research article an electre approach for multicriteria...

Research ArticleAn ELECTRE Approach for MulticriteriaInterval-Valued Intuitionistic Trapezoidal FuzzyGroup Decision Making Problems

Sireesha Veeramachaneni and Himabindu Kandikonda

Department of Applied Mathematics GITAM Institute of Science GITAM University Visakhapatnam 530045 India

Correspondence should be addressed to Sireesha Veeramachaneni vsirisha80gmailcom

Received 24 November 2015 Revised 2 February 2016 Accepted 18 February 2016

Academic Editor Ibrahim Ozkan

Copyright copy 2016 S Veeramachaneni and H Kandikonda This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The Multiple Criteria Decision Making (MCDM) is acknowledged as the most useful branch of decision making It provides aneffective framework for comparison based on the evaluation of multiple conflicting criteria In this paper a method is proposedto work out multiple attribute group decision making (MAGDM) problems with interval-valued intuitionistic trapezoidal fuzzynumbers (IVITFNs) using Elimination and Choice Translation Reality (ELECTRE) method A new ranking function based onvalue and ambiguity is introduced to compare the IVITFNs which overcomes the limitations of existing methods An illustrativenumerical example is solved to verify the efficiency of the proposed method to select the better alternative

1 Introduction

The fuzzy sets (FSs) introduced by Zadeh [1] which arecharacterized by a membership function have acquiredsuccessful applications in various fields In fuzzy sets themembership of an element is defined to be a number fromthe interval [0 1] and the nonmembership is simply itscomplement But in reality this hypothesis does not matchwith human intuition Thus in 1986 Atanassov [2] extendedthe concept of fuzzy sets to intuitionistic fuzzy set (IFS) bycharacterizing amembership function and a nonmembershipfunction such that the sum of both values is less than or equalto one However it is often difficult for experts to exactlyquantify their opinions as exact numbers in the interval [0 1]and hence it is more suitable to represent them in intervalform [3] Atanassov and Gargov [4] further generalized theconcept of IFS and introduced interval-valued intuitionisticfuzzy sets (IVIFSs) by combining IFS concept with interval-valued fuzzy set concept IFSs and IVIFSs have been appliedto many different fields such as decision making supplierselection and investment options [5] But the domain inIFSs and IVIFSs is a discrete set therefore their membership

degrees and the nonmembership degrees can only expressfuzzy concept in terms of ldquoexcellentrdquo or ldquogoodrdquo To over-come this limitation Shu et al [6] defined intuitionistictriangular fuzzy numbers (ITFNs) such that the domain isa consecutive set Later on Wang [7] extended intuitionistictriangular fuzzy number to intuitionistic trapezoidal fuzzynumber Wan [8] introduced the concepts of interval-valuedintuitionistic trapezoidal fuzzy numbers (IVITFNs) wherethe membership and nonmembership values are intervalsrather than exact numbers Hence there is a need to studythe MAGDM using the notion of IVITFNs

MCDM is well known branch of decision making whichdeals with decision problems under the presence of a numberof decision criteria In many cases the decision makerinformation is vague or fuzzy in natureThe classical MCDMmethods cannot effectively handle the problems under suchimprecision Bellman and Zadeh [9] introduced the conceptof decision making in fuzzy environment Many researcherscombined the fuzzy theory with classical MCDM techniquesand proposed fuzzy MCDM methods to solve the problemswith imprecision [10ndash21] The concept of intuitionistic fuzzyenvironment was given by Angelov [22] Li and Wan [23]

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 1956303 17 pageshttpdxdoiorg10115520161956303

2 Advances in Fuzzy Systems

developed new fuzzy linear programming techniques forsolving multiple attribute decision making (MADM) prob-lems with multiple types of attribute values and incompleteweight information Wu and Liu [24] proposed an approachfor MAGDM problems with IVITFNs by taking into accountthe expertrsquos risk attitude Zhang and Wei [25] developed theE-VIKOR method and TOPSIS method to solve the MCDMproblems with hesitant fuzzy set information

Many MCDM models use ldquooutranking relationsrdquo to ranka set of alternatives The important feature of outrankingapproach is its noncompensatory nature The most popularoutranking approach amongst the family of outrankingapproaches is ELECTRE The ELECTRE approach was firstintroduced by Benayoun et al in 1966 [26] Soon after theintroduction of the first version of ELECTRE I [27] thisapproach was evolved into a number of other variants such asELECTRE II III IV IS and TRI [28] Sevkli [29] proposedfuzzy ELECTRE for supplier selection problem to deal withimprecise and vague nature of linguistic assessment Vahdaniet al [30] extended ELECTRE with interval weights anddata to deal with uncertainty Later in 2011 Vahdani andHadipour extended the ELECTREmethod based on interval-valued fuzzy sets [3]Wei et al [31] investigated theMAGDMproblems in which attribute values take the form of IVITFNsand both the attribute weights and the expert weights take theform of real numbers Since fuzzy numbers and intuitionisticfuzzy numbers are special cases of IVITFNs it is significant todevelop the most accepted outranking method ldquoELECTRErdquoby taking both attribute values and weights of criteria ofMCDM problem in the form of IVITFNs To do this weproposed a ranking method for IVITFNs which takes theexpertrsquos risk attitude into consideration

The paper is organized as follows In Section 2 basic defi-nitions and arithmetic operations of IVITFNs are reviewedIn Section 3 a new ranking of IVITFNs based on valueand ambiguity functions is proposed to compare IVITFNsand its advantages over existing methods are presentedIn Section 4 the proposed algorithm of ELECTRE methodbased on IVITFNs is presented Section 5 provides illustrativeexamples Conclusions and future work are presented inSection 6

2 Preliminaries

In this section we briefly introduce some basic conceptsrelated to interval-valued intuitionistic trapezoidal fuzzynumbers from Xu and Chen [20] and De and Das [32]

Definition 1 (intuitionistic fuzzy set) An intuitionistic fuzzyset over universe of discourse119883 is of the form

119860 = ⟨119909 120583119860 (119909) 120592119860 (119909)⟩ 119909 isin 119883 (1)

where 120583119860

denotes membership function and 120592119860

denotesnonmembership function with the condition 0 le 120583

119860(119909) +

120592119860(119909) le 1 120583

119860(119909) 120592119860(119909) isin [0 1] for all 119909 isin 119883

Definition 2 (interval-valued intuitionistic fuzzy set) Aninterval-valued intuitionistic fuzzy set in119860 over119883 is an objecthaving the form

119860

= ⟨119909 [120583119871

119860(119909) 120583

119880

119860(119909)] [120592

119871

119860(119909) 120592

119880

119860(119909)]⟩ 119909 isin 119883

(2)

where

120583119871

119860 120583119880

119860 120592119871

119860 120592119880

119860 119883 997888rarr [0 1] 120583

119871

119860le 120583119880

119860 120592119871

119860le 120592119880

119860 (3)

Definition 3 (IVIFNs score function) Let =

([120583119871

120583119880

] [120592119871

120592119880

]) be an interval-valued intuitionistic

fuzzy number The score function of 119878119883() is represented

as

119878119883() =

120583119871

+ 120583119880

minus 120592119871

minus 120592119880

2 119878119883() isin [minus1 1] (4)

Definition 4 (IVIFNs accuracy function) Let =

([120583119871

120583119880

] [120592119871

120592119880


fuzzy number The accuracy function of 119867119883() is

represented as

119867119883() =

120583119871

+ 120583119880

+ 120592119871

+ 120592119880

2 119867119883() isin [0 1] (5)

