

þ ðW x ðX j Þ W z ðZ j ÞÞ2 þ ðW y ðY j Þ W z ðZ j ÞÞ2 :

ð9Þ

However, numerical experiments show that for this paper data minimization of objective function (9) does not provide the stability of set A elements (6) determination. The following analog of the objective function (9) also did not yield stable results: W1 ¼

N X

½jW x ðX j Þ W y ðY j Þj

j¼1

þ jW x ðX j Þ W z ðZ j Þj þ jW y ðY j Þ W z ðZ j Þj:

ð10Þ

In the current research the calculations realizing the following algorithm were conducted: (i) Algorithm parameters are determined: 103 " 101 ;

102 max iter 106 ; 20% w px;y;z 30% ðw px;y;z ¼ 100% w x;y;z Þ:

(ii) The initial vector of weights satisfying conditions (3) is chosen: 0 0 0 0 0 0 0 0 0 w 0 ¼ ðw x1 ; w x2 ; . . . ; w xn ; w y1 ; w y2 ; . . . ; w yn ; w z1 ; w z2 ; . . . ; w zn Þ: x y z

(iii) The set A0 of the best elements (6) is determined and the number of set elements nA0 is calculated. (iv) Randomly selected direction vector is w ¼ ðw x1 ; . . . ; w xn x ; w y1 ; . . . ; w yn y ; w z1 ; . . . ; w zn z Þ:


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(v) The vector w 1 ¼ w 0 þ "w is calculated. If it does not satisfy restrictions (3), the corrections are carried out: (a) (b) (c) (d)

1 1 if w x;y;z;i < 0, change: w x;y;z;i ¼ 0; 1 1 1 1 if w x;y;z;i < w x;y;z;iþ1 , change: w x;y;z;i ¼ w x;y;z;iþ1 ; 1 P n x ;n y ;n z 1 w x;y;z;i 1 if s ¼ i¼1 w x;y;z;i 6¼ 1, change: w x;y;z;i ¼ s ; the number of iterations iter is calculated.

(vi) If iter > max iter , then algorithm is ¯nishing the calculations. (vii) The set A1 of the best elements (6) is determined and the number of set elements nA1 is calculated. (viii) If nA1 > nA0 , change w 0 ¼ w 1 ; A0 ¼ A1 and go to point (iv) of algorithm. (ix) If nA1 nA0 , go to point (iv) of algorithm. Comments. (1) Calculating the values w x ; w y ; w z , the following inequalities must be satis¯ed: 100 jA xw x j w px ; N

100 jA yw y j w py ; N

100 jA zw z j w pz ; N

as the best w px % respondents according to criterion X, the best w py % according to Y and the best w pz % according to Z will be included in the set A. (2) In the points (viii) and (ix) of algorithm, the objective function (7) can be changed respectively by (8)–(10). Let us consider few steps of computations re°ecting the proposed method. Suppose we have six alternatives and three groups of criteria: X ¼ ðX1; X2; X3Þ; Y ¼ ðY 1; Y 2Þ; Z ¼ ðZ 1; Z 2; Z 3; Z 4Þ. Suppose that priorities of criteria in each criterion group are X1 X2 X3;

Y 2 Y 1;

Z 2 Z 4 Z 3 Z 1:

Consequently, we will look for criteria weights satisfying the corresponding conditions: wx1 wx2 wx3; wy2 wy1; wz2 wz4 wz3 wz1; 3 2 4 X X X wxi ¼ wyj ¼ wzk ¼ 1: i¼1

j¼1

k¼1

Our decision-making matrix is as follows: No.

X1

X2

X3

Y2

Y1

Z2

Z4

Z3

Z1

1 2 3 4

0.2 0.8 0.2 0.1

0.7 0.2 0.6 0.4

0.5 0.5 0.2 1.0

0.3 0.0 0.08 0.68

0.8 0.9 0.15 0.7

0.25 0.5 0.28 0.14

0.55 0.13 0.23 0.84

0.35 1.0 0.45 0.68

0.33 0.65 0.34 0.72

ð11Þ


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A. Krylovas et al. (Continued ) No.

X1

X2

X3

Y2

Y1

Z2

Z4

Z3

Z1

5 6

1.0 0.4

0.6 0.3

0.35 0.3

0.34 0.44

0.4 0.55

0.34 0.13

1.0 0.33

0.51 0.16

0.2 1.0

(i) Choose the parameters: w x;y;z ¼ 0:35; " ¼ 101 . (ii) Choose the initial weights vector satisfying conditions (11): w 0 ¼ ð0:4; 0:31; 0:29; 0:85; 0:15; 0:4; 0:3; 0:2; 0:1Þ:

ð12Þ

(iii) For each alternative calculate Wx; Wy; Wz by formulas (4). Conduct calculations for the alternative 1: Wx ¼ 0:2 0:4 þ 0:7 0:31 þ 0:5 0:29 ¼ 0:442; Wy ¼ 0:3 0:85 þ 0:8 0:15 ¼ 0:375; Wz ¼ 0:25 0:4 þ 0:55 0:3 þ 0:35 0:2 þ 0:33 0:1 ¼ 0:368: Since Wx; Wy; Wz > 0:35, alternative 1 belongs to the set A. Calculations for all alternatives are presented below: No.

Wx

Wy

Wz

Ax

Ay

Az

A

B

1 2 3 4 5 6

0.442 0.527 0.324 0.454 0.688 0.340

0.375 0.135 0.091 0.683 0.349 0.457

0.368 0.504 0.305 0.516 0.558 0.283

1 1 0 1 1 0

1 0 0 1 0 1

1 1 0 1 1 0

1 0 0 1 0 0

0 1 0 0 1 1

The set A consists of two alternatives, the set B of three alternatives. (iv) Randomly select the direction vector w ¼ ð0:5; 0:3; 0:1; 0:2; 0:2; 0:4; 0:4; 0:2; 0:2Þ: (v) Calculate w 1 ¼ w 0 þ "w ¼ ð0:45; 0:28; 0:3; 0:83; 0:17; 0:36; 0:34; 0:22; 0:12Þ: Perform the necessary corrections. Since wx2 < wx3, let wx2 ¼ wx3 ¼ 0:29: w 1 ¼ ð0:45; 0:29; 0:29; 0:83; 0:17; 0:36; 0:34; 0:22; 0:12Þ: P3 P3 P4 Since i¼1 wxi ¼ 1:03; j¼1 wyj ¼ 1; k¼1 wzk ¼ 1:04, for vectors normalization divide vectors X-coordinates by 1.03 and vectors Z -coordinates by 1.04 and round numbers to two digits: w 1 ¼ ð0:44; 0:28; 0:28; 0:83; 0:17; 0:35; 0:33; 0:21; 0:11Þ:

ð13Þ


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(vi) Calculate Wx; Wy; Wz; A; B; W 1 ¼ minðWx; Wy; WzÞ; W 2 ¼ Wx þ Wy þ Wz: No.

Wx

Wy

Wz

Ax

Ay

Az

A

B

W1

W2

1 2 3 4 5 6

0.425 0.547 0.313 0.438 0.704 0.344

0.385 0.153 0.092 0.683 0.350 0.459

0.379 0.502 0.307 0.550 0.576 0.302

1 1 0 1 1 0

1 0 0 1 1 1

1 1 0 1 1 0

1 0 0 1 1 0

0 1 0 0 0 1

0.379 0.153 0.092 0.438 0.350 0.302

1.189 1.202 0.711 1.671 1.630 1.105

The set A now consists of three alternatives, the set B has two alternatives. (vii) Set w 0 ¼ w 1 and go to point (iv) and so on. If we could not ¯nd better weights vector than (13) with bigger nA value and/or lower nB value, then we rank the best three alternatives according to the value of objective function W 2 : a4 a5 a1. Note that the alternative criterion of \good" weights vector in the steps (iii) and (v) could be minimization of functions' (9), (10) values. For example, calculate W 2 for the weights vectors (12) and (13). For weights vector (12) we obtain W 2 ¼ ð0:442 0:375Þ2 þ ð0:442 0:368Þ2 þ ð0:375 0:368Þ2 þ ð0:527 0:135Þ2 þ þ ð0:457 0:283Þ2 ¼ 0:7074: For weights vector (13), W 2 ¼ 0:7004; 0:7004 < 0:7074, so vector (13) is preferable compared to (12). Then go to the point (vii) and so on. 4. Numerical Experiments Participants. The participants of the experiment were 87 randomly selected 21–23year-old (standard deviation is 1.21), Second–third year students (men) from different faculties of Vilnius Gediminas Technical University (VGTU). Procedure. Participants completed the questionnaire during a large practice lecture that took place at the beginning of the fall semester. Evaluation criteria. The selected students evaluated the weights of criteria describing 11 life goals (values) suggested by Chua42 (Table 1). The students evaluate the criteria (life goals) according to the following rules: Ranking (x): Rank all the general life goal criteria in our life from 1 to 11 (1: the least important; 11: the most important). Present rate (): Rate our satisfaction life goal criteria level today in each of the every slices of our life goals above, using the following scale: from 1 ¼ Totally Dissatis¯ed to 10 ¼ Totally Satis¯ed.


