ingenioussolutionfortherankreversalproblemof...

Research ArticleIngenious Solution for the Rank Reversal Problem ofTOPSIS Method

Wenguang Yang 12

1College of Science North China Institute of Science and Technology Beijing 101601 China2School of Automation Science and Electrical Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Wenguang Yang yangwenguangncisteducn

Received 1 September 2019 Revised 9 November 2019 Accepted 8 January 2020 Published 30 January 2020

Academic Editor Gordon Huang

Copyright copy 2020 Wenguang Yang is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Although the classic TOPSIS method is very practical there may be a problem of rank reversal in the addition deletion orreplacement of the candidate set which makes its credibility greatly compromised Based on the understanding of the classicalTOPSIS method this paper establishes a new improved TOPSIS method called NR-TOPSIS Firstly the historical maximum andminimum values of all attribute indicators from a global perspective during the evaluation process are determined Secondlyaccording to whether the attributes belong to the benefit attribute or cost attribute standardization is carried out And then in thecase where the historical values of attributes are determined we re-fix the positive ideal solution and the negative ideal solution Atthe same time this paper gives the definition of ranking stable and proves that the NR-TOPSIS proposed satisfies ranking stablewhich theoretically guarantees that the rank reversal phenomenon does not exist Finally in the verification of examples theresults are consistent with the theoretical analysis which further support the theoretical analysis e NR-TOPSIS methodovercomes rank reversal which is not only obviously superior to the classical TOPSIS method but also relatively superior to theR-TOPSIS method which has also overcome rank reversal It is also superior to other reference methods due to itssimple calculation

1 Introduction

Over the last thirty years the research on multiple attribute(criteria) decision-making (MADMMCDM) has become ahot issue in different fields of natural science and socialscience [1ndash16] e technique for order preference by simi-larity to the ideal solution (TOPSIS) is a useful and powerfulmethod for dealing with MADM problems which is proposedby Hwang and Yoon [17 18] TOPSIS is a general method forsolving MADM problems which takes into account bothpositive and negative ideal solutions Many scholars havecombined the TOPSIS method with other intelligent com-puting methods resulting in many cross-research results andsolving many problems in real life [19ndash33] ese cross-combination approaches include data envelopment analysis(DEA) [19] hesitant fuzzy correlation coefficient [20] ana-lytic hierarchy process (AHP) [21] fuzzy AHP [22ndash25]intuitionistic fuzzy number [26 27] triangular fuzzy number

[28] vague sets [29] analytic network process (ANP) [30]weighted grey relational coefficient [31] and neutrosophicsets [32 33] ese hybrid methods make full use of theadvantages of the TOPSIS method which can quantitativelycharacterize the difference between the alternatives and thepositive and negative ideal solutions However they do notdiscuss the fact that if the TOPSIS method has a reversal oforder the credibility of the hybrid approach will be severelyreduced When the decision-making object changes on theoriginal basis especially increasing or reducing the evaluationobject or replacing a certain evaluation object the traditionalTOPSIS method often has the phenomenon of rank reversale quality of evaluation methods often depends on thestability and consistency of evaluation results at is to saywhen the evaluation object is changed the correspondingresults should not be inconsistent is is what is commonlyreferred to as rank reversal In fact many MADM methodshave the problem of rank reversal such as analytic hierarchy

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9676518 12 pageshttpsdoiorg10115520209676518

process (AHP) [34ndash41] VIseKriterijumska Optimizacija IKOmpromisno Resenje (VIKOR) [42] preference rankingorganization method for enrichment of evaluations(PROMETHEE) [43] and evaluation based on distance fromaverage solution (EDAS) [44] Achieving the rank preser-vation of MADMMCDMmethods has turned into the focusof many scholars in applied study [45ndash51]

e problem of rank reversal in the TOPSIS method hasreceived great attention and different scholars have tried tofind satisfactory solutions from an experimental or theo-retical level [52ndash57] By collecting 130 related papers pub-lished in international journals from 1980 to 2015 Aires andFerreira provided an extensive literature review on MCDMmethodologies and rank reversals including the TOPSISmethod [52] Zavadskas et al [53] also reviewed the de-velopment of the TOPSIS method from 2000 to 2015 but didnot discuss the solution of rank reversal Ren et al [54]proposed a novel M-TOPSIS method which can solve theproblems of TOPSIS such as rank reversals and evaluationfailure when alternatives are symmetrical but there is notheoretical proof Wang and Luo [45] used the counterex-ample to illustrate the fact that the TOPSIS method has aproblem of rank reversal but did not give a solution Garcıa-Cascales and Lamata [55] considered two aspects to improvethe rank reversal problem of the TOPSIS method On theone hand a new norm is used for normalization but it is notsufficient On the other hand the absolute mode is used torewrite the positive ideal solution and the negative idealsolution e fixed instance verification is feasible but theindicator attribute is not considered and the improvedmethod does not have the validity in all cases Once the costindicator data are encountered it will be invalid Based onthe understanding of data dispersion degree Yang and Wu[56] proposed a new strategy combining improved greyrelational analysis and TOPSIS method and verified thecontrollability of order reversal in the process of case ver-ification In the process of evaluation the idea of attributevariable weight was adopted with the change of evaluationknowledge Aires and Ferreira [57] analyzed that most of theliterature on rank reversal of TOPSIS was limited to casestudies and then developed an improved TOPSIS methodwith domain parameters called R-TOPSIS method whichdemonstrated the effectiveness of the method from a sta-tistical point of view But in theory there is no reasonableproof and explanation for this method e key to solvingthe problem of rank reversal is to analyze the problem from aglobal perspective and always use the same scale to measurethe data conversion In addition some scholars have useddifferent integrated MADM methods to focus on the searchof the best option and also to determine which methodrsquos bestoption was better and weakened the attention of the rankreversal problem [46 58ndash61] In doing so there is nosubstantive solution to the problem of rank reversal

In brief this paper will focus on the following mainresearch (i) we propose a novel extended TOPSIS method toovercome the rank reversal problem which is suitable forMADM problems with arbitrary attribute indicators (ii) wegive the definition of ranking stable and ranking unstableand strictly prove in theory that the NR-TOPSIS method

proposed in this paper is ranking stable and the R-TOPSISmethod is also ranking stable (iii) we use this method toverify two classic cases which highlight the effectiveness andconsistency of this method and provide a new simpleTOPSIS improvement scheme for solvingMADMproblems

e remainder of the paper is organized as followsSection 2 presents the traditional TOPSIS methodR-TOPSIS method and a new improvement to solve rankreversal problem in the TOPSIS method which is called theNR-TOPSIS method Section 3 analyzes the theoreticalmechanism of rank reversal problem and proves that theNR-TOPSIS method as well as the R-TOPSIS method isranking stable Section 4 describes two numerical cases toverify the validity and consistency of the NR-TOPSISmethod Finally Section 5 provides some importantconclusions

2 Rank Reversal in TOPSIS Method andIts Improvement

e proposal of the TOPSIS method in a sense conforms tothe logic of human decision-making and judgment and canreasonably compare the proposed scheme with the positiveideal solution and the negative ideal solution quantitativelythus giving the order closest to the ideal solution In thissection we will show the classical TOPSIS method and theimproved TOPSIS method named R-TOPSIS that hasovercome the rank reversal Furthermore a new improvedmethod of TOPSIS to overcome rank reversal based onstatistical laws will be proposed

Formally consider an MADM problem with m alter-natives and n attributes and the weights of the attributes areknown which are denoted by w1 w2 wn with1113936

nj1wj 1 wj gt 0 j 1 2 n Use the symbol X to

represent the original decision matrix which is expressed asthe following form

C1 C2 Cn

hellip

hellip

hellip hellip hellip helliphellip

hellip

hellipA1 x12

x22

xm2

x1n

x2n

xmn mtimesn

x11

x21

xm1

A2

Am

X = (1)

where A1 A2 Am are alternatives C1 C2 Cn areattributes and xij are original attribute values

21 ampe Traditional TOPSIS Method We briefly explain theimplementation of the classic TOPSIS method as follows

Step 1 Compute the normalized decision-makingmatrix Y (yij)mtimesn where yij are normalized attributevalues expressed as

yij xij

1113936

mi1x

2ij

1113969 i 1 2 m j 1 2 n (2)

