A Novel Multicriteria Decision-Making Method Based on...

Research ArticleA Novel Multicriteria Decision-Making Method Based onDistance Similarity and Correlation DSC TOPSIS

Semra Erpolat TaGabat 12

1Faculty of Science Literature Department of Statistics Mimar Sinan Fine Art University Istanbul Turkey2Graduate School of Educational Sciences Bahcesehir University Istanbul Turkey

Correspondence should be addressed to Semra Erpolat Tasabat semraerpolatmsgsuedutr

Received 27 October 2018 Revised 12 February 2019 Accepted 11 March 2019 Published 30 April 2019

Guest Editor Danielle Morais

Copyright copy 2019 Semra Erpolat Tasabat This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Decision-making briefly defined as choosing the best among the possible alternatives within the possibilities and conditionsavailable is a far more comprehensive process than instant While in the decision-making process there are often a lot of criteriaas well as alternatives In this case methods referred to as Multicriteria Decision-Making (MCDM) are applied The main purposeof the methods is to facilitate the decision-makers job to guide the decision-maker and help him to make the right decisions ifthere are too many options In cases where there are many criteria effective and useful decisions have been taken for granted atthe beginning of the 1960s for the first time and supported by day-to-day work A variety of methods have been developed for thispurposeThe basis of some of these methods is based on distance measuresThemost known method in the literature based on theconcept of distance is of course a method called Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) Inthis study a new MCDMmethod that uses distance similarity and correlation measures has been proposed This new method isshortly called DSCTOPSIS to include the initials of distance similarity and correlationwords respectively prefix of TOPSIS nameIn the method Euclidean was used as distancemeasure cosine was used as similarity measure and Pearson correlation was used asrelationmeasure Using the positive ideal and negative-ideal values obtained from these measures respectively a common positiveideal value and a common negative-ideal value were obtained AfterwardDSC TOPSIS is discussed in terms of standardization andweighting The study also proposed three different new ranking indexes from the ranking index used in the traditional TOPSISmethod The proposed method has been tested on the variables showing the development levels of the countries that have a veryimportant place today The results obtained were compared with the Human Development Index (HDI) value developed by theUnited Nations

1 Introduction

Decision-making with the simplest definition is the processof making choices from the available alternatives Although itwas expressed in different forms basically a decision-makingprocess involves identification of the objective selection ofthe criteria selection of the alternatives selection of theweighting methods determination of aggregation methodand making decisions according to the results

To be able to adapt to rapidly changing environmentalconditions and make effective decisions in parallel with thischange can only be possible by using scientific methods thatcan evaluate a large number of qualitative and quantitative

factors in the decision-making process [1] Many problemsencountered in real life fit the definition of multicriteriadecision-making People find their individual preferenceswhile they are present in evaluative judgments in multicri-teria decision-making problems It may not be difficult todecide when there are few criteria or few alternatives How-ever as the subject becomes more complex the informationprocessing capacity of people is restricted decision-makingbecomes more difficult and help may be needed In suchcases instead of trying to integrate too much knowledge andtrying to decide applying simple rules and procedures andevaluating the problem gradually will make it easier to decideSuch approaches will also facilitate decision-makers to make

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9125754 20 pageshttpsdoiorg10115520199125754

2 Mathematical Problems in Engineering

rational decisions and the decision will be appropriate withinthe constraints [1]

Taking more than one criterion choosing the mostappropriate one among the alternatives or alternating sortingproblems is called Multicriteria Decision-Making (MCDM)problems The MCDM could have been appropriate for thepurposes of evaluating the alternatives in a particular orderor alternatively in order to determine the alternatives [2]MCDM methods which are used in a wide range of fieldsfrom personal selection problems to economic industrialfinancial political decision problems have begun to developtogether with the increasingly complex decision-makingprocess from the beginning of 1960s In particular MCDMmethods are being used to control the decision-makingmechanism and to obtain the decision result as quicklyand as easily as possible provided that the target to beachieved is explained by a number of criteria and each ofthe alternatives has its own advantages MCDM methodscan involve different decision-makers in the decision-makingprocess and allow many factors related to the decision-making problem to be evaluated simultaneously at differentlevels [1]

TOPSIS one of the MCDM methods developed byHwang and Yoon [3] is a technique to evaluate the perfor-mance of alternatives through the similarity with the idealsolution According to this technique the best alternativewould be one that is closest to the positive-ideal solutionand farthest from the negative-ideal solution The positive-ideal solution is one that maximizes the benefit criteriaand minimizes the cost criteria The negative-ideal solutionmaximizes the cost criteria andminimizes the benefit criteriaIn summary the positive-ideal solution is composed ofall best values attainable of criteria and the negative-idealsolution consists of all the worst values attainable of criteria[4ndash6]

The basis of this study is based on the report presented inthe statistics conference (istkon 2017) held in Turkey in 2017[7] In the report it was emphasized that TOPSIS methodcould be formed in different structures by drawing attentionto the unit of measurement In addition it was noted thatranking indexes may also be different This study was carriedout with the consideration of the issues discussed in thecongress and much more

In this study a new MCDM method based on thetraditional TOPSIS method is proposed In the proposedmethod ideal positive solution approximation or ideal nega-tive solution distance is calculated based on the distance sim-ilarity and correlation (DSC)measures unlike the traditionalTOPSIS method For this reason this new unit of measureproposed to rank alternatives is named as DSC and the newMCDMmethod developed is called DSC TOPSIS The mainadvantage of the proposed unit of measurement is that itdoes not only rank the alternatives according to the conceptof distance but also according to the concepts of similarityand correlation In other words the major advantage ofthe proposed method is that it proposes a stronger unit ofmeasure by considering the three basic concepts that can beused to compare units distance similarity and correlationAnother advantage of the proposed new MCDM method is

that it suggests three new different methods that can be usedto rank alternatives according to their importance levels Inaddition the impact of the proposed method on the caseswhere the decision matrix is dealt with by row columnand double standardization methods is discussed in detailThese methods are applied to the results obtained by theproposed method Thus the traditional TOPSIS method andthe sorting technique used in this method are compared withthe proposed MCDM method and the proposed three newsorting methods The functioning of the method has beentested to determine the order of development of countriesFor this purpose the indicators of Human DevelopmentIndex (HDI) calculated by UN Development Programme(UNDP) have been utilized The results were compared withHDI and traditional TOPSIS values

There are no studies in the literature that use the con-cepts of distance similarity and correlation together In thiscontext this study will be the first The closest one to theproposed method in this study is Dengrsquos method which waspublished in the study entitled ldquoA Similarity-Based Approachto Ranking Multicriteriardquo in 2007 With this study Deng pre-sented a similarity-based approach to ranking multicriteriaalternatives for solving discrete multicriteria problems ThenSafari and et al [5] modified Dengrsquos similarity-based methodand they proposed a newMCDMmethod based on similarityand TOPSIS And then Safari and Ebrahimi [8] used asimilarity-based technique by Deng [4] to rank countries interms of HDI They also proposed a solution for resolving aproblem which exists in Dengs method Although the issuediscussed in the application part of Safari and Ebrahimi [8]and this study is on HDI data the ways of handling thisdata are different The most important difference betweenthe two studies is the selection of criteria While Safari andEbrahimi [8] preferred to use ldquolife expectancy at birthrdquoldquomean years of schooling ldquoexpected years of schoolingrdquo andldquolog GNIrdquo which are the indicator of HDI as criteria inthis study it was preferred to use ldquoLiferdquo ldquoEducationrdquo andldquoIncomerdquo indexes that expressed the dimensions of HDI Atthis point Safari and Ebrahimis [8] method can be criticizedon their preferred criteria That is HDIrsquos two indicators ofthe education dimension are considered as separate criteriain their study in fact it gave more weight to the dimensionof education than the other dimensions in terms of relativeimportance In this context with this study a suggestionhas been made to this situation mentioned above Anotherdifference is undoubtedly that theHDI data discussed in bothstudies are of different years

In the following sections the concepts of distance sim-ilarity and correlation will be mentioned first (Section 2)Then in Section 3 respectively the operation of the tradi-tional TOPSIS method will be described the steps of theMCDMmethod proposed in the study will be given and theproposed sorting methods will be explained In Section 4the functioning of the proposedMCDMmethod and rankingtechniques will be tested on the variables that indicate thelevel of development of the countries For this purposebrief information about HDI will be given first in thissection In Section 5 evaluation of the results obtained willbe made

Mathematical Problems in Engineering 3

2 Distance Similarity and Correlation

Since the new unit of measurement proposed in the studywhich is developed as an alternative to the unit of measure-ment used in the traditional TOPSIS method is based on theconcepts of distance similarity and correlation these andrelated concepts will be briefly explained in the followingsubsections Thus a better understanding of the proposedunit of measurement will be provided

21 Metrics the Euclidean Distance Any function 119889119894119895 isdefined as a metric in case it satisfies the following fourconditions (metric axioms) for all points [9ndash11]

(i) If 119894 = 119895 then 119889119894119895 =0(ii) If 119894 = 119895 then 119889119894119895 gt 0(iii) 119889119894119895 = 119889119895119894 (symmetry axiom)(iv) 119889119894119895 + 119889119894119896 ge 119889119895119896 for any triple 119894 119895 119896 of points (triangle

inequality axiom)

In mathematics distance and metric expressions are used inthe same sense [12] The most important and special caseof a family of functions metrics or distances is known asEuclidian distance Euclidean distance is defined as the lineardistance between two points in the simplest sense

22 Dissimilarity Any function dij is defined as ldquodissimilar-ityrdquo if the first three of the metric axioms described aboveare satisfied Thus dissimilarity is more general concept Theupper and lower bound of the most dissimilarity functionsare 1 and 0 respectively (0 le dij le 1)

23 Similarity Themost common measure used to comparetwo cases is similarity The most important reason whysimilarity is more preferable than distance and dissimilarityis that it is easier for people to find similar aspects whencomparing two things Similar to dissimilarity the concept ofsimilarity varies between 0 and 1 (0 le sij le 1) Similarity is thecomplement of the dissimilarity This relationship betweenthe two concepts is shown in

119904119894119895 = 1 minus 119889119894119895 (1)

There are a variety of transforms methods for achieving adistance from similarity Including transformation given in(2) the most preferred ones are 119889119894119895 = 1 minus 119904119894119895 119889119894119895 = radic1 minus 119904119894119895119889119894119895 = (1 minus 119904119894119895)119904119894119895 119889119894119895 = radic2(1 minus 1199042119894119895) 119889119894119895 = arccos 119904119894119895 119889119894119895 =minus ln 119904119894119895 etc [11]

The transformation given in (2) contains specialmeaningThat is if many coefficients are converted according to thisformula in the range of [0 1] the structure to be obtainedwill be a metric or even Euclidean [10]

119889119894119895 = radic1 minus 119904119894119895 (2)

24 Correlation and Association The concepts of distancedissimilarity and similarity can be interpreted in geometrical

terms because they express the relative positions of points inmultidimensional space On the other hand the associationand correlation concepts reveal the relations between the axesof the same space based on the coordinates of the points

Except for covariancemost of association and correlationcoefficients measure the strength of the relationship in theinterval of [ndash1 1] These coefficients can be easily convertedto Euclidean by using (2) [10]