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A. Krylovas et al. Table 1.

Evaluation of the life goals criteria (case study).

No. Life goals 1

2

3 4 5 6

7 8

9 10 11

ISSN: 0219-6220

Business/Career/Studies Usually the key segment in our lives. Business is for entrepreneurs, career for employees and studies for students. Finance/Wealth How rich we are. The amount of wealth, assets and material possessions we have. Health/Fitness Our state of health as well as our lifestyle. Diet, sleep and exercise falls here. Social/Friends How we are faring in our social circle. Family Our parents, siblings, next of kin, relatives or even our guardians. Love The amount of love we feel in our life. While it can represent the status of our relationship with our spouse/boyfriend/girlfriend, it does not have to be the case. Love here does not refer to romantic love, but about universal love. Recreation/Fun Our recreation and enjoyment in life. Contribution How we are giving back to the society. Social cause. Humanitarian activities. Personal growth Our personal development as a whole. Spiritual Our connection with the universe. Some call it higher power/God/higher self. Self-image How we see Ourself.

Ranking (x) Present rate () Future rate () 3

6

9

2

5

8

1

7

10

7

6

7

5

6

8

4

7

8

8

7

7

9

6

6

6

6

6

10

4

5

11

6

7

Future rate (): Repeat rate of the exercise in number, this time assigning percentages that we would desire to be true of how we allocate our time in the future. Initial data are normalized by the formula (1). In Table A.1 of Appendix A we normalized the measurement values x ij ; y kj ; z lj for 87 students and integer parts of numbers E ¼ 1000 e xi ð1000 e yk ; 1000 e zl Þ are presented. In each criteria group X; Y ; Z we set the priority order of criteria by the descending entropy value (E xi ; E yk ; E zl ). Columns in Table A.1 are organized in this order. Since this task requires large-scale calculations, the calculations were carried out in C++ program. The results of the ¯rst experiment were obtained for di®erent initial weights values w 0 , when algorithm was realized by maximizing the number of set A elements nA and values of parameters were: max iter ¼ 106 , " ¼ 0:1, w px;y;z ¼ 30%. The results of the ¯rst experiment are presented in Table 2. Maximum value of nA ¼ 11 was reached in the Experiment No. 4. Other group of experiments includes minimization of value nB, i.e., algorithm realization with another objective function (8). In Table 3 the best results of the second experiment calculations are presented, when the initial weights vectors w 0 are taken from the Experiment Nos. 1–10.


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Weight Balancing by Application of Entropy and KEMIRA Table 2. Initial weights w 0 , calculated weights w 1 , the set A and values nA; nB for the ¯rst numerical experiment.

No. 1

No. 2

No. 3

No. 4

No. 5

No. 6

No. 7

No. 8

No. 9

No. 10

w0 1.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0 1.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0 1.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0 w1 0.3907; 0.1236; 0.0885; 0.0885; 0.0736; 0.0736; 0.0554; 0.0554; 0.0506; 0.0000; 0.0000 0.2656; 0.0996; 0.0996; 0.0996; 0.0996; 0.0996; 0.0996; 0.0946; 0.0423; 0.0000; 0.0000 0.2859; 0.0859; 0.0859; 0.0859; 0.0859; 0.0859; 0.0859; 0.0859; 0.0859; 0.0133; 0.0133 A nA nB {2; 12; 18; 27; 38; 40; 42; 48; 50; 72} 10 41 1 w 0.2171; 0.1064; 0.1064; 0.0810; 0.0810; 0.0810; 0.0810; 0.0736; 0.0736; 0.0541; 0.0451 0.1076; 0.1076; 0.1076; 0.1076; 0.1076; 0.1076; 0.1076; 0.1061; 0.1061; 0.0344; 0.0000 0.1484; 0.1088; 0.1088; 0.1088; 0.0933; 0.0920; 0.0920; 0.0920; 0.0920; 0.0410; 0.0231 A nA nB {2; 12; 38; 42; 45; 48; 50; 71; 72} 9 48 w1 0.0949; 0.0949; 0.0949; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894 0.1220; 0.1220; 0.1220; 0.1220; 0.1220; 0.0772; 0.0772; 0.0772; 0.0772; 0.0491; 0.0321 0.1356; 0.1356; 0.1356; 0.1260; 0.1148; 0.1148; 0.0778; 0.0778; 0.0272; 0.0272; 0.0272 A nA nB {12; 13; 23; 38; 43; 45; 50; 52; 65; 71} 10 48 1 w 0.3221; 0.0839; 0.0839; 0.0839; 0.0609; 0.0609; 0.0609; 0.0609; 0.0609; 0.0609; 0.0609 0.2308; 0.1011; 0.0832; 0.0832; 0.0832; 0.0832; 0.0832; 0.0832; 0.0820; 0.0820; 0.0049 0.1529; 0.1375; 0.1375; 0.1375; 0.1375; 0.1272; 0.0907; 0.0600; 0.0193; 0.0000; 0.0000 A nA nB {2; 12; 18; 27; 38; 40; 42; 48; 50; 61; 72} 11 44 w0 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.0 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.0 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.0 w1 0.0942; 0.0942; 0.0942; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897 0.1268; 0.1268; 0.1268; 0.1268; 0.1268; 0.0732; 0.0732; 0.0732; 0.0732; 0.0732; 0.0000 0.1412; 0.1412; 0.1210; 0.0969; 0.0969; 0.0969; 0.0969; 0.0969; 0.0474; 0.0440; 0.0210 A nA nB {12; 13; 23; 38; 43; 45; 50; 52; 65; 71} 10 45 w1 0.1993; 0.1174; 0.1071; 0.0814; 0.0814; 0.0814; 0.0814; 0.0814; 0.0814; 0.0438; 0.0438 0.1087; 0.1087; 0.1087; 0.1087; 0.1087; 0.1087; 0.1020; 0.1020; 0.1020; 0.0363; 0.0058 0.1109; 0.1109; 0.1073; 0.1073; 0.1073; 0.1073; 0.1073; 0.0710; 0.0622; 0.0622; 0.0463 A nA nB {2; 12; 27; 38; 45; 48; 50; 71; 72} 9 49 1 w 0.0954; 0.0919; 0.0919; 0.0919; 0.0919; 0.0919; 0.0919; 0.0919; 0.0873; 0.0873; 0.0868 0.1825; 0.1216; 0.1120; 0.0913; 0.0913; 0.0913; 0.0913; 0.0913; 0.0913; 0.0207; 0.0152 0.1150; 0.1150; 0.1150; 0.0972; 0.0972; 0.0972; 0.0972; 0.0939; 0.0891; 0.0416; 0.0416 A nA nB {12; 13; 23; 38; 40; 43; 45; 52; 65; 71} 10 46 w0 0.2; 0.2; 0.2; 0.2; 0.2; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0 0.2; 0.2; 0.2; 0.2; 0.2; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0 0.2; 0.2; 0.2; 0.2; 0.2; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0 w1 0.1904; 0.1101; 0.0907; 0.0856; 0.0748; 0.0748; 0.0748; 0.0748; 0.0748; 0.0748; 0.0748 0.1573; 0.1573; 0.0920; 0.0920; 0.0920; 0.0920; 0.0815; 0.0815; 0.0815; 0.0364; 0.0364 0.1117; 0.1117; 0.1117; 0.1117; 0.0959; 0.0959; 0.0959; 0.0667; 0.0667; 0.0667; 0.0652 A nA nB {2; 12; 27; 38; 40; 42; 48; 50; 57; 72} 10 46 1 w 0.0951; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905 0.1696; 0.1330; 0.1330; 0.0961; 0.0961; 0.0961; 0.0926; 0.0926; 0.0906; 0.0000; 0.0000 0.1463; 0.1463; 0.0987; 0.0866; 0.0866; 0.0866; 0.0866; 0.0866; 0.0866; 0.0866; 0.0025 A nA nB {12; 23; 38; 40; 43; 45; 50; 52; 65; 71} 10 44 1 w 0.1978; 0.1264; 0.0874; 0.0813; 0.0813; 0.0804; 0.0804; 0.0804; 0.0804; 0.0521; 0.0521 0.1164; 0.0929; 0.0929; 0.0929; 0.0879; 0.0879; 0.0879; 0.0879; 0.0879; 0.0847; 0.0808 0.1239; 0.1239; 0.0960; 0.0960; 0.0960; 0.0960; 0.0960; 0.0960; 0.0622; 0.0569; 0.0569 A nA nB {2; 12; 27; 42; 45; 48; 57; 71; 72} 9 47