Step 2 Calculate the weighted normalized decision-making matrix R (rij)mtimesn e weighted normalizedattribute value rij is computed by

2 Mathematical Problems in Engineering

rij wjyij i 1 2 m j 1 2 n (3)

Step 3 Determine the positive ideal solution r+j and the

negative ideal solution rminusj respectively

If Cj is a benefit attribute then

r+j max

irij

rminusj min

irij

⎧⎪⎨

⎪⎩

i 1 2 m j 1 2 n

(4)

If Cj is a the cost attribute then

r+j min

irij

rminusj max

irij

⎧⎪⎨

⎪⎩

i 1 2 m j 1 2 n

(5)

Step 4 Calculate the Euclidean distances of each al-ternative with the positive ideal solution and thenegative ideal solution respectively

S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

i 1 2 m

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(6)

Step 5 Compute the evaluation result Si for alternativeAi defined as follows

Si Sminus

i

S+i + Sminus

i

i 1 2 m (7)

Step 6 Rank alternatives according to their evaluationresults in descending order e larger the Si the betterthe alternative Ai

22 R-TOPSIS Method Although the TOPSIS method isvery popular and practical it has a fatal defect that is it isprone to the problem of rank reversal when the evaluationalternative changes which leads to the untrustworthyevaluation results Aires and Ferreira [57] put forward a newapproach named R-TOPSIS and proposed the use of anadditional input parameter to the TOPSIS method calledldquodomainrdquo which proved to be robust in experiments but didnot give a theoretical proof that avoids rank reversal eR-TOPSIS method is composed mainly of the followingseven steps

Step 1 Determine a domain matrix D (dkj)2timesn

according to decision makers experts or interviewees[57] d1j(j 1 2 n) and d2j(j 1 2 n) arerespectively the minimum value and the maximumvalue of Dj(j 1 2 n) where Dj(j 1 2 n)

is the domain of attribute Cj(j 1 2 n)

Step 2 Calculate the normalized decision matrix Y

(yij)mtimesn based on Max or Max-Min wayMax way

yij xij

d2j

i 1 2 m j 1 2 n (8)

Max-Min way

yij xij minus d1j

d2j minus d1j

i 1 2 m j 1 2 n (9)

Step 3 Calculate the weighted normalized decision-making matrix R (rij)mtimesn where the weighted nor-malized attribute value rij is also computed by equation(3)Step 4 Determine the positive ideal solution r+

j and thenegative ideal solution rminus

j respectivelyIf Cj is a benefit attribute then

r+j wj

rminusj

d1j

d2j

wj

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

i 1 2 m j 1 2 n

(10)

If Cj is a cost attribute then

r+j

d1j

d2j

wj

rminusj wj

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

i 1 2 m j 1 2 n

(11)

Because d1j lt d2j obviously in this case if Cj is abenefit attribute then rminus

j lt r+j otherwise rminus

j gt r+j

j 1 2 nStep 5 Calculate the Euclidean distances of each al-ternative with the positive ideal solution and thenegative ideal solution respectively

S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(12)

Step 6 Calculate the closeness for every alternative Ai tothe ideal solution as

Mathematical Problems in Engineering 3

RSi Sminus

i

S+i + Sminus

i

i 1 2 m (13)

Step 7 e alternatives are ranked in descending orderaccording to the scores between each alternative andideal solution then the final evaluation result isobtained

Remark 1 In the process of normalization the R-TOPSISmethod selects Max or Max-Min way and does not considerthe difference between the benefit attribute and the costattribute but considers it in the fourth step We will analyzethe rationality of the two normalized ways as follows

221 MaxWay If Cj is a cost-type criteria then the less thevalue of xij the better the performance Because yij xijd2j

and rij wjyij(i 1 2 m j 1 2 n) yij and rij

are also as small as possible Consider two extreme cases oneis to let xij d1j and the other is to let xij d2j If xij d1jthen yij d1jd2j and rij (d1jd2j)wj r+

j If xij d2jthen yij 1 and rij wj rminus

j For any attribute value xijthe condition d1j le xij led2j is satisfied We can get thefollowing conclusion r+

j le rij le rminusj and the smaller the dis-

tance between rij and r+j the better the evaluation result and

the larger the distance between rij and rminusj the better the

evaluation resulte fourth step of the R-TOPSIS method isa clever combination of cost-type criteria and positive andnegative ideal solutions in line with peoplersquos thinking logic

If Cj is a benefit attribute we also consider the next twocases If xij d1j then yij d1jd2j and rij (d1jd2j)wj rminus

j If xij d2j then yij 1 and rij wj r+j We

can easily get the result that rminusj le rij le r+

j

222 Max-Min Way If Cj is a benefit attribute then theattribute value xij is as big as possible and yij and rij are alsoas big as possible Also calculate the two extremes one is tolet xij d1j and the other is to let xij d2j If xij d1j thenyij (d1j minus d1j)(d2j minus d1j) 0 and rij 0 In fact in thecase of xij d1j and rminus

j (d1jd2j)wj the result should berij rminus

j but this result is not necessarily true in R-TOPSISunless d1j 0 such a design is defective If xij d2j thenyij 1 and rij wj r+

j If Cj is a cost-type criteria we can also find some flaws

For example if xij d1j then yij 0 by equation (9) andrij 0 Similarly rij 0ne r+

j is is also inconsistent withour logical understanding If xij d2j then yij 1 andrij wj r+

j In summary for the normalization of the second step of

the R-TOPSIS method we recommend using Max wayinstead of Max-Min way We can be sure that in Max-Minway for the case of xij d1j the R-TOPSIS method en-counters unexplained contradictions while Max way has noproblems

23 ANew ImprovedMethod of TOPSIS Aires and Ferreira[57] prove that the R-TOPSIS method can overcome rankreversal through various experimental tests and is animproved TOPSIS scheme with high credibility In factwe can also consider the standardized method and thedetermination of positive and negative ideal solutionsand establish a new improved TOPSIS method toovercome rank reversal A new improved TOPSISmethod based on this knowledge will be given below withseven steps Since the new improved TOPSIS method cansolve rank reversal problem in this paper based on thehistorical maximum value of indicator data it is ab-breviated as the NR-TOPSIS method e establishmentof the following method belongs to an absolute mode[55] which is an evaluation with comprehensiveand accumulated historical knowledge of the evaluationproblem

Step 1 Determine the minimum value mj and maxi-mum value Mj of each attribute Cj according to thestatistical law of attribute values at is to say for anyattribute value xij the following conditionmj lexij leMj is satisfied At the same time the con-dition mj lexij leMj is still satisfied when the scheme isincreased decreased or replacedStep 2 In order to eliminate the influence of dimensionon data decision-making the original decision-makingmatrix X (xij)mtimesn is standardized and transformed togenerate standardized decision-making matrixY (yij)mtimesn where yij are normalized attribute valuesIf Cj is a benefit attribute then

yij xij minus mj

Mj minus mj

(14)


yij Mj minus xij

Mj minus mj

(15)

Step 3 Calculate the weighted normalized decision-making matrix R (rij)mtimesn Compared with the tra-ditional TOPSIS method the weighted normalizedattribute value rij has the same calculation equation(3) which is omitted hereStep 4 Determine the positive ideal solution r+


j respectively

r+j wj

rminusj 0

⎧⎨

⎩

j 1 2 n

(16)

Step 5 Compute the Euclidean distances S+i and Sminus

i forevery alternative Ai between the positive ideal solutionand the negative ideal solution respectively


S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(17)

Step 6 Calculate the closeness for every alternativeAi tothe ideal solution as

NSi Sminus

i

S+i + Sminus

i

i 1 2 m (18)


Remark 2 e extended NR-TOPSIS method established inthis paper is based on the global perspective of evaluationproblems and the normalization of two types of attributeindicators benefit type and cost type It completely avoids thedefects of R-TOPSIS in Max-Min way Equations (14) and(15) are commonly used linear normalization transformationmeasures for benefit-type and cost-type indicators respec-tivelyWe will analyze the rationality of the two cases in whichthe indicator is a benefit type or cost type as follows

