New KEMIRA Method for Determining Criteria Priority

International Journal of Information Technology & Decision Making Vol. 13, No. 6 (2014) 1119–1133 c World Scienti¯c Publishing Company ° DOI: 10.1142/S0219622014500825

Int. J. Info. Tech. Dec. Mak. 2014.13:1119-1133. Downloaded from www.worldscientific.com by WSPC on 01/30/15. For personal use only.

New KEMIRA Method for Determining Criteria Priority and Weights in Solving MCDM Problem

Aleksandras Krylovas Department of Mathematical Modelling Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania Department of Mathematical Modelling Mykolas Romeris University Ateities st. 20, Vilnius, Lithuania

Edmundas Kazimieras Zavadskas Research Institute of Smart Building Technologies Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania

Natalja Kosareva Department of Mathematical Modelling Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania

Stanislav Dadelo Department of Physical Training Vilnius Gediminas Technical University Sauletekio av. 11, Vilnius, Lithuania Published 28 October 2014 This study presents a new KEmeny Median Indicator Ranks Accordance (KEMIRA) method for determining criteria priority and selection criteria weights in the case of two separate groups of criteria for solving multiple criteria decision making (MCDM) problem. Kemeny median method is proposed to generalize experts' opinion. Medians are calculated applying three different measure functions. Criteria weights are calculated and alternatives ranking accomplished by solving optimization problem minimization of ranks discrepancy function calculated for two groups of criteria. A numerical example for solving speci¯c task of elite selection from security personnel is given to illustrate the proposed method. Keywords: Kemeny median; preference feature; multiple criteria decision making; optimizing algorithm.

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1. Introduction The determination of the criteria weights for solving Multiple Criteria Decision Making (MCDM) problems is very important issue for the accuracy of the evaluation results. The subjective weight determination methods include Delphi method, expert judgment method, Analytic Hierarchy Process (AHP), step-wise weight assessment ratio analysis (SWARA), Factor Relationship (FARE) and other methods. SWARA which allows including experts' opinion about signi¯cance ratio of the attributes in the process of rational decision determination proposed in the paper.1 SWARA method was applied for selecting an appropriate location to establish a new shopping mall in order to prioritize and calculating the relative importance of the criteria.2 An original approach towards the development of multi-criteria assessment and ranking technique for alternatives of technology in construction proposed.3 In this paper three hybrid methods SWARA-TOPSIS, SWARA-ELECTRE III, SWARA-VIKOR were used to solve this problem. Priority of considered alternatives was determined based on the average of alternatives performance rank. A novel hybrid MADM method including SWARA and Vlse Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) were applied in decision-making process for agile supplier selection.4 Research5 presents a new framework for special situations using Yin-Yang balance (YYB) theory in producing and designing products for international producers and industries. SWARA method is applied for prioritizing important criteria of this issue. The paper6 is focused on selection of the optimal method for mechanical longitudinal ventilation of tunnel pollutants from four presented models. The authors used SWARA and VIKOR methods for managerial decision making in complex situations with multiple and varied measures. SWARA method was applied for evaluating and prioritizing sustainability assessment indicators of energy system.7 Two MADM methods, SWARA and complex proportional assessment of alternatives with gray relations (COPRAS-G), were applied for machine tool evaluation and selection.8 Integrated fuzzy MCDM model used with application of SWARA and additive ratio assessment (ARAS) method with fuzzy numbers (ARAS-F) for architect selection.9 Di®erent MCDM methods evaluate objects from di®erent aspects and thus they may produce distinct rankings of objects. The goal of the paper10 is to propose an approach to resolve disagreements among MCDM methods based on Spearman's rank correlation coe±cient. The other way to resolve disagreement in MCDM methods and help DMs pick the most suitable classi¯ers is the proposed fusion approach to produce a weighted compatible MCDM ranking of multiclass classi¯cation algorithms.11 The method for evaluation criteria weights is FARE method.12 It is based on the relationships between all the criteria describing the phenomenon considered. It allows to considerably reduce the amount of expert work and increase the accuracy of calculations.13 At ¯rst stage, a minimal amount of the initial data about the relationships between a part of the set of criteria, as well as their strength and direction,