13


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A. Krylovas et al. Table 3. Initial weights w 0 , calculated weights w 1 , the sets A; B and values nA; nB for the second numerical experiment. w0 0.0942; 0.0942; 0.0942; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897; 0.0897 0.1268; 0.1268; 0.1268; 0.1268; 0.1268; 0.0732; 0.0732; 0.0732; 0.0732; 0.0732; 0.0000 0.1412; 0.1412; 0.1210; 0.0969; 0.0969; 0.0969; 0.0969; 0.0969; 0.0474; 0.0440; 0.0210 No. w1 0.0950; 0.0935; 0.0932; 0.0901; 0.0901; 0.0901; 0.0901; 0.0901; 0.0892; 0.0892; 0.0892 11 0.1250; 0.1250; 0.1250; 0.1250; 0.1250; 0.0756; 0.0756; 0.0756; 0.0756; 0.0684; 0.0043 0.1425; 0.1395; 0.1144; 0.0988; 0.0988; 0.0988; 0.0988; 0.0911; 0.0479; 0.0479; 0.0214 A nA B nB   2; 4; 7; 8; 9; 10; 11; 14; 15; 17; 18; 19       20; 21; 22; 24; 25; 27; 32; 33; 34; 36; 37 {12; 13; 23; 38; 43; 45; 50; 52; 65; 71} 10 44 39; 40; 41; 42; 46; 48; 49; 56; 57; 59; 60       64; 68; 70; 72; 74; 80; 82; 83; 86; 87 w0 0.0954; 0.0919; 0.0919; 0.0919; 0.0919; 0.0919; 0.0919; 0.0919; 0.0873; 0.0873; 0.0868 0.1825; 0.1216; 0.1120; 0.0913; 0.0913; 0.0913; 0.0913; 0.0913; 0.0913; 0.0207; 0.0152 0.1150; 0.1150; 0.1150; 0.0972; 0.0972; 0.0972; 0.0972; 0.0939; 0.0891; 0.0416; 0.0416 No. w1 0.0957; 0.0933; 0.0933; 0.0902; 0.0902; 0.0902; 0.0902; 0.0902; 0.0890; 0.0890; 0.0886 12 0.1787; 0.1218; 0.1144; 0.0976; 0.0937; 0.0937; 0.0909; 0.0909; 0.0904; 0.0163; 0.0116 0.1136; 0.1136; 0.1055; 0.1008; 0.1008; 0.0999; 0.0995; 0.0953; 0.0843; 0.0433; 0.0433 A nA B nB   2; 4; 7; 8; 9; 10; 11; 14; 15; 17; 18; 19       20; 21; 22; 24; 25; 27; 32; 33; 34; 36; 37 {12; 13; 23; 38; 40; 43; 45; 50; 52; 65; 71} 11 44 39; 41; 42; 46; 48; 49; 55; 56; 57; 59; 60       64; 68; 70; 72; 74; 80; 82; 83; 86; 87 w0 0.0951; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905; 0.0905 0.1696; 0.1330; 0.1330; 0.0961; 0.0961; 0.0961; 0.0926; 0.0926; 0.0906; 0.0000; 0.0000 0.1463; 0.1463; 0.0987; 0.0866; 0.0866; 0.0866; 0.0866; 0.0866; 0.0866; 0.0866; 0.0025 No. w1 0.0947; 0.0934; 0.0934; 0.0901; 0.0901; 0.0901; 0.0901; 0.0901; 0.0893; 0.0893; 0.0893 13 0.1617; 0.1342; 0.1328; 0.0978; 0.0978; 0.0967; 0.0923; 0.0923; 0.0923; 0.0020; 0.0000 0.1436; 0.1430; 0.0977; 0.0873; 0.0873; 0.0873; 0.0873; 0.0873; 0.0873; 0.0873; 0.0047 A nA B nB 2; 4; 7; 8; 9; 10; 11; 14; 15; 17; 18; 19 20; 21; 22; 24; 25; 27; 32; 33; 34; 36; 37 {12; 13; 23; 38; 40; 43; 45; 50; 52; 65; 71} 11 43 39; 41; 42; 46; 48; 49; 56; 57; 59; 60; 64 68; 70; 72; 74; 80; 82; 83; 86; 87 w0 0.0949; 0.0949; 0.0949; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894; 0.0894 0.1220; 0.1220; 0.1220; 0.1220; 0.1220; 0.0772; 0.0772; 0.0772; 0.0772; 0.0491; 0.0321 0.1356; 0.1356; 0.1356; 0.1260; 0.1148; 0.1148; 0.0778; 0.0778; 0.0272; 0.0272; 0.0272 No. w1 0.0951; 0.0935; 0.0935; 0.0902; 0.0902; 0.0902; 0.0897; 0.0897; 0.0897; 0.0897; 0.0884 14 0.1271; 0.1259; 0.1210; 0.1210; 0.1158; 0.0835; 0.0782; 0.0760; 0.0760; 0.0460; 0.0294 0.1441; 0.1391; 0.1270; 0.1205; 0.1078; 0.1057; 0.0789; 0.0778; 0.0330; 0.0330; 0.0330 A nA B nB 2; 4; 7; 8; 9; 10; 11; 14; 15; 17; 18; 19 20; 21; 22; 24; 25; 27; 32; 33; 34; 36; 37 {12; 13; 23; 38; 43; 45; 50; 52; 65; 71} 10 45 39; 40; 41; 42; 46; 48; 49; 56; 57; 59; 60 61; 64; 68; 70; 72; 74; 80; 82; 83; 86; 87


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Table 4. The results of the Experiment No. 13 with reduced parameters' w px;y;z values. w px;y;z (%)

A

25 27 29

f23; 38; 65g f12; 23; 38; 43; 45; 52; 65g f12; 13; 23; 38; 43; 45; 52; 65; 71g

nA

nB

3 7 9

51 47 45

Thus, in the Experiment No. 13 we obtained a local maximum of the objective function (7) of nA ¼ 11 and a local minimum of the objective function (8) of nB ¼ 43 corresponding to the threshold values of w px;y;z ¼ 30%. We observe the stability of the set A elements in all experiments of the second group. However, we cannot signi¯cantly reduce the number of set B elements, the so-called \doubtful" items. Having the goal to lower the number of elements nA tightens requirements for respondents by reducing the parameters' w px;y;z values. The results of an experiment when the initial weights vector w 0 is taken from the Experiment No. 13 is presented in Table 4. 5. Results and Discussion Take the weights of the point No. 13 and calculate criteria (4) values: 1 ð10x 1 þ 9ðx 2 þ x 3 þ x 4 þ x 5 þ x 6 þ x 7 þ x 8 þ x 9 þ x 10 þ x 11 ÞÞ; 100 1 ð16y 1 þ 13ðy 2 þ y 3 Þ þ 10ðy 4 þ y 5 þ y 6 Þ þ 9ðy 7 þ y 8 þ y 9 Þ þ y 10 Þ; W y ðY Þ ¼ 100 1 ð14ðz 1 þ z 2 Þ þ 10z 3 þ 9ðz 4 þ z 5 þ z 6 þ z 7 þ z 8 þ z 9 Þ þ 8z 10 Þ: W z ðZ Þ ¼ 100 ð14Þ W x ðXÞ ¼