(1) If Cj is a benefit attribute then the attribute value xij

is as big as possible and yij and rij are also as big aspossible Consider two extreme cases one is to letxij mj and the other is to let xij Mj If xij mjthen yij (mj minus mj)(Mj minus mj) 0 and rij 0 Inthis case rminus

j 0 so rij rminusj which is consistent with

peoplersquos understanding If xij Mj then yij 1 andrij wj r+

j For any attribute value xij we caneasily find that rminus

j le rij le r+j

(2) If Cj is a cost-type criteria then the attribute valuexij is as small as possible and yij and rij are also assmall as possible If xij d1j thenyij (Mj minus mj)(Mj minus mj) 1 and rij wj In thiscase rij wj r+

j If xij Mj then yij 0 andrij 0 rminus

j

In conclusion the NR-TOPSIS method overcomes thepossible deficiencies of the R-TOPSIS method in the se-lection of normalization e following section will focus onthe ability to maintain order theoretically

3 Theoretical Discussion on Improving theTOPSIS Method to Overcome RankReversal Problem

is section tries to give the theoretical basis of R-TOPSISand NR-TOPSIS to overcome the rank reversal At the same

time we also try to give the explanation that the traditionalTOPSIS method is prone to rank reversal e final step inthe three TOPSIS methods mentioned above is to evaluatethe value ranking For the evaluation results if there is norank reversal under any circumstances the ranking of thealgorithm is stable otherwise the ranking is unstable

Definition 1 Let A A1 A2 Am1113864 1113865 be an alternative setand the ranking after evaluation by some evaluationmethods is A(1) ≻A(2) ≻ middot middot middot ≻A(m) whereA(i) i 1 2 m is an alternative in A In the case ofadding deleting or replacing alternatives under the originalalternative set A a new alternative set B B1 B2 Bl1113864 1113865 isobtained After ranking by the evaluation method theranking result is B(1) ≻B(2) ≻ middot middot middot ≻B(l) whereB(k) k 1 2 l is an alternative in B For any two al-ternatives Bp Bq isin AcapB if there is no rank reversal thenthe evaluation method is ranking stable otherwise theranking is unstable

Theorem 1 Let A A1 A2 Am1113864 1113865 be an alternative setthe R-TOPSIS method is ranking stable

Proof Let A A1 A2 Am1113864 1113865 be an alternative set andthe ranking after evaluation by the R-TOPSIS method isA(1) ≻A(2) ≻ middot middot middot ≻A(m) where A(i) i 1 2 m is analternative in A In the case of adding deleting or replacingalternatives under the original alternative set A a new al-ternative set B B1 B2 Bl1113864 1113865 is obtained After rankingby the evaluation method the ranking result isB(1) ≻B(2) ≻ middot middot middot ≻B(l) where B(k) k 1 2 l is an al-ternative in B

For any two alternatives Bp Bq isin AcapB there arematching alternatives Ae Af isin A subject toAe Bp andAf Bq with Ae xe1 xe2 xen1113864 1113865 andAf xf1 xf2 xfn1113966 1113967

Obviously ye1 ye2 yen1113864 1113865 and yf1 yf2 yfn1113966 1113967 areinvariant by equation (8) or (9) In the third step of thecalculation the obtained weighted normalized decision at-tribute values re1 re2 ren1113864 1113865 and rf1 rf2 rfn1113966 1113967 arealso invariant by equation (3)

For the positive ideal solution r+j and the negative ideal

solution rminusj they only depend on whether the indicators are

benefit indicators or cost indicators When the indicators aredetermined they are also fixed j 1 2 n

Furthermore the final evaluation results of attributes Ae

and Af determined by equations (12) and (13) are RSe andRSf which are also fixed

For any two alternatives that exist before and after thechange their evaluation values are constant and the ordermust be consistent erefore the R-TOPSIS method isranking stable is completes the proof

Theorem 2 Let A A1 A2 Am1113864 1113865 be an alternative setthe NR-TOPSIS method is ranking stable

Proof Let A A1 A2 Am1113864 1113865 be an alternative set andthe ranking after evaluation by the NR-TOPSIS method isA(1) ≻A(2) ≻ middot middot middot ≻A(m) where A(i) i 1 2 m is an


alternative in A In the case of adding deleting orreplacing alternatives under the original alternative set Aa new alternative set B B1 B2 Bl1113864 1113865 is obtained Afterranking by evaluation method the ranking result is B(1) ≻B(2) ≻ middot middot middot ≻B(l) where B(k) k 1 2 l is an alternativein B

For any two alternatives Bp Bq isin AcapB there arematching alternatives Ae Af isin A subject to Ae

Bp andAf Bq with Ae xe1 xe2 xen1113864 1113865 andAf

xf1 xf2 xfn1113966 1113967Because theminimum valuemj andmaximum valueMj of

each attribute Cj are determinate then ye1 ye2 yen1113864 1113865 andyf1 yf2 yfn1113966 1113967 are also invariant by equations (14) and(15)

en it is clear that the final evaluation results for Ae andAf obtained through Steps 3ndash6 are also determined to beconstant erefore under the action of the NR-TOPSISmethod for the alternatives that exist before and after thechange the evaluation results are unchanged and theranking naturally remains unchanged is completes theproof

Remark 3 e proof process of eorems 1 and 2 isstraightforward In particular the maximum value of theindicator used in NR-TOPSIS is the historical maximumvalue and it is also the global maximum value which is aconstant not limited to the maximum value of the indicatorin each evaluation What we can theoretically determine isthat the R-TOPSIS and NR-TOPSIS methods ensure theconsistency of the evaluation results and are thereforecredible In contrast the traditional TOPSIS method maychange both the positive ideal solution and the negative idealsolution after the change of the alternative set resulting inthe change of the evaluation result of the alternative In thisway it is easy to cause ranking unstable and rank reversal

4 Numerical Results

In this section two classical examples will be used to verifythe effectiveness of the NR-TOPSIS method presentedabove e first simple example comes from the literature[45] which confirms with the theory to verify that themethod can avoid the phenomenon of rank reversal esecond example is used to illustrate the comprehensive ef-fectiveness of the NR-TOPSIS method A classical weaponperformance evaluation problem is selected Consideringthat the problem includes both the benefit attribute and costattribute the test of the evaluation method is more con-vincing According to the analysis of the theoretical part inthe following comparative experiment the R-TOPSISmethod adopts the Max way

Example 1 Table 1 will show an MADM example in whichfour alternatives with respect to four benefit attributes aregiven Consider the four attribute weights denoted byW 16 13 13 16 It is not difficult for us to determinethe historical maximum of each attribute We use m torepresent the historical minimum value vector of each at-tribute and M to represent the historical maximum value

vector of each attribute where m 24 40 43 70 and M

36 50 50 100 in the example When the scheme is in-creased decreased or replaced the conditionmj le xij leMj(j 1 2 3 4) must be met

e weighted normalized decision matrix can be cal-culated as

C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

01667

R2

R3

R4

X = (19)

Also the positive ideal solution vector and the negativeideal solution vector can be determined by equation (16) as

R+ 16131316

1113882 1113883

Rminus 0 0 0 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)

Figure 1 illustrates the comparison curves of the attri-butes of each alternative with the positive ideal solution andnegative ideal solution after weight normalization FromFigure 1 it can be found that A3 of weight normalization hasthe smallest synthetic distance with the positive ideal so-lution and the largest synthetic distance with the negativeideal solution so it is the best among the four alternativesfollowed by A2 and A1 is at the end e ranking is differentfrom the traditional TOPSIS method and R-TOPSISmethod

If A4 is removed from the original alternative set it iseasy to find that A3 is also the best which is consistent withprevious ranking results However at this time the tradi-tional TOPSIS method has already undergone rank reversalen we add a new alternative A5 36 50 45 70 to theoriginal alternative set we can also find that A3 is also thebest and the ranking results can be shown as

A3 ≻A5 ≻A2 ≻A4 ≻A1 (21)

Whether it is a deletion alternative or an addition al-ternative the experimental verification results of NR-TOPSIS indicate that the retained alternative evaluationresults are unchanged and the ranking is stable In starkcontrast to this the traditional TOPSIS method still has theproblem of rank reversal At the same time we also find thatthe NR-TOPSIS method is quite different from theR-TOPSIS method under the new alternative A5 eweighted normalized decision matrix with a new alternativeA5 can be calculated as