New KEMIRA Method for MCDM Problem

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is elicited from experts. Then, the relations between other criteria of the set and their direction are determined analytically in compliance with those established at the ¯rst stage. The methodology for estimating the weights or saliences of sub criteria (attributes) in a composite criterion measure proposed by Srinivasan and Shocker.14 Each object is de¯ned by the set of attribute values and the set of paired comparison dominance judgments on the objects made by a single expert in terms of the global criterion is given. A criterion of ¯t is developed and its optimization performed via linear programming. Weighted least-square method is utilized to obtain the weights of belonging of each member to the fuzzy set.15 Saaty and Shang developed an innovative AHP-based structure to capture the relationship between various levels of activities contributed by people to society.16 The derivation of a priority vector from a pair-wise comparison matrix (PCM) is an important issue in the AHP. A cosine maximization method (CM) based on similarity measure is proposed for the priority vector derivation in AHP.17 The adapted Hadamard product model to identify and adjust the cardinally inconsistent element(s) in a PCM is proposed in the paper.18 In addition to the traditional approach of structuring criteria into multiple clusters, the alternatives of a decision are also organized into the lowest multiple levels of that hierarchy. A new method of attribute weight assignment based on rough set is proposed by Bao and Liu,19 authors declared that this approach can improve the ability of generalization and interpretation. Herrera et al. developed an aggregation process for dealing with nonhomogeneous contexts, for example combining numerical, interval valued and linguistic information.20 Wang and Fan presented two optimization aggregation approaches to determine the relative weights of individual fuzzy preference relations so that they can be aggregated into a collective fuzzy preference relation in an additively optimal manner.21 Methods based on entropy theory are the objective weight determination methods. The concept of entropy as the amount of disorder or chaos in a system was introduced by Shannon.22 Structure entropy weight method was analyzed by Cheng.23 This method combines qualitative analysis and quantitative analysis to con¯rm the weight of evaluating index. The quantitative order conclusion of weight is obtained by the entropy theory. Han and Liu were solving hybrid multiple attributes decision-making problems under risk with unknown attribute weights.24 A new decision approach based on entropy weight and TOPSIS is proposed in the paper. Application of entropy method proposed in the work of Kildien_e et al.25 The main goal of the work was to group investigated European countries applying the COPRAS method and evaluating relevance of six criteria, describing the construction sector, via entropy method. The paper of Sušinskas et al.26 presents the process of selection the pilecolumns instalment alternative. Criteria weights were determined by applying entropy method. The solution of problem was made by applying ARAS method. Wang and Lee developed a fuzzy TOPSIS approach based on subjective weights and objective weights.27 Subjective weights were assigned by DM and objective weights were adopted as the end-user ratings based on Shannon's entropy theory.


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Ustinovichius et al.28 applied the developed quantitative multiple criteria decisionmaking (MCDM-1) method to the analysis of investments in construction. Subjective and objective integrated approach to determine attribute weights for solving MADM problems presented by Ma et al.29 The resultant attribute weights and rankings of alternatives re°ect both the subjective considerations of a decision maker (DM) and the objective information. Several versions of a mathematical programming model which determine attribute weights were proposed by Pekelman and Sen.30 Comparison of di®erent designs of building or its structure and selection of the best alternative using criteria of optimality was analyzed.31 Design solutions can be characterized by quantitative and qualitative criteria which possibly have di®erent weight, dimension and direction of optimization. Wei32 investigated the case with the attribute values taking the form of intuitionistic fuzzy numbers. An optimization model based on the basic idea of gray relational analysis (GRA) method, by which the attribute weights were determined, proposed in the paper. Detailed research of the variety of methods for aggregating opinions of experts group is proposed in the study.33 Solutions of this problem for the tasks raised in the construction sector were analyzed in the papers.34–36 The current paper is mostly concerned with method of Kemeny median for adjustment of ranked data for establishing of criteria priority and weights. The Kemeny– Young method was developed by John Kemeny37 to identify the most popular choices in an election. Kemeny median was applied in prioritizing the development of enterprises,38 for solving MCDM problems in sports,39 in the situation when there is a large number of indicators by remoteness from ideal trajectory of alternatives.40 An e±cient approach for the rank aggregation problem is proposed in the paper.41 This paper presents a measure of similarity between partial rankings. Orlov42 proposed a computation-friendly variation of Kemeny median, which avoids the \center of the bagel hole" e®ect. The sample average values in di®erent spaces were examined.43 The author introduced the weighted average values of type I corresponding to the sample, and type II, corresponding to the set of order statistics. Method for adjustment clustered rankings is proposed, allowing to \drive" the contradictions inside specially constructed clusters (groups).44 The distance between the ordered and disordered partitions was treated.45,46 Axiom system for similarity measures proposed.47 The advantage of the proposed KEMIRA method is that it allows simultaneous identi¯cation of the weights of criteria and accomplish procedure of ranking of the alternatives. This method is especially e±cient when there are separate groups of criteria and we must determine criteria weights in each group. For example, in the situation when there is combination of certain objects external and internal evaluation criteria.

2. Problem Formulation and Previous Results Current paper deals with alternative optimal solution ¯nding technique for MCDM problem formulated by Dadelo et al.48 The task under investigation is not a classic



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MCDM problem, because in this approach there are two separate groups of criteria. In general case it may be more than two groups of criteria. Qualitative (external evaluation) criteria values and quantitative measurements (internal evaluation) are given for each object. Additionally, information about experts' independently set preferences for criteria in each criterion group is known. According to this information we need to select 10% of best objects. At ¯rst stage of the problem solution criteria preferences which re°ect the aggregated expert opinion must be determined in each group, then criteria weights are estimated and ¯nally MCDM methods applied for selecting the best objects. Each problem solution phase could be performed in a variety of methods. Suppose that there are known certain measurements or expert evaluations of ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ K test takers T ðjÞ , j ¼ 1; 2; . . . ; K : x 1 ; x 2 ; . . . ; x n ; y 1 ; y 2 ; . . . ; y m , 0 6 x i , ðjÞ y i 6 1. Say ðj 1 Þ

T ðj 1 Þ º T ðj 2 Þ ;

if ð8 iÞ x i

ðj 2 Þ

> xi

ðj 1 Þ

& yi

ðj Þ

> yi 2 :

ð2:1Þ

Suppose that w xi , w yi are weighted coe±cients: 0 6 w xi ;

w yi 6 1;

n X

w xi ¼

i¼1

m X

w yi ¼ 1:

ð2:2Þ

i¼1

Then under conditions (2.1) the following unequalities will take place: V xðj 1 Þ > V xðj 2 Þ & V yðj 1 Þ > V yðj 2 Þ :