Coe±cients in formulas (14) were selected by performing small initial values perturbations and satisfying the condition w 1 þ w 2 þ þ w 11 ¼ 1. Calculate criteria (14) values for all respondents j ¼ 1; 2; . . . ; 87; minfW x ; W y ; W z g (according to this attribute the respondent enters set A) and generalized criterion W x þ W y þ W z (respondents are ranked according to this criterion). The results for all 87 respondents presented in Table B.1 of Appendix B. Numbers of respondents and corresponding minfW x ; W y ; W z g values in the table are denoted in bold font, if minfW x ; W y ; W z g 0:495. These objects are treated as the best. When W x þ W y þ W z 1:95; corresponding numbers are outlined by a frame. According to this criteria, objects can be ranked as follows: a 12 a 41 a 64 a 50 a 82 a 52 a 65 a 23 a 34 a 13 a 49 a 42 a 2 : The table makes it comfortable to analyze data. The ¯rst condition distinguishes the set A of the \best." If we want to select even \better" items from the set A (those


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which satisfy the additional requirements), the criterion can be either minfW x ; W y ; W z g or other criteria [for example, maxfW x ; W y ; W z g ]. Here, threshold values and are lower if we want to select more objects, and higher if we want to select fewer objects. Formulas (14) are not \sensitive" to small coe±cient perturbations they do not change or slightly change constructed sets, almost all elements of these sets were obtained in all numerical experiments. For example, if we calculate results for all 87 respondents with the following weights: w x ½11 ¼ f0:10; 0:10; 0:09; 0:09; 0:09; 0:09; 0:09; 0:09; 0:09; 0:09; 0:08g; w y ½11 ¼ f0:15; 0:13; 0:13; 0:10; 0:10; 0:10; 0:09; 0:09; 0:09; 0:01; 0:01g; w z ½11 ¼ f0:14; 0:14; 0:10; 0:09; 0:09; 0:09; 0:09; 0:09; 0:09; 0:08; 0:00g and the selection criteria are minfW x ; W y ; W z g 0:490 and W x þ W y þ W z 1:90, then the ranking results will change only slightly: a 12 a 64 a 82 a 50 a 52 a 65 a 23 a 25 a 34 a 13 a 42 a 2 a 49 : Note that the Experiment No. 13 gives the local extreme of the objective function (8). We have received a steady and satisfactory solution of the optimization problem. However, other local extremes as solutions of this problem can also exist. This research is the case study. Therefore, the statements we formulate are hypotheses whose con¯rmation requires statistical analysis. Especially interesting subject is the comparison of proposed weight balancing methodology with other widely used weight determining methods. In the article,43 Monte Carlo experiments were elaborated to compare the e±ciencies of the whole group of weights balancing methods, which are called WEBIRA (KEMIRA is one of them) and other weights determining methods: (a) Simple Additive Weighting (SAW) with equal weights and (b) entropy method. All three methods were compared for four di®erent data normalizations (max, min-max, sum and vector21). The design of experiment is as follows. Coordinate only the best alternatives, other alternatives are not under control. The bigger the set of the best alternatives, the more di±cult is the alignment. In Ref. 43 we limited to only one of the best alternatives. Random matrices which imitate expert assessments are being generated while one alternative has the bigger probability of being the best. Unlike the methods (a) and (b), whose results can be either true or false setting of the best alternative, WEBIRA sometimes cannot determine the best alternative when the maximum number of iterations is exceeded. Meanwhile, the number of false best alternative settings is signi¯cantly lower in the case of WEBIRA method. The study revealed that WEBIRA has signi¯cantly higher e±ciency for all considered numbers of alternatives (3–50) and four data normalization methods. E±ciency of all three methods decreased with increasing number of alternatives, but WEBIRA is still applicable, while application of other methods is e®ectless as the number of alternatives is greater than 11.


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WEBIRA is the least a®ected by the data normalization, while entropy method is the most a®ected. 6. Conclusions This study was conducted to present entropy–KEMIRA approach for determining criteria priority and weights when solving MCDM problem. In the case study there were three groups of criteria 11 criteria in each group. The goal was to determine and rank the best students according to their ranking, evaluating present and future rates of 11 life goals. Entropy method was applied for establishing priority of the criteria in each of the three groups of criteria. Then KEMIRA method was applied for discovering criteria weights. For this purpose the original algorithm was proposed. Please note that the overall number of criteria (33) is very big. It is rather di±cult to solve such complicated MCDM problem. KEMIRA method allows to ¯nd a good enough solution by solving the optimization problem to maximize the size of the set of \best" respondents and minimize the set of \doubtful" respondents. Thus, the study shows that the set of \best" respondents is established to be su±ciently stable. We see that measurement results of respondents according to the three concerned criteria are not well aligned to each other Pareto set of solutions is the empty set, as it usually happens in MCDM tasks. It is the reason for which the process of obtaining solution of the problem is not trivial. KEMIRA method is suitable for very important and economically bene¯cial carrying MCDM tasks where natural groups of attributes arise. For example, while solving MCDM problems of sustainable development researchers must pay attention to three main components such as economic development, social development and environmental protection. Such problems could be solved by applying entropy–KEMIRA approach for three sets of attributes economic, social and environmental components of sustainable development. Our next investigations will be carried out in two directions. The ¯rst is to compare entropy–KEMIRA method with other existing methods used for solving MCDM problems with natural groups of attributes. The second is data analysis by other methods instead of using entropy method authors will apply IDOCRIW method presented by Zavadskas and Podvezko.21


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Appendix A

Table A.1. Normalized initial data x ij ; y kj ; z lj and integer parts of entropies multiplied by 1000 ðE xi ; E yk ; E zl Þ. j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

x ij1

x ij2

x ij3

x ij4

x ij5

x ij6

x ij7

x ij8

x ij9

x ij10

x ij11

0:200 0:800 0:900 0:500 1:0 0:600 1:0 0:900 0:0 0:700 0:400 0:800 0:500 0:600 0:500 0:300 0:200 0:700 0:200 0:200 0:500 0:900 0:600 0:700 0:600 0:100 0:700 1:0 0:500 0:500 0:500 1:0 1:0 0:300 0:300 0:300 1:0 0:800 0:600 0:700 0:700 0:700 0:500 0:400 0:400

0:700 0:700 0:600 0:700 0:600 0:300 0:700 0:600 0:900 0:200 0:500 0:300 0:400 0:900 0:800 0:800 0:700 0:500 0:700 0:700 0:700 0:600 0:100 0:100 0:700 0:800 0:800 0:700 0:800 0:800 0:800 0:400 0:400 0:900 0:900 0:500 0:200 0:100 0:800 0:600 0:0 0:800 0:700 0:100 0:200

0:300 0:500 0:300 0:400 0:400 0:400 0:400 0:400 0:300 0:600 0:200 0:600 0:600 0:400 0:0 0:600 0:800 0:0 0:100 0:500 0:100 0:300 0:500 0:400 0:500 0:400 0:500 0:500 0:700 0:700 0:400 0:200 0:100 0:100 0:0 0:200 0:500 0:0 0:200 0:100 0:300 0:200 0:300 0:0 0:300

0:0 0:111 0:777 0:111 0:666 0:666 0:888 1:0 0:555 0:222 1:0 0:666 0:666 0:666 0:666 0:444 0:222 0:888 0:888 0:555 1:0 1:0 0:666 0:444 0:333 0:555 0:333 0:222 0:333 0:222 1:0 0:888 0:888 0:777 0:555 0:777 0:666 0:888 0:666 0:777 0:888 0:444 0:777 0:666 0:777

0:500 0:300 0:700 0:600 0:200 0:100 0:0 0:100 0:100 0:0 0:600 0:100 0:300 0:300 0:200 1:0 0:500 0:200 0:500 0:400 0:400 0:200 0:0 0:300 0:100 0:300 0:200 0:400 0:600 0:400 0:300 0:300 0:0 0:500 0:500 0:0 0:300 0:300 0:300 0:300 0:600 0:300 0:400 0:300 0:600