25 Some Distances and Similarities The power (119901 119903) dis-tance is a distance onR119899 defined by

( 119898sum119894

1003816100381610038161003816119909119894 minus 1199101198941003816100381610038161003816119901)1119903

(3)

For p = r ge1 it is the 119897119901-metric including the EuclideanManhattan (or magnitude or city block) and Chebyshev (ormaximum value dominance) metrics for p = 2 1 and infinrespectivelyThe case (p r) = (2 1) corresponds to the squaredEuclidean distance [11]

The covariance similarity is a similarity onR119899 defined by

cov 119904119894119895 = sum119898119894 (119909119894 minus 119909) (119910119894 minus 119910)119899 = sum119898119894 119909119894119910119894119899 minus 119909119910 (4)

The correlation similaritywhich is also referred to as Pearsoncorrelation or Pearson product-moment correlation linearcoefficient is a similarity onR119899 defined by

119888119900119903119903 119904119894119895 = sum119898119894 (119909119894 minus 119909) (119910119894 minus 119910)radic(sum119898119894 (119909119894 minus 119909)2) (sum119898119894 (119910119894 minus 119910)2) (5)

The dissimilarities

1 minus (119888119900119903119903 119904119894119895) (6)

1 minus (119888119900119903119903 119904119894119895)2 (7)

are called the Pearson correlation distance and squared Pear-son distance respectively

The cosine similarity (or Orchini similarity angular simi-larity normalized dot product) is a similarity onR119899 definedby

cos 119904119894119895 = sum119898119894 (119909119894 minus 119910119894)2radic(sum119898119894 (119909119894)2) (sum119898119894 (119910119894)2) = ⟨119909 119910⟩1199092 100381710038171003817100381711991010038171003817100381710038172= cos 120601

(8)

were 120601 is the angle between vectors x and yAccording to this the cosine dissimilarity or cosine dis-

tance are defined by

1 minus (cos 119904119894119895) (9)

26 Global Distance The ldquoglobal distancerdquo is a measure inwhich the result of combining the various distance measuresis called ldquolocal distancerdquo with different methods The mostcommon of these methods used to combine local distancesare ldquototal sum weighted sum and weighted average (eggeometric mean)rdquo [13]


3 Multicriteria Decision-Making

MCDM problem is a problem in which the decision-makerintends to choose one out of several alternatives on the basisof a set of criteria MCDM constitutes a set of techniqueswhich can be used for comparing and evaluating the alterna-tives in terms of a number of qualitative andor quantitativecriteria with different measurement units for the purpose ofselecting or ranking [8]

MCDM problems are divided into Multiattribute Deci-sion-Making (MADM) and Multiobjective Decision-Making(MODM) problems The MADM problems have a predeter-mined number of alternatives and the aim is to determine thesuccess levels of each of these alternatives Decisions in theMADM problems are made by comparing the qualities thatexist for each alternative On the other hand in the MODMproblems the number of alternatives cannot be determinedin advance and the aimof themodel is to determine the ldquobestrdquoalternative [14]There are different methods used in the liter-ature for the solution of MCDM problems [15ndash17] and noneof these methods gives a complete advantage over othersThemost important advantage of these methods is that they allowus to evaluate quantitative and qualitative criteria togetherThe most well-known MCDM methods are Weighted SumMethod (WSM) Weighted Product Method (WPM) Ana-lytic Network Process (ANP) method Analytical HierarchyProcess (AHP) method ELimination Et Choix Traduisantla REalite (ELECTRE) the Technique for Order Preferenceby Similarity to Ideal Solution (TOPSIS) Vise KriterijumskaOptimizacija I Kompromisno Resenje (VIKOR) PreferenceRanking Organization Method for Enrichment Evaluations(PROMETHEE) Superiority and Inferiority Ranking (SIR)and so on

Regardless of the type of decision-making problem thedecision-making process generally consists of the followingfour basic steps

(i) determination of criteria and alternatives(ii) assignment of numerical measures of relative impor-

tance to criteria(iii) assigning numerical measures to alternatives accord-

ing to each criterion(iv) numerical values for sorting alternatives

The MCDM methods have been developed to effectivelycarry out the fourth stage of this process There are differentmethods used in the literature for the solution of MCDMproblemsThe differences between themethods are due to theapproaches that they recommend to make decisions In factno one has a complete superiority over the other

In this study a new MCDM method based on thetraditional TOPSIS method is proposed In the proposedmethod ideal positive solution approximation or ideal neg-ative solution distance is calculated based on the distancesimilarity and correlation measures unlike the traditionalTOPSIS method For this reason the method is called DSCTOPSIS Three different new methods are also proposed inorder to rank the alternatives according to their importancelevels in DSC TOPSIS or similar MCDM methods There

are several studies in the literature that modify the TOPSISmethod Some of them can be listed as follows Hepu Deng[4] in his paper named as ldquoA Similarity-Based Approach toRanking Multi-Criteria Alternativesrdquo presented a similarity-based approach to ranking multicriteria alternatives forsolving discrete multicriteria problems Ren and et al [18]introduced a novel modified synthetic evaluation methodbased on the concept of original TOPSIS and calculatedthe distance between the alternatives and ldquooptimized idealreference pointrdquo in the D+ Dminus plane and constructing the Pvalue to evaluate quality of alternative in their study titledldquoComparative Analysis of a Novel M-TOPSIS Method andTOPSISrdquo Cha [19] built the edifice of distancesimilaritymeasures by enumerating and categorizing a large varietyof distancesimilarity measures for comparing nominal typehistograms at the paper titled ldquoComprehensive Survey on Dis-tanceSimilarity Measures between Probability Density Func-tionsrdquo Chakraborty S and YehC-H [20] in the study namedas ldquoA Simulation Comparison of Normalization Proceduresfor TOPSISrdquo compare four commonly known normalizationprocedures in terms of their ranking consistency and weightsensitivity when used with TOPSIS to solve the generalMADM problem with various decision settings Chang etal [21] adopted the concepts of ldquoIdealrdquo and ldquoAnti-Idealrdquosolutions as suggested by Hwang and Yoon [3] and studiedthe extended TOPSIS method using two different ldquodistancerdquoideas namely ldquoMinkowskirsquos L120582 metricrdquo and ldquoMahalanobisrdquodistances in their study titled ldquoDomestic Open-End EquityMutual Fund Performance Evaluation Using Extended TopsisMethod with Different Distance Approachesrdquo Hossein and etal [5] in their paper named as ldquoANewTechnique forMulti Cri-teria Decision-Making Based on Modified Similarity Methodrdquomodified Dengrsquos similarity-based method Omosigho andOmorogbe [22] in their study named as ldquoSupplier SelectionUsingDifferentMetric Functionsrdquo examined the deficiencies ofusing only one metric function in TOPSIS and proposed theuse of spherical metric function in addition to the commonlyused metric functions Kuo [23] in his paper named asldquoA Modified TOPSIS with a Different Ranking Indexrdquo byproposing119908minus and119908+ as theweights of the ldquocostrdquo criterion andthe ldquobenefitrdquo criterion respectively he defined a new rankingindex

Since the proposed method is based on the TOPSISmethod the steps of the traditional TOPSIS method willfirst be explained in the following subsections And then thedetails of the proposed MCDMmethod will be discussed

31 TOPSIS Method

Step 1 Determining the decision matrix

119883 = [[[[[[

11990911 11990912 sdot sdot sdot 119909111989811990921 11990922 sdot sdot sdot 1199092119898sdot sdot sdot sdot sdot sdot 119909119894119895 sdot sdot sdot1199091198991 1199091198992 sdot sdot sdot 119909119899119898]]]]]]

(10)

In a decision matrix lines represent alternatives 119860 119894 (119894 =1 2 119899) and columns refer to criteria 119862119895 (119895 = 1 2 119898)


Step 2 Determining the weighting vector as follows

119882 = (1199081 sdot sdot sdot 119908119895 sdot sdot sdot 119908119898) (11)

in which the relative importance of criterion Cj with respectto the overall objective of the problem is represented as wj

Step 3 Normalizing the decision matrix through Euclideannormalization

1199091015840119894119895 = 119909119894119895(sum119899119895=1 1199092119894119895)12 (12)

As a result a normalized decision matrix can be determinedas

1198831015840 = [[[[[[[

119909101584011 119909101584012 sdot sdot sdot 11990910158401119898119909101584021 119909101584022 sdot sdot sdot 11990910158402119898sdot sdot sdot sdot sdot sdot 1199091015840119894119895 sdot sdot sdot11990910158401198991 11990910158401198992 sdot sdot sdot 1199091015840119899119898

]]]]]]](13)

Step 4 Calculating the performance matrixThe weighted performance matrix which reflects the

performance of each alternative with respect to each criterionis determined bymultiplying the normalized decision matrix(13) by the weight vector (11)

119884 = [[[[[[[


]]]]]]]= [[[[[[


(14)

Step 5 Determining the PIS and the NISThe positive-ideal solution (PIS) and the negative-ideal

solution (NIS) consist of the best or worst criteria valuesattainable from all the alternatives Deng [4] enumerated theadvantages of using these two concepts as their simplicityand comprehensibility their computational efficiency andtheir ability to measure the relative performance of thealternatives in a simple mathematical form [8]

For PIS (119868+)119868+ = (max

119894119910119894119895 | 119895 isin 119869) (min

119894119910119894119895 | 119895 isin 1198691015840 (15)

119868+ = 119868+1 119868+2 119868+119899 (16)

And for NIS (119868minus)119868minus = (min

119894119910119894119895 | 119895 isin 119869) (max

119894119910119894119895 | 119895 isin 1198691015840 (17)

119868minus = 119868minus1 119868minus2 119868minus119899 (18)

At both formulas 119869 shows benefit (maximization) and 1198691015840shows loss (minimization) value

Step 6 Calculating the degree of distance of the alternativesbetween each alternative and the PIS and the NIS

The D+ and the Dminus formulas are given in (19) and (20)respectively by using Euclidean distance

119863+119894 = radic 119898sum119895=1

(119910119894119895 minus 119868+119895 )2 (19)

119863minus119894 = radic 119898sum119895=1

(119910119894119895 minus 119868minus119895 )2 (20)

Step 7 Calculating the overall performance index for eachalternative across all criteria

1198751119894 = 119863minus119894119863minus119894 + 119863+119894 (21)

0 le 119875119894 le 1 The Pi value indicates the absolute closeness ofthe ideal solution If 119875119894 = 1 then Ai is the PIS if 119875119894 = 0 thenAi is the NIS

Step 8 Ranking the alternatives in the descending order ofthe performance index value