Table 1 Decision matrix of four alternatives with respect to fourattributes [45]

Alternative C1 C2 C3 C4

A1 36 42 43 70A2 25 50 45 80A3 28 45 50 75A4 24 40 47 100


C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

0166701667 03333 00952 0

R2

R3

R4

R5

R = (22)

Moreover we make a comparison between the weightednormalized alternatives curve and the positive and negativeideal solution under the new alternative A5 and fromFigure 2 it is easy to observe that A1 is the worst ereforewe believe that the NR-TOPSIS method based on globalunderstanding constructed in this paper is better than theR-TOPSIS method although the R-TOPSIS method alsoovercomes the problem of rank reversal

e specific ranking results of the three methods areshown in Table 2 Table 2 gives the ranking results of threeTOPSIS methods under the original alternative set deletionalternative A4 and addition alternative A5 and the nor-malized decision data obtained by NR-TOPSIS method In asense the TOPSIS method has positive meaning and valuebut it also has the problem of rank reversal Both the NR-TOPSIS and R-TOPSIS methods inherit the advantages ofTOPSIS while overcoming the shortcomings of rank re-versal In all three cases the NR-TOPSIS method andTOPSIS method have the same ranking in one caseAccording to the previous analysis we believe that A3 is thebest and NR-TOPSIS keeps the ranking consistency beforeand after change erefore the NR-TOPSIS method is abetter improvement of the TOPSIS method

Example 2 In this example seven surface-to-air missileweapon system alternatives X1 X2 X7 with eleven at-tributes C1 C2 C11 [62] will be evaluated by the NR-TOPSIS method ese attributes include the maximumspeed of missile (C1) the maximum speed of target (C2) the

maximum overload of target (C3) the highest boundary ofkilling range (C4) the farthest boundary of killing range(C5) the number of targets that can simultaneously be shotby one weapon system (C6) the single-shot kill probabilityof missiles (C7) the reaction time of missile weapon system(C8) the lowest boundary of killing range (C9) thelaunching weight of missiles (C10) and the nearestboundary of killing range (C11) [62] Benefit attributesinclude C1 C2 C3 C4 C5 C6 and C7 and cost attributesinclude C8 C9 C10 and C11 Assuming that according to thestatistical law and expert knowledge we have determined thehistorical lower limit value of each index and thus deter-mined the historical maximum value Mj and historicalminimum value mj of each attribute Cj as follows

m 2 400 1 3 8 1 07 10 0025 85 05

M 6 2300 6 27 100 8 08 40 1 2375 8 1113896 (23)

en the normalized decision matrix data by equations(14) and (15) are listed in Table 3 According to the refer-ences it is assumed that the weights of 11 attributes are thesame that is wj 111 j 1 2 11

Determine the weighted normalized multiattribute de-cision matrix data by (3) as listed in Table 4

e evaluation score obtained by the NR-TOPSISmethod is as follows

R1 06887

R2 07872

R3 06315

R4 02735

R5 04424

R6 04965

R7 04509

(24)

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue minus

R1R2R3

R+

Rminus

R5

R4

Figure 2 Weighted normalized attribute values with a new al-ternative and the ideal solutions

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue

R+

RminusR1R2

R3

R4

Figure 1 e weighted normalized attribute values and the idealsolutions


where Si is the evaluation score of Xi i 1 2 7 usthe ranking is

X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)

Table 5 shows the evaluation results under four TOPSISmethods including R-TOPSIS method the referencemethod [56] the classical TOPSIS method and NR-TOPSISmethod established in this paper Figure 3 also shows thedistribution of evaluation results under the four TOPSISmethods It is not difficult to find that the NR-TOPSISmethod proposed in this paper is consistent with the ref-erence method [56] and the R-TOPSIS method is consistent

with the classical TOPSIS method there is only a smalldifference between the two e ranking results of theR-TOPSIS method and the classical TOPSIS method are

X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)

e difference between the two kinds of methods is thatthe order of alternatives X6 and X7 is different Carefullycompare the weighted normalized data of alternative X6 andX7 in Table 4 seven attributes of the two alternatives are thesame and four attributes are different According to theprinciple that the larger the value of weighted normalizeddata is the closer it is to the positive ideal solution and the

Table 2 Normalized decision matrix by NR-TOPSIS and comparison of evaluation results of three TOPSIS methods

AlternativeNormalized decision

matrix NR-TOPSISevaluation results

NR-TOPSISrank

R-TOPSISevaluation results

R-TOPSISrank

TOPSISevaluationresults

TOPSISrank

C1 C2 C3 C4

e original alternativeset

A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4

Drop A4 out ofalternative set

A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1

Add a new alternative A5to the original alternative

setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1

Table 3 e normalized multiattribute decision matrix data

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1

Table 4 e weighted normalized multiattribute decision matrix data

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909


farther it is from the negative ideal solution it can be judgedthat the alternativeX6 is better than the alternativeX7 so theevaluation result of NR-TOPSIS is better

For the above original multiattribute missile selectionproblem we will delete or add alternative to evaluatewhether there is a rank reversal phenomenon Generally analternative is often deleted to test the rank reversal whichresults in the limitation of the discussion Next we considerdeleting three alternatives X4 X5 and X6 from the originalalternative set In this case we will compare the results of theNR-TOPSIS method proposed in this paper with otherTOPSIS methods especially to check whether the rank re-versal phenomenon occurs e main results are shown inTable 6

In this case we can easily get the following results withdropping X4X5 and X6 out of the original alternative set bythe NR-TOPSIS method

R1 06887

R2 07872

R3 06315

R7 04509

(27)

us the ranking of the case with dropping X4 X5 andX6 out of the original alternative set is

X2 ≻X1 ≻X3 ≻X7 (28)

Table 5 e comparison between the NR-TOPSIS method and reference method

Evaluation results R1 R2 R3 R4 R5 R6 R7

NR-TOPSIS method 06887 07872 06315 02735 04424 04965 04509e reference method [56] 06242 06842 05564 02604 04213 04894 04342R-TOPSIS method 07043 07644 06329 02550 04534 04887 04928Classical TOPSIS 06640 07589 06207 02026 04833 05347 05392

1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults

NR-TOPSIS methodThe reference method

R-TOPSIS methodClassical TOPSIS

Figure 3 Evaluation results and trend

Table 6 e evaluation ranking results of four TOPSIS

e reference method [56] Rank NR-TOPSIS Rank R-TOPSIS Rank Classical TOPSIS RankDrop X4 X5 and X6 out of the original alternative setX1 04916 2 06887 2 07043 2 04745 4X2 05892 1 07872 1 07644 1 06694 1X3 04230 3 06315 3 06329 3 05316 2X7 03757 4 04509 4 04928 4 05046 3Add a new alternative X8 to the changed alternative set discussed aboveX1 05158 3 06887 2 07043 2 04826 5X2 06125 1 07872 1 07644 1 07049 1X3 04597 4 06315 4 06329 4 05677 3X7 03927 5 04509 5 04928 5 05465 4X8 05526 2 06463 3 06512 3 05937 2


It can be determined quickly and the evaluation resultand ranking of the scheme to be selected are unchanged andno rank reversal occurs

At last a new alternative X8 2 2300 6 27 100 8 08

25 0025 1000 75 will be added to the changed alternativeset discussed above e new comparison results are alsoshown in Table 6

At the last case we can obtain the following ranking forthe NR-TOPSIS method

X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)

e example further verifies the theory showing that theNR-TOPSIS method established in this paper provides aninvariable measurement scale which can ensure that theevaluation results of the alternatives remain unchangederefore whether deleting the original alternatives oradding new alternatives the rank stability will be guaranteedand the problem of rank reversal will not arise Corre-spondingly the traditional TOPSIS method has an orderreversal phenomenon when deleting and adding new al-ternatives such as X1 and X3 In addition there is no rankreversal between the reference method [56] and theR-TOPSIS method which are also superior to the classicalTOPSIS method but the reference method is too compli-catede ranking of R-TOPSIS and NR-TOPSISmethods isconsistent in Example 2 while the ranking of the referencemethod [56] is slightly different from that of R-TOPSIS andNR-TOPSIS In view of the analysis of various situationsunder the above two examples we believe that the NR-TOPSIS method constructed in this paper is consideredfrom a global perspective taking into account the differencesof attributes in normalization overcoming the problem ofrank reversal and is more in line with human thinking logic