0:777 0:555 0:111 0:777 0:444 0:777 0:555 0:222 0:444 0:333 0:777 0:444 1:0 1:0 0:555 0:888 0:555 0:555 0:222 0:222 0:888 0:444 0:888 0:888 0:777 0:666 0:222 0:555 0:222 0:555 0:555 0:444 0:444 0:555 0:777 0:666 0:333 0:555 0:333 0:444 0:333 0:555 0:111 0:444 0:666

0:600 0:100 1:0 0:0 0:100 0:0 0:100 0:200 0:200 0:100 0:100 0:200 0:100 0:100 0:300 0:0 0:100 0:400 0:600 0:0 0:200 0:100 0:300 0:200 0:300 0:500 0:0 0:100 0:100 0:200 0:200 0:100 0:200 0:400 0:400 0:400 0:0 0:400 0:500 0:200 0:500 0:400 0:100 0:200 1:0

0:500 0:500 0:125 0:375 0:375 0:250 0:625 0:625 1:0 1:0 0:375 0:500 0:250 0:625 0:500 0:250 0:500 0:125 1:0 0:125 0:375 0:500 0:500 0:750 0:0 0:0 0:125 0:250 0:250 0:125 0:125 1:0 0:375 0:250 0:125 0:125 0:750 0:625 0:0 0:500 0:125 0:125 0:750 0:750 0:625

0:0 0:0 0:0 0:100 0:0 0:500 0:300 0:0 0:400 0:500 0:0 0:0 0:0 0:0 0:100 0:100 0:0 0:300 0:0 1:0 0:0 0:0 0:200 0:0 0:200 0:200 0:600 0:0 0:0 0:0 0:0 0:0 0:600 0:0 0:200 0:900 0:100 0:200 0:100 0:0 0:200 0:0 0:0 0:800 0:900

0:900 1:0 0:500 0:900 0:800 1:0 0:800 0:700 1:0 0:900 0:900 0:900 0:900 0:800 0:900 0:400 0:900 0:800 0:400 0:900 0:800 0:700 0:800 1:0 0:900 0:900 0:900 0:800 1:0 1:0 0:900 0:600 0:700 1:0 1:0 0:600 0:900 1:0 0:900 1:0 0:800 1:0 0:900 1:0 0:100

1:0 0:900 0:400 1:0 0:900 0:900 0:200 0:800 0:700 1:0 0:700 1:0 0:800 0:200 1:0 0:700 1:0 1:0 1:0 0:800 0:600 0:800 1:0 0:800 1:0 1:0 1:0 0:900 0:900 0:900 0:700 0:700 0:800 0:700 0:700 1:0 0:800 0:700 1:0 0:900 1:0 0:900 1:0 0:900 0:0


7:12:38pm

WSPC/173-IJITDM

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ISSN: 0219-6220


19

Table A.1. (Continued ) x ij1

x ij2

x ij3

x ij4

x ij5

x ij6

x ij7

x ij8

x ij9

x ij10

x ij11

1:0 0:500 0:700 0:500 0:800 0:400 0:100 0:100 0:700 0:600 0:200 0:700 0:700 0:100 0:200 0:800 0:500 0:700 0:500 0:100 0:800 0:500 0:500 0:200 0:700 0:500 1:0 0:100 0:600 0:200 0:700 0:800 0:300 0:500 0:400 0:800 0:600 0:500 0:700 0:600 0:300 0:700 497

0:400 0:100 0:800 0:300 0:0 0:0 0:700 0:200 0:300 0:800 0:600 0:800 0:800 0:900 0:700 0:0 0:700 0:800 0:700 0:300 0:700 0:800 0:700 0:500 0:900 0:100 0:500 1:0 0:700 0:700 0:400 0:600 1:0 0:700 0:300 0:700 0:800 0:600 0:500 0:400 0:700 0:200 491

0:200 0:600 0:500 0:200 0:700 0:900 0:200 0:600 0:400 0:500 0:0 0:200 0:300 0:400 0:100 0:400 0:300 0:300 0:200 0:0 0:600 0:400 0:100 1:0 0:500 0:200 0:800 0:200 0:500 0:400 0:200 0:300 0:200 0:900 0:100 0:300 0:0 0:200 0:600 0:300 0:100 0:400 485

0:888 0:666 0:555 0:777 0:444 0:222 0:222 0:777 0:555 0:666 0:444 0:444 0:888 0:444 0:888 0:555 0:333 0:555 0:222 0:888 0:888 0:222 0:777 0:888 1:0 0:333 0:888 0:666 0:111 0:444 0:777 0:666 0:666 0:333 1:0 0:555 0:444 0:666 0:111 0:777 0:555 0:777 484

0:300 0:200 0:200 0:0 0:200 0:200 0:500 0:300 0:100 0:400 0:300 0:0 0:400 0:600 0:500 0:100 0:200 0:400 0:600 0:800 0:200 0:700 0:300 0:800 0:800 0:800 0:400 0:900 0:100 0:600 0:600 0:400 0:800 0:300 0:200 0:500 0:300 0:400 0:300 0:200 0:500 0:100 480

0:444 0:222 0:333 0:555 0:222 1:0 0:888 0:333 0:777 0:222 0:777 0:333 0:555 0:777 0:222 0:444 0:777 0:444 0:777 0:333 0:333 0:555 0:333 0:333 0:111 0:555 0:222 0:333 0:777 0:777 0:222 0:444 0:888 0:111 0:777 0:0 0:666 0:777 0:777 0:444 0:777 0:222 467

0:100 0:800 0:100 0:400 0:100 0:700 0:400 0:500 0:200 0:100 0:400 0:300 0:500 0:200 0:600 0:700 0:0 0:200 0:0 0:500 0:300 0:200 0:200 0:100 0:100 0:900 0:200 0:0 0:0 0:300 0:500 0:0 0:400 0:0 0:500 0:200 0:400 0:300 0:100 0:0 0:400 0:500 457

1:0 0:500 0:0 0:125 0:750 0:125 0:750 0:0 0:0 0:250 0:125 0:125 0:250 0:375 1:0 0:375 0:125 0:125 0:500 0:875 0:125 0:125 0:0 0:375 0:500 0:875 0:0 0:625 0:500 0:125 0:0 0:125 0:125 0:750 0:750 0:500 0:125 0:125 0:500 0:125 0:0 0:0 453

0:0 0:0 0:300 0:700 0:400 0:800 0:0 1:0 0:500 0:0 0:700 0:600 0:100 0:0 0:0 0:200 0:600 0:0 0:100 0:600 0:0 0:0 0:600 0:0 0:0 1:0 0:100 0:300 0:300 0:0 0:100 0:200 0:0 0:100 0:0 0:0 0:200 0:0 0:0 0:900 0:200 0:600 409

0:600 1:0 0:900 1:0 1:0 0:500 1:0 0:900 0:900 1:0 0:900 1:0 0:0 1:0 0:400 1:0 0:900 1:0 0:900 0:200 0:500 1:0 1:0 0:700 0:600 0:0 0:600 0:600 0:900 1:0 1:0 0:900 0:600 1:0 0:900 0:900 0:900 0:900 0:900 0:700 0:900 0:900 388

0:700 0:900 1:0 0:900 0:900 0:600 0:800 0:700 1:0 0:900 1:0 0:900 1:0 0:700 1:0 0:900 1:0 0:900 1:0 1:0 1:0 0:900 0:900 0:600 0:300 0:300 0:700 0:800 1:0 0:900 0:900 1:0 0:500 0:800 0:700 1:0 1:0 1:0 1:0 1:0 1:0 1:0 376

j

y kj 1

y kj 2

y kj 3

y kj 4

y kj 5

y kj 6

y kj 7

y kj 8

y kj 9

y kj 10

y kj 11

1 2 3 4 5 6

0:400 0:500 0:800 0:800 0:100 0:300

0:368 0:894 0:263 0:789 0:578 0:473

0:600 0:700 0:600 0:750 0:600 0:900

0:400 0:500 0:600 0:200 0:100 0:400

0:400 0:200 0:100 0:500 0:010 0:600

0:789 1:0 0:368 0:052 1:0 0:684

0:800 0:900 0:100 0:100 0:500 0:700

0:400 0:600 0:300 0:900 0:500 0:700

0:777 0:777 0:555 0:555 0:222 0:777

0:500 1:0 0:750 0:687 0:375 1:0

0:428 0:714 0:571 1:0 0:571 0:714

j 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 E xi


20

7:12:58pm

WSPC/173-IJITDM

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ISSN: 0219-6220

A. Krylovas et al. Table A.1. (Continued )

j 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

y kj 1

y kj 2

y kj 3

y kj 4

y kj 5

y kj 6

y kj 7

y kj 8

y kj 9

y kj 10

y kj 11

0:700 0:500 0:100 0:150 0:200 0:900 0:800 0:100 0:600 0:050 0:610 0:600 0:700 0:900 0:500 0:100 0:600 0:100 0:800 0:600 0:600 0:300 0:050 0:300 0:100 0:200 0:200 0:900 0:700 0:600 0:200 0:600 0:700 0:700 0:900 0:800 0:500 0:300 0:300 0:200 0:800 0:700 0:800 0:900 0:200 0:800 0:500 0:100 0:200 0:900 0:200 0:400