ð2:3Þ

Here V xðjÞ ¼

n X

ðjÞ

w xi x i ;

i¼1

V yðjÞ ¼

m X

ðjÞ

w yi y i :

ð2:4Þ

i¼1 ðj Þ

ðj Þ

In general, the results of the measurements could be as follows: x i 1 1 > x i 1 2 and ðj Þ ðj Þ x i 2 1 < x i 2 2 . Then it is not possible to apply the criteria (2.1) for selecting better alternative, nevertheless weighted criteria (2.4) could be applied with any values of weights (2.2). It is clear that weights w xi , w yi could not be chosen arbitrarily. Usually they are determined by expert assessments. However, such assessments do not ðjÞ ðjÞ guarantee proximity of values V x and V y , mostly we can only expect their sigðjÞ ðjÞ ðjÞ ni¯cant correlation. Let us denote R x and R y positive integers R x;y 2 f1; 2; . . . ; K g ðj 1 Þ ðj 2 Þ ðj 1 Þ ðj 2 Þ satisfying the condition: R x;y < R x;y , when V x;y > V x;y , i.e., they are ranks of ðjÞ numbers V x;y . Let us de¯ne X K x and Y K y as subsets of the set f1; 2; . . . ; K g: X K x ¼ ffj 1 ; j 2 ; . . . ; j K x g : R xðj i Þ 6 K x g;

Y K y ¼ ffj 1 ; j 2 ; . . . ; j K y g : R yðj i Þ 6 K y g:

X K x and Y K y are sets of the best alternatives according to one of the criteria (2.4) and having K x and K y elements respectively. Note that X 0 ¼ Y 0 ¼ , X K ¼ Y K ¼ f1; 2; . . . ; K g. The best alternatives selection task is equivalent to the task of ¯nding the set A K x ;K y ¼ X K x \ Y K y , which has the required number of elements jA K x ;K y j. If the set A K x ;K y was successfully found and numbers K x , K y are not

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big (for example, 0:1 K), it means that we succeeded to ¯nd best elements according to both weighted criteria (2.4). If the number of elements jA K x ;K y j insu±cient, it is necessary to increase numbers K x and K y . In this case the number of the set B K x ;K y ¼ ðX K x [ Y K y ÞnA K x ;K y elements jB K x ;K y j, which shows how many elements are the best only by one of the criteria (2.4), will increase. In the author's paper48 the personnel selection task has being solved when priority of criteria x i ; i ¼ 1; 2; . . . ; 6 and y i ; i ¼ 1; 2; . . . ; 9 was estimated by 22 experts. Experts sorted criteria in the priority descending order: ðsÞ

ðsÞ

ðsÞ

ðsÞ

ðsÞ

ðsÞ

x i1 x i2 x i6 ; y i1 y i2 y i9 ;

ð2:5Þ

here s 2 f1; 2; . . . ; 22g is the number of expert. Table 1 presents evaluation results. Criterion x i for the ¯rst place (2.5) is given 6 points, criterion y i – 9 points, for the second place respectively 5 and 8 points, for the last places one point. For the case when weighted coe±cients (2.2) are determined in proportion to the collected score (criteria x i and y i total scores are x ¼ 1þ6 2 6 22 ¼ 462, 1þ9 y ¼ 2 9 22 ¼ 990) the obtained values of weights are: 114 41 0:247; . . . ; w x i ¼ 0:089; 6 462 462 190 25 0:192; . . . ; w y i 9 ¼ 0:025 ¼ 990 990

wxi ¼ 1

wyi1

Table 1. Criteria X and Y components preferences established by experts. Expert 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

x1

x2

x3

x4

x5

x6

y1

y2

y3

y4

y5

y6

y7

y8

y9

6 5 5 6 6 5 4 6 5 5 4 6 6 6 4 4 6 6 4 6 5 4 114

1 2 1 1 1 1 2 2 4 4 1 2 4 3 1 1 1 3 1 2 2 1 41

5 3 3 5 5 4 3 5 6 6 3 4 5 5 3 3 4 4 5 4 6 6 97

3 1 2 3 2 2 1 3 3 1 2 1 1 1 2 2 2 1 2 1 3 3 42

2 6 6 4 3 6 6 4 2 2 6 5 3 4 6 6 3 5 6 5 4 2 96

4 4 4 2 4 3 5 1 1 3 5 3 2 2 5 5 5 2 3 3 1 5 72

7 7 5 5 7 5 4 6 7 6 5 6 5 7 5 7 6 6 4 5 6 6 127

9 9 9 9 9 9 8 8 9 7 8 9 9 8 9 9 8 9 9 9 8 9 190

4 5 4 7 6 6 5 5 4 4 6 5 6 6 7 8 7 8 8 6 7 7 131

8 6 6 8 8 8 9 9 8 9 9 7 8 9 8 6 9 7 6 7 9 8 172

5 8 8 6 5 7 7 7 6 8 7 8 7 5 6 5 4 4 7 8 5 5 138

3 2 1 3 3 3 2 3 3 3 2 4 3 4 3 4 5 5 1 3 4 4 68

6 3 7 4 4 4 6 4 5 5 4 3 4 3 2 2 2 2 5 4 3 2 84

2 4 3 2 2 2 3 2 1 2 3 2 2 2 4 3 3 3 3 2 2 3 55

1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 25


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and sets X K x , Y K y , when K x ¼ K y ¼ 12 are as follows: X 12 ¼ f1; 6; 21; 22; 34; 46; 47; 56; 76; 81; 84; 96g; Y 12 ¼ f8; 21; 24; 26; 36; 45; 54; 56; 76; 81; 91; 101g: The set A 12;12 ¼ f21; 56; 76; 81g has only four common elements. This means that it is not possible to perform the best personnel selection according to the both criteria in such way. Increasing of numbers K x ¼ 12 and K y ¼ 12 do not help, because the set Int. J. Info. Tech. Dec. Mak. 2014.13:1119-1133. Downloaded from www.worldscientific.com by WSPC on 01/30/15. For personal use only.