5 Conclusions

Faced with the complicated evaluation problems in real lifethe validity and credibility of the evaluation methods arereflected in the consistency of the evaluation results and thestability of the evaluation ranking However the classicalTOPSIS method is not credible because of the possiblephenomenon of rank reversal so it is necessary to carry outtheoretical research and process improvement of TOPSIS toresist rank reversal In order to make the assessment resultsfair reasonable and consistent we need to consider andevaluate the problem from a global perspective and ensurethat the scale of measurement is always the same Based onthis understanding this paper constructs a new improvedTOPSIS method which not only inherits the advantages ofthe TOPSIS method but also makes reasonable improve-ments taking into account the attribute characteristics ofindicators making the improved TOPSIS method more inline with human cognition and logic is paper establishesthe definition of ranking stability and theoretically states thatthe NR-TOPSIS method constructed in this paper and theR-TOPSIS method established in the reference literature areranking stable while the classical TOPSIS method is rankingunstable At the same time when using the R-TOPSISmethod for normalization we analyze that Max way is more

reasonable and Max-Min way has certain flaws In the ex-ample verification process we used two classic cases anddiscussed various situations such as deleting and increasingthe alternatives e verification results are completelyconsistent with the theory Compared with other types ofTOPSIS methods the NR-TOPSIS method is more rea-sonable and effective For a given evaluation problem theNR-TOPSISmethod can ensure that the evaluation results ofthe alternative remain unchanged thus ensuring that theranking is stable In the future we will continue to study theuse of NR-TOPSIS to solve more complex fuzzy problemsinvolving semantic uncertainty looking for a newmethod tosolve uncertain MADM problems

Data Availability

Data used to support the findings of this study are includedwithin the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (11801173 and 11702094) and theNational Key Research Project (2018YFC0808306) isstudy was also supported by Key Discipline of Probabilityeory and Mathematical Statistics of North China Instituteof Science and Technology (06DV09) and Hebei IoTMonitoring Engineering Technology Research Center(3142018055)

References

[1] E TriantaphyllouMulti-Criteria Decision Making Methods AComparative Study Springer Science amp Business MediaBerlin Germany 1st edition 2000

[2] E Triantaphyllou B Shu S Nieto Sanchez and T RayldquoMulti-criteria decision making an operations research ap-proachrdquo Encyclopedia of Electrical and Electronics Engineer-ing vol 15 pp 175ndash186 1998

[3] E Triantaphyllou and A Sanchez ldquoA sensitivity analysisapproach for some deterministic multi-criteria decision-making methodsrdquo Decision Sciences vol 28 no 1pp 151ndash194 1997

[4] E Triantaphyllou and S H Mann ldquoAn examination of theeffectiveness of multi-dimensional decision-making methodsa decision-making paradoxrdquo Decision Support Systems vol 5no 3 pp 303ndash312 1989

[5] E Triantaphyllou ldquoTwo new cases of rank reversals when theAHP and some of its additive variants are used that do notoccur with the multiplicative AHPrdquo Journal of Multi-CriteriaDecision Analysis vol 10 no 1 pp 11ndash25 2001

[6] P Liu and P Wang ldquoMultiple-attribute decision-makingbased on Archimedean Bonferroni operators of q-rungorthopair fuzzy numbersrdquo IEEE Transactions on Fuzzy Sys-tems vol 27 no 5 pp 834ndash848 2019

[7] H Zhang Y Dong and X Chen ldquoe 2-rank consensusreaching model in the multigranular linguistic multiple-at-tribute group decision-makingrdquo IEEE Transactions on


Systems Man and Cybernetics Systems vol 48 no 12pp 2080ndash2094 2018

[8] G Wei H Gao and Y Wei ldquoSome q-rung orthopair fuzzyheronian mean operators in multiple attribute decisionmakingrdquo International Journal of Intelligent Systems vol 33no 7 pp 1426ndash1458 2018

[9] P Ziemba ldquoNeat F-PROMETHEEmdasha new fuzzy multiplecriteria decision making method based on the adjustment ofmapping trapezoidal fuzzy numbersrdquo Expert Systems withApplications vol 110 pp 363ndash380 2018

[10] H Liao Z Xu E Herrera-Viedma and F Herrera ldquoHesitantfuzzy linguistic term set and its application in decisionmaking a state-of-the-art surveyrdquo International Journal ofFuzzy Systems vol 20 no 7 pp 2084ndash2110 2018

[11] A Baykasoglu K Subulan and F S Karaslan ldquoA new fuzzylinear assignment method for multi-attribute decisionmakingwith an application to spare parts inventory classificationrdquoApplied Soft Computing vol 42 pp 1ndash17 2016

[12] S Pramanik and R Mallick ldquoTODIM strategy for multi-at-tribute group decision making in trapezoidal neutrosophicnumber environmentrdquo Complex amp Intelligent Systems vol 5no 4 pp 379ndash389 2019

[13] S Pramanik and R Mallick ldquoVIKOR basedMAGDM strategywith trapezoidal neutrosophic numbersrdquo Neutrosophic Setsand Systems vol 22 pp 1ndash13 2018

[14] P Biswas S Pramanik and B C Giri ldquoTOPSIS strategy formulti-attribute decision making with trapezoidal neu-trosophic numbersrdquo Neutrosophic Sets and Systems vol 19pp 29ndash39 2018

[15] P Biswas S Pramanik and B C Giri ldquoDistance measurebased MADM strategy with interval trapezoidal neutrosophicnumbersrdquo Neutrosophic Sets and Systems vol 19 pp 40ndash462018

[16] S Pramanik P Biswas and B C Giri ldquoHybrid vector sim-ilarity measures and their applications to multi-attributedecision making under neutrosophic environmentrdquo NeuralComputing and Applications vol 28 no 5 pp 1163ndash11762017

[17] K Yoon Systems Selection by Multiple Attribute DecisionMaking PhD Dissertation Kansas State University Man-hattan KS USA 1980

[18] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Vol 186 Springer Hei-delberg Germany 1981

[19] Z Wang H Hao F Gao Q Zhang J Zhang and Y ZhouldquoMulti-attribute decision making on reverse logistics based onDEA-TOPSIS a study of the Shanghai End-of-life vehiclesindustryrdquo Journal of Cleaner Production vol 214 pp 730ndash737 2019

[20] G Sun X Guan X Yi and Z Zhou ldquoAn innovative TOPSISapproach based on hesitant fuzzy correlation coefficient andits applicationsrdquo Applied Soft Computing vol 68 pp 249ndash267 2018

[21] M P Amiri ldquoProject selection for oil-fields development byusing the AHP and fuzzy TOPSIS methodsrdquo Expert Systemswith Applications vol 37 no 9 pp 6218ndash6224 2010

[22] C-C Sun ldquoA performance evaluation model by integratingfuzzy AHP and fuzzy TOPSIS methodsrdquo Expert Systems withApplications vol 37 no 12 pp 7745ndash7754 2010

[23] T Paksoy N Y Pehlivan and C Kahraman ldquoOrganizationalstrategy development in distribution channel managementusing fuzzy AHP and hierarchical fuzzy TOPSISrdquo ExpertSystems with Applications vol 39 no 3 pp 2822ndash2841 2012

[24] A C Kutlu and M Ekmekccedilioglu ldquoFuzzy failure modes andeffects analysis by using fuzzy TOPSIS-based fuzzy AHPrdquoExpert Systems with Applications vol 39 no 1 pp 61ndash672012

[25] F Torfi R Z Farahani and S Rezapour ldquoFuzzy AHP todetermine the relative weights of evaluation criteria and FuzzyTOPSIS to rank the alternativesrdquo Applied Soft Computingvol 10 no 2 pp 520ndash528 2010

[26] Z Yue ldquoTOPSIS-based group decision-making methodologyin intuitionistic fuzzy settingrdquo Information Sciences vol 277pp 141ndash153 2014