0:684 0:684 0:789 0:789 0:578 0:894 0:684 0:263 0:789 0:473 0:726 0:473 0:578 0:684 0:789 0:473 0:894 0:052 0:578 0:052 0:684 0:421 0:0 0:368 0:263 0:789 0:789 0:684 0:473 0:368 0:578 0:894 0:578 0:789 0:789 0:473 0:684 0:578 0:684 0:789 0:263 0:473 0:842 0:684 0:368 0:684 0:473 0:052 0:473 0:684 0:789 0:368

0:700 0:600 1:0 0:500 0:800 1:0 1:0 1:0 0:800 0:700 0:660 0:800 0:400 0:800 0:500 0:600 0:800 0:300 0:800 1:0 0:800 0:400 0:100 0:400 0:600 0:950 0:800 0:800 0:900 0:200 0:600 0:700 0:700 0:700 0:700 0:900 0:700 0:700 1:0 0:950 0:800 0:200 0:900 0:800 0:150 0:800 0:600 0:700 0:300 0:800 0:700 0:500

0:900 0:500 0:800 0:100 0:300 0:800 0:800 0:900 0:500 0:700 0:400 0:750 0:600 0:600 0:500 0:400 0:600 0:300 0:700 0:200 0:700 0:500 0:200 0:500 0:300 0:500 0:400 0:600 0:200 0:100 0:800 0:400 0:500 0:300 0:800 0:600 0:600 0:400 0:800 0:500 0:300 0:500 0:400 1:0 0:300 0:900 0:500 0:150 0:600 0:600 0:700 0:600

0:200 0:200 0:750 0:700 0:100 0:600 0:100 0:100 0:500 0:500 0:100 0:300 0:200 0:200 0:500 0:100 0:500 0:100 0:600 0:100 0:800 0:100 0:050 0:300 0:100 0:400 0:600 0:200 0:200 0:800 0:500 0:400 0:500 0:100 0:700 0:500 0:600 0:300 0:500 0:400 0:800 0:800 0:750 0:200 0:100 0:400 0:100 0:300 0:200 1:0 0:800 0:100

1:0 0:578 0:894 0:789 1:0 1:0 0:021 0:473 0:894 0:789 0:473 0:578 0:263 1:0 0:578 1:0 0:684 0:052 0:789 0:894 0:894 0:473 0:0 0:473 0:473 0:894 0:578 1:0 0:789 0:368 0:578 0:894 0:684 0:894 0:684 0:789 0:684 0:473 0:368 0:894 0:263 0:789 0:684 0:684 0:473 0:684 0:473 0:052 0:368 1:0 0:894 1:0

0:900 0:600 0:100 0:800 0:300 1:0 0:600 0:600 1:0 0:500 0:210 0:800 0:800 0:700 0:500 0:600 0:700 0:100 0:700 0:100 0:800 0:800 0:100 0:500 0:200 0:800 0:800 0:700 0:600 0:250 0:800 0:800 0:600 0:500 1:0 0:700 0:700 0:400 0:800 0:800 0:100 0:500 0:500 0:900 0:400 0:900 0:800 0:200 0:700 0:900 0:700 0:900

0:200 0:700 0:900 0:100 0:800 0:700 0:900 0:500 0:900 0:900 0:920 0:700 0:600 0:900 0:700 0:500 0:700 0:300 0:800 0:300 0:900 0:400 0:150 0:800 0:600 0:600 0:200 0:900 0:800 0:0 0:500 0:600 0:500 0:800 0:800 0:500 0:600 0:800 0:600 0:600 0:900 0:800 0:500 0:900 0:800 0:700 0:700 0:300 0:500 0:800 0:200 0:700

0:222 0:777 0:555 1:0 0:888 1:0 0:777 0:333 1:0 0:555 0:333 0:666 0:777 0:666 0:666 0:666 0:777 0:0 1:0 0:777 1:0 0:888 0:111 0:333 0:111 0:666 0:333 0:666 0:666 0:555 0:555 0:666 0:777 0:888 1:0 0:777 0:888 0:555 1:0 0:666 0:555 1:0 0:944 0:777 0:666 1:0 0:222 0:222 1:0 1:0 1:0 1:0

0:875 0:375 0:625 1:0 0:750 1:0 0:750 0:750 1:0 0:500 0:062 1:0 0:625 1:0 0:375 0:375 0:625 0:125 1:0 0:875 1:0 0:125 0:0 0:750 0:250 0:375 1:0 1:0 1:0 0:875 0:875 0:875 1:0 0:750 1:0 1:0 0:625 0:875 1:0 0:375 1:0 1:0 0:375 1:0 0:250 1:0 0:500 0:875 0:750 1:0 0:812 0:062

0:857 0:285 1:0 0:0 0:285 0:714 0:714 1:0 0:714 0:428 0:642 0:857 0:571 0:857 0:285 0:714 0:571 0:285 1:0 0:857 1:0 0:0 0:0 0:714 0:714 0:571 0:428 0:857 0:714 0:571 0:285 0:285 1:0 0:714 0:714 1:0 0:571 0:428 1:0 0:571 0:285 1:0 0:428 0:714 0:571 0:571 0:857 0:428 0:428 1:0 0:857 1:0


7:13:17pm

WSPC/173-IJITDM

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ISSN: 0219-6220


21

Table A.1. (Continued ) j 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 E yk j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

y kj 1

y kj 2

y kj 3

y kj 4

y kj 5

y kj 6

y kj 7

y kj 8

y kj 9

y kj 10

y kj 11

0:500 0:400 1:0 0:400 0:700 1:0 0:600 0:200 0:400 0:700 0:500 0:100 0:300 0:500 0:100 0:100 0:450 0:600 0:0 0:100 0:300 0:500 0:400 0:800 0:500 0:200 0:500 0:700 0:900 547

0:578 0:368 1:0 0:473 0:157 1:0 0:684 0:789 0:526 0:263 0:789 1:0 0:473 0:473 0:263 0:578 0:315 0:052 0:157 0:263 0:578 0:684 0:526 0:684 0:473 0:473 0:263 0:789 0:789 531

0:800 0:500 0:0 0:700 0:800 0:700 0:800 0:950 0:300 0:800 0:500 1:0 0:850 0:500 0:500 1:0 0:800 0:500 0:900 0:500 0:700 0:850 0:300 0:800 0:800 0:800 0:600 0:900 1:0 521

0:500 0:600 0:0 0:500 0:700 1:0 0:700 0:500 0:500 0:400 0:500 0:300 0:900 1:0 0:100 0:0 0:500 0:500 0:500 0:100 0:400 0:400 0:500 0:800 0:600 0:030 0:800 0:700 0:700 519

0:100 0:100 0:0 0:500 0:100 0:900 0:800 0:400 0:100 0:700 0:500 0:100 0:400 0:500 0:050 0:700 0:200 0:400 0:300 0:050 0:300 0:050 0:100 0:900 0:500 0:010 0:500 0:900 0:500 518

0:473 1:0 1:0 0:684 0:789 0:578 0:894 0:894 0:578 1:0 0:578 1:0 0:894 1:0 0:157 0:578 0:526 0:789 1:0 0:157 0:473 0:894 0:578 1:0 0:789 0:684 0:526 0:989 1:0 509