Definition 5 (IVITFS) Let be an interval-valued intu-itionistic trapezoidal fuzzy set (IVITFS) its interval-valuedmembership function is

120583119880

(119909) =

119909 minus 119886

119887 minus 119886120583119880

119886 le 119909 lt 119887

120583119880

119887 le 119909 le 119888

119889 minus 119909

119889 minus 119888120583119880

119888 lt 119909 le 119889

0 others

120583119871

(119909) =

119909 minus 119886

119887 minus 119886120583119871

119886 le 119909 lt 119887

120583119871

119887 le 119909 le 119888

119889 minus 119909

119889 minus 119888120583119871

119888 lt 119909 le 119889

0 others

(6)

Advances in Fuzzy Systems 3

Its interval-valued nonmembership function is

120592119880

(119909) =

119887 minus 119909 + 120592119880

(119909 minus 119886)

119887 minus 119886 119886 le 119909 lt 119887

120592119880

119887 le 119909 le 119888

119909 minus 119888 + 120592119880

(119889 minus 119909)

119889 minus 119888 119888 lt 119909 le 119889

0 others

120592119871

(119909) =

119887 minus 119909 + 120592119871

(119909 minus 119886)

119887 minus 119886 119886 le 119909 lt 119887

120592119871

119887 le 119909 le 119888

119909 minus 119888 + 120592119871

(119889 minus 119909)

119889 minus 119888 119888 lt 119909 le 119889

0 others

(7)

where 0 le 120583119871

le 120583119880

le 1 0 le 120592

119871

le 120592119880

le 1 0 le

120583119880

+ 120592119880

le 1 0 le 120583

119871

+ 120592119871

le 1 119886 119887 119888 119889 isin 119877 Then

= ([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880

]) is called interval-valued

intuitionistic trapezoidal fuzzy set (IVITFS)

Definition 6 (arithmetic operation law of IVITFS) Let1

= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]) and 2

=

([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

]) be two interval-valuedintuitionistic trapezoidal fuzzy numbers then

1oplus 2= ([1198861+ 1198862 1198871+ 1198872 1198881+ 1198882 1198891+ 1198892]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

1minus 2= ([1198861minus 1198862 1198871minus 1198872 1198881minus 1198882 1198891minus 1198892]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(8)

for 1gt 0

2gt 0 consider

1otimes 2= ([1198861sdot 1198862 1198871sdot 1198872 1198881sdot 1198882 1198891sdot 1198892]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

1

2

= ([1198861

1198892

1198871

1198882

1198881

1198872

1198891

1198862

]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(9)

Definition 7 (120572-cut set of IVITFN) 120572-cut set of an IVITFN = ([119886 119887 119888 119889] [120583

119871

120583119880

] [120592119871

120592119880

]) is a crisp subset of 119877

which is defined as 120572= 119909120583

(119909) ge 120572 where 0 le 120572 le 120583

is

a closed interval denoted by

120572= [119871(120572) 119877

(120572)]

= [119886 +120572

119878119909()

(119887 minus 119886) 119889 +120572

119878119909()

(119888 minus 119889)] (10)

where 119878119909() = (120583

119871

+ 120583119880

minus 120592119871

minus 120592119880

)2 is the score function of


119871

120583119880

] [120592119871

120592119880



(119909) le 120573 where 120592

le 120573 le 1 is


120573= [119871(120573) 119877

(120573)]

= [(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

(1 minus 120573) 119888 + (120573 minus 119867119909()) 119889

1 minus 119867119909()

]

(11)

where119867119909() = (120583

119871

+120583119880

+120592119871

+120592119880

)2 is the accuracy function

of

21 ExistingWu and Liu Ranking of IVITFN For an interval-valued intuitionistic trapezoidal fuzzy number Wu and Liu[24] ranked the IVITFN using score and expected functionThe score expected function 119868(119878

119909()) and accurate expected

function of are given by

119868 (119878119909()) =

119878119909()

2[(1 minus 120597) (119886 + 119887) + 120597 (119888 + 119889)]

119868 (119867119909()) =

119867119909()

2[(1 minus 120597) (119886 + 119887) + 120597 (119888 + 119889)]

(12)

where 119878119909() and 119867

119909() are the score and accuracy function

of and 120597 isin [0 1] indicates the risk tolerance of the expertRanking is defined as follows

(1) The larger the value of 119868(119878119909()) the more the degree

of score of

(2) If score expected functions of two IVITFNs are thesame then find the accurate expected functions thelarger the value of 119868(119867

119909()) the more the degree of

accuracy of

(3) If the values of score and accurate expected functionare the same then the IVITFNs are said to be equal


3 Proposed Ranking of IVITFNs Based onValue and Ambiguity

In this section we propose a method to rank IVITFN basedon value and ambiguity defined by Delgado et al [33] usingalpha-cuts and beta-cuts The parameter ldquovaluerdquo allows us torepresent an IVITFN as a real value It assesses the ill-definedmagnitude represented by the fuzzy number ldquoAmbiguityrdquomeasures how much vagueness is present in the ill-definedmagnitude of the fuzzy number Hence the relation is similarto mean and variance in statistics

Definition 9 (value of IVITFN) The values of membershipfunction 120583

(119909) and nonmembership function 120592

(119909) for the

IVITFN are respectively defined as

119881120583() = int

119878119909()

0

119871(120572) + 119877

(120572)

2119891 (120572) 119889120572

119881120592() = int

1

119867119883()

119871(120573) + 119877

(120573)

2119892 (120573) 119889120573

(13)

where the function 119891(120572) is a nonnegative and nondecreasingfunction on the interval [0 119878

119909()] with 119891(0) = 0 and

int119878119909()

0119891(120572)119889120572 = 119878

119909() the function 119892(120573) is a nonnegative

and nonincreasing function on the interval [119867119909() 1] with

119892(120573) = 1 and int1

119867119883()

119892(120573)119889120573 = 1 minus 119867119909()

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1 minus 120573)(1 minus 119867

119909())

Consider

119881120583() = int

119878119909()

0

1

2[119886 +

120572 (119887 minus 119886)

119878119909()

+ 119889 +120572 (119888 minus 119889)

119878119909()

]

sdot2120572

119878119909()

119889120572 =1

119878119909()

[int119878119909()

0

(119886 + 119889) 120572 119889120572]

+1

119878119909

2 ()[int119878119909()

0

(119887 + 119888 minus 119886 minus 119889) 1205722119889120572]

=119886 + 2119887 + 2119888 + 119889

6119878119909()

119881120592() = int

1

119867119909()

1

2[(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

+(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)

+ 120573 (119886 + 119889) minus 119867119909() (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)

minus (1 minus 120573) (119886 + 119889) + (1 minus 119867119909()) (119886 + 119889)] (1

minus 120573) 119889120573 =1

(1 minus 119867119909())2

sdot int1

119867119909()

[(1 minus 120573) (119888 + 119887) minus (1 minus 120573) (119886 + 119889)

+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887 minus 119886 minus 119889)

+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())

(14)

Definition 10 (ambiguity of IVITFN) The ambiguities ofmembership function 120583

(119909) and nonmembership function

120592(119909) for the IVITFN are respectively defined as

119860120583() = int

119878119909()

0

(119877(120572) minus 119871

(120572)) 119891 (120572) 119889120572

119860120592() = int

1

119867119883()

(119877(120573) minus 119871

(120573)) 119892 (120573) 119889120573

(15)

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1minus120573)(1minus119867

119909()) Consider

119860120583() = int

119878119909()

0

[(119889 +120572 (119888 minus 119889)

119878119909()

)

minus (119886 +120572 (119887 minus 119886)