[27] T Wang J Liu J Li and C Niu ldquoAn integrating OWA-TOPSIS framework in intuitionistic fuzzy settings for multipleattribute decision makingrdquo Computers amp Industrial Engi-neering vol 98 pp 185ndash194 2016

[28] Z Pei ldquoA note on the TOPSIS method in MADM problemswith linguistic evaluationsrdquo Applied Soft Computing vol 36pp 24ndash35 2015

[29] S Zhou W Liu and W Chang ldquoAn improved TOPSIS withweighted hesitant vague informationrdquo Chaos Solitons ampFractals vol 89 pp 47ndash53 2016

[30] H-J Shyur ldquoCOTS evaluation using modified TOPSIS andANPrdquo Applied Mathematics and Computation vol 177 no 1pp 251ndash259 2006

[31] Z Zhang Y Y Wang and Z X Wang ldquoA grey TOPSISmethod based on weighted relational coefficientrdquo Journal ofGrey System vol 26 no 2 pp 112ndash123 2014

[32] P Biswas S Pramanik and B C Giri ldquoNeutrosophic TOPSISwith group decision makingrdquo Fuzzy Multi-Criteria Decision-Making Using Neutrosophic Sets vol 369 pp 543ndash585 2018

[33] P Biswas S Pramanik and B C Giri ldquoTOPSIS method formulti-attribute group decision-making under single-valuedneutrosophic environmentrdquo Neural Computing and Appli-cations vol 27 no 3 pp 727ndash737 2016

[34] I Millet and T L Saaty ldquoOn the relativity of relative measures- accommodating both rank preservation and rank reversalsin the AHPrdquo European Journal of Operational Researchvol 121 no 1 pp 205ndash212 2000

[35] S Zahir ldquoNormalisation and rank reversals in the additiveanalytic hierarchy process a new analysisrdquo InternationalJournal of Operational Research vol 4 no 4 pp 446ndash4672009

[36] W C Wedley B Schoner and E U Choo ldquoClusteringdependence and ratio scales in AHP rank reversals and in-correct priorities with a single criterionrdquo Journal of Multi-Criteria Decision Analysis vol 2 no 3 pp 145ndash158 1993

[37] H Maleki and S Zahir ldquoA comprehensive literature review ofthe rank reversal phenomenon in the analytic hierarchyprocessrdquo Journal of Multi-Criteria Decision Analysis vol 20no 3-4 pp 141ndash155 2013

[38] Y-M Wang and T M S Elhag ldquoAn approach to avoidingrank reversal in AHPrdquo Decision Support Systems vol 42no 3 pp 1474ndash1480 2006

[39] J Barzilai and B Golany ldquoAHP rank reversal normalizationand aggregation rulesrdquo INFOR Information Systems andOperational Research vol 32 no 2 pp 57ndash64 1994

[40] Y B Shin and S Lee ldquoNote on an approach to preventingrank reversals with addition or deletion of an alternative inanalytic hierarchy processrdquo US-China Education Review Avol 3 no 1 pp 66ndash72 2013

[41] S H Zyoud and D Fuchs-Hanusch ldquoA bibliometric-basedsurvey on AHP and TOPSIS techniquesrdquo Expert Systems withApplications vol 78 pp 158ndash181 2017


[42] S H Mousavi-Nasab and A Sotoudeh-Anvari ldquoA new multi-criteria decision making approach for sustainable materialselection problem a critical study on rank reversal problemrdquoJournal of Cleaner Production vol 182 pp 466ndash484 2018

[43] B Mareschal Y De Smet and P Nemery ldquoRank reversal inthe PROMETHEE II method some new resultsrdquo in Pro-ceedings of the IEEE International Conference on IndustrialEngineering and Engineering Management pp 959ndash963Singapore December 2008

[44] M Keshavarz-Ghorabaee M Amiri E K Zavadskas et al ldquoAcomparative analysis of the rank reversal phenomenon in theEDAS and TOPSIS methodsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 52 no 3pp 121ndash134 2018

[45] Y-M Wang and Y Luo ldquoOn rank reversal in decisionanalysisrdquo Mathematical and Computer Modelling vol 49no 5-6 pp 1221ndash1229 2009

[46] S Mufazzal and S M Muzakkir ldquoA new multi-criteriondecision making (MCDM) method based on proximityindexed value for minimizing rank reversalsrdquo Computers ampIndustrial Engineering vol 119 pp 427ndash438 2018

[47] D M Buede D T Maxwell and R disagreement ldquoRankdisagreement a comparison of multi-criteria methodologiesrdquoJournal of Multi-Criteria Decision Analysis vol 4 no 1pp 1ndash21 1995

[48] S Aouadni A Rebai and Z Turskis ldquoe meaningful mixeddata TOPSIS (TOPSIS-MMD) method and its application insupplier selectionrdquo Studies in Informatics Control vol 26no 3 pp 353ndash363 2017

[49] E K Zavadskas Z Turskis T Vilutiene and N LepkovaldquoIntegrated group fuzzy multi-criteria model case of facilitiesmanagement strategy selectionrdquo Expert Systems with Appli-cations vol 82 pp 317ndash331 2017

[50] A Jahan and K L Edwards ldquoA state-of-the-art survey on theinfluence of normalization techniques in ranking improvingthe materials selection process in engineering designrdquo Ma-terials amp Design (1980ndash2015) vol 65 pp 335ndash342 2015

[51] U Pinter and I Psunder ldquoEvaluating construction projectsuccess with use of the M-TOPSIS methodrdquo Journal of CivilEngineering and Management vol 19 no 1 pp 16ndash23 2013

[52] R F D F Aires and L Ferreira ldquoe rank reversal problem inmulti-criteria decision making a literature reviewrdquo PesquisaOperacional vol 38 no 2 pp 331ndash362 2018

[53] E K Zavadskas A Mardani Z Turskis A Jusoh andK M Nor ldquoDevelopment of TOPSIS method to solvecomplicated decision-making problemsmdashan overview ondevelopments from 2000 to 2015rdquo International Journal ofInformation Technology amp Decision Making vol 15 no 3pp 645ndash682 2016

[54] L Ren Y Zhang Y Wang and Z Sun ldquoComparative analysisof a novel M-TOPSIS method and TOPSISrdquo Applied Math-ematics Research eXpress vol 2007 Article ID abm00510 pages 2007

[55] M S Garcıa-Cascales and M T Lamata ldquoOn rank reversaland TOPSIS methodrdquo Mathematical and Computer Model-ling vol 56 no 5-6 pp 123ndash132 2012

[56] W Yang and Y Wu ldquoA novel TOPSIS method based onimproved grey relational analysis for multiattribute decision-making problemrdquo Mathematical Problems in Engineeringvol 2019 Article ID 8761681 10 pages 2019

[57] R F D F Aires and L Ferreira ldquoA new approach to avoidrank reversal cases in the TOPSIS methodrdquo Computers ampIndustrial Engineering vol 132 pp 84ndash97 2019

[58] E K Zavadskas J Antucheviciene Z Turskis and H AdelildquoHybrid multiple-criteria decision-making methods a reviewof applications in engineeringrdquo Scientia Iranica vol 23 no 1pp 1ndash20 2016

[59] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-g and topsis grey techniquesrdquo Journal of Business Economicsand Management vol 11 no 1 pp 34ndash55 2010

[60] E K Zavadskas Z Turskis and S Kildiene ldquoState of artsurveys of overviews on MCDMMADM methodsrdquo Tech-nological and Economic Development of Economy vol 20no 1 pp 165ndash179 2014

[61] Z Turskis and B Juodagalviene ldquoA novel hybrid multi-cri-teria decision-making model to assess a stairs shape fordwelling housesrdquo Journal of Civil Engineering and Manage-ment vol 22 no 8 pp 1078ndash1087 2016