0:100 0:900 0:0 0:500 1:0 0:800 0:800 0:800 0:700 0:800 0:500 0:300 1:0 1:0 0:400 0:900 0:400 0:900 1:0 0:400 0:400 0:200 0:700 0:900 0:800 0:800 0:200 0:800 1:0 508

0:700 0:700 1:0 0:500 1:0 0:900 0:900 0:600 0:500 0:800 0:700 0:800 0:600 0:500 0:800 0:750 0:600 0:300 0:600 0:800 0:800 0:700 0:500 0:500 0:600 0:700 0:600 0:500 0:800 487

0:555 1:0 1:0 1:0 0:555 0:777 1:0 0:666 0:722 1:0 0:666 0:444 0:666 1:0 0:111 0:555 0:888 0:777 1:0 0:111 0:555 0:777 0:722 0:888 0:333 0:0 0:222 0:888 1:0 479

0:875 0:062 1:0 0:750 0:750 1:0 1:0 0:375 1:0 1:0 0:375 1:0 0:750 0:875 0:375 0:937 0:937 0:500 1:0 0:375 0:875 0:875 1:0 1:0 0:937 0:375 0:625 1:0 1:0 451

0:714 1:0 1:0 0:571 0:571 1:0 1:0 0:571 0:857 0:571 0:285 1:0 0:857 0:285 0:571 0:857 0:642 0:285 0:714 0:571 0:428 0:714 0:857 0:714 0:642 0:428 0:642 0:857 0:428 439

z lj1

z lj2

z lj3

z lj4

z lj5

z lj6

z lj7

z lj8

z lj9

z lj10

z lj11

0:285 0:795 0:081 0:489 0:183 0:591 0:285 0:285 0:540 0:693 0:183 0:591 0:285 0:081 0:693 0:489 0:081 0:489 0:183

0:473 0:473 0:684 0:894 0:263 0:263 0:473 0:473 0:263 0:105 0:473 0:894 0:789 0:157 0:473 0:368 0:736 0:684 0:684

0:444 0:888 0:333 0:888 0:333 0:444 0:444 0:222 0:444 0:444 0:777 0:888 0:888 0:333 0:444 0:666 0:888 0:444 0:611

0:444 0:555 0:333 0:444 0:0 0:333 0:055 0:222 0:444 0:0 0:444 0:444 0:777 0:111 0:444 0:666 0:722 0:666 0:500

0:777 0:777 0:777 1:0 0:555 0:333 0:444 0:444 0:555 0:0 0:444 0:666 1:0 1:0 0:666 0:777 0:888 0:888 0:744

0:750 1:0 0:375 1:0 1:0 0:750 0:750 0:375 1:0 0:375 0:750 1:0 0:937 0:375 0:500 0:750 0:937 0:750 0:250

0:500 0:700 0:700 0:800 0:900 0:900 1:0 0:500 0:700 0:700 0:800 1:0 1:0 1:0 0:700 0:800 0:850 0:900 0:450

0:500 0:800 0:700 0:800 0:700 0:900 0:900 0:800 0:400 0:500 0:700 0:900 1:0 1:0 0:600 0:800 0:800 0:800 0:600

0:800 1:0 0:200 0:700 0:700 0:800 0:800 0:500 0:900 0:500 0:800 1:0 1:0 0:900 0:800 0:700 0:950 1:0 0:800

0:857 1:0 0:571 0:857 0:714 0:857 0:428 0:714 0:714 1:0 1:0 1:0 1:0 0:571 1:0 0:714 0:714 0:857 0:771

0:800 1:0 0:800 1:0 1:0 1:0 1:0 0:800 0:800 1:0 0:800 1:0 1:0 1:0 1:0 0:400 0:0 1:0 0:400


22

7:13:37pm

WSPC/173-IJITDM

1750027

ISSN: 0219-6220

A. Krylovas et al. Table A.1. (Continued )

j 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

z lj1

z lj2

z lj3

z lj4

z lj5

z lj6

z lj7

z lj8

z lj9

z lj10

z lj11

0:285 0:489 0:081 1:0 0:081 0:591 0:081 0:795 0:387 0:030 0:285 0:081 0:591 0:795 0:183 0:591 1:0 0:591 0:489 0:489 0:489 1:0 0:693 0:591 0:285 0:387 0:591 0:183 1:0 0:897 0:285 0:285 0:387 0:081 0:591 0:387 0:693 0:693 0:132 0:081 0:132 0:081 0:489 0:081 0:897 0:795 0:591 0:183 0:693 0:489 0:081 0:744

0:894 0:473 0:368 0:894 0:368 0:684 0:578 0:473 0:263 0:0 0:473 0:052 0:368 0:578 0:894 0:473 1:0 0:157 0:684 0:736 0:578 0:684 0:789 0:578 0:263 0:368 0:368 0:157 0:789 0:789 0:894 0:368 0:842 0:578 0:473 0:368 0:684 0:842 0:368 0:578 0:368 1:0 0:368 0:578 1:0 0:473 0:368 0:473 0:684 0:473 0:052 0:263

0:888 0:111 0:444 0:777 0:222 0:666 0:111 0:666 0:277 0:055 0:666 0:222 0:444 1:0 0:666 0:444 0:222 0:666 1:0 0:611 0:777 0:555 0:555 0:888 0:555 0:666 0:444 0:666 0:444 0:777 1:0 0:888 0:666 0:777 0:055 0:777 0:444 0:777 0:277 0:777 0:277 1:0 0:444 0:222 0:222 0:666 0:444 0:555 0:222 0:111 0:444 0:666

0:444 0:222 0:444 0:222 0:0 0:666 0:333 0:555 0:222 0:0 0:555 0:333 0:500 0:333 0:777 0:444 0:0 0:777 0:555 0:444 0:666 0:777 0:666 0:666 0:555 0:555 0:500 0:0 0:666 0:444 0:888 0:888 0:888 0:777 0:222 0:444 0:444 0:444 0:444 0:777 0:444 1:0 0:444 1:0 0:888 0:555 0:500 0:444 0:444 0:222 0:777 0:555

0:888 0:444 0:555 0:666 0:111 1:0 0:777 1:0 0:222 0:222 0:777 0:555 0:555 0:777 0:888 0:444 1:0 0:777 0:666 1:0 0:777 0:777 1:0 0:833 0:444 1:0 0:555 0:444 0:666 0:722 0:777 0:888 0:777 0:888 0:555 0:777 1:0 0:944 1:0 1:0 1:0 1:0 0:666 0:666 1:0 1:0 0:555 0:888 0:333 0:444 1:0 0:444

0:375 0:250 1:0 0:875 0:125 0:625 0:750 0:625 0:375 0:0 0:500 0:375 0:937 1:0 1:0 0:500 1:0 0:875 0:750 0:875 0:625 0:750 0:875 0:812 0:625 0:375 0:937 0:625 0:750 0:750 1:0 0:625 0:750 0:625 0:625 0:750 0:375 0:812 1:0 0:750 1:0 1:0 0:875 0:750 0:500 0:625 0:937 0:562 1:0 0:250 1:0 1:0

0:900 0:500 0:700 0:900 0:500 0:900 1:0 0:900 0:400 0:200 0:600 0:600 0:800 1:0 0:800 0:800 1:0 0:600 0:900 0:700 0:950 0:800 0:800 0:800 0:700 1:0 0:800 0:200 1:0 0:950 0:900 0:400 0:900 0:600 0:700 0:500 0:700 0:750 0:500 0:900 0:500 0:0 0:700 0:800 0:500 0:900 0:800 0:500 0:700 0:500 1:0 0:900

0:900 0:500 0:200 0:900 0:500 0:800 0:300 0:800 0:500 0:500 0:700 0:700 0:500 0:900 0:900 0:500 0:800 1:0 0:700 0:800 0:700 1:0 0:700 0:850 0:650 0:800 0:500 0:700 0:800 0:600 1:0 0:600 1:0 0:500 0:150 0:800 0:800 0:800 0:700 0:800 0:700 0:0 0:500 0:600 1:0 0:800 0:500 1:0 0:700 0:500 0:500 0:800

0:800 0:500 0:600 0:900 0:500 0:900 0:800 0:900 0:500 0:300 0:500 0:600 0:800 1:0 0:900 0:700 1:0 1:0 1:0 0:800 0:900 0:800 0:900 0:900 0:700 0:800 0:800 0:900 0:900 0:600 0:900 0:900 1:0 0:800 0:800 1:0 0:700 0:900 1:0 0:700 1:0 0:0 0:500 1:0 0:900 0:900 0:800 0:800 0:800 0:500 0:500 0:800