32 Proposed Method The approaches used in the steps ofldquonormalizing the decision matrixrdquo ldquocalculating the distanceof positive and negative-ideal solutionsrdquo and ldquocalculatingthe overall performance index for each alternative acrossall criteriardquo of the traditional TOPSIS method are open tointerpretation and can be examined improved or modifiedApproaches that can be recommended in these steps arebriefly summarized below

ldquoNormalizing the decision matrixrdquo step (Step 3 at thetraditional TOPSIS)The normalization method used to nor-malize the decision matrix can also be achieved with differentnormalization formulas Also for this step standardizationrather than normalization is one of the methods that can beapplied

ldquoCalculating the distance of positive- and negative-idealsolutionsrdquo step (Step 6 at the traditional TOPSIS) alternativeto the distance of Euclidean (Minkowski L2) which is used tocalculate the distance of PIS and NIS is also possible to usemany different distancemeasures such as linear [15] spherical[22] Hamming [22] Chebyshev [21] Dice [24] Jaccard [24]and cosine (Liao and Xu 2015) The concept of similarity isalso considered as an alternative (Zhongliang 2011 and [4])

ldquoCalculating the overall performance index for eachalternative across all criteriardquo step (Step 7 at the traditionalTOPSIS) different approaches such as the concept of distancecan be used instead of the simple ratio recommended in thetraditional TOPSIS method since it is considered to be a verysimple way by most researchers

To offer solutions to the weaknesses listed above in thisstudy instead of


(i) ldquonormalization approachrdquo applied in Step 3 ofthe traditional TOPSIS method column and row-standardization approach

(ii) ldquoeuclidean distancerdquo used in Step 6 a new measurebased on the concept of ldquodistance similarity and cor-relationrdquo (DSC)

(iii) ldquosimple ratiordquo used in Step 7 new sorting approachesbased on the concept of distance have been proposed

This method which could be an alternative to MCDMmeth-ods especially TOPSIS due to the new measure proposed iscalled DSC TOPSIS The main objective of

(i) using different standardization methods is to empha-size what can be done for different situations that canbe encountered in real life problems

(ii) developing a newmeasure for the traditional TOPSISmethod in this study is to improve the approach ofldquoevaluating the two alternatives only based on theEuclidean distance to the PIS and NIS valuesrdquo andmake it more valid

(iii) developing new sorting methods is to criticize themethod used in traditional TOPSIS because it is basedon simple rate calculation

At this point an important issue mentioned earlier must beremembered again That is the fact that none of the MCDMmethods is superior to the other Therefore the new MCDMmethod proposed in this studywill certainly not provide a fulladvantage over other methods But a different method will begiven to the literature

Basically there are three approaches used to comparevectors These are distance similarity and correlation Theconcept of distance is the oldest known comparison approachand is the basis of many sorting or clustering algorithms Onthe other hand both similarity and correlation are two otherimportant concepts that should be used in vector compar-isons According to Deng [4] mathematically comparing twoalternatives in the form of two vectors is better represented bythe magnitude of the alternatives and the degree of conflictbetween each alternative and the ideal solution rather thanjust calculating the relative distance between them [8] Thedegree of conflict between each alternative and the idealsolution is calculated by ldquocosine similarityrdquo On the otherhand it is possible to compare alternatives according totheir relationship In this case the concept of correlationthat evaluates from another point of view and thereforecorrelation similarity will be introduced Briefly Euclideandistance cosine similarity and correlation similarity arise asthe basic metrics that can be used for ranking the alternatives

In order to express an MCDM problem in ldquo119898-dimensional real spacerdquo alternatives can be representedby Ai vector and PIS and NIS can be represented by 119868+119895and 119868minus119895 vectors respectively In this case the angle betweenAi and 119868+119895 (119868minus119895 ) in the m-dimensional real space which isshown by 120579+119894 (120579minus119894 ) is a good measure of conflict between thevectors [8] These vectors (119868+119895 119868minus119895 ) and the degree of conflict(cos 120579119894) between them are shown in Figure 1 by Deng [4]

minusi +

i

x3

x1

x2

Iminusj

I+j

Ai

y

x

y

Figure 1 The degree of conflict

The situation of conflict occurs when 120579119894 = 0 that is whenthe gradients of Ai and 119868+119895 (119868minus119895 ) are not coincident Thus theconflict index is equal to ldquoonerdquo as the corresponding gradientvectors lie in the same direction and the conflict index isldquozerordquo when 120579119894 = 1205872 which indicates that their gradientvectors have the perpendicular relationship with each other[4 8]

In the light of the above-mentioned explanations andcauses the three approaches of ldquodistance similarity andcorrelationrdquo that can be used to compare vectors have beenexploited in order to combine them in a common pavilionto develop a stronger comparison measure Thus the stepsdetailed below have been carried out

321 Proposed Step 3 Standardize the Decision MatrixUnlike the traditional TOPSIS method this step has beendeveloped on the basis of standardization The differences ofcolumn and row-standardization were emphasized

Standardization vs Normalization Data preprocessingwhich is one of the stages of data analysis is a very importantprocess One of the first steps of data preprocessing is thenormalization of data This step is particularly importantwhen working with variables that contain different units andscales

All parameters should have the same scale for a faircomparison between them when using the Euclidean dis-tance and similar methods Two methods are usually wellknown for rescaling data These are ldquonormalizationrdquo andldquostandardizationrdquo Normalization is a technique which scalesall numeric variables in the range [0 1] In addition to (12)some other possible formulas of normalization are givenbelow and the other is given in (59)

1199091015840119894119895 = 119909119894119895119898119890119889119894119886119899119883 (22)

1199091015840119894119895 = 119909119894119895max (119909119894119895) (23)


On the other hand standardization is a method whichtransforms the variables to have zeromean and unit variancefor example using the equation below

1199091015840119894119895 = 119909119894119895 minus 120583119883120590119883 (24)

Both of these techniques have their drawbacks If youhave outliers in your data set normalizing your data willcertainly scale the ldquonormalrdquo data to a very small intervalAnd generally most of data sets have outliers Furthermorethese techniques can have negative effects For example ifthe data set dealt with outlier values the normalization ofthis data will cause the data to be scaled at much smallerintervals On the other hand using standardization is notbounded data set (unlike normalization) Therefore in thisstudy standardization method was preferred to use

Standardization can be performed in three ways column-standardization (variable or criteria) row-standardization(observation or alternative) and double standardization(both row and column standardization) [25] In column-standardization variables are taken separately and observa-tions for each variable are taken as a common measurewhereas in row-standardization the opposite is the case Thatis the observations are handled one by one and the variablesare brought to the samemeasurement for each observation Inaddition itmay be the case that both column-standardizationand row-standardization are done togetherThis is referred toas ldquodouble-standardizationrdquo

Selection Appropriate Standardization Method Differenttransformations of the data allow the researchers to examinedifferent aspects of the underlying basic structure Structuremay examine ldquowholisticallyrdquo with main effects interactionand error present or the main effects can be ldquoremovedrdquo toexamine interaction (and error) [25]

A row or column effect can be removed with a transfor-mation that sets rowor columnmeans (totals) to equal valuesCentering or standardization of row or column variablesresults in a partial removal of the main effects by settingone set of means to zero Double centering and doublestandardization remove both sets of means Removal of theldquomagnituderdquo or ldquopopularityrdquo dimension allows the researcherto examine the data for patterns of ldquointeractiverdquo structure[25]

Centering or standardization within rows removes dif-ferences in row means but allows differences in columnmean to remain Thus the ldquoconsensusrdquo pattern among rowscharacterized by differences in the column means remainsrelatively unaffected Column-standardization also removesonly one set of means column means are set to zeroCentering of data usually performed on column or rowvariables is analogous to analyzing a covariance matrix Dataare sometimes double-centered to remove the magnitude orpopularity dimension (Green 1973)

These are approaches that can be used in sorting and clus-tering alternatives According to the structure of the decisionproblem it is necessary to determine which standardizationmethod should be applied That is

(i) column standardization if observations are impor-tant

(ii) row-standardization if it is important to bring thevariables to the same unit

(iii) double standardization where both observations andvariables are important to be independent of the unitpreferable

322 Proposed Step 6 Calculating the Degree of Distance ofthe Alternatives between Each Alternative and the PIS and theNIS In the context described above and in Section 25 (usinggeometric mean approach to combine local distances) theproposed new measure formula is constructed as follows

Pr119900119901119900119904119890119889 119889119894 = 3radic(119904119902119906119886119903119890119889 119889119894) (119888119900119903119903 119889119894) (cos 119889119894) (25)

The three terms in the formula are ldquothe squared Euclideandistance ((3) the case (119901 119903) = (2 1))rdquo ldquothe correlation distance(6)rdquo and ldquothe cosine distance (9)rdquo respectively

As mentioned before the methods that can be usedto determine the differences between vectors are distancecorrelation and similarity In (25) these concepts werecombined with the geometric mean and a stronger measurethan the traditional TOPSIS method used to determine thedifferences between the vectors

Below column-standardization and row-standardizationcases of this proposed measure were discussed respectivelyIn the literature the effects of row-standardization on theTOPSIS method have not been addressed at all For thisreason the effects of row-standardization on the TOPSISmethod in this study were also investigated

For Column-Standardization In the case of standardizing thedata according to the columns in other words accordingto the alternatives there is no change or reduction in theformulas of the squared Euclidean correlation and cosinedistances used in the proposed measure For this reason (25)can be used exactly in the case of column-standardization

Thus the degree of distance of the alternatives betweeneach alternative and the PIS and the NIS for column-standardized data are as follows

Pr119900119901119900119904119890119889 119863+119894= 3radic(119904119902119906119886119903119890119889 119889+119894 ) (119888119900119903119903 119889+119894 ) (cos 119889+119894 ) (26)

Pr119900119901119900119904119890119889 119863minus119894= 3radic(119904119902119906119886119903119890119889 119889minus119894 ) (119888119900119903119903 119889minus119894 ) (cos 119889minus119894 ) (27)

For square Euclidean distance cosine distance and corre-lation distance the D+ and Dminus values and for correlation


similarity and cosine similarity the S+ and Sminus values arerespectively as follows

Square Euclidean (Sq Euc) Distance for Column-StandardizedData

119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

Correlation Similarity (Corr s) for Column-StandardizedData

119878+119894 = sum119898119895 (119910119895 minus 119910) (119868+119895 minus 119868+119895 )radic(sum119898119895 (119910119895 minus 119910)2) (sum119898119895 (119868+119895 minus 119868+119895 )2) (30)

119878minus119894 = sum119898119895 (119910119895 minus 119910) (119868minus119895 minus 119868minus119895 )radic(sum119898119895 (119910119895 minus 119910)2) (sum119898119895 (119868minus119895 minus 119868minus119895 )2) (31)

Cosine Similarity (Cos s) for Column-Standardized Data

119878+119894 = sum119898119895 (119910119895 minus 119868+119895 )2radic(sum119898119895 (119910119895)2) (sum119898119895 (119868+119895 )2) (32)

119878minus119894 = sum119898119895 (119910119895 minus 119868minus119895 )2radic(sum119898119895 (119910119895)2) (sum119898119895 (119868minus119895 )2) (33)

Correlation Distance (Cor d) for Column-Standardized Data

119863+119894 = 1 minus 119878+119894= 1 minus sum119898119895 (119910119895 minus 119910) (119868+119895 minus 119868+119895 )

radic(sum119898119895 (119910119895 minus 119910)2) (sum119898119895 (119868+119895 minus 119868+119895 )2)(34)

119863minus119894 = 1 minus 119878minus119894= 1 minus sum119898119895 (119910119895 minus 119910) (119868minus119895 minus 119868minus119895 )

radic(sum119898119895 (119910119895 minus 119910)2) (sum119898119895 (119868minus119895 minus 119868minus119895 )2)(35)