[62] H Gu and B Song ldquoStudy on effectiveness evaluation ofweapon systems based on grey relational analysis andTOPSISrdquo Journal of Systems Engineering and Electronicsvol 20 no 1 pp 106ndash111 2009


process (AHP) [34ndash41] VIseKriterijumska Optimizacija IKOmpromisno Resenje (VIKOR) [42] preference rankingorganization method for enrichment of evaluations(PROMETHEE) [43] and evaluation based on distance fromaverage solution (EDAS) [44] Achieving the rank preser-vation of MADMMCDMmethods has turned into the focusof many scholars in applied study [45ndash51]

e problem of rank reversal in the TOPSIS method hasreceived great attention and different scholars have tried tofind satisfactory solutions from an experimental or theo-retical level [52ndash57] By collecting 130 related papers pub-lished in international journals from 1980 to 2015 Aires andFerreira provided an extensive literature review on MCDMmethodologies and rank reversals including the TOPSISmethod [52] Zavadskas et al [53] also reviewed the de-velopment of the TOPSIS method from 2000 to 2015 but didnot discuss the solution of rank reversal Ren et al [54]proposed a novel M-TOPSIS method which can solve theproblems of TOPSIS such as rank reversals and evaluationfailure when alternatives are symmetrical but there is notheoretical proof Wang and Luo [45] used the counterex-ample to illustrate the fact that the TOPSIS method has aproblem of rank reversal but did not give a solution Garcıa-Cascales and Lamata [55] considered two aspects to improvethe rank reversal problem of the TOPSIS method On theone hand a new norm is used for normalization but it is notsufficient On the other hand the absolute mode is used torewrite the positive ideal solution and the negative idealsolution e fixed instance verification is feasible but theindicator attribute is not considered and the improvedmethod does not have the validity in all cases Once the costindicator data are encountered it will be invalid Based onthe understanding of data dispersion degree Yang and Wu[56] proposed a new strategy combining improved greyrelational analysis and TOPSIS method and verified thecontrollability of order reversal in the process of case ver-ification In the process of evaluation the idea of attributevariable weight was adopted with the change of evaluationknowledge Aires and Ferreira [57] analyzed that most of theliterature on rank reversal of TOPSIS was limited to casestudies and then developed an improved TOPSIS methodwith domain parameters called R-TOPSIS method whichdemonstrated the effectiveness of the method from a sta-tistical point of view But in theory there is no reasonableproof and explanation for this method e key to solvingthe problem of rank reversal is to analyze the problem from aglobal perspective and always use the same scale to measurethe data conversion In addition some scholars have useddifferent integrated MADM methods to focus on the searchof the best option and also to determine which methodrsquos bestoption was better and weakened the attention of the rankreversal problem [46 58ndash61] In doing so there is nosubstantive solution to the problem of rank reversal

In brief this paper will focus on the following mainresearch (i) we propose a novel extended TOPSIS method toovercome the rank reversal problem which is suitable forMADM problems with arbitrary attribute indicators (ii) wegive the definition of ranking stable and ranking unstableand strictly prove in theory that the NR-TOPSIS method

proposed in this paper is ranking stable and the R-TOPSISmethod is also ranking stable (iii) we use this method toverify two classic cases which highlight the effectiveness andconsistency of this method and provide a new simpleTOPSIS improvement scheme for solvingMADMproblems

e remainder of the paper is organized as followsSection 2 presents the traditional TOPSIS methodR-TOPSIS method and a new improvement to solve rankreversal problem in the TOPSIS method which is called theNR-TOPSIS method Section 3 analyzes the theoreticalmechanism of rank reversal problem and proves that theNR-TOPSIS method as well as the R-TOPSIS method isranking stable Section 4 describes two numerical cases toverify the validity and consistency of the NR-TOPSISmethod Finally Section 5 provides some importantconclusions

2 Rank Reversal in TOPSIS Method andIts Improvement

e proposal of the TOPSIS method in a sense conforms tothe logic of human decision-making and judgment and canreasonably compare the proposed scheme with the positiveideal solution and the negative ideal solution quantitativelythus giving the order closest to the ideal solution In thissection we will show the classical TOPSIS method and theimproved TOPSIS method named R-TOPSIS that hasovercome the rank reversal Furthermore a new improvedmethod of TOPSIS to overcome rank reversal based onstatistical laws will be proposed

Formally consider an MADM problem with m alter-natives and n attributes and the weights of the attributes areknown which are denoted by w1 w2 wn with1113936

nj1wj 1 wj gt 0 j 1 2 n Use the symbol X to

represent the original decision matrix which is expressed asthe following form

C1 C2 Cn

hellip

hellip

hellip hellip hellip helliphellip

hellip

hellipA1 x12

x22

xm2

x1n

x2n

xmn mtimesn

x11

x21

xm1

A2

Am

X = (1)

where A1 A2 Am are alternatives C1 C2 Cn areattributes and xij are original attribute values

21 ampe Traditional TOPSIS Method We briefly explain theimplementation of the classic TOPSIS method as follows

Step 1 Compute the normalized decision-makingmatrix Y (yij)mtimesn where yij are normalized attributevalues expressed as

yij xij

1113936

mi1x

2ij

1113969 i 1 2 m j 1 2 n (2)

Step 2 Calculate the weighted normalized decision-making matrix R (rij)mtimesn e weighted normalizedattribute value rij is computed by






r+j max

irij

rminusj min

irij

⎧⎪⎨

⎪⎩

i 1 2 m j 1 2 n

(4)


r+j min

irij

rminusj max

irij

⎧⎪⎨

⎪⎩

i 1 2 m j 1 2 n

(5)


S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

i 1 2 m

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(6)


Si Sminus

i

S+i + Sminus

i

i 1 2 m (7)








yij xij

d2j

i 1 2 m j 1 2 n (8)

Max-Min way

yij xij minus d1j

d2j minus d1j

i 1 2 m j 1 2 n (9)




r+j wj

rminusj

d1j

d2j

wj

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

i 1 2 m j 1 2 n

(10)


r+j

d1j

d2j

wj

rminusj wj

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

i 1 2 m j 1 2 n

(11)



j gt r+j


S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(12)



RSi Sminus

i

S+i + Sminus

i

i 1 2 m (13)















j











yij xij minus mj

Mj minus mj

(14)


yij Mj minus xij

Mj minus mj

(15)



j respectively

r+j wj

rminusj 0

⎧⎨

⎩

j 1 2 n

(16)




S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(17)


NSi Sminus

i

S+i + Sminus

i

i 1 2 m (18)








j le rij le r+j



j


























4 Numerical Results






C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

01667

R2

R3

R4

X = (19)


R+ 16131316

1113882 1113883

Rminus 0 0 0 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



A3 ≻A5 ≻A2 ≻A4 ≻A1 (21)




A1 36 42 43 70A2 25 50 45 80A3 28 45 50 75A4 24 40 47 100


C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

0166701667 03333 00952 0

R2

R3

R4

R5

R = (22)





m 2 400 1 3 8 1 07 10 0025 85 05

M 6 2300 6 27 100 8 08 40 1 2375 8 1113896 (23)




R1 06887

R2 07872

R3 06315

R4 02735

R5 04424

R6 04965

R7 04509

(24)

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue minus

R1R2R3

R+

Rminus

R5

R4


1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue

R+

RminusR1R2

R3

R4




X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)



X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)





NR-TOPSISrank


R-TOPSISrank


TOPSISrank

C1 C2 C3 C4


A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4


A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1


setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909





R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References







































































r+j max

irij

rminusj min

irij

⎧⎪⎨

⎪⎩

i 1 2 m j 1 2 n

(4)


r+j min

irij

rminusj max

irij

⎧⎪⎨

⎪⎩

i 1 2 m j 1 2 n

(5)


S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

i 1 2 m

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(6)


Si Sminus

i

S+i + Sminus

i

i 1 2 m (7)








yij xij

d2j

i 1 2 m j 1 2 n (8)

Max-Min way

yij xij minus d1j

d2j minus d1j

i 1 2 m j 1 2 n (9)




r+j wj

rminusj

d1j

d2j

wj

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

i 1 2 m j 1 2 n

(10)


r+j

d1j

d2j

wj

rminusj wj

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

i 1 2 m j 1 2 n

(11)



j gt r+j


S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(12)



RSi Sminus

i

S+i + Sminus

i

i 1 2 m (13)















j











yij xij minus mj

Mj minus mj

(14)


yij Mj minus xij

Mj minus mj

(15)



j respectively

r+j wj

rminusj 0

⎧⎨

⎩

j 1 2 n

(16)