1:0 0:0 0:571 1:0 0:142 1:0 1:0 1:0 0:857 0:285 0:571 0:571 0:642 0:714 0:928 0:714 0:857 0:714 0:857 0:857 0:857 1:0 0:857 0:714 0:571 1:0 0:642 0:142 1:0 0:857 1:0 0:857 1:0 0:285 1:0 1:0 0:857 1:0 1:0 0:714 1:0 1:0 1:0 0:428 0:571 1:0 0:642 0:714 1:0 0:0 0:714 1:0

1:0 0:0 0:600 1:0 0:0 1:0 0:800 1:0 0:400 0:200 0:800 0:200 0:500 1:0 1:0 1:0 1:0 0:800 1:0 1:0 1:0 1:0 1:0 0:600 0:800 1:0 0:500 1:0 1:0 0:800 1:0 0:600 1:0 0:400 1:0 1:0 1:0 1:0 0:500 1:0 0:500 1:0 1:0 0:600 1:0 1:0 0:500 1:0 1:0 0:0 1:0 0:400


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Table A.1. (Continued ) j 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 E zl

z lj1

z lj2

z lj3

z lj4

z lj5

z lj6

z lj7

z lj8

z lj9

z lj10

z lj11

0:489 0:030 0:693 0:336 0:489 0:591 0:030 0:285 0:081 0:183 1:0 0:693 0:0 0:489 0:897 0:693 554

0:473 0:578 0:157 0:473 0:789 0:052 0:578 0:263 0:684 0:473 0:473 0:684 0:052 0:631 0:684 1:0 537

0:444 0:222 0:666 0:277 0:0 0:111 0:222 0:555 0:611 0:555 0:555 0:666 0:666 0:333 0:777 0:444 522

0:444 0:888 1:0 0:500 0:111 0:333 0:888 0:555 0:222 0:444 0:888 0:666 0:555 0:611 0:444 0:888 521

0:888 0:444 0:444 0:666 0:444 0:777 0:444 0:444 0:722 0:888 0:555 0:555 0:444 0:777 0:888 0:888 486

0:875 0:750 0:750 0:500 0:937 1:0 0:750 0:625 0:875 0:562 0:875 0:625 0:625 0:625 1:0 0:750 478

0:900 0:700 1:0 0:800 0:600 0:600 0:700 0:700 0:500 0:500 1:0 0:900 0:800 0:500 0:900 1:0 476

1:0 0:700 1:0 0:700 0:800 0:800 0:700 0:650 0:900 1:0 1:0 0:800 0:800 0:800 0:800 0:400 456

1:0 0:600 0:800 0:700 0:900 1:0 0:600 0:700 0:900 0:800 0:900 0:700 0:600 0:300 0:900 0:500 426

1:0 0:571 0:857 0:928 0:857 1:0 0:571 0:571 1:0 0:714 0:857 0:714 0:714 0:0 0:857 1:0 418

1:0 0:400 1:0 1:0 0:900 1:0 0:400 0:800 0:900 1:0 1:0 0:900 1:0 0:600 1:0 1:0 325

Appendix B

Table B.1. Criteria (14), minfW x ; W y ; W z g and W x þ W y þ W z values for all 87 respondents. j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

W x ðX j Þ

W y ðY j Þ

W z ðZ j Þ

minfW x ; W y ; W z g

Wx þ Wy þ Wz

0:495 0:500 0:496 0:497 0:504 0:500 0:511 0:508 0:504 0:507 0:504 0:504 0:501 0:509 0:502 0:496 0:495 0:499 0:507 0:488 0:506 0:508 0:506

0:532 0:672 0:440 0:550 0:394 0:601 0:630 0:565 0:639 0:531 0:538 0:883 0:651 0:464 0:763 0:540 0:507 0:629 0:548 0:731 0:577 0:468 0:697

0:558 0:781 0:464 0:778 0:500 0:594 0:540 0:441 0:574 0:423 0:604 0:828 0:833 0:507 0:622 0:648 0:724 0:728 0:545 0:721 0:363 0:468 0:825

0:495 0:500 0:440 0:497 0:394 0:500 0:511 0:441 0:504 0:423 0:504 0:504 0:501 0:464 0:502 0:496 0:495 0:499 0:507 0:488 0:363 0:468 0:506

1:585 1:953 1:400 1:825 1:398 1:695 1:681 1:514 1:717 1:461 1:646 2:215 1:985 1:480 1:887 1:684 1:726 1:856 1:600 1:940 1:446 1:444 2:028


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A. Krylovas et al. Table B.1. (Continued ) j 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

W x ðX j Þ

W y ðY j Þ

W z ðZ j Þ


Wx þ Wy þ Wz

0:509 0:493 0:489 0:491 0:498 0:491 0:491 0:498 0:517 0:506 0:496 0:494 0:495 0:509 0:509 0:492 0:504 0:497 0:495 0:503 0:504 0:505 0:517 0:499 0:492 0:496 0:504 0:494 0:501 0:488 0:496 0:495 0:492 0:493 0:501 0:496 0:507 0:501 0:494 0:495 0:500 0:505 0:498 0:491 0:492 0:497 0:503 0:506 0:497 0:498 0:500 0:492

0:144 0:751 0:467 0:781 0:451 0:078 0:430 0:300 0:627 0:526 0:731 0:605 0:378 0:549 0:667 0:625 0:639 0:818 0:683 0:651 0:498 0:660 0:627 0:552 0:625 0:717 0:767 0:357 0:763 0:487 0:238 0:455 0:850 0:644 0:581 0:497 0:581 0:580 0:572 0:633 0:862 0:781 0:627 0:472 0:704 0:577 0:565 0:651 0:690 0:268 0:557 0:519

0:253 0:765 0:540 0:754 0:387 0:142 0:545 0:371 0:598 0:800 0:765 0:555 0:803 0:681 0:744 0:717 0:711 0:813 0:776 0:747 0:509 0:660 0:598 0:384 0:805 0:748 0:837 0:636 0:797 0:570 0:509 0:648 0:667 0:791 0:596 0:671 0:596 0:601 0:576 0:582 0:764 0:754 0:598 0:582 0:653 0:363 0:550 0:692 0:719 0:520 0:704 0:563

0:144 0:493 0:467 0:491 0:387 0:078 0:430 0:300 0:517 0:506 0:496 0:494 0:378 0:509 0:509 0:492 0:504 0:497 0:495 0:503 0:498 0:505 0:517 0:384 0:492 0:496 0:504 0:357 0:501 0:487 0:238 0:455 0:492 0:493 0:501 0:496 0:507 0:501 0:494 0:495 0:500 0:505 0:498 0:472 0:492 0:363 0:503 0:506 0:497 0:268 0:500 0:492

0:906 2:009 1:496 2:026 1:336 0:711 1:466 1:169 1:742 1:832 1:992 1:654 1:676 1:739 1:920 1:834 1:854 2:128 1:954 1:901 1:511 1:825 1:742 1:435 1:922 1:961 2:108 1:487 2:061 1:545 1:243 1:598 2:009 1:928 1:678 1:664 1:684 1:682 1:642 1:710 2:126 2:040 1:723 1:545 1:849 1:437 1:618 1:849 1:906 1:286 1:761 1:574


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Table B.1. (Continued ) j 76 77 78 79 80 81 82 83 84 85 86 87

W x ðX j Þ

W y ðY j Þ

W z ðZ j Þ


Wx þ Wy þ Wz

0:493 0:497 0:496 0:499 0:510 0:499 0:495 0:497 0:501 0:496 0:492 0:493

0:520 0:561 0:268 0:498 0:573 0:472 0:807 0:600 0:409 0:473 0:797 0:859

0:589 0:587 0:520 0:509 0:619 0:582 0:800 0:699 0:475 0:515 0:811 0:760

0:493 0:497 0:268 0:498 0:510 0:472 0:495 0:497 0:409 0:473 0:492 0:493

1:602 1:645 1:284 1:506 1:702 1:553 2:102 1:796 1:385 1:484 2:100 2:112

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