Cosine Distance (Cos d) for Column-Standardized Data

119863+119894 = 1 minus 119878+119894 = 1 minus sum119898119895 (119910119895 minus 119868+119895 )2radic(sum119898119895 (119910119895)2) (sum119898119895 (119868+119895 )2) (36)

119863minus119894 = 1 minus 119878minus119894 = 1 minus sum119898119895 (119910119895 minus 119868minus119895 )2radic(sum119898119895 (119910119895)2) (sum119898119895 (119868minus119895 )2) (37)

For Row-Standardization If the vectors are standardizedaccording to the rows some reductions are concerned in theformula of the recommendedmeasure It is detailed below (inthe following it is expressed as Case 4)

Let X and Y be two vectors When 120583119883 and 120583119884 are themeans ofX andY respectively and120590119883 and120590119884 are the standarddeviations of X and Y the correlation between X and Y isdefined as

119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)

The numerator of the equation is called the covariance of Xand Y and is the difference between the mean of the productof X and Y subtracted from the product of the means If Xand Y are row-standardized they will each have amean of ldquo0rdquoand a standard deviation of ldquo1rdquo so the (38) reduces to

119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)

While Euclidean distance is the sum of the squared differ-ences correlation is basically the average product From theEuclidean distance formula it can be seen that there is furtherrelationship between them

119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)

If X and Y are standardized at (40) the sum of squares willbe equal to 119899 (sum119899119895 1199092119895 = 119899 and sum119899119895 1199102119895 = 119899) So (40) reduced to(42)

119864119906119888 119889lowast119894 = radic119899 + 119899 minus 2 119899sum119895

119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)

In this case if the square of the Euclidean distance is taken(the square distance) and some adjustments are made (42)reduced to the formula for the correlation coefficient asfollows

1198892119894 lowast = 2119899 (1 minus 119888119900119903119903 119904119894lowast) (43)

In this way for row-standardized data the correlationbetween X and Y can be written in terms of the squareddistance between them

119888119900119903119903 119904119894lowast = 1 minus 1198892119894 lowast2119899 (44)


And the correlation distance is as

119888119900119903119903 119889119894lowast = 1 minus 119888119900119903119903 119904119894lowast = 1198892119894 lowast2119899 (45)

On the other hand in the case 119909 = 0 and 119910 = 0 the correlationsimilarity becomes ⟨119909 119910⟩(1199092 1199102)which is the formula ofcosine similarity So the formula of correlation distance is theformula of cosine distance at the same time

cos 119889119894lowast = 119888119900119903119903 119889119894lowast = 1198892119894 lowast2119899 (46)

Considering the relationships described above for row-standardized data correlation and cosine similarities aretransformed into square distance value Thus these threeconcepts used tomeasure the differences between the vectorsare expressed in the same wayThus (25) is converted into theform given in

Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)

So calculating the degree of distance of the alternativesbetween each alternative and the PIS and the NIS for row-standardized data are as follows

Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)

Pr119900119901119900119904119890119889 119863minus119894 = 13radic41198992 119889119894minus2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)

In row-standardization case the D+ D- and S+ S- valuesfor square Euclidean distance correlation similarity cosinesimilarity correlation distance and cosine distance are asfollows Differently here the reduced states of the correlationand cosine formulas are used (for column-standardizationwas explained at Proposed Step 6 with (30)-(37)) For thisreason the correlation S+ and S- formulas are equal to theformulas of cosine S+ and S- respectively

Square Euclidean (Sq Euc) Distance for Row-StandardizedData The D+ and D- values of the square Euclidean distancefor row-standardized data are equal to formulas of (28)-(29)respectively (the D+ and D- values of the square Euclideandistance for column-standardized data)

Correlation Similarity (Corr s) or Cosine Similarity (Cos s) forRow-Standardized Data

119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)

Correlation Distance (Corr d) or Cosine Distance (Cos d) forRow-Standardized Data

119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)

119863minus119894 = 1 minus 119878minus119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868minus119895) (54)

Circumstance Critique although the Euclidean distanceformula (28)-(29) is not affected by column or row-standardization the cosine (34)-(35) and correlation (36)-(37) distance formulas are affected In order to apply (53)-(54) it is necessary that the performance matrix has a rowbased average of ldquo0rdquo and a standard deviation of ldquo1rdquo To ensurethis condition see the following

For the row-standardization case either the criteria (vari-ables) are not to be weighted so the weight vector must betreated as a ldquounit vectorrdquo or the performancematrix obtainedafter weighting the criteria has to be standardized again on arow basis

For the column-standardization case whether the crite-rion weighting is done or not the performance matrix hasto be standardized again on a row basis

For both cases the PIS and NIS vectors must be standard-ized on a row basis

323 Proposed Step 7 Calculating the Overall PerformanceIndex (Ranking Index 119875119894) for Each Alternative across AllCriteria In this study three different performance indexeswere proposed The significance of these three new differentPi values were tried to emphasize by taking into account ldquothePi value suggested by the traditional TOPSIS methodrdquo (21)and ldquothe Pi value proposed by Ren et al [18]rdquo

1198752119894 = radic[119863+119894 minusmin (119863+119894 )]2 + [119863minus119894 minus max (119863minus119894 )]2 (55)

As can be understood from the above formula Ren and et al[18] calculated the Pi values using the Euclidean distance Forthis reason this value will be called the Euclidean 119875119894 value inthe study

Euclidean119875119894 value the D+ D- plane is established with D+ atthe x-axis and D- at the y-axis The point (119863+119894 119863minus119894 ) representseach alternative (119894 = 1 2 n) and the (min(119863+119894 ) max(119863minus119894 ))point to be the ldquooptimized ideal reference pointrdquo (in Figure 2point 119860) then the distance from each alternative to point 119860is calculated by using (55)The graph of the distance is shownin Figure 2 And finally for ranking the preference order thevalue obtained for each alternative is sorted from ldquosmall tobigrdquo because the formulas are based on distance concepts

Proposed 119875119894 values The ranking index of the traditionalTOPSIS method is obtained as the ratio of the NIS to the sum


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )

Figure 2 Explanation of the Euclidean distance

A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )

Figure 3 Manhattan distance

of the NIS and PIS (see (21)) Although it seems reasonableits validity is questionable and debatable in that it is basedon a simple rate calculation For this reason researchers havetried various methods to rank alternatives One of them is theEuclidean Pi value detailed above

In this study alternative to the Euclidean Pi value (1198752119894 )and of course to the traditional Pi value (1198751119894 ) three dif-ferent methods have been proposed below The first twoproposed methods include the distances of ldquoManhattanrdquo andldquoChebyshevrdquo in the Lq distance family just like the Euclideandistance The latter method is a new global distance measureconsisting of the geometric mean of these three distancesin the Lq family The Euclidean Manhattan and Chebyshevdistances are detailed in Section 24 For this reason inthis section it is mentioned that these measures and thenew global distance measure are adapted to the (119863+119894 119863minus119894 )and [min(119863+119894 )max(119863minus119894 )] points so that they can sort thealternatives from the best to the less good ones Just asEuclidean 119875119894 (1198752119894 ) for ranking the preference order the valueobtained for each alternative is sorted from ldquosmall to bigrdquo

(1) Manhattan 119875119894 value (1198753119894 ) the formula for the proposedindex based on the Manhattan distance is given in (56)and the graph of the distance is shown with thick lines inFigure 3

1198753119894 = 1003816100381610038161003816119863+119894 minusmin (119863+119894 )1003816100381610038161003816 + 1003816100381610038161003816119863minus119894 minusmax (119863minus119894 )1003816100381610038161003816 (56)

A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )

Figure 4 Chebyshev distance

(2) Chebyshev 119875119894 value (1198754119894 ) the proposed index based onthe Chebyshev distance is given in (57) The graph of thedistance is shown with thick lines in Figure 4

1198754119894 = max [1003816100381610038161003816119863+119894 minusmin (119863+119894 )1003816100381610038161003816 1003816100381610038161003816119863minus119894 minusmax (119863minus119894 )1003816100381610038161003816] (57)

(3) Global Distance 119875119894 value (1198755119894 ) in global distancevalue Euclidean Manhattan and Chebyshev are consideredas local distance The global distance measure is based on thegeometric average of these three local distance measures (seeSection 25)

1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)

33 Formulations of the DSC TOPSIS Method According tothe standardization method used (Structure of the Perfor-mance Matrix) and whether the criteria are weighted or notthe distance and similarity values mentioned above can becalculated by means of which formulas (Proposed MethodFormulas) are shown in Table 1

Explanations about the cases in Table 1

(i) Case 1 Case 2 Case 3 and Case 4 are the caseswhere only one of the column or row-standardizationmethods are applied while the remaining last fourcases Case 5 Case 6 Case 7 and Case 8 showed theconditions under which ldquodouble standardizationrdquo areapplied

(ii) In Case 1 Case 3 Case 5 and Case 7 the ldquostandardizedmatrixrdquo will be also ldquoperformance matrixrdquo becauseany weighting method is not used

(iii) In Case 3 Case 4 Case 5 and Case 6 afterthe row-standardization the ldquoPIS and NIS valuesrdquoshould be also standardized according to ldquorow-standardizationrdquo This correction is needed since thePIS and NIS are considered as a single value inthe column-standardization and as a vector in row-standardization


Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX

Long and Healthy Life

Life expectancyat birth

Life expectancy index

Knowledge A Descent Standart of Living

Expected years of schooling

Mean years ofschooling

GNI per capita(PPP $)

Education index GNI index

HUMAN DEVELOPMENT INDEX(HDI)

Figure 5 The dimensions and indicators of HDI

Note In the application part of the study Case 5 Case 6 Case7 and Case 8 where double standardization is applied willnot be included because they are partly addressed in Case 1Case 2 Case 3 and Case 4 where one-way (row or column)standardization is applied

The application types that can be used for the casesdetailed in Table 1 can be summarized as follows

Case 1 and Case 2 in these two cases where column-standardization is used alternatives for each criterion arebrought to the same unit In Case 1 the criteria are handledwithout weighting and in Case 2 they are weighted In thisstudy equalweighting method was preferred for comparisonIn Case 2 different weighting methods such as Saaty pointbest-worst entropy swara etc can also be used

Case 3 and Case 4 for these two cases where row-standardization is used the criteria for each alternative arebrought to the sameunit InCase 3 the criteria are unweightedand in Case 4 they are weighted For Case 4 it is possible toapply different weighting methods just as in Case 2 At thesecases the PIS and NIS values obtained after standardizationof the decision matrix are standardized according to row-standardization

Case 5 Case 6 Case 7 and Case 8 these are thecases where double standardization is used In Case 5 andCase 6 following the standardization of the decision matrixaccording to double standardization (first by row second bycolumn) PIS and NIS values are also standardized accord-ing to the rows because the latest row-standardization isapplied

4 Case Study of the Proposed Methods

In this section all of the proposed methods were appliedon the HDI data Therefore in the following subsectionsinformation about HDI will be given first and then imple-mentation steps of the proposedmethod will show on criteriaand alternatives that constitute HDI Finally the resultsobtained will be compared with the HDI value