B 12;12 ¼ f1; 6; 8; 22; 24; 26; 34; 36; 45; 46; 47; 54; 84; 91; 96; 101g; i.e., the set of elements (employees), for which there is doubt, will be unacceptably large. The idea of method proposed in the paper48 do not calculate weights w xi , w yi in formulas (2.4) a priori, but rather to solve optimization problems max jA K x ;K y j;

W x ;W y

min jB K x ;K y j;

W x ;W y

ð2:6Þ

where coe±cients W x ¼ ðw x1 ; w x2 ; . . . ; w x6 Þ, W y ¼ ðw y1 ; w y2 ; . . . ; w y9 Þ are consistent with the conditions (2.5). In this case, the weights satisfy the following inequalities: w x1 > w x3 > w x5 > w x6 > w x4 > w x2 ; w y2 > w y4 > w y5 > w y3 > w y1 > w y7 > w y6 > w y8 > w y9 :

ð2:7Þ

The solution of optimization problem (2.6) under restrictions (2.7) in Ref. 48 was found as follows: A 21;21 ¼ f21; 34; 36; 47; 56; 76; 77; 81; 91; 102; 106; 112g:

ð2:8Þ

3. Determining Priority of Criteria by Kemeny Median Method Suppose that X ¼ ðx 1 ; x 2 ; . . . ; x n Þ and permutations ðj 1 ; j 2 ; . . . ; j n Þ of the set of natural numbers f1; 2; . . . ; ng determine priorities of vector (criteria) X components: x j 1 x j 2 x j n . This means that for the set fx 1 ; x 2 ; . . . ; x n g a strict order relationship is de¯ned R ¼ fðx j 1 ; x j 2 Þ; ðx j 1 ; x j 3 Þ; . . . ; ðx j 1 ; x j n Þ; ðx j 2 ; x j 3 Þ; ðx j 2 ; x j 4 Þ; . . . ; ðx j 2 ; x j n Þ; . . . ; ðx j n1 ; x j n Þg: The set R corresponds to the square matrix A R ¼ jja ij jj, which elements are: 1; if ðx i ; x j Þ 2 R; a ij ¼ 0; if ðx i ; x j Þ 62 R: Notice that a ii ¼ 0 – relationship is antire°ective ir a ij ¼ 1 a ji , for i 6¼ j – antisymmetric. De¯ne function A ðRð1Þ ; Rð2Þ Þ ¼

n X n X i¼1 j¼1

ð1Þ

ð2Þ

ja ij a ij j:

ð3:9Þ

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Function (3.9) is a certain measure of di®erence between two relationships and its values coincide with values of Kemeny distance function Ref. 33. For example, Rð1Þ ¼ fð1; 2Þ; ð1; 3Þ; ð1; 4Þ; ð2; 3Þ; ð2; 4Þ; ð3; 4Þg;


Rð2Þ ¼ fð4; 1Þ; ð4; 2Þ; ð4; 3Þ; ð3; 1Þ; ð3; 2Þ; ð2; 1Þg: Then corresponding matrices 0 0 B 0 Að1Þ ¼ B @0 0

are: 1 0 0 0

1 1 0 0

1 1 1C C; 1A 0

0

Að2Þ

0 B1 ¼B @1 1

0 0 1 1

0 0 0 1

1 0 0C C: 0A 0

Function (3.9) in this case is gaining value A ðRð1Þ ; Rð2Þ Þ ¼ 12. Suppose that m experts established priorities Rð1Þ ; Rð2Þ ; . . . ; RðmÞ . Most consistent with these estimates will be priority R A , which is called median: R A ¼ arg min R

m X

A ðR; RðjÞ Þ:

ð3:10Þ

j¼1

Notice that relationships RðjÞ are uniquely de¯ned by permutations ðj 1 ; j 2 ; . . . ; j n Þ. For example, relationships Rð1Þ ir Rð2Þ are de¯ned by respective permutations ð1; 2; 3; 4Þ ir ð4; 3; 2; 1Þ. We will analyze functions' A analogues M and I : n X ð1Þ ð2Þ M ðRð1Þ ; Rð2Þ Þ ¼ jj i j i j; ð3:11Þ i¼1

I ðRð1Þ ; Rð2Þ Þ ¼ Invers

ð1Þ

j2 . . . jn

ð2Þ

j2 . . . jn

j1 j1

ð1Þ

ð1Þ

ð2Þ

ð2Þ

! ;