S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(17)


NSi Sminus

i

S+i + Sminus

i

i 1 2 m (18)








j le rij le r+j



j


























4 Numerical Results






C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

01667

R2

R3

R4

X = (19)


R+ 16131316

1113882 1113883

Rminus 0 0 0 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



A3 ≻A5 ≻A2 ≻A4 ≻A1 (21)




A1 36 42 43 70A2 25 50 45 80A3 28 45 50 75A4 24 40 47 100


C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

0166701667 03333 00952 0

R2

R3

R4

R5

R = (22)





m 2 400 1 3 8 1 07 10 0025 85 05

M 6 2300 6 27 100 8 08 40 1 2375 8 1113896 (23)




R1 06887

R2 07872

R3 06315

R4 02735

R5 04424

R6 04965

R7 04509

(24)

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue minus

R1R2R3

R+

Rminus

R5

R4


1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue

R+

RminusR1R2

R3

R4




X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)



X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)





NR-TOPSISrank


R-TOPSISrank


TOPSISrank

C1 C2 C3 C4


A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4


A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1


setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909





R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References



































































RSi Sminus

i

S+i + Sminus

i

i 1 2 m (13)















j











yij xij minus mj

Mj minus mj

(14)


yij Mj minus xij

Mj minus mj

(15)



j respectively

r+j wj

rminusj 0

⎧⎨

⎩

j 1 2 n

(16)




S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(17)


NSi Sminus

i

S+i + Sminus

i

i 1 2 m (18)








j le rij le r+j



j


























4 Numerical Results






C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

01667

R2

R3

R4

X = (19)


R+ 16131316

1113882 1113883

Rminus 0 0 0 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



A3 ≻A5 ≻A2 ≻A4 ≻A1 (21)




A1 36 42 43 70A2 25 50 45 80A3 28 45 50 75A4 24 40 47 100


C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

0166701667 03333 00952 0

R2

R3

R4

R5

R = (22)





m 2 400 1 3 8 1 07 10 0025 85 05

M 6 2300 6 27 100 8 08 40 1 2375 8 1113896 (23)




R1 06887

R2 07872

R3 06315

R4 02735

R5 04424

R6 04965

R7 04509

(24)

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue minus

R1R2R3

R+

Rminus

R5

R4


1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue

R+

RminusR1R2

R3

R4




X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)



X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)





NR-TOPSISrank


R-TOPSISrank


TOPSISrank

C1 C2 C3 C4


A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4


A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1


setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909





R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References



































































S+i

1113944

n

j1rij minus r+

j1113872 11138732

11139741113972

Sminusi

1113944

n

j1rij minus rminus

j1113872 11138732

11139741113972

i 1 2 m

(17)


NSi Sminus

i

S+i + Sminus

i

i 1 2 m (18)








j le rij le r+j



j


























4 Numerical Results






C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

01667

R2

R3

R4

X = (19)


R+ 16131316

1113882 1113883

Rminus 0 0 0 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



A3 ≻A5 ≻A2 ≻A4 ≻A1 (21)




A1 36 42 43 70A2 25 50 45 80A3 28 45 50 75A4 24 40 47 100


C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

0166701667 03333 00952 0

R2

R3

R4

R5

R = (22)





m 2 400 1 3 8 1 07 10 0025 85 05

M 6 2300 6 27 100 8 08 40 1 2375 8 1113896 (23)




R1 06887

R2 07872

R3 06315

R4 02735

R5 04424

R6 04965

R7 04509

(24)

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue minus

R1R2R3

R+

Rminus

R5

R4


1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue

R+

RminusR1R2

R3

R4




X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)



X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)





NR-TOPSISrank


R-TOPSISrank


TOPSISrank

C1 C2 C3 C4


A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4


A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1


setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909





R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References










































































4 Numerical Results






C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

01667

R2

R3

R4

X = (19)


R+ 16131316

1113882 1113883

Rminus 0 0 0 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)



A3 ≻A5 ≻A2 ≻A4 ≻A1 (21)




A1 36 42 43 70A2 25 50 45 80A3 28 45 50 75A4 24 40 47 100


C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

0166701667 03333 00952 0

R2

R3

R4

R5

R = (22)





m 2 400 1 3 8 1 07 10 0025 85 05

M 6 2300 6 27 100 8 08 40 1 2375 8 1113896 (23)




R1 06887

R2 07872

R3 06315

R4 02735

R5 04424

R6 04965

R7 04509

(24)

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue minus

R1R2R3

R+

Rminus

R5

R4


1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue

R+

RminusR1R2

R3

R4




X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)



X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)





NR-TOPSISrank


R-TOPSISrank


TOPSISrank

C1 C2 C3 C4


A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4


A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1


setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909





R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References



































































C1 C2 C3 C4R1

01667

00139

00556

0

00667

03333

01667

0

0

00952

03333

01905

0

00556

00278

0166701667 03333 00952 0

R2

R3

R4

R5

R = (22)





m 2 400 1 3 8 1 07 10 0025 85 05

M 6 2300 6 27 100 8 08 40 1 2375 8 1113896 (23)




R1 06887

R2 07872

R3 06315

R4 02735

R5 04424

R6 04965

R7 04509

(24)

1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue minus

R1R2R3

R+

Rminus

R5

R4


1 15 2 25 3 35 40

005

01

015

02

025

03

035

04

045

Attribute index

The w

eigh

ted

norm

aliz

ed at

trib

ute v

alue

R+

RminusR1R2

R3

R4




X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)



X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)





NR-TOPSISrank


R-TOPSISrank


TOPSISrank

C1 C2 C3 C4


A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4


A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1


setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909





R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References




































































X2 ≻X1 ≻X3 ≻X6 ≻X7 ≻X5 ≻X4 (25)



X2 ≻X1 ≻X3 ≻X7 ≻X6 ≻X5 ≻X4 (26)





NR-TOPSISrank


R-TOPSISrank


TOPSISrank

C1 C2 C3 C4


A1 1 06 03 0 02815 4 03971 3 04184 3A2 00833 1 05 03333 05362 2 05027 1 04858 1A3 03333 075 1 01667 06078 1 04836 2 04634 2A4 0 05 07 1 03881 3 03889 4 03915 4


A1 1 06 03 0 02815 3 03971 3 04319 3A2 00833 1 05 03333 05362 2 05027 1 04742 2A3 03333 075 1 01667 06078 1 04836 2 05007 1


setA1 1 06 03 0 02815 5 03971 4 04047 5A2 00833 1 05 03333 05362 3 05027 2 04862 2A3 03333 075 1 01667 06078 1 04836 3 04639 3A4 0 05 07 1 03881 4 03889 5 04061 4A5 05 065 0 05 05696 2 05937 1 05793 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 1 01842 1 0875 1 1 05 06667 07179 06004 06667X2 0875 1 08 1 08913 1 1 08333 1 06004 04X3 06 04211 08 1 07283 07143 06 06667 1 03105 04X4 025 00105 0 08958 02609 0 05 0 0 0 0X5 0 0 02 00833 00435 02857 05 1 05128 0941 09333X6 005 0 02 0 0 02857 1 09867 09744 1 09333X7 005 00053 02 0 0 02857 0 1 09744 1 1


C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

X1 00909 00167 00909 00795 00909 00909 00455 00606 00653 00546 00606X2 00795 00909 00727 00909 0081 00909 00909 00758 00909 00546 00364X3 00545 00383 00727 00909 00662 00649 00545 00606 00909 00282 00364X4 00227 0001 0 00814 00237 0 00455 0 0 0 0X5 0 0 00182 00076 0004 0026 00455 00909 00466 00855 00848X6 00045 0 00182 0 0 0026 00909 00897 00886 00909 00848X7 00045 00005 00182 0 0 0026 0 00909 00886 00909 00909





R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References






































































R1 06887

R2 07872

R3 06315

R7 04509

(27)


X2 ≻X1 ≻X3 ≻X7 (28)




1 2 3 4 5 6 702

03

04

05

06

07

08

Index

Eval

uatio

n re

sults











X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References







































































X2 ≻X1 ≻X8 ≻X3 ≻X7 (29)


5 Conclusions



Data Availability




Acknowledgments


References



































































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