41 Human Development Index Human development orthe human development approach is about expanding therichness of human life rather than simply the richness of theeconomy in which human beings live [26] Developed byan economist Mahbub ul Haq this approach emphasizes thatmonetary indicators such as Gross National Income (GNI)per capita alone are not enough to measure the level ofdevelopment of countries InHDR the termGNI is defined asldquoaggregate income of an economy generated by its productionand its ownership of factors of production less the incomespaid for the use of factors of production owned by therest of the world converted to international dollars usingpurchasing power parity (PPP) rates divided by mid-yearpopulationrdquo [27]

It has been pointed out that the opportunities and choicesof people have a decisive role in the measurement These areto live long and healthy to be educated and to have accessto resources for a decent standard of living [28] The humandevelopment approach was transformed into an index witha project supported by UNDP in 1989 and named HDI Inthis way it is aimed to measure the development levels ofthe world states better After introducing the HDI the UNDPpublished a report in 1990 in which the index was computedfor each country as a measure of the nations human develop-ment Since then UNDP has continued publishing a series ofannual Human Development Reports (HDRs) [5]

The first Human Development Report introduced theHDI as a measure of achievement in the basic dimensionsof human development across countries This somewhatcrude measure of human development remains a simpleunweighted average of a nationrsquos longevity education andincome and is widely accepted in development discourseBefore 2010 these indicators are used to measure HDI Overthe years some modifications and refinements have beenmade to the index In HDI 20th anniversary edition in 2010the indicators calculating the index were changed Figure 5shows the dimensions and indicators of the HDI index [26]

As can be seen from Figure 1 HDI is a compositeindex of three dimensions which are a decent standard


Table 2 The first and last five countries with high HDI ILife IEducation and IIncome values

No HDI ILife IEducation IIncome

1 Norway Hong Kong China (SAR) Australia Liechtenstein2 Switzerland Japan Denmark Singapore3 Australia Italy New Zealand Qatar4 Germany Singapore Norway Kuwait5 Singapore Switzerland Germany Brunei Darussalam 184 Eritrea Guinea-Bissau Guinea Madagascar185 Sierra Leone Mozambique Ethiopia Togo186 South Sudan Nigeria Sudan Mozambique187 Mozambique Angola Mali Malawi188 Guinea Chad Djibouti Guinea

of living knowledge and long and healthy life A decentstandard of living dimension which contained GNI percapita create ldquoGNI Index (Income Index-IIncome)rdquo Likewiselife expectancy at birth at the long and healthy life generateldquoLife Expectancy Index (Life Index-ILife)rdquo And finally bothmean years of schooling and expected years of schoolingwhich is in dimension knowledge create ldquoEducation Index(IEducation)rdquo

In order to calculate the HDI first these three coredimensions are put on a common (0 1) scale For this purposethe following equation is used

1199091015840119894119895 = 119909119894119895 minusmin (119909119894119895)max (119909119894119895) minus min (119909119894119895) (59)

After that the geometricmeanof the dimensions is calculatedto produce the HDI (UNDP 2011) This formula is given in

119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)

The world countries are ranked according to the HDI valuescalculeted from (60)

42 Implementation of the Proposed Method In this sectionthe degree of similarity of the alternatives between each alter-native and the PIS and the NIS is calculated with proposed119863+ and D- value at (26)-(27) for column-standardization andat (49)-(50) for row-standardization respectively The overallperformance index for each alternative across all criteria iscalculated with proposed Pi values at (56)-(57)-(58) used torank countries in terms of HDI The HDI value is basicallybased on the criteria of life expectancy at birth mean years ofschooling expected years of schooling andGNIper capita asdetailed above These criteria were used to calculate the HDIvalue by being reduced to three dimensions Income IndexLife Index and Education Index

From this point of view in order to operate the DSCTOPSIS method proposed in this study the developmentlevel of the preferred countries and the problem of deciding

Table 3 The original data for the first five countries Decisionmatrix

Country ILife IEducation IIncome

Afghanistan 0626 0398 0442Albania 0892 0715 0699Algeria 0847 0658 0741Andorra 0946 0718 0933Angola 0503 0482 0626 Weights 0333 0333 0333Mean 0790 0639 0687St Dev 0127 0174 0180

to create a new HDI were discussed through these threedimensions Since 188 countries were considered in the HDRreport published in 2016 (prepared with 2015 values) theinitial decision matrix created in the implementation part ofthe study was formed from 188 alternatives Income IndexLife Index and Education Index are considered as criteriaso that the initial decision matrix was formed from 188alternatives and 3 criteria For the tables not to take up toomuch space the data and the results of the first five countriesin alphabetical order from 188 countries are given In termsof giving an idea before the steps of the DSC TOPSIS methodare implemented the first and last five countries with thehighest and lowest Income Index Life Index and EducationIndex values in addition to HDI value are given in Table 2respectively

Step 1 (determining the decision matrix) In this matrix 188countries which have been investigated in the 2016HDR areconsidered to be alternatives and the three HD dimensionformed the criteria The decision matrix is shown at Table 3for the first ten countries


Table 4 Standardized data (or performance matrix of unweighting data) for the first five countries

Country Column-Standardization Row-StandardizationILife IEducation IIncome Mean St Dev ILife IEducation IIncome Mean St Dev

Afghanistan -1289 -1386 -1363 -1346 0041 1391 -0918 -0473 0000 1000Albania 0801 0439 0069 0436 0299 1410 -0614 -0797 0000 1000Algeria 0448 0111 0303 0287 0138 1271 -1172 -0099 0000 1000Andorra 1226 0456 1372 1018 0402 0768 -1412 0644 0000 1000Angola -2255 -0902 -0338 -1165 0805 -0535 -0866 1401 0000 1000 Weights 0333 0333 0333 0333 0333 0333Mean 0000 0000 0000 1025 -0780 -0245St Dev 1000 1000 1000 0595 0604 0750

Table 5 Performance matrix for the first five countries



Step 2 (determining the weighting vector) Since the purposeis to explain the operation of the model rather than theweighting the criteria are handled without any weighting(unweighted) and with ldquoequally weightedrdquo in the study Just asin the traditional TOPSIS method in the proposed methodcan also be given different weights for the criteria (will beconsidered as the future research topic)

ldquoEqual weightingrdquo is amethod of giving equal importanceto the criteria being considered Since three criteria areconsidered in the study the weight of each criterion isdetermined as ldquo13 = 0333rdquo The weights for the criteria areshown in the bottom row of Table 3

Step 3 (standardized the decision matrix) Before rankingthe countries to put data on a common scale standard-ization method are used column-standardization and row-standardization It was performed with (24) The result isshown in Table 4

In the column-standardization part of Table 4 while thealternatives (countries) are brought to the same unit foreach criterion (Life Education and Income) in the row-standardization part the criteria for each alternative arebrought to the same unit

Step 4 (calculating the performance matrix) The perfor-mance matrix is obtained by multiplying the weight vectorby the matrix of the standardized values (Table 5)

It should be reminded again at this point that if the crite-rion weighting is not performed then the matrix obtained asa result of column or row-standardization (standardized dataTable 4) would also be considered as a ldquoperformance matrixrdquo

Step 5 (determining the PIS and the NIS) The PIS and theNIS are attainable from all the alternatives (188 countries)across all three criteria according to (15) and (17) respectivelyThese values are shown in Table 6

Step 6 (calculating the degree of distance of the alternativesbetween each alternative and the PIS and the NIS) After thecolumn and row-standardizations in this step first the degreeof similarities and distances of the alternatives between eachalternative and the PIS and the NIS are calculated And thenfrom these values the degree of proposed distances of thealternatives between each alternative and the PIS and the NISis calculated (Table 7)

Step 7 (calculating the overall performance index) Theoverall proposed performance index for each alternative


Table 6 PIS (I+) and NIS (Iminus) value

Weighting Status PIS and NIS Column-Standardization Row-StandardizationILife IEducation IIncome Mean St Dev ILife IEducation IIncome Mean St Dev

Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002

Table 7 PIS (I+) and NIS (Iminus) value for Case 4

Row-StandardizationILife IEducation IIncome Mean St Dev0958 -1380 0422 0000 10001414 -0705 -0710 0000 1000

across all three criteria is calculated based on (21)-(55)-(56)-(57)-(58) The results obtained by applying the DSCTOPSIS method according to the ldquocasesrdquo detailed above andthe results of ldquorankingrdquo performed in Step 8 are given inTables 8 9 10 and 11 respectivelyThe tables are constructedto contain proposed Pi values (1198753119894 -(56) 1198754119894 -(56) 1198755119894 -(56)) andtheir ranking as well as the values and ranking of 1198751119894 (21)1198752119894 (55) HDI and traditional TOPSIS method (P) Thus allmethods are provided to be implemented together (Tables 89 10 and 11)

Table 8 shows the results obtained from the applicationof Case 1 In Case 1 after the column-standardization of thedecision matrix was carried out (in other words after thecountries are brought to comparable level for Life Educationand Income criteria respectively) the proposed DSC TOPSISmethod was applied without any weighting to the criteria

The application results of Case 2 were given in Table 9Unlike Case 1 in Case 2 after the column-standardizationof the decision matrix was performed the proposed DSCTOPSIS method was applied with equal weight (differentweighting methods can also be used) to the criteria

Table 10 shows the results obtained from the applica-tion of Case 3 Unlike Case 1 in Case 3 after the row-standardization of the decision matrix was made (in otherwords after Life Education and Income criteria are broughtto comparable level for each country) the proposed DSCTOPSIS method was applied without any weight to thecriteria

The application results of Case 4 were given in Table 11In Case 4 unlike Case 3 after the row-standardization of thedecision matrix was performed the proposed DSC TOPSISmethod was applied by giving equal weight to the criteria Asin Case 2 in Case 4 weighting methods other than the equalweighting method can be used

Step 8 (ranking the alternatives) The alternatives could beranked in the descending order of the proposed Pi indexesvalues In Tables 12 and 13 the first five countries withthe highest value and the last five countries with the lowestvalue are listed according to the ranking results obtained

for Case 1 Case 2 Case 3 Case 4 and Case 5 respectivelyin alphabetical order Additionally in Figures 6 and 7 thevalues of all the indexes computed during the study mainlyHDI are handled together for all countriesThus it is possibleto compare development levels of countries according todifferent indexes In the graphs the traditional TOPSIS indexis denoted as ldquoPrdquo and 1198751119894 1198752119894 1198753119894 1198754119894 1198755119894 are denoted as ldquoP1P2 P3 P4 P5rdquo respectively In accordance with the equalweighting method used in Case 2 although all the Pi valuesare different for Case 1 and Case 2 the results are giventogether because the ranks are the same

If Table 12 is examined Norway for HDI0 P and 1198751119894 France for 1198752119894 and 1198755119894 Japan for 1198753119894 and Korea for 1198754119894 havebecome the most developed countries Other results in thetable can be interpreted similarly

Figure 6 shows that although there are some differencesin the ranking according to development levels of countriesboth HDI and other indexes are similar

According to Table 13 and Figure 7 it is observed thatindices give similar results even though there are somedifferences

In Figures 6 and 7 the 119909-axis represents the countries(order numbers 1 to 188 of the 188 world countries sorted inalphabetical order) and the y-axis represents a range of 0 and1 for each index (for each line in the figures)