ð3:12Þ ð1Þ

ð1Þ

here Invers ð Þ means the number of inversions49 of the substitute ðj 1 ; j 2 ; . . . ; ð1Þ ð2Þ ð2Þ ð2Þ j n Þ ! ðj 1 ; j 2 ; . . . ; j n Þ. For the current example, M ðRð1Þ ; Rð2Þ Þ ¼ j1 4j þ j2 3j þ j3 2j þ j4 1j ¼ 8; I ðRð1Þ ; Rð2Þ Þ ¼ 3 þ 2 þ 1 ¼ 6: The number of inversions of the substitute ð1; 2; 3; 4Þ ! ð4; 3; 2; 1Þ was calculated hereby. First of all inversions of the ¯rst element 4 of the permutation ð4; 3; 2; 1Þ with the right-hand elements are calculated comparing with the natural permutation ð1; 2; 3; 4Þ. We get three inversions. Then inversions of the second (3) and third (2) elements are calculated we get respectively two and one inversions. By analogy with (3.10) applying formulas (3.11) and (3.12) medians R M and R I sought: R M ¼ arg min R

m X j¼1

M ðR; RðjÞ Þ:

ð3:13Þ


R I ¼ arg min R

m X

I ðR; RðjÞ Þ:

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ð3:14Þ

j¼1

It is notable that number of medians can be greater than 1. We will look for solutions which coincide with medians calculated by all three methods: R A , R M and R I .


4. Calculation of Weights by Indicator Ranks Accordance Method De¯nition. Say that criteria convolution ’ has priority feature ðj 1 ; j 2 ; . . . ; j n Þ, if all inequalities are valid: ð8 i < k & > 0Þ ’ð. . . ; x i þ ; . . . ; x k ; . . .Þ > ’ð. . . ; x i ; . . . ; x k þ ; . . .Þ; when j i > j k . Notice that all convolutions constructed on the base of weighted averages have priority feature. For example, convolution ’ðXÞ ¼

w1x1 þ w2x2 þ w3x3 þ w4x4 ; w1 þ w2 þ w3 þ w4

while w 2 > w 3 > w 1 > w 4 > 0 has priority feature ð2; 3; 1; 4Þ. Suppose that criteria-referenced assessments are known for certain alternatives: ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ X ðkÞ ¼ ðx 1 ; x 2 ; . . . ; x n Þ, Y ðkÞ ¼ ðy 1 ; y 2 ; . . . ; y m Þ, k ¼ 1; 2; . . . ; K and criteria X, Y priorities are determined: ðj 1 ; j 2 ; . . . ; j n Þ, ði 1 ; i 2 ; . . . ; i m Þ. The task is to rank all K objects and select the best 10%. We must ¯nd the set of common solutions of two MCDM problems. Suppose that ’ and are criteria X and Y convolutions having corresponding priority features. Denote S ðj 1 ;j 2 ;...;j n Þ class of all convolutions having priority feature ðj 1 ; j 2 ; . . . ; j n Þ in which we will look for functions ’ (analogous ). We will restrict class of such functions to weighted averages having priority features: xj1 xj2 xjn ;

yi1 yi2 yim :

ð4:15Þ

According to the established criteria X, Y priorities (4.15) functions ’ðXÞ and ðY Þ are chosen as follows: ’ðXÞ ¼

n X

jr xjr ;

j1 > j2 > > jn > 0

is yis ;

i 1 > i 2 > > i m > 0:

r¼1

and ðY Þ ¼

m X s¼1

Require that the weighting coe±cients j and i satisfy normalizing condition n X r¼1

jr ¼

m X s¼1

i s ¼ 1:

ð4:16Þ


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For each alternative ðX ðkÞ ; Y ðkÞ Þ we will calculate values of each criteria convolutions ’ðX ðkÞ Þ, ðY ðkÞ Þ. Numbering them in ascending order we get ranks of alternatives: ðkÞ ðkÞ R x , R y . Notice that the lower the rank, the better is alternative. Denote A K x and B K y sets of the best alternatives, i.e., those subsets of the set ðkÞ ðkÞ f1; 2; . . . ; Kg, whose elements ranks satisfy the inequalities R x 6 K x and R y 6 K y . In the set A K x there are K x the best alternatives according to criteria X, similarly in the set B K y there are K y the best alternatives according to criteria Y . Numbers K x and K y are chosen so that the intersection of sets A K x \ B K y have not less than 10% of the best alternatives according to both criteria. Denote jA K x \ B K y j number of elements of the sets A K x and B K y intersection and formulate the optimization problem: max

’2S ðj 1 ;j 2 ;...;j n Þ 2S ði 1 ;i 2 ;...;i m Þ

jA K x \ B K y j:

ð4:17Þ

Condition (4.17) means that we must select convolutions ’ and , which maximize criteria X and Y compatibility, i.e., number of elements of sets A K x and B K y intersection should be the maximum. The number of convolutions ’ and , maximizing (4.17) can be great, therefore additional optimization problem must be formulated. Denote RðkÞ ðÞ and RðkÞ ðÞ ranks of the numbers f’ðX ð1Þ Þ; ’ðX ð2Þ Þ; . . . ; ’ðX ðK Þ Þg and f ðY ð1Þ Þ; ðY ð2Þ Þ; . . . ; ðY ðK Þ Þg respectively (k ¼ 1; 2; . . . ; K ). Weighting coe±cients j and i are sought by minimizing ranks discrepancy function, i.e., sum of squares of the highest ranks di®erences according to criteria X and Y : X CR K x ;K y ð; Þ ¼ min ðRðkÞ ðÞ RðkÞ ðÞÞ2 : ð4:18Þ ’2S ðj 1 ;j 2 ;...;j n Þ k2f1; 2; . . . ; K g: 2S ði 1 ;i 2 ;...;i m Þ RðkÞ ðÞ 6 K x RðkÞ ðÞ 6 K y