119878119909()

)]2120572

119878119909()

119889120572

=2

119878119909()

[int119878119909()

0

(119889 minus 119886) 120572 119889120572]

+2

119878119909

2 ()[int119878119909()

0

(119888 minus 119889 + 119886 minus 119887) 1205722119889120572]

=(119889 minus 119886) minus 2 (119887 minus 119888)

3119878119909()

119860120592() = int

1

119867119909()

[(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()

minus(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)

+ 120573 (119889 minus 119886) minus 119867119909() (119889 minus 119886)] (1 minus 120573) 119889120573

=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)

minus (1 minus 120573) (119889 minus 119886) + (1 minus 119867119909()) (119889 minus 119886)] (1

minus 120573) 119889120573 =2

(1 minus 119867119909())2

sdot int1

119867119909()

[(1 minus 120573) (119888 minus 119887 + 119886 minus 119889)

+ (1 minus 119867119909()) (119889 minus 119886)] (1 minus 120573) 119889120573


3(1 minus 119867

119909())

(16)

Definition 11 (value index of IVITFN) Based on the valuesof membership function and nonmembership function thevalue index of IVITFN = ([119886 119887 119888 119889] [120583

119871

120583119880

] [120592119871

120592119880

]) is

defined as

119881() = 119896119881120583() + (1 minus 119896)119881120592 ()

= 119896 [119886 + 2119887 + 2119888 + 119889

6119878119909()] + (1 minus 119896)

sdot [119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())]

= [119886 + 2 (119887 + 119888) + 119889

6]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(17)

where 119896 isin [0 1] varies according to the decision makerrsquos riskattitude The expert is said to be risk-averse if 119896 lt 05 risk-prone if 119896 gt 05 and risk-neutral if 119896 = 05

For 119896 = 05

119881() =119886 + 2 (119887 + 119888) + 119889

12(1 + 119878

119909() minus 119867

119909()) (18)

If [120583119871 120583119880

] = [1 1] and [120592

119871

120592119880

] = [0 0] then the IVITFN

degenerates to a trapezoidal fuzzy number = [119886 119887 119888 119889] Inthis case for 119896 = 05

119881() =119886 + 2119887 + 2119888 + 119889

12 (19)

Definition 12 (ambiguity index of IVITFN) Based onthe ambiguities of membership function and nonmem-bership function the ambiguity index of IVITFN =

([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880

]) is defined as

119860() = 119896119860120583() + (1 minus 119896)119860120592 ()

= 119896 [(119889 minus 119886) minus 2 (119887 minus 119888)

3119878119909()] + (1 minus 119896)

sdot [(119889 minus 119886) minus 2 (119887 minus 119888)

3(1 minus 119867

119909())]

= [(119889 minus 119886) minus 2 (119887 minus 119888)

3]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(20)

where 119896 isin [0 1] varies according to the decision makerrsquos riskattitude

For 119896 = 05

119860() =(119889 minus 119886) minus 2 (119887 minus 119888)

6(1 + 119878

119909() minus 119867

119909()) (21)

Based on the value index function 119881() and the ambiguityindex function 119860() the following ranking procedure isproposed

For two interval-valued intuitionistic trapezoidal fuzzynumbers

1= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

])

2= ([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(22)

(1) if 119881(1) lt 119881(

2) then

1lt 2

(2) if 119881(1) gt 119881(

2) then

1gt 2

(3) if 119881(1) = 119881(

2) then find 119860(

1) and 119860(

2)

(i) if 119860(1) lt 119860(

2) then

1lt 2

(ii) if 119860(1) gt 119860(

2) then

1gt 2

(iii) if 119860(1) = 119860(

2) then

1= 2

Remark 13 Throughout the paper we discuss the methodol-ogy by assuming that the decision maker is risk-neutral Thesame can be discussed in other two cases also

Advantages of Proposed Ranking The advantage of the pro-posedmethod is shownby comparisonwith existingmethodsin the literature

Example 14 Consider two IVITFNs

= ([02 03 04 05] [04 06] [02 03])

= ([04 05 06 07] [03 05] [02 03])

(23)


the score and accurate expected values of and byWu andLiu [24] are

119868 (119878119909()) = 00875

119868 (119878119909()) = 00825

(24)

and hence gt And by proposed ranking we get

119881() = 004375

119881 () = 004125

(25)

and therefore gt

Example 15 Consider

= ([03 04 05 06] [1 1] [0 0])

= ([02 03 06 07] [1 1] [0 0])

= ([01 04 05 08] [1 1] [0 0])

(26)

the score and accurate expected values of and by Wuand Liu are

119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

997904rArr = =

(27)

which is not true by intuitionBut by using the proposed method we have

119881() = 0225

119881 () = 0225

119881 () = 0225

119860 () = 00833

119860 () = 01833

119860 () = 015

997904rArr gt gt

(28)

Example 16 Consider

= ([05 06 07 075] [1 1] [0 0])

= ([045 065 07 075] [1 1] [0 0])

(29)

Then

119868 (119878119909()) = 06375

119868 (119878119909()) = 06375

119868 (119867119909()) = 06375

119868 (119867119909()) = 06375

997904rArr =

(30)

By proposed ranking

119881() = 03208

119881 () = 0325

(31)

and hence we get gt From these examples it is proved that the proposed

method can rank IVITFNs effectively when compared to Wuand Liu

4 Proposed Algorithm of ELECTRE Methodfor IVITFNs

ELECTRE is the most popular outranking approach amongstthe family of outranking approaches It is used to rankthe set of alternatives in many MCDM problems In theproposed method criteria values of each alternative andcriteria weights are considered as IVITTFNs This represen-tation gives an opportunity to decision maker to define themembership and nonmembership in the form of an intervalas well as discussing the problem on a consecutive set

Let 1198601 1198602 1198603 119860

119898be 119898 possible alternatives and

let 1198621 1198622 1198623 119862

119899be 119899 criteria with which alternativesrsquo

performance is measured Let 119894119895be the performance of

alternative with respect to criterion which is expressed asIVITFN represented by

119894119895= ([119886119894119895 119887119894119895 119888119894119895 119889119894119895] [120583minus

119894119895

120583+

119894119895

] [120592minus

119894119895

120592+

119894119895

]) (32)

Let = [119896

119895]119896times119899

be the weight matrix where

119896

119895= ([119908

119896

1119895 119908119896

2119895 119908119896

3119895 119908119896

4119895] [120583119871

119896

119895

120583119880

119896

119895

] [120592119871

119896

119895

120592119880

119896

119895

]) (33)


is theweight of the criterion119862119895which is also an IVITFNThen

the average weight of each criterion is calculated using theequation

119895=

1

119896[1

119895oplus 2

119895oplus sdot sdot sdot oplus

119896

119895] (34)

here 119896

119895is the assessment of the 119896th decision maker

Step 1 (construction of decision matrix) Let 119896 decisionmakers be involved in the decision making The rating of thealternate with respect to each criterion can be calculated as

119894119895=

1

119896[1

119894119895oplus 2


119896

119894119895] (35)

where 119896

119894119895is the assessment of the 119896th decision maker and

oplus is the sum operator applied to the IVITFNs as defined inDefinition 6

Then the decision matrix for a multicriteria decisionmaking problem (MCDMP) is defined as 119863 = [

119894119895]119898times119899

Step 2 (calculation of normalized decision matrix) Thenormalization of the decision matrix 119863 is as follows

119894119895=

119894119895

radicsum119898

119894=12

119894119895

(36)