43 Discussion of Results Findings obtained from the studyin DSC TOPSIS method standardization weighting andperformance indexes were found to be important in orderto rank the alternatives It is possible to interpret the resultsobtained in the application part of the study as follows

(i) In order to sort and compare the alternatives HDIvalues the results of the traditional TOPSIS methodand the results of the proposed DSC TOPSIS methodwere given together and a general comparison wasmadeWith the DSC TOPSIS method a newmeasurethat is more sensitive and stronger than the measureused in the traditional TOPSISmethod was proposedThe reason why the proposed unit of measurementis expressed as more sensitive and stronger is the useof similarity and correlation distances in addition tothe Euclidean distance used in the traditional TOPSISmethod in order to rank the alternatives In otherwords DSC TOPSIS uses three units that can beused to compare alternatives distance similarity andcorrelation From the results of the application it wasobserved that HDI TOPSIS and DSC TOPSIS resultsare not exactly the same and show some differences


Table 8 The overall performance index Pi values for the first five countries-Case 1

Country HDI 119875 1198751119894 1198752119894 1198753119894 1198754119894 1198755119894Value Rank Value Rank Value Rank Value Rank Value Rank Value Rank Value Rank

Afghanistan 0479 169 0281 172 0061 182 7035 185 9938 185 5197 172 7136 186Albania 0764 75 0697 70 0787 78 1520 32 2119 40 1241 27 1587 32Algeria 0745 83 0667 83 0793 75 1761 46 2468 53 1402 41 1827 47Andorra 0858 32 0819 33 0879 52 1496 29 1948 32 1386 40 1593 33Angola 0533 150 0338 155 0166 153 5396 138 6304 128 5303 176 5650 142

Table 9 The overall performance index Pi value for the first five countries-Case 2






This is the expected result because the measurementunits used in each method are different At this pointwhich method is preferred depends on the decision-makers opinion

(ii) In the DSC TOPSIS method although the perfor-mance indexes for Case 1 and Case 2 which have theobjective to bring the alternatives to the same unitfor each criterion have different values the rankingvalues are the same This is expected because theequal weighting method is actually equivalent to thesituation where there is no weighting Of course ifdifferent weights are used for the criteria in Case 2it is highly probable that both different performanceindex values and different rankings can be obtained

(iii) Case 3 andCase 4 which bring the criteria to the sameunit for each alternative refer to situations that areperformed without weighting and equal weightingrespectively The situation described above is alsovalid here and the results of the indexes are the same

(iv) Which standardization method is preferred dependson the decision-makers opinion and the nature ofthe problem being addressed The decision problemin this study was the calculation of HDI and theproposed solution method was DSC TOPSIS When

the structure of the decision problem is examined itcan be concluded that it is more reasonable to preferCase 1 and Case 2 where column-standardizationis used Considering the advantage of the fact thatdifferent weighting methods can be tried as well asequal weighting in Case 2 it is obvious that the mostreasonable solution would be Case 2

(v) Which performance index is preferred to rank thealternatives depends on the decision-maker At thispoint it is expected that P5 which is the averageof all performance indexes proposed in the study ispreferred

5 Conclusions

There are lots of efficient MCDMmethods which are suitablefor the purpose of ranking alternatives across a set of criteriaNone of MCDM methods in the literature gives a completesuperiority over the other Each of them can give differentresults according to their main purposes For this reasonthe method to be applied should be decided according tothe structure of the decision problem and the situationto be investigated In some cases the results obtained bytrying different MCDMmethods are compared and the mostreasonable solution is selected





Table 12 Top and bottom five countries with high and low level of development-Case 1 and Case 2

No HDI 119875 1198751119894 1198752119894 1198753119894 1198754119894 11987551198941 Norway Norway Norway France Japan Korea

(Republic of) France

2 Australia Switzerland Finland Korea(Republic of) France France Korea

(Republic of)

3 Switzerland Australia Netherlands Japan Korea(Republic of) Israel Japan

4 Germany Netherlands Switzerland Israel Israel Japan Israel5 Denmark Germany United States Spain Spain Spain Spain

184 Burundi Guinea-Bissau

CentralAfricanRepublic

Burkina Faso Burkina Faso Chad Burkina Faso

185 Burkina Faso Mozambique Guyana Afghanistan Afghanistan Benin Gambia186 Chad Sierra Leone South Sudan Gambia Gambia Guyana Afghanistan

187 Niger Chad Chad Niger Niger Guinea-Bissau Niger

188CentralAfricanRepublic


Congo Guinea Guinea Congo Guinea

Undoubtedly the oldest and valid method used to rankthe alternatives is distance The concept of distance forms thebasis of the TOPSIS method (it was called ldquotraditional TOP-SISrdquo in the study) InTOPSIS (the steps of implementation aregiven in detail in Section 31) the best result and the worstresult are determined first Then each of the alternativesdiscussed determines the distance between the best resultand the worst result A general ranking value is calculated byevaluating the best solution distance and the worst solutiondistance values and the alternatives are sorted according tothe size of these values It is desirable that the alternatives inthe method are as close as possible to the best solution andthe worst solution is away from the other side

As previously stated ldquonormalizing the decision matrix(Step 3)rdquo ldquocalculating the distance of positive and negative-ideal solutions (Step 6)rdquo and ldquocalculating the overall perfor-mance index for each alternative across all criteria (Step 7)rdquoof the traditional TOPSIS method steps can be examinedimproved or modified The aim of the study is to drawattention to the above-mentioned weaknesses of the tradi-tional TOPSIS method and to propose new solutions and

approaches For this purpose ldquocolumn row and double stan-dardizationrdquo were substituted for ldquonormalization approachrdquoldquoa new measure based on the concepts of distance similar-ity and correlation (DSC)rdquo was substituted for ldquoEuclideandistancerdquo and ldquonew sorting approaches based on the con-cept of distance (performance index)rdquo were substituted forldquosimple ratiordquo This new MCDM method is called DSCTOPSIS

The DSC TOPSIS method is particularly remarkable inits new measure and new performance indexes Because inaddition to the distance method used in the traditionalTOPSIS this new measure also includes similarity andcorrelation effects for sorting alternatives Just as in thedistance method alternatives are compared with positiveand negative-ideal solutions in the similarity method Inthe correlation method alternatives are compared accordingto the concept of relationship It has been tried to achievea stronger evaluation criterion by combining these threemeasurement techniques In the proposed new performanceindexes distance concepts were used Thus more powerfulsorting techniques have been tried to be obtained


Table 13 Top and bottom ten countries with high and low level of development-Case3 and Case 4

No HDI 119875 1198751119894 1198752119894 1198753119894 1198754119894 11987551198941 Norway Azerbaijan Austria Angola Angola Angola Angola

2 Australia France Hong KongChina (SAR) United States United

StatesUnitedStates

UnitedStates

3 Switzerland Lesotho Azerbaijan Nigeria Nigeria Nigeria Nigeria4 Germany Finland Libya Botswana Botswana Botswana Botswana

5 Denmark Spain Chad EquatorialGuinea

EquatorialGuinea

EquatorialGuinea

EquatorialGuinea

184 Burundi Niger Israel Malawi Palau Argentina Slovenia

185 Burkina Faso El Salvador ArgentinaBolivia

(PlurinationalState of)

Latvia Nepal Armenia

186 Chad Cabo Verde Nepal PalestineState of Estonia Ghana Liberia

187 Niger Dominica Ghana Denmark Fiji Haiti Burundi


Ethiopia Haiti Uganda Ukraine Lithuania Hungary

HDI P P1 P2 P3 P4 P5

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives

Figure 6 Comparison of the development levels of countries for all indices Case 1 and Case 2



5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives


The formulation of the DSC TOPSIS method is detailedfor the eight different cases defined (Table 1) The first fourcases are exemplified on the HDI data in the ldquoImplementationof the Proposed Methodrdquo subsection (Section 42) becausethey represent the basis of the DSC TOPSIS method (theycontain the last four cases partly) HDI is an index developedto rank countries according to their level of developmentThe index is basically based on Income Index Life Indexand Education Index variables and is calculated by geometricmean One of the most important reasons for considering theHDI in the study is that it is not very desirable to calculate theindex based on the geometric mean and that there are variouscriticisms on it With this paper the order of development forthe countries of the world has been tried to be obtained by adifferent method In a more explicit way a different proposalwas made for the calculation method of HDI calculated byUNDP It was concluded that Case 2 and P5 performanceindex could be preferred when using DSC TOPSIS methodin the calculation of HDI

In conclusion the effect of the proposed DSC TOPSISmethod on the following concepts is discussed in all aspects

(i) different standardization methods

(ii) different weighting methods

(iii) different performance indexes

6 Limitations and Future Research

The proposed DSC TOPSIS method does not have anylimitations and the method can be easily applied to alldecision problems where traditional TOPSIS can be applied

In the study the standardization of the decision matrix indifferent ways is also emphasized According to the preferredstandardization type it is emphasized that the decisionmatrix will have different meanings and different results willbe obtainedTheonly issue to be considered in the applicationof the method is to decide the standardization method to beapplied The following topics can be given as examples forfuture applications

(i) establishing different versions of the new measure-ment proposed by the DSC TOPSIS method egcombining the concepts of distance similarity andcorrelation with a different technique instead ofa geometric mean or experimenting with differentcombinations of these concepts together (distanceand similarity distance and correlation similarity andcorrelation etc)

(ii) suggesting different ranking index (Pi) formulas tosort alternatives

(iii) experiment of different weighting techniques in DSCTOPSIS method

(iv) blurring the DSC TOPSIS method fuzzy DSC TOP-SIS

Data Availability

Data are obtained from ldquohttphdrundporgenhumandevUnited Nations Development Programme Human Develop-ment Reportsrdquo website


Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] H A Taha Operations Research Pearson Education Inc Fayet-teville 1997

[2] Y Chen D M Kilgour and K W Hipel ldquoAn extreme-distance approach to multiple criteria rankingrdquo Mathematicaland Computer Modelling vol 53 no 5-6 pp 646ndash658 2011

[3] C LHwang andK P YoonMultiple AttributesDecisionMakingMethods and Applications Springer Berlin Germany 1981

[4] H Deng ldquoA Similarity-Based Approach to Ranking Multicrite-ria Alternativesrdquo in Proceedings of the International Conferenceon Intelligent Computing ICIC 2007 Intelligent ComputingTheories and Applications with Aspects of Artificial Intelligencepp 253ndash262 2007

[5] H Safari E Khanmohammadi A Hafezamini and S SAhangari ldquoA new technique for multi criteria decision makingbased on modified similarity methodrdquo Middle-East Journal ofScientific Research vol 14 no 5 pp 712ndash719 2013

[6] R A Krohling and A G C Pacheco ldquoA-TOPSIS ndash an approachbased on toPSIS for ranking evolutionary algorithmsrdquo inProceedings of the 3rd International Conference on InformationTechnology and Quantitative Management ITQM rsquo15 vol 55 ofProcedia Computer Science pp 308ndash317 2015