Here, K x and K y are chosen so that the number of elements in the intersection A K x \ B K y will be equal to the desired number of selected objects. 5. Case Study The present problem has been solved in the paper.48 The obtained results will be compared with previous results. About 118 security guards were randomly selected from the company G4S Lietuva. A total of 22 leader managers (experts) ranked the competences described below in the article. Personnel elite 10% the best employees of the private security company are selected according to six internal assessment criteria ðx 1 x 6 Þ and nine external evaluation criteria ðy 1 y 9 Þ. 6 internal evaluation criteria are objective tests and measurements: x 1 is employee's theoretical and practical preparation; x 2 professional activity, x 3 mental qualities; x 4 physical development; x 5 motor



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abilities (personal physical conditions allowing to carry out physical tasks); x6 ¯ghting e±ciency. 9 external evaluation criteria are evaluation of subordinate specialty knowledge, professionalism, y 2 diliby his immediate superior: y 1 gence and positive attitude to work, y 3 behavior with colleagues and supervisors; y4 reliability at work; y 5 quality of work; y 6 workload performance; y 7 image; y 8 development rate; y 9 being promising (potential to make a career). Security guards internal and external evaluation criteria values are presented in the paper.48 First of all criteria values were transformed to values x ij ; y ij , belonging to the range ½0; 1 by these transformations: x ij ¼

j x ij x min j x max

j x min

;

y ij ¼

j y ij y min j j y max y min

:

22 independent experts determined preferences separately for internal and external evaluation criteria. Criteria preferences established by experts are presented in Table 1. A higher grade means that the criterion is more important. Criteria X priority feature is determined from expert estimates of the form ðj 1 ; j 2 ; . . . ; j 6 Þ. For example, ¯rst expert set criteria X components priorities as follows: ðx 1 ; x 3 ; x 6 ; x 4 ; x 5 ; x 2 Þ. Here x 1 is the most important and x 2 the least important component. Criterion Y priority feature is established similarly. For example, ¯rst expert set them in this order: ðy 2 ; y 4 ; y 1 ; y 7 ; y 5 ; y 3 ; y 6 ; y 8 ; y 9 Þ. In Ref. 48 criteria priority feature (generalized opinion of experts) is determined by the sum (or average) of all experts evaluations. The result is presented by inequalities (2.7). The current paper proposes a new criteria priority feature establishing methodology applying three methods for calculating median with the respective metrics (3.9), (3.11) and (3.12) to calculate distances between priorities. In the present managerial problem calculation of medians could be performed by full re-selection of options, which will be respectively 6! ¼ 720 and 9! ¼ 362880. For criterion X components x j medians were calculated by three methods. Two solutions minimizing corresponding functions (3.10), (3.13) and (3.14) obtained: ð1; 5; 3; 6; 4; 2Þ, ð1; 3; 5; 6; 4; 2Þ. Criterion Y components y i priority features were determined uniquely for all three medians: ð2; 4; 5; 3; 1; 7; 6; 8; 9Þ. Therefore, criteria X and Y components order was established respectively: x1 x5 x3 x6 x4 x2

or x 1 x 3 x 5 x 6 x 4 x 2

ð5:19Þ

and y2 y4 y5 y3 y1 y7 y6 y8 y9 :

ð5:20Þ

From the priority features (5.19) and (5.20) it follows, that it is necessary to search for functions ’ and in the form: ’ 1 ðXÞ ¼ 1 x 1 þ 5 x 5 þ 3 x 3 þ 6 x 6 þ 4 x 4 þ 2 x 2 ; 1 > 5 > 3 > 6 > 4 > 2 > 0;

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or ’ 2 ðXÞ ¼ 1 x 1 þ 3 x 3 þ 5 x 5 þ 6 x 6 þ 4 x 4 þ 2 x 2 ; 1 > 3 > 5 > 6 > 4 > 2 > 0; and


ðY Þ ¼ 2 y 2 þ 4 y 4 þ þ 9 y 9 ;

2 > 4 > 5 > > 8 > 9 > 0:

De¯ne parameter values for our problem: number of investigated objects K ¼ 118, number of criterion X components n ¼ 6, number of criterion Y components m ¼ 9. Values K x ¼ 21 and K y ¼ 21 were chosen to ful¯ll the condition jA K x \ B K y j ¼ 12, since the goal is to select the top 12 security guards. In the case of functions ’ 1 ðXÞ and ðY Þ the following values of weighted coe±cients were found:

The best 12 security guards, belonging to the intersection of sets A 21 \ B 21 : a 21 , a 34 , a 36 , a 47 , a 56 , a 76 , a 77 , a 81 , a 91 , a 102 , a 106 , a 111 . In the case of functions ’ 2 ðXÞ and ðY Þ the lowest criterion (4.18) value was obtained with these weighted coe±cients values:

The same 12 security guards got to the intersection of sets A 21 \ B 21 : a 21 , a 34 , a 36 , a 47 , a 56 , a 76 , a 77 , a 81 , a 91 , a 102 , a 106 , a 111 . Applying an additional condition, when the solutions were sought between the functions that minimize the number of \doubtful" cases, i.e., having a high rank by one and low by another criterion, in the case of ’ 1 ðXÞ and ðY Þ, the set of solutions changed by the only one element instead of a 111 in the top 12 fell a 112 . The obtained results coincide with the results got in Ref. 48, since in the case of ’ 2 ðXÞ and ðY Þ criteria priorities were set in the same order as in the mentioned article.