Step 3 (calculation of the weighted normalized decisionmatrix) By taking into account the weight of each criterionthe weighted normalized decision matrix denoted by 119877 isconstructed as

119877 = [119894119895]119898times119899

(37)

where 119894119895

= 119894119895

otimes 119895and otimes is the multiplication operator

applied to the IVITFNs as defined in Definition 6

Step 4 (estimating the concordance and discordance IVITFNssets) The set of given IVITFN indicators are divided intotwo different sets of concordance and discordance IVITFNsets For each pair of alternatives 119860

119896and 119860

119897 where 119896 119897 =

1 2 3 119898 and 119896 = 119897 the IVITFN concordance set 119862119896119897

is

the set which contains all the criteria for which the alternative119860119896is superior to the alternative 119860

119897 and it is represented by

119862119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

ge ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

(38)

and the IVITFN discordance set 119863119896119897 the complement of the

set 119862119896119897 is given by

119863119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

lt ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

= 119869

minus 119862119896119897

(39)

Step 5 (calculation of IVITFN concordance matrix) Theconcordance index 119862

119896119897reflects the relative importance of

119860119896with respect to 119860

119897 It is equal to the sum of IVITFN

weights corresponding to the criteria which are contained inthe concordance set119862

119896119897Thus the concordance index is given

by

119862119896119897

= ([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

= sum119895isin119862119896119897

([1199081119895 1199082119895 1199083119895 1199084119895] [120583119871

119895 120583119880

119895] [120592119871

119895 120592119880

119895])

(40)

The successive values of the concordance indices 119862119896119897(119896 119897 =

1 2 3 119898 and 119896 = 119897) form the concordance matrix 119862119898times119898

Hence the asymmetrical concordance IVITF matrix is asfollows

119862119898times119898

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

sdot sdot sdot 1198621119898

11986221

minus 11986223


sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

1198621198981

1198621198982

1198621198983

sdot sdot sdot 119862119898119898

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(41)

Step 6 (calculation of IVITFN discordance matrix) Thediscordance index 119863119868

119896119897reflects the degree to which 119860

119896is

worse than119860119897 It is calculated for each element of discordance

IVITFN set 119863119896119897 using the members of weighted normalized

matrix 119877 as follows

119863119868119896119897

=max119895isin119863119896119897

10038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871

119896119895 120583119880

119896119895] [120592119871

119896119895 120592119880

119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

10038161003816100381610038161003816

max119895isin119869

100381610038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871119897119895 120583119880119897119895] [120592119871119897119895 120592119880119897119895])

100381610038161003816100381610038161003816 (42)


These computations convert IVITFNs to crisp numbersHence we get the discordance matrix with the crisp valuesThe successive values of the discordance indices 119863119868

119896119897(119896 119897 =

1 2 3 119898 and 119896 = 119897) form the discordance matrix 119863119898times119898

which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898

Step 7 (determining the concordance IVITF dominancematrix) The concordance IVITF dominancematrix is calcu-lated by comparing the values of concordance IVITF matrix(119862119898times119898

) with threshold value (1198621015840) It indicates alternative

119860119896rsquos chance of dominating alternative119860

119897The threshold is the

average of concordance IVITF index that is

1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)

On the basis of this threshold value 1198621015840 a Boolean matrix 119865

is constructed as follows

119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)

Element 1 in the Boolean matrix 119865 represents the dominanceof one alternative with respect to the other

Step 8 (determining the discordance IVITF dominancematrix) In a similar way the discordance IVITF dominancematrix can be calculated with the help of discordance indicesand the threshold value119863119868

1015840 which is given as follows

1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897

of the Boolean matrix 119866 are calculated asfollows

119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)


Step 9 (estimating the aggregate dominance matrix) Theaggregate dominance matrix 119879 is the intersection of concor-dance IVITF dominance matrix 119865 and discordance IVITFdominance matrix 119866 The elements 119905

119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)

The aggregate dominance matrix indicates the partialpreference ordering of the alternatives If 119905

119896119897= 1 then

119860119896is preferred to 119860

119897in terms of both concordance criteria

and discordance criteria In this case the alternative 119860119897

is eliminated However 119860119896may be dominated by other

alternatives Hence the condition which makes alternative119860119896more effective is defined as follows

119905119896119897

= 1 for at least one 119897 for 119897 = 1 2 119898 119896 = 119897

119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example

In this section the proposed IVITF-ELECTRE method isillustrated by taking two real world problems discussed in theliterature

Example 1 The proposed method is applied to find thebest green supplier for one of the key elements in themanufacturing process of a food company presented byWu and Liu [24] After preevaluation the company hasselected three suppliers (alternatives) for further evaluationand evaluated them based on four criteria product quality(1198621) technology capability (119862

2) pollution control (119862

3) and

environmental management (1198624) Three decision makers

namelyDM1 DM2 andDM

3 are chosen from three different

departments namely production purchasing and qualityinspection Assessments of three suppliers by three decisionmakers based on each criterion are given respectively inTables 1 2 and 3

Weights of each criterion are given as

1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)

The decision matrix 119863 for the given data using Step 1 iscalculated and presented in Table 4

The normalized decision matrix and weighted normal-ized decision matrices are calculated using (36) and (37) andrespectively given in Tables 5 and 6

Concordance and discordance sets are estimated using(38) and (39) and are as follows

11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601


Table 1 Assessment by DM1

Criteria Suppliers1198601

1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)

Based on the concordance sets obtained and using (40)the concordance index and concordance IVITF matrix arecalculated and are given below

For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)

and the concordance IVITF matrix is

1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)

Calculation of discordance IVITF matrix is as followsThe discordance index and discordance IVITF are calcu-

lated using (42) and are given as follows

11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)

and the discordance IVITF matrix is

1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)

Next concordance dominance matrix is computed using thethreshold value

The threshold value of concordance index using (44) is

1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])

Table 4 Decision matrix119863

Criteria Alternatives1198601

1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])

Hence the concordance dominance matrix is as follows

119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)

On the other hand discordance dominance matrix is calcu-lated using threshold value

The threshold value of discordance index is

1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)

Hence the discordance dominance matrix is as follows

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)

Finally the aggregate dominance matrix 119879 = 119865 sdot 119866 is

119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)

Hence by (49) and matrix 119879 the alternatives can be orderedas

1198602gt 1198601gt 1198603 (62)

Example 2 An MCDM problem concerned with a customerwho intends to buy an air conditioner is solved usingproposed method taken from [34] Three types of air condi-tioners (alternatives 119860

119894(119894 = 1 2 3)) are to be evaluated on

four criteria (attributes) (1) quality (1198621) (2) design (119862

2) (3)

price (1198623) and (4) level of after-sale service (119862

4) The crisp

weighting vector of the criteria given in [34] is converted to

suitable IVITFN in order to apply the proposed methodTheassessments of three alternatives by a decision maker basedon each criterion are given in Table 7 and the weights of eachcriterion are taken as

1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)

The normalized decision matrix and weighted normalizeddecision matrices are calculated using (36) and (37) andrespectively given in Tables 8 and 9


11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])

Accordingly concordance and discordance matrices are cal-culated and are as follows

The concordance IVITF matrix is

1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)

Furthermore the concordance dominancematrix anddiscor-dance dominance matrices are computed using the thresholdvalue


1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)

and the threshold value of discordance index is

1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)

Hence the concordance dominance and discordance domi-nance matrices are respectively obtained as

119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)

Therefore the alternatives can be ordered as

1198601gt 1198603gt 1198602 (71)