[7] E S Tasabat ldquoA new multi criteria decision making methodbased on distance similarity and correlationrdquo in Proceedings ofthe 10th International Statistics Congress ISC rsquo17 p 42 AnkaraTurkey 2017 httpwwwistkonnet

[8] H Safari and E Ebrahimi ldquoUsing Modified Similarity Multiplecriteria Decision Making technique to rank countries in termsof Human Development Indexrdquo Journal of Industrial Engineer-ing and Management vol 7 no 1 pp 254ndash275 2014

[9] J C Gower and P Legendre ldquoMetric and Euclidean propertiesof dissimilarity coefficientsrdquo Journal of Classification vol 3 no1 pp 5ndash48 1986

[10] Chapter 3 Distance similarity correlation httprameteltehusimpodani3-Distance20similaritypdf December 2017

[11] M M Deza and E Deza Encyclopedia of Distances SpringerBerlin Germany 2009

[12] B Mirkin Mathematical classification and clustering KAP1996

[13] Distances in Classification httpwwwieeemauaesbpdfdis-tances-in-classificationpdf July 2018

[14] E Triantaphyllou ldquoMulticriteria decision making methodsrdquo inA Comparative Study Springer 2000

[15] D L Olson ldquoComparison of weights in TOPSIS modelsrdquoMathematical and Computer Modelling vol 40 no 7-8 pp 721ndash727 2004

[16] C-H Yeh ldquoA problem-based selection of multiattributedecision-making methodsrdquo International Transactions in Oper-ational Research vol 9 no 2 pp 169ndash181 2002

[17] S H Zanakis A Solomon N Wishart and S Dublish ldquoMulti-attribute decision making a simulation comparison of selectmethodsrdquo European Journal of Operational Research vol 107no 3 pp 507ndash529 1998

[18] L Ren Y Zhang YWang andZ Sun ldquoComparativeAnalysis ofa Novel M-TOPSIS Method and TOPSISrdquo Applied MathematicsResearch Express Article ID abm005 10 pages 2007

[19] S H Cha ldquoComprehensive survey on distancesimilarity mea-sures between probability density functionsrdquo International Jour-nal of Mathematical Models and Methods in Applied Sciencesvol 1 no 4 pp 300ndash307 2007

[20] S Chakraborty and C-H Yeh ldquoA simulation comparison ofnormalization procedures for TOPSISrdquo in Proceedings of theInternational Conference on Computers and Industrial Engineer-ing CIE rsquo09 pp 1815ndash1820 2009

[21] C Chang J Lin J Lin and M Chiang ldquoDomestic open-endequity mutual fund performance evaluation using extendedTOPSIS method with different distance approachesrdquo ExpertSystems with Applications vol 37 no 6 pp 4642ndash4649 2010

[22] SOmosigho andDOmorogbe ldquoSupplier selectionusingdiffer-ent metric functionsrdquo Yugoslav Journal of Operations Researchvol 25 no 3 pp 413ndash423 2015

[23] K Ting ldquoA Modified TOPSIS with a different ranking indexrdquoEuropean Journal of Operational Research vol 260 pp 152ndash1602017

[24] J Ye ldquoMulticriteria group decision-making method usingvector similarity measures for trapezoidal intuitionistic fuzzynumbersrdquo Group Decision and Negotiation vol 21 no 4 pp519ndash530 2012

[25] S C Weller and A K Romney ldquoMetric Scaling Correspon-dence Analysis (P) Correspondence Analysis (QuantitativeApplications in the Social Sciences)rdquo a Sage University Paper1990

[26] UNDP (2016) Human Development Report 1990 UNDPNew York httphdrundporgenhumandev - United NationsDevelopment Programme Human Development Reports)

[27] httpswwweuieuResearchLibraryResearchGuidesEconom-icsStatisticsDataPortalHDR 2019

[28] UNDP (1990) Human Development Report 1990 UNDP NewYork httphdrundporgenreportsglobal

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rational decisions and the decision will be appropriate withinthe constraints [1]

Taking more than one criterion choosing the mostappropriate one among the alternatives or alternating sortingproblems is called Multicriteria Decision-Making (MCDM)problems The MCDM could have been appropriate for thepurposes of evaluating the alternatives in a particular orderor alternatively in order to determine the alternatives [2]MCDM methods which are used in a wide range of fieldsfrom personal selection problems to economic industrialfinancial political decision problems have begun to developtogether with the increasingly complex decision-makingprocess from the beginning of 1960s In particular MCDMmethods are being used to control the decision-makingmechanism and to obtain the decision result as quicklyand as easily as possible provided that the target to beachieved is explained by a number of criteria and each ofthe alternatives has its own advantages MCDM methodscan involve different decision-makers in the decision-makingprocess and allow many factors related to the decision-making problem to be evaluated simultaneously at differentlevels [1]

TOPSIS one of the MCDM methods developed byHwang and Yoon [3] is a technique to evaluate the perfor-mance of alternatives through the similarity with the idealsolution According to this technique the best alternativewould be one that is closest to the positive-ideal solutionand farthest from the negative-ideal solution The positive-ideal solution is one that maximizes the benefit criteriaand minimizes the cost criteria The negative-ideal solutionmaximizes the cost criteria andminimizes the benefit criteriaIn summary the positive-ideal solution is composed ofall best values attainable of criteria and the negative-idealsolution consists of all the worst values attainable of criteria[4ndash6]

The basis of this study is based on the report presented inthe statistics conference (istkon 2017) held in Turkey in 2017[7] In the report it was emphasized that TOPSIS methodcould be formed in different structures by drawing attentionto the unit of measurement In addition it was noted thatranking indexes may also be different This study was carriedout with the consideration of the issues discussed in thecongress and much more

In this study a new MCDM method based on thetraditional TOPSIS method is proposed In the proposedmethod ideal positive solution approximation or ideal nega-tive solution distance is calculated based on the distance sim-ilarity and correlation (DSC)measures unlike the traditionalTOPSIS method For this reason this new unit of measureproposed to rank alternatives is named as DSC and the newMCDMmethod developed is called DSC TOPSIS The mainadvantage of the proposed unit of measurement is that itdoes not only rank the alternatives according to the conceptof distance but also according to the concepts of similarityand correlation In other words the major advantage ofthe proposed method is that it proposes a stronger unit ofmeasure by considering the three basic concepts that can beused to compare units distance similarity and correlationAnother advantage of the proposed new MCDM method is

that it suggests three new different methods that can be usedto rank alternatives according to their importance levels Inaddition the impact of the proposed method on the caseswhere the decision matrix is dealt with by row columnand double standardization methods is discussed in detailThese methods are applied to the results obtained by theproposed method Thus the traditional TOPSIS method andthe sorting technique used in this method are compared withthe proposed MCDM method and the proposed three newsorting methods The functioning of the method has beentested to determine the order of development of countriesFor this purpose the indicators of Human DevelopmentIndex (HDI) calculated by UN Development Programme(UNDP) have been utilized The results were compared withHDI and traditional TOPSIS values

There are no studies in the literature that use the con-cepts of distance similarity and correlation together In thiscontext this study will be the first The closest one to theproposed method in this study is Dengrsquos method which waspublished in the study entitled ldquoA Similarity-Based Approachto Ranking Multicriteriardquo in 2007 With this study Deng pre-sented a similarity-based approach to ranking multicriteriaalternatives for solving discrete multicriteria problems ThenSafari and et al [5] modified Dengrsquos similarity-based methodand they proposed a newMCDMmethod based on similarityand TOPSIS And then Safari and Ebrahimi [8] used asimilarity-based technique by Deng [4] to rank countries interms of HDI They also proposed a solution for resolving aproblem which exists in Dengs method Although the issuediscussed in the application part of Safari and Ebrahimi [8]and this study is on HDI data the ways of handling thisdata are different The most important difference betweenthe two studies is the selection of criteria While Safari andEbrahimi [8] preferred to use ldquolife expectancy at birthrdquoldquomean years of schooling ldquoexpected years of schoolingrdquo andldquolog GNIrdquo which are the indicator of HDI as criteria inthis study it was preferred to use ldquoLiferdquo ldquoEducationrdquo andldquoIncomerdquo indexes that expressed the dimensions of HDI Atthis point Safari and Ebrahimis [8] method can be criticizedon their preferred criteria That is HDIrsquos two indicators ofthe education dimension are considered as separate criteriain their study in fact it gave more weight to the dimensionof education than the other dimensions in terms of relativeimportance In this context with this study a suggestionhas been made to this situation mentioned above Anotherdifference is undoubtedly that theHDI data discussed in bothstudies are of different years

In the following sections the concepts of distance sim-ilarity and correlation will be mentioned first (Section 2)Then in Section 3 respectively the operation of the tradi-tional TOPSIS method will be described the steps of theMCDMmethod proposed in the study will be given and theproposed sorting methods will be explained In Section 4the functioning of the proposedMCDMmethod and rankingtechniques will be tested on the variables that indicate thelevel of development of the countries For this purposebrief information about HDI will be given first in thissection In Section 5 evaluation of the results obtained willbe made






inequality axiom)




119904119894119895 = 1 minus 119889119894119895 (1)



119889119894119895 = radic1 minus 119904119894119895 (2)





( 119898sum119894

1003816100381610038161003816119909119894 minus 1199101198941003816100381610038161003816119901)1119903

(3)






The dissimilarities

1 minus (119888119900119903119903 119904119894119895) (6)

1 minus (119888119900119903119903 119904119894119895)2 (7)




(8)



1 minus (cos 119904119894119895) (9)














31 TOPSIS Method


119883 = [[[[[[


(10)







1199091015840119894119895 = 119909119894119895(sum119899119895=1 1199092119894119895)12 (12)


1198831015840 = [[[[[[[


]]]]]]](13)



119884 = [[[[[[[


]]]]]]]= [[[[[[


(14)



For PIS (119868+)119868+ = (max

119894119910119894119895 | 119895 isin 119869) (min

119894119910119894119895 | 119895 isin 1198691015840 (15)

119868+ = 119868+1 119868+2 119868+119899 (16)


119894119910119894119895 | 119895 isin 119869) (max

119894119910119894119895 | 119895 isin 1198691015840 (17)





119863+119894 = radic 119898sum119895=1

(119910119894119895 minus 119868+119895 )2 (19)

119863minus119894 = radic 119898sum119895=1

(119910119894119895 minus 119868minus119895 )2 (20)


1198751119894 = 119863minus119894119863minus119894 + 119863+119894 (21)



















minusi +

i

x3

x1

x2

Iminusj

I+j

Ai

y

x

y







1199091015840119894119895 = 119909119894119895119898119890119889119894119886119899119883 (22)

1199091015840119894119895 = 119909119894119895max (119909119894119895) (23)



1199091015840119894119895 = 119909119894119895 minus 120583119883120590119883 (24)











Pr119900119901119900119904119890119889 119889119894 = 3radic(119904119902119906119886119903119890119889 119889119894) (119888119900119903119903 119889119894) (cos 119889119894) (25)






Pr119900119901119900119904119890119889 119863+119894= 3radic(119904119902119906119886119903119890119889 119889+119894 ) (119888119900119903119903 119889+119894 ) (cos 119889+119894 ) (26)






119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

















119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)


119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)


119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)



119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)











Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018













inequality axiom)




119904119894119895 = 1 minus 119889119894119895 (1)