6. Conclusions In this paper a new method of establishing criteria components preferences and determining of criteria weights the novel KEMIRA method was proposed. Criteria components preferences were established applying Kemeny median



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method. Criteria weights were determined by minimizing sum of squares of ranks discrepancies of two criteria groups. There is variety of other methods to determine criteria priority features. It was shown that method of Kemeny medians gives similar results comparing with the method based on the sum (or average) of all experts evaluations. This suggests that the proposed method is reasonable. Top 12 security guards were selected by maximizing number of elements which are the best according to the both groups of criteria and minimizing sum of squared discrepancies of ranks. Further our research object will be searching of other methods of criteria components priority setting and weighting coe±cients establishing. In many ¯elds there are not uni¯ed employee evaluation criteria and selection methods algorithms. Stability of emerging global political and economic space requires uni¯cation of procedural standards. The proposed algorithm de¯nes the employee selection process guidelines. This methodology allows to weight and synthesize subjective (managers assessments) and objective (the candidates skills) indicators. This opens up the new opportunities of application and development of decision-making methods not only in the selection of personnel. It is suggested that this type of research could be extended to other areas of human activities where MCDM problems arise (business, manufacturing, trade and etc.) References 1. V. Keršulien_e, E. K. Zavadskas and Z. Turskis, Selection of rational dispute resolution method by applying new step-wise weight assessment ratio analysis (SWARA), Journal of Business Economics and Management 11(2) (2010) 243–258. 2. S. Hashemkhani Zolfani, M. H. Aghdaie, A. Derakhti, E. K. Zavadskas and M. H. Varzandeh, Decision making on business issues with foresight perspective; an application of new hybrid MCDM model in shopping mall locating, Expert Systems with Applications 40 (17) (2013) 7111–7121. 3. E. K. Zavadskas, Z. Turskis, R. Volvačiovas and S. Kildiene, Multi-criteria assessment model of technologies, Studies in Informatics and Control 22(4) (2013) 249–258. 4. M. Alimardani, S. Hashemkhani Zolfani, M. H. Aghdaie and J. Tamošaitien_e, A novel hybrid SWARA and VIKOR methodology for supplier selection in an agile environment, Technological and Economic Development of Economy 19(3) (2013) 533–548. 5. S. Hashemkhani Zolfani, E. K. Zavadskas and Z. Turskis, Design of products with both international and local perspectives based on Yin-Yang balance theory and SWARA method, Economska Istra z ivanja Economic Research 26(2) (2013) 153–166. 6. S. Hashemkhani Zolfani, M. H. Esfahani, M. Bitarafan, E. K. Zavadskas and S. L. Are¯, Developing a new hybrid MCDM method for selection of the optimal alternative of mechanical longitudinal ventilation of tunnel pollutants during automobile accidents, Transport 28(1) (2013) 89–96. 7. S. Hashemkhani Zolfani and J. Saparauskas, New application of SWARA method in prioritizing sustainability assessment indicators of energy system, In z inerine_ Ekonomika Engineering Economics 24(5) (2013) 408–414. 8. M. H. Aghdaie, S. Hashemkhani Zolfani and E. K. Zavadskas, Decision making in machine tool selection: An integrated approach with SWARA and COPRAS-G methods, In z inerine_ Ekonomika Engineering Economics 24(1) (2013) 5–17.


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9. V. Keršulien_e and Z. Turskis, Integrated fuzzy multiple criteria decision making model for architect selection, Technological and Economic Development of Economy 17(4) (2011) 645–666. 10. G. Kou, Y. Peng and Y. Shi, Evaluation of classi¯cation algorithms using MCDM and rank correlation, International Journal of Information Technology & Decision Making 11(1) (2012) 197–225. 11. Y. Peng, G. Kou, G. Wang and Y. Shi, FAMCDM: A fusion approach of MCDM methods to rank multiclass classi¯cation algorithms, Omega 39(6) (2011) 677–689. 12. R. Ginevičius and V. Podvezko, The in°uence of complex evaluation methods on rating the alternative solutions of wall insulation, in The 9th Int. Conf. \Modern Building Materials, Structures and Techniques": Selected Papers (Vilnius, Lithuania 2007), pp. 248–251. 13. R. Ginevičius, A new determining method for the criteria weights in multicriteria evaluation, International Journal of Information Technology & Decision Making 10(6) (2011) 1067–1095. 14. V. Srinivasan and A. D. Shocker, Estimating the weights for multiple attributes in a composite criterion using pairwise judgments, Psychometrika 38(4) (1973) 473–493. 15. A. T. W. Chu, R. E. Kalaba and K. Spingarn, A comparison of two methods for determining the weights of belonging to fuzzy sets, Journal of Optimization Theory and Applications 27(4) (1979) 531–538. 16. T. L. Saaty and J. S. Shang, An innovative orders-of-magnitude approach to AHP-based mutli-criteria decision making: Prioritizing divergent intangible humane acts, European Journal of Operational Research 214(3) (2011) 703–715. 17. G. Kou and C. Lin, A cosine maximization method for the priority vector derivation in AHP, European Journal of Operational Research 235(1) (2014) 225–232. 18. G. Kou, D. Ergu and J. Shang, Enhancing data consistency in decision matrix: Adapting hadamard model to mitigate judgment contradiction, European Journal of Operational Research 236(1) (2014) 261–271. 19. X. Z. Bao and C. Liu, A new method of ascertaining attribute weight based on rough sets theory, Chinese Journal of Management 17(3) (2009) 131–135. 20. F. Herrera, L. Martinez and P. J. Sanchez, Managing non-homogeneous information in group decision making, European Journal of Operational Research 166(1) (2005) 115–132. 21. Y. M. Wang and Z. P. Fan, Fuzzy preference relations: Aggregation and weight determination, Computers & Industrial Engineering 53(1) (2007) 163–172. 22. C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (Urbana: University of Illinois Press, 1949). 23. Q. Cheng, Structure entropy weight method to con¯rm the weight of evaluating index, Systems Engineering-Theory & Practice 30(7) (2010) 1225–1228. 24. Z. Han and P. Liu, A fuzzy multi-attribute decision-making method under risk with unknown attribute weights, Technological and Economic Development of Economy 17(2) (2011) 246–258. 25. S. Kildien_e, A. Kaklauskas and E. K. Zavadskas, COPRAS based comparative analysis of the european country management capabilities within the construction sector in the time of crisis, Journal of Business Economics and Management 12(2) (2011) 417–434. 26. S. Sušinskas, E. K. Zavadskas and Z. Turskis, Multiple criteria assessment of pile-columns alternatives, Baltic Journal of Road & Bridge Engineering 6(3) (2011) 145–152. 27. T. C. Wang and H. D. Lee, Developing a fuzzy TOPSIS approach based on subjective weights and objective weights, Expert Systems with Applications 36(5) (2009) 8980–8985.