This means that 1198601is the best alternative which agrees

with human intuition as 1198601rsquos performance is better when

compared to other alternatives in the criteria1198623and119862

4which

are given more weight in decision making process

6 Conclusion

In amulticriteria group decisionmakingmethod the expertsoften quantify their opinion in the form of an interval on aconsecutive set Hence it is more suitable to represent andstudy the decision making problems using IVITFNs Basedon this fact we proposed ELECTRE method to find thebest alternative when the criterion and weights of criterionare IVITFNs The advantage of the proposed method overexisting aggregated methods is that the proposed methodretains the fuzziness of criterionweights during computationwhile the aggregated methods use the weights and convertthem into crisp numbers in the initial step itself Furthermorethe proposed ranking method of IVITFNs using value andambiguity index of membership and nonmembership allowsstudying the IVITFNs in a statistical manner and also takesexpertrsquos risk attitude into consideration while comparing thefuzzy numbers It is also observed that the proposed rankingcan rank the IVITFNs more accurately when compared toexisting methods of ranking IVITFNs Finally the proposedmethod was illustrated by applying it to real world problemsfinding the best green supplier for one of the key elements inthe manufacturing process of a food company and selectingthe best air conditioner from a customer perspective Thecomparison reveals that the proposed method can rank thealternatives more strictly compared to the method ofWu andLiu In the future we will focus on applying the proposedmethod to other ELECTRE methods


Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K Atanassov ldquoIntuitonistic fuzzy setsrdquo Fuzzy Sets and Systemsvol 20 pp 87ndash96 1986

[3] B Vahdani and H Hadipour ldquoExtension of the ELECTREmethod based on interval-valued fuzzy setsrdquo Soft Computingvol 15 no 3 pp 569ndash579 2011

[4] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[5] J Wu and Q-w Cao ldquoSame families of geometric aggrega-tion operators with intuitionistic trapezoidal fuzzy numbersrdquoApplied Mathematical Modelling vol 37 no 1-2 pp 318ndash3272013

[6] M-H Shu C-H Cheng and J-R Chang ldquoUsing intuitionisticfuzzy sets for fault-tree analysis on printed circuit boardassemblyrdquo Microelectronics Reliability vol 46 no 12 pp 2139ndash2148 2006

[7] J Q Wang ldquoOverview on fuzzy multi-criteria decision makingapproachrdquo Control and Decision vol 23 pp 601ndash606 2008

[8] S P Wan ldquoMulti-attribute decision making method basedon interval-valued intuitionistic trapezoidal fuzzy numberrdquoControl and Decision vol 26 no 6 pp 857ndash860 2011

[9] R E Bellman and L A Zadeh ldquoDecision making in fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[10] S-M Chen and L-W Lee ldquoFuzzy multiple criteria hierarchicalgroup decision-making based on interval type-2 fuzzy setsrdquoIEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 5 pp 1120ndash1128 2010

[11] Z-W Gong L-S Li J Forrest and Y Zhao ldquoThe optimalpriority models of the intuitionistic fuzzy preference relationand their application in selecting industries with higher mete-orological sensitivityrdquo Expert Systems with Applications vol 38no 4 pp 4394ndash4402 2011

[12] E Herrera-Viedma F Chiclana F Herrera and S AlonsoldquoGroup decision-making model with incomplete fuzzy prefer-ence relations based on additive consistencyrdquo IEEETransactionson SystemsMan and Cybernetics Part B Cybernetics vol 37 no1 pp 176ndash189 2007

[13] H-W Liu and G-J Wang ldquoMulti-criteria decision-makingmethods based on intuitionistic fuzzy setsrdquo European Journalof Operational Research vol 179 no 1 pp 220ndash233 2007

[14] P Liu and F Jin ldquoA multi-attribute group decision-makingmethod based on weighted geometric aggregation operators ofinterval-valued trapezoidal fuzzy numbersrdquoAppliedMathemat-ical Modelling vol 36 no 6 pp 2498ndash2509 2012

[15] J-Q Wang and Z Zhang ldquoMulti-criteria decision-makingmethod with incomplete certain information based on intu-itionistic fuzzy numberrdquoControl and Decision vol 24 no 2 pp226ndash230 2009

[16] J-Q Wang K-J Li and H-Y Zhang ldquoInterval-valued intu-itionistic fuzzy multi-criteria decision-making approach based

on prospect score functionrdquo Knowledge-Based Systems vol 27pp 119ndash125 2012

[17] GWWei X F Zhao and H J Wang ldquoAn approach to multipleattribute group decision making with interval intuitionistictrapezoidal fuzzy informationrdquo Technological and EconomicDevelopment of Economy vol 18 no 2 pp 317ndash330 2012

[18] Z S Xu ldquoIntuitionistic preference relations and their applica-tion in group decision makingrdquo Information Sciences vol 177no 11 pp 2363ndash2379 2007

[19] Z S Xu ldquoApproaches to multiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[20] Z S Xu and J Chen ldquoAn approach to group decision makingbased on interval-valued intuitionistic judgment matricesrdquoSystem EngineermdashTheory and Practice vol 27 pp 126ndash133 2007

[21] N Yayala andM Karacasu ldquoA decision support model to incor-porate public and expert opinions for assessing the privatizationof public bus transit system application of ELECTRE for the bussystem in Eskisehir Turkeyrdquo Scientific Research and Essays vol6 no 21 pp 4657ndash4664 2011

[22] P P Angelov ldquoOptimization in an intuitionistic fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 86 no 3 pp 299ndash306 1997

[23] D-F Li and S-P Wan ldquoFuzzy linear programming approach tomultiattribute decision making with multiple types of attributevalues and incomplete weight informationrdquo Applied Soft Com-puting Journal vol 13 no 11 pp 4333ndash4348 2013

[24] J Wu and Y Liu ldquoAn approach for multiple attribute groupdecision making problems with interval-valued intuitionistictrapezoidal fuzzy numbersrdquo Computers amp Industrial Engineer-ing vol 66 no 2 pp 311ndash324 2013

[25] N Zhang and G Wei ldquoExtension of VIKOR method fordecision making problem based on hesitant fuzzy setrdquo AppliedMathematical Modelling vol 37 no 7 pp 4938ndash4947 2013

[26] R Benayoun B Roy and N Sussman Manual de Referencedu Programme Electre Note de Synthese et Formation No 25Direction Scientific SEMA Paris France 1966

[27] J Mustajoki R P Hamalainen and A Salo ldquoDecision supportby interval SMARTSWINGmdashincorporating imprecision in theSMART and SWINGmethodsrdquoDecision Sciences vol 36 no 2pp 317ndash339 2005

[28] J Figueria V Mousseau and B Roy ldquoELECTRE methodsrdquo inMultiple Criteria Decision Analysis State of the Art Surveys JFiguera S Greco and M Ehrgott Eds pp 133ndash153 SpringerBoston Mass USA 2005

[29] M Sevkli ldquoAn application of the fuzzy ELECTRE method forsupplier selectionrdquo International Journal of Production Researchvol 48 no 12 pp 3393ndash3405 2010

[30] B Vahdani A H K Jabbari V Roshanaei and M ZandiehldquoExtension of the ELECTRE method for decision-makingproblems with interval weights and datardquo International Journalof Advanced Manufacturing Technology vol 50 no 5ndash8 pp793ndash800 2010

[31] G Wei X Zhao and H Wang ldquoAn approach to multipleattribute group decision making with interval intuitionistictrapezoidal fuzzy informationrdquo Technological and EconomicDevelopment of Economy vol 18 no 2 pp 317ndash330 2012