119889119894119895 = radic1 minus 119904119894119895 (2)





( 119898sum119894

1003816100381610038161003816119909119894 minus 1199101198941003816100381610038161003816119901)1119903

(3)






The dissimilarities

1 minus (119888119900119903119903 119904119894119895) (6)

1 minus (119888119900119903119903 119904119894119895)2 (7)




(8)



1 minus (cos 119904119894119895) (9)














31 TOPSIS Method


119883 = [[[[[[


(10)







1199091015840119894119895 = 119909119894119895(sum119899119895=1 1199092119894119895)12 (12)


1198831015840 = [[[[[[[


]]]]]]](13)



119884 = [[[[[[[


]]]]]]]= [[[[[[


(14)



For PIS (119868+)119868+ = (max

119894119910119894119895 | 119895 isin 119869) (min

119894119910119894119895 | 119895 isin 1198691015840 (15)

119868+ = 119868+1 119868+2 119868+119899 (16)


119894119910119894119895 | 119895 isin 119869) (max

119894119910119894119895 | 119895 isin 1198691015840 (17)





119863+119894 = radic 119898sum119895=1

(119910119894119895 minus 119868+119895 )2 (19)

119863minus119894 = radic 119898sum119895=1

(119910119894119895 minus 119868minus119895 )2 (20)


1198751119894 = 119863minus119894119863minus119894 + 119863+119894 (21)



















minusi +

i

x3

x1

x2

Iminusj

I+j

Ai

y

x

y







1199091015840119894119895 = 119909119894119895119898119890119889119894119886119899119883 (22)

1199091015840119894119895 = 119909119894119895max (119909119894119895) (23)



1199091015840119894119895 = 119909119894119895 minus 120583119883120590119883 (24)











Pr119900119901119900119904119890119889 119889119894 = 3radic(119904119902119906119886119903119890119889 119889119894) (119888119900119903119903 119889119894) (cos 119889119894) (25)






Pr119900119901119900119904119890119889 119863+119894= 3radic(119904119902119906119886119903119890119889 119889+119894 ) (119888119900119903119903 119889+119894 ) (cos 119889+119894 ) (26)






119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

















119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)


119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)


119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)



119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)











Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018




















31 TOPSIS Method


119883 = [[[[[[


(10)







1199091015840119894119895 = 119909119894119895(sum119899119895=1 1199092119894119895)12 (12)


1198831015840 = [[[[[[[


]]]]]]](13)



119884 = [[[[[[[


]]]]]]]= [[[[[[


(14)



For PIS (119868+)119868+ = (max

119894119910119894119895 | 119895 isin 119869) (min

119894119910119894119895 | 119895 isin 1198691015840 (15)

119868+ = 119868+1 119868+2 119868+119899 (16)


119894119910119894119895 | 119895 isin 119869) (max

119894119910119894119895 | 119895 isin 1198691015840 (17)





119863+119894 = radic 119898sum119895=1

(119910119894119895 minus 119868+119895 )2 (19)

119863minus119894 = radic 119898sum119895=1

(119910119894119895 minus 119868minus119895 )2 (20)


1198751119894 = 119863minus119894119863minus119894 + 119863+119894 (21)



















minusi +

i

x3

x1

x2

Iminusj

I+j

Ai

y

x

y







1199091015840119894119895 = 119909119894119895119898119890119889119894119886119899119883 (22)

1199091015840119894119895 = 119909119894119895max (119909119894119895) (23)



1199091015840119894119895 = 119909119894119895 minus 120583119883120590119883 (24)











Pr119900119901119900119904119890119889 119889119894 = 3radic(119904119902119906119886119903119890119889 119889119894) (119888119900119903119903 119889119894) (cos 119889119894) (25)






Pr119900119901119900119904119890119889 119863+119894= 3radic(119904119902119906119886119903119890119889 119889+119894 ) (119888119900119903119903 119889+119894 ) (cos 119889+119894 ) (26)






119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

















119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)


119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)


119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)



119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)











Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018













1199091015840119894119895 = 119909119894119895(sum119899119895=1 1199092119894119895)12 (12)


1198831015840 = [[[[[[[


]]]]]]](13)



119884 = [[[[[[[


]]]]]]]= [[[[[[


(14)



For PIS (119868+)119868+ = (max

119894119910119894119895 | 119895 isin 119869) (min

119894119910119894119895 | 119895 isin 1198691015840 (15)

119868+ = 119868+1 119868+2 119868+119899 (16)


119894119910119894119895 | 119895 isin 119869) (max

119894119910119894119895 | 119895 isin 1198691015840 (17)





119863+119894 = radic 119898sum119895=1

(119910119894119895 minus 119868+119895 )2 (19)

119863minus119894 = radic 119898sum119895=1

(119910119894119895 minus 119868minus119895 )2 (20)


1198751119894 = 119863minus119894119863minus119894 + 119863+119894 (21)



















minusi +

i

x3

x1

x2

Iminusj

I+j

Ai

y

x

y







1199091015840119894119895 = 119909119894119895119898119890119889119894119886119899119883 (22)

1199091015840119894119895 = 119909119894119895max (119909119894119895) (23)



1199091015840119894119895 = 119909119894119895 minus 120583119883120590119883 (24)











Pr119900119901119900119904119890119889 119889119894 = 3radic(119904119902119906119886119903119890119889 119889119894) (119888119900119903119903 119889119894) (cos 119889119894) (25)






Pr119900119901119900119904119890119889 119863+119894= 3radic(119904119902119906119886119903119890119889 119889+119894 ) (119888119900119903119903 119889+119894 ) (cos 119889+119894 ) (26)






119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

















119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)


119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)


119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)



119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)











Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018



















minusi +

i

x3

x1

x2

Iminusj

I+j

Ai

y

x

y







1199091015840119894119895 = 119909119894119895119898119890119889119894119886119899119883 (22)

1199091015840119894119895 = 119909119894119895max (119909119894119895) (23)



1199091015840119894119895 = 119909119894119895 minus 120583119883120590119883 (24)











Pr119900119901119900119904119890119889 119889119894 = 3radic(119904119902119906119886119903119890119889 119889119894) (119888119900119903119903 119889119894) (cos 119889119894) (25)






Pr119900119901119900119904119890119889 119863+119894= 3radic(119904119902119906119886119903119890119889 119889+119894 ) (119888119900119903119903 119889+119894 ) (cos 119889+119894 ) (26)






119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

















119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)


119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)


119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)



119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)











Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018










1199091015840119894119895 = 119909119894119895 minus 120583119883120590119883 (24)











Pr119900119901119900119904119890119889 119889119894 = 3radic(119904119902119906119886119903119890119889 119889119894) (119888119900119903119903 119889119894) (cos 119889119894) (25)






Pr119900119901119900119904119890119889 119863+119894= 3radic(119904119902119906119886119903119890119889 119889+119894 ) (119888119900119903119903 119889+119894 ) (cos 119889+119894 ) (26)






119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

















119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)


119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)


119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)



119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)











Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018











119863+119894 = 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (28)

119863minus119894 = 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (29)

















119888119900119903119903 119904119883119884 = (1119899)sum119899119895 119909119895119910119895 minus 120583119883120583119884120590119883120590119884 (38)


119888119900119903119903 119904lowast119883119884 = 1119899119899sum119895

119909119895119910119895 (39)


119864119906119888 119889119894 = radic 119899sum119895

(119909119895 minus 119910119895)2

= radic 119899sum119895

1199092119895 + 119899sum119895

1199102119895 minus 2 119899sum119895

119909119895119910119895(40)



119909119895119910119895 (41)

= radic2(119899 minus 119899sum119895

119909119895119910119895) (42)











Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018














Pr119900119901119900119904119890119889 119889119894 = 3radic1198892119894 11988921198942119899 11988921198942119899 (47)

= 13radic41198992 1198892119894 (48)


Pr119900119901119900119904119890119889 119863+119894 = 13radic41198992 119889119894+2 = 1

3radic41198992 119899sum119895=1

(119910119894119895 minus 119868+119895 )2 (49)


3radic41198992 119899sum119895=1

(119910119894119895 minus 119868minus119895 )2 (50)




119878+119894 = 1119899119899sum119895=1

119910119894119895119868+119895 (51)

119878minus119894 = 1119899119899sum119895=1

119910119894119895119868minus119895 (52)


119863+119894 = 1 minus 119878+119894 = 1 minus (1119899119899sum119895=1

119910119894119895119868+119895) (53)


119910119894119895119868minus119895) (54)











A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018









A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )


A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )






A

B C

D

D+

Dminus

min(D+i )

min(Dminusi )

max(D+i )

max(Dminusi )





1198755119894 = 3radic1198752119894 1198753119894 1198754119894 (58)







Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018









Table1Im

plem

entatio

nof

thep

ropo

sedmetho

d

Cases

Structureo

fthe

Perfo

rmance

Matrix

Prop

osed

Metho

dFo

rmulas

Colum

n-Standardized

Row-

Standardized

Weigh

ting

SqEu

cCorrs

Cos

sCos

dCos

dProp

osed

D

Case1

Used

Unu

sed

Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case2

Used

Unu

sed

Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case3

Unused

119880119904119890119889lowastlowast

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case4

Unused

119880119904119890119889lowastlowast

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case5

Used

(First)

119880119904119890119889lowastlowast

(Second)

Unused

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case6

Used

(First)

119880119904119890119889lowastlowast

(Second)

Used

Equa

tions

(28)-(2

9)Eq

uatio

ns(51)-(52)

Equa

tions

(51)-(5

2)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(53)-(5

4)Eq

uatio

ns(49)-(5

0)

Case7

Used

(Secon

d)Usedlowast

(Firs

t)Unu

sed

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

Case8

Used

(Secon

d)Usedlowast

(Firs

t)Used

Equatio

ns(28)-(29)

Equatio

ns(30)-(31)

Equatio

ns(32)-(33)

Equatio

ns(34)-(35)

Equatio

ns(36)-(37)

Equatio

ns(26)-(27)

lowastStand

ardize

criteria

accordingto

row-stand

ardizatio

nlowastlowastFirs

tstand

ardize

criteria

accordingto

row-stand

ardizatio

nandthen

stand

ardize

PISandNIS

values

accordingto

row-stand

ardizatio

n


DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018









DIMENSIONS

INDICATORS

DIMENSION INDEX






























119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018
















119867119863119868 = 3radic119868119871119894119891119890119868119864119889119906119888119886119905119894119900119899119868119868119899119888119900119898119890 (60)




























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018



























Unweighted I+ 1548 1728 1746 1674 0089 1414 1358 1401 1391 0024I- -2711 -2491 -2338 -2513 0153 -1404 -1414 -1414 -1411 0005

Weighted I+ 0516 0576 0582 0558 0030 0471 0453 0467 0464 0008I- -0904 -0830 -0779 -0838 0051 -0468 -0471 -0471 -0470 0002
































5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018
























5 Conclusions










(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018
















(Republic of)



















StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018












StatesUnitedStates

UnitedStates



EquatorialGuinea

EquatorialGuinea

EquatorialGuinea










1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185

Alternatives




5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018










5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133 137 141 145 149 153 157 161 165 169 173 177 181 1851

Alternatives














Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018











References




































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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