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28. L. Ustinovichius, E. K. Zavadskas and V. Podvezko, The Application of a quantitative multiple criteria decision making (MCDM-1) approach to the analysis of investments in construction, Control and Cybernetics 36(1) (2007) 251–268. 29. J. Ma, Z. P. Fan and L. H. Huang, A subjective and objective integrated approach to determine attribute weights, European Journal of Operational Research 112(2) (1999) 397–404. 30. D. Pekelman and S. K. Sen, Mathematical programming models for the determination of attribute weights, Management Science 20(8) (1974) 1217–1229. 31. J. Šaparauskas, E. K. Zavadskas and Z. Turskis, Selection of facade's alternatives of commercial and public buildings based on multiple criteria, International Journal of Strategic Property Management 15(2) (2011) 189–203. 32. G. Wei, GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting, Knowledge-Based Systems 23(3) (2010) 243–247. 33. B. G. Mirkin, The Problem of Group Selection (\Nauka", Moscow, 1974), p. 256. (in Russian) 34. E. K. Zavadskas, Systems Engineering Evaluation of Technological Solutions of Building Production (\Stroyizdat", Leningrad, 1991), p. 256. (in Russian) 35. E. K. Zavadskas, Integrated Resource Assessment and Selection Decisions in Construction (Mokslas, Vilnius, 1987), p. 210. (in Russian) 36. K. Fiedler, F. Peldschus and E. K. Zavadskas, Decision-Making Methods in Construction (Wiss. Berichte der Technischen Hochschule Leipzig, Heft 17) (1986), pp. 1–56. (in German) 37. J. G. Kemeny, Mathematics Without Numbers, Daedalus 88(4) (1959) 577–591. 38. A. I. Gluchov and A. K. Pogodajev, Kemeny Median in prioritizing the development of enterprises, UBS 14 (2006), 40–45. (in Russian) 39. L. E. Sadovskii and A. L. Sadovskii, Mathematics and Sports (American Mathematical Society, Providence, 1993). 40. G. I. Perminov and N. V. Leonova, The method of determining importance of criteria in a multicriteria decision problem, China-USA Business Review 11(9) (2012) 1267–1276. 41. L. P. Dinu and F. Manea, An e±cient approach for the rank aggregation problem, Theoretical Computer Science 359(1) (2006) 455–461. 42. A. I. Orlov, On mean values, UBS 46 (2013) 88–117. (in Russian) 43. A. I. Orlov, Average values and rules of large numbers in the spaces of arbitrary origin, Scienti¯c Journal of KubSAU 89(5) (2013) 556–586. (in Russian) 44. V. G. Gorskii, A. A. Gritsenko and A. I. Orlov, The method of agreement of clustered rankings, Avtomat. i Telemekh 3 (2000) 159–167. (in Russian) 45. B. G. Mirkin and L. Chiornyi, On measuring the proximity between di®erent partitions of a ¯nite set of objects, Avtomat. i Telemekh 5 (1970), 120–127. (in Russian) 46. B. G. Mirkin and L. Chiornyi, Some properties of partitions spaces, Mathematical Analysis of Economic Models (1972) 126–147. (in Russian) 47. G. V. Rauschenbach, Proximity and similarity measures, Analysis of Non-Numerical Information in Sociological Research (1985) 169–203. (in Russian) 48. S. Dadelo, A. Krylovas, N. Kosareva, E. K. Zavadskas and R. Dadelien_e, Algorithm of maximizing the set of common solutions for several MCDM problems and it's application for security personnel scheduling, International Journal of Computers, Communications & Control 9(2) (2014) 140–148. 49. M. G. Kendall, Rank Correlation Methods (Charles Gri±n & Co., London, 1970).

New KEMIRA Method for Determining Criteria Priority

New KEMIRA Method for Determining Criteria Priority

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