[32] P K De and D Das ldquoA study on ranking of trapezoidalintuitionistic fuzzy numbersrdquo International Journal of ComputerInformation System and Industrial Management Applicationsvol 6 pp 437ndash444 2014


[33] M Delgado M A Vila and W Voxman ldquoOn a canonicalrepresentation of fuzzy numbersrdquo Fuzzy Sets and Systems vol93 no 1 pp 125ndash135 1998

[34] M Tian and J Liu ldquoSome aggregation operators with interval-valued intuitionistic trapezoidal fuzzy numbers and theirapplication in multiple attribute decision makingrdquo AdvancedModeling and Optimization vol 15 no 2 2013

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


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Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

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ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



developed new fuzzy linear programming techniques forsolving multiple attribute decision making (MADM) prob-lems with multiple types of attribute values and incompleteweight information Wu and Liu [24] proposed an approachfor MAGDM problems with IVITFNs by taking into accountthe expertrsquos risk attitude Zhang and Wei [25] developed theE-VIKOR method and TOPSIS method to solve the MCDMproblems with hesitant fuzzy set information

Many MCDM models use ldquooutranking relationsrdquo to ranka set of alternatives The important feature of outrankingapproach is its noncompensatory nature The most popularoutranking approach amongst the family of outrankingapproaches is ELECTRE The ELECTRE approach was firstintroduced by Benayoun et al in 1966 [26] Soon after theintroduction of the first version of ELECTRE I [27] thisapproach was evolved into a number of other variants such asELECTRE II III IV IS and TRI [28] Sevkli [29] proposedfuzzy ELECTRE for supplier selection problem to deal withimprecise and vague nature of linguistic assessment Vahdaniet al [30] extended ELECTRE with interval weights anddata to deal with uncertainty Later in 2011 Vahdani andHadipour extended the ELECTREmethod based on interval-valued fuzzy sets [3]Wei et al [31] investigated theMAGDMproblems in which attribute values take the form of IVITFNsand both the attribute weights and the expert weights take theform of real numbers Since fuzzy numbers and intuitionisticfuzzy numbers are special cases of IVITFNs it is significant todevelop the most accepted outranking method ldquoELECTRErdquoby taking both attribute values and weights of criteria ofMCDM problem in the form of IVITFNs To do this weproposed a ranking method for IVITFNs which takes theexpertrsquos risk attitude into consideration

The paper is organized as follows In Section 2 basic defi-nitions and arithmetic operations of IVITFNs are reviewedIn Section 3 a new ranking of IVITFNs based on valueand ambiguity functions is proposed to compare IVITFNsand its advantages over existing methods are presentedIn Section 4 the proposed algorithm of ELECTRE methodbased on IVITFNs is presented Section 5 provides illustrativeexamples Conclusions and future work are presented inSection 6

2 Preliminaries

In this section we briefly introduce some basic conceptsrelated to interval-valued intuitionistic trapezoidal fuzzynumbers from Xu and Chen [20] and De and Das [32]

Definition 1 (intuitionistic fuzzy set) An intuitionistic fuzzyset over universe of discourse119883 is of the form

119860 = ⟨119909 120583119860 (119909) 120592119860 (119909)⟩ 119909 isin 119883 (1)

where 120583119860

denotes membership function and 120592119860

denotesnonmembership function with the condition 0 le 120583

119860(119909) +

120592119860(119909) le 1 120583

119860(119909) 120592119860(119909) isin [0 1] for all 119909 isin 119883

Definition 2 (interval-valued intuitionistic fuzzy set) Aninterval-valued intuitionistic fuzzy set in119860 over119883 is an objecthaving the form

119860

= ⟨119909 [120583119871

119860(119909) 120583

119880

119860(119909)] [120592

119871

119860(119909) 120592

119880

119860(119909)]⟩ 119909 isin 119883

(2)

where

120583119871

119860 120583119880

119860 120592119871

119860 120592119880

119860 119883 997888rarr [0 1] 120583

119871

119860le 120583119880

119860 120592119871

119860le 120592119880

119860 (3)

Definition 3 (IVIFNs score function) Let =

([120583119871

120583119880

] [120592119871

120592119880


fuzzy number The score function of 119878119883() is represented

as

119878119883() =

120583119871

+ 120583119880

minus 120592119871

minus 120592119880

2 119878119883() isin [minus1 1] (4)

Definition 4 (IVIFNs accuracy function) Let =

([120583119871

120583119880

] [120592119871

120592119880


fuzzy number The accuracy function of 119867119883() is

represented as

119867119883() =

120583119871

+ 120583119880

+ 120592119871

+ 120592119880

2 119867119883() isin [0 1] (5)

Definition 5 (IVITFS) Let be an interval-valued intu-itionistic trapezoidal fuzzy set (IVITFS) its interval-valuedmembership function is

120583119880

(119909) =

119909 minus 119886

119887 minus 119886120583119880

119886 le 119909 lt 119887

120583119880

119887 le 119909 le 119888

119889 minus 119909

119889 minus 119888120583119880

119888 lt 119909 le 119889

0 others

120583119871

(119909) =

119909 minus 119886

119887 minus 119886120583119871

119886 le 119909 lt 119887

120583119871

119887 le 119909 le 119888

119889 minus 119909

119889 minus 119888120583119871

119888 lt 119909 le 119889

0 others

(6)



120592119880

(119909) =

119887 minus 119909 + 120592119880

(119909 minus 119886)

119887 minus 119886 119886 le 119909 lt 119887

120592119880

119887 le 119909 le 119888

119909 minus 119888 + 120592119880

(119889 minus 119909)

119889 minus 119888 119888 lt 119909 le 119889

0 others

120592119871

(119909) =

119887 minus 119909 + 120592119871

(119909 minus 119886)

119887 minus 119886 119886 le 119909 lt 119887

120592119871

119887 le 119909 le 119888

119909 minus 119888 + 120592119871

(119889 minus 119909)

119889 minus 119888 119888 lt 119909 le 119889

0 others

(7)

where 0 le 120583119871

le 120583119880

le 1 0 le 120592

119871

le 120592119880

le 1 0 le

120583119880

+ 120592119880

le 1 0 le 120583

119871

+ 120592119871

le 1 119886 119887 119888 119889 isin 119877 Then

= ([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880




= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]) and 2

=

([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2


1oplus 2= ([1198861+ 1198862 1198871+ 1198872 1198881+ 1198882 1198891+ 1198892]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])


min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(8)

for 1gt 0

2gt 0 consider


min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

1

2

= ([1198861

1198892

1198871

1198882

1198881

1198872

1198891

1198862

]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(9)


119871

120583119880

] [120592119871

120592119880



(119909) ge 120572 where 0 le 120572 le 120583

is


120572= [119871(120572) 119877

(120572)]

= [119886 +120572

119878119909()

(119887 minus 119886) 119889 +120572

119878119909()

(119888 minus 119889)] (10)

where 119878119909() = (120583

119871

+ 120583119880

minus 120592119871

minus 120592119880



119871

120583119880

] [120592119871

120592119880



(119909) le 120573 where 120592

le 120573 le 1 is


120573= [119871(120573) 119877

(120573)]

= [(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

(1 minus 120573) 119888 + (120573 minus 119867119909()) 119889

1 minus 119867119909()

]

(11)

where119867119909() = (120583

119871

+120583119880

+120592119871

+120592119880


of




119868 (119878119909()) =

119878119909()

2[(1 minus 120597) (119886 + 119887) + 120597 (119888 + 119889)]

119868 (119867119909()) =

119867119909()

2[(1 minus 120597) (119886 + 119887) + 120597 (119888 + 119889)]

(12)

where 119878119909() and 119867




of score of



accuracy of







(119909) for the


119881120583() = int

119878119909()

0

119871(120572) + 119877

(120572)

2119891 (120572) 119889120572

119881120592() = int

1

119867119883()

119871(120573) + 119877

(120573)

2119892 (120573) 119889120573

(13)


119909()] with 119891(0) = 0 and

int119878119909()

0119891(120572)119889120572 = 119878



119892(120573) = 1 and int1

119867119883()

119892(120573)119889120573 = 1 minus 119867119909()

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1 minus 120573)(1 minus 119867

119909())

Consider

119881120583() = int

119878119909()

0

1

2[119886 +

120572 (119887 minus 119886)

119878119909()

+ 119889 +120572 (119888 minus 119889)

119878119909()

]

sdot2120572

119878119909()

119889120572 =1

119878119909()

[int119878119909()

0

(119886 + 119889) 120572 119889120572]

+1

119878119909

2 ()[int119878119909()

0

(119887 + 119888 minus 119886 minus 119889) 1205722119889120572]

=119886 + 2119887 + 2119888 + 119889

6119878119909()

119881120592() = int

1

119867119909()

1

2[(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

+(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)

+ 120573 (119886 + 119889) minus 119867119909() (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)


minus 120573) 119889120573 =1

(1 minus 119867119909())2

sdot int1

119867119909()


+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()


+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())

(14)




119860120583() = int

119878119909()

0

(119877(120572) minus 119871

(120572)) 119891 (120572) 119889120572

119860120592() = int

1

119867119883()

(119877(120573) minus 119871

(120573)) 119892 (120573) 119889120573

(15)

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1minus120573)(1minus119867

119909()) Consider

119860120583() = int

119878119909()

0

[(119889 +120572 (119888 minus 119889)

119878119909()

)

minus (119886 +120572 (119887 minus 119886)

119878119909()

)]2120572

119878119909()

119889120572

=2

119878119909()

[int119878119909()

0

(119889 minus 119886) 120572 119889120572]

+2

119878119909

2 ()[int119878119909()

0

(119888 minus 119889 + 119886 minus 119887) 1205722119889120572]


3119878119909()

119860120592() = int

1

119867119909()

[(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()


119909()) 119886

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


minus 120573) 119889120573 =2

(1 minus 119867119909())2

sdot int1

119867119909()




3(1 minus 119867

119909())

(16)


119871

120583119880

] [120592119871

120592119880

]) is

defined as

119881() = 119896119881120583() + (1 minus 119896)119881120592 ()

= 119896 [119886 + 2119887 + 2119888 + 119889

6119878119909()] + (1 minus 119896)

sdot [119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())]

= [119886 + 2 (119887 + 119888) + 119889

6]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(17)


For 119896 = 05

119881() =119886 + 2 (119887 + 119888) + 119889

12(1 + 119878

119909() minus 119867

119909()) (18)

If [120583119871 120583119880

] = [1 1] and [120592

119871

120592119880



119881() =119886 + 2119887 + 2119888 + 119889

12 (19)


([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880

]) is defined as

119860() = 119896119860120583() + (1 minus 119896)119860120592 ()


3119878119909()] + (1 minus 119896)


3(1 minus 119867

119909())]


3]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(20)


For 119896 = 05


6(1 + 119878

119909() minus 119867

119909()) (21)



1= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

])

2= ([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(22)

(1) if 119881(1) lt 119881(

2) then

1lt 2

(2) if 119881(1) gt 119881(

2) then

1gt 2

(3) if 119881(1) = 119881(


1) and 119860(

2)

(i) if 119860(1) lt 119860(

2) then

1lt 2

(ii) if 119860(1) gt 119860(

2) then

1gt 2

(iii) if 119860(1) = 119860(

2) then

1= 2




= ([02 03 04 05] [04 06] [02 03])

= ([04 05 06 07] [03 05] [02 03])

(23)



119868 (119878119909()) = 00875

119868 (119878119909()) = 00825

(24)


119881() = 004375

119881 () = 004125

(25)

and therefore gt

Example 15 Consider

= ([03 04 05 06] [1 1] [0 0])

= ([02 03 06 07] [1 1] [0 0])

= ([01 04 05 08] [1 1] [0 0])

(26)


119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

997904rArr = =

(27)


119881() = 0225

119881 () = 0225

119881 () = 0225

119860 () = 00833

119860 () = 01833

119860 () = 015

997904rArr gt gt

(28)

Example 16 Consider

= ([05 06 07 075] [1 1] [0 0])

= ([045 065 07 075] [1 1] [0 0])

(29)

Then

119868 (119878119909()) = 06375

119868 (119878119909()) = 06375

119868 (119867119909()) = 06375

119868 (119867119909()) = 06375

997904rArr =

(30)

By proposed ranking

119881() = 03208

119881 () = 0325

(31)





Let 1198601 1198602 1198603 119860


let 1198621 1198622 1198623 119862




119894119895= ([119886119894119895 119887119894119895 119888119894119895 119889119894119895] [120583minus

119894119895

120583+

119894119895

] [120592minus

119894119895

120592+

119894119895

]) (32)

Let = [119896

119895]119896times119899


119896

119895= ([119908

119896

1119895 119908119896

2119895 119908119896

3119895 119908119896

4119895] [120583119871

119896

119895

120583119880

119896

119895

] [120592119871

119896

119895

120592119880

119896

119895

]) (33)




119895=

1

119896[1

119895oplus 2


119896

119895] (34)

here 119896



119894119895=

1

119896[1

119894119895oplus 2


119896

119894119895] (35)

where 119896




119894119895]119898times119899


119894119895=

119894119895

radicsum119898

119894=12

119894119895

(36)


119877 = [119894119895]119898times119899

(37)

where 119894119895

= 119894119895




119896and 119860

119897 where 119896 119897 =


is



119862119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

ge ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

(38)


set 119862119896119897 is given by

119863119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

lt ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

= 119869

minus 119862119896119897

(39)



119860119896with respect to 119860




by

119862119896119897

= ([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

= sum119895isin119862119896119897

([1199081119895 1199082119895 1199083119895 1199084119895] [120583119871

119895 120583119880

119895] [120592119871

119895 120592119880

119895])

(40)




119862119898times119898

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213


11986221

minus 11986223



1198621198981

1198621198982

1198621198983

sdot sdot sdot 119862119898119898

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(41)



119896is




119863119868119896119897

=max119895isin119863119896119897

10038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871

119896119895 120583119880

119896119895] [120592119871

119896119895 120592119880

119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

10038161003816100381610038161003816

max119895isin119869

100381610038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871119897119895 120583119880119897119895] [120592119871119897119895 120592119880119897119895])

100381610038161003816100381610038161003816 (42)



119896119897(119896 119897 =


which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898






1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)



119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)




1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897


119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)



119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)


119896119897= 1 then

119860119896is preferred to 119860





119905119896119897


119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example




3) and


namelyDM1 DM2 andDM




1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)




11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601




1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120592119880

(119909) =

119887 minus 119909 + 120592119880

(119909 minus 119886)

119887 minus 119886 119886 le 119909 lt 119887

120592119880

119887 le 119909 le 119888

119909 minus 119888 + 120592119880

(119889 minus 119909)

119889 minus 119888 119888 lt 119909 le 119889

0 others

120592119871

(119909) =

119887 minus 119909 + 120592119871

(119909 minus 119886)

119887 minus 119886 119886 le 119909 lt 119887

120592119871

119887 le 119909 le 119888

119909 minus 119888 + 120592119871

(119889 minus 119909)

119889 minus 119888 119888 lt 119909 le 119889

0 others

(7)

where 0 le 120583119871

le 120583119880

le 1 0 le 120592

119871

le 120592119880

le 1 0 le

120583119880

+ 120592119880

le 1 0 le 120583

119871

+ 120592119871

le 1 119886 119887 119888 119889 isin 119877 Then

= ([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880




= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]) and 2

=

([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2


1oplus 2= ([1198861+ 1198862 1198871+ 1198872 1198881+ 1198882 1198891+ 1198892]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])


min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(8)

for 1gt 0

2gt 0 consider


min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

1

2

= ([1198861

1198892

1198871

1198882

1198881

1198872

1198891

1198862

]

min [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

]

max [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(9)


119871

120583119880

] [120592119871

120592119880



(119909) ge 120572 where 0 le 120572 le 120583

is


120572= [119871(120572) 119877

(120572)]

= [119886 +120572

119878119909()

(119887 minus 119886) 119889 +120572

119878119909()

(119888 minus 119889)] (10)

where 119878119909() = (120583

119871

+ 120583119880

minus 120592119871

minus 120592119880



119871

120583119880

] [120592119871

120592119880



(119909) le 120573 where 120592

le 120573 le 1 is


120573= [119871(120573) 119877

(120573)]

= [(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

(1 minus 120573) 119888 + (120573 minus 119867119909()) 119889

1 minus 119867119909()

]

(11)

where119867119909() = (120583

119871

+120583119880

+120592119871

+120592119880


of




119868 (119878119909()) =

119878119909()

2[(1 minus 120597) (119886 + 119887) + 120597 (119888 + 119889)]

119868 (119867119909()) =

119867119909()

2[(1 minus 120597) (119886 + 119887) + 120597 (119888 + 119889)]

(12)

where 119878119909() and 119867




of score of



accuracy of







(119909) for the


119881120583() = int

119878119909()

0

119871(120572) + 119877

(120572)

2119891 (120572) 119889120572

119881120592() = int

1

119867119883()

119871(120573) + 119877

(120573)

2119892 (120573) 119889120573

(13)


119909()] with 119891(0) = 0 and

int119878119909()

0119891(120572)119889120572 = 119878



119892(120573) = 1 and int1

119867119883()

119892(120573)119889120573 = 1 minus 119867119909()

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1 minus 120573)(1 minus 119867

119909())

Consider

119881120583() = int

119878119909()

0

1

2[119886 +

120572 (119887 minus 119886)

119878119909()

+ 119889 +120572 (119888 minus 119889)

119878119909()

]

sdot2120572

119878119909()

119889120572 =1

119878119909()

[int119878119909()

0

(119886 + 119889) 120572 119889120572]

+1

119878119909

2 ()[int119878119909()

0

(119887 + 119888 minus 119886 minus 119889) 1205722119889120572]

=119886 + 2119887 + 2119888 + 119889

6119878119909()

119881120592() = int

1

119867119909()

1

2[(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

+(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)

+ 120573 (119886 + 119889) minus 119867119909() (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)


minus 120573) 119889120573 =1

(1 minus 119867119909())2

sdot int1

119867119909()


+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()


+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())

(14)




119860120583() = int

119878119909()

0

(119877(120572) minus 119871

(120572)) 119891 (120572) 119889120572

119860120592() = int

1

119867119883()

(119877(120573) minus 119871

(120573)) 119892 (120573) 119889120573

(15)

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1minus120573)(1minus119867

119909()) Consider

119860120583() = int

119878119909()

0

[(119889 +120572 (119888 minus 119889)

119878119909()

)

minus (119886 +120572 (119887 minus 119886)

119878119909()

)]2120572

119878119909()

119889120572

=2

119878119909()

[int119878119909()

0

(119889 minus 119886) 120572 119889120572]

+2

119878119909

2 ()[int119878119909()

0

(119888 minus 119889 + 119886 minus 119887) 1205722119889120572]


3119878119909()

119860120592() = int

1

119867119909()

[(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()


119909()) 119886

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


minus 120573) 119889120573 =2

(1 minus 119867119909())2

sdot int1

119867119909()




3(1 minus 119867

119909())

(16)


119871

120583119880

] [120592119871

120592119880

]) is

defined as

119881() = 119896119881120583() + (1 minus 119896)119881120592 ()

= 119896 [119886 + 2119887 + 2119888 + 119889

6119878119909()] + (1 minus 119896)

sdot [119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())]

= [119886 + 2 (119887 + 119888) + 119889

6]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(17)


For 119896 = 05

119881() =119886 + 2 (119887 + 119888) + 119889

12(1 + 119878

119909() minus 119867

119909()) (18)

If [120583119871 120583119880

] = [1 1] and [120592

119871

120592119880



119881() =119886 + 2119887 + 2119888 + 119889

12 (19)


([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880

]) is defined as

119860() = 119896119860120583() + (1 minus 119896)119860120592 ()


3119878119909()] + (1 minus 119896)


3(1 minus 119867

119909())]


3]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(20)


For 119896 = 05


6(1 + 119878

119909() minus 119867

119909()) (21)



1= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

])

2= ([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(22)

(1) if 119881(1) lt 119881(

2) then

1lt 2

(2) if 119881(1) gt 119881(

2) then

1gt 2

(3) if 119881(1) = 119881(


1) and 119860(

2)

(i) if 119860(1) lt 119860(

2) then

1lt 2

(ii) if 119860(1) gt 119860(

2) then

1gt 2

(iii) if 119860(1) = 119860(

2) then

1= 2




= ([02 03 04 05] [04 06] [02 03])

= ([04 05 06 07] [03 05] [02 03])

(23)



119868 (119878119909()) = 00875

119868 (119878119909()) = 00825

(24)


119881() = 004375

119881 () = 004125

(25)

and therefore gt

Example 15 Consider

= ([03 04 05 06] [1 1] [0 0])

= ([02 03 06 07] [1 1] [0 0])

= ([01 04 05 08] [1 1] [0 0])

(26)


119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

997904rArr = =

(27)


119881() = 0225

119881 () = 0225

119881 () = 0225

119860 () = 00833

119860 () = 01833

119860 () = 015

997904rArr gt gt

(28)

Example 16 Consider

= ([05 06 07 075] [1 1] [0 0])

= ([045 065 07 075] [1 1] [0 0])

(29)

Then

119868 (119878119909()) = 06375

119868 (119878119909()) = 06375

119868 (119867119909()) = 06375

119868 (119867119909()) = 06375

997904rArr =

(30)

By proposed ranking

119881() = 03208

119881 () = 0325

(31)





Let 1198601 1198602 1198603 119860


let 1198621 1198622 1198623 119862




119894119895= ([119886119894119895 119887119894119895 119888119894119895 119889119894119895] [120583minus

119894119895

120583+

119894119895

] [120592minus

119894119895

120592+

119894119895

]) (32)

Let = [119896

119895]119896times119899


119896

119895= ([119908

119896

1119895 119908119896

2119895 119908119896

3119895 119908119896

4119895] [120583119871

119896

119895

120583119880

119896

119895

] [120592119871

119896

119895

120592119880

119896

119895

]) (33)




119895=

1

119896[1

119895oplus 2


119896

119895] (34)

here 119896



119894119895=

1

119896[1

119894119895oplus 2


119896

119894119895] (35)

where 119896




119894119895]119898times119899


119894119895=

119894119895

radicsum119898

119894=12

119894119895

(36)


119877 = [119894119895]119898times119899

(37)

where 119894119895

= 119894119895




119896and 119860

119897 where 119896 119897 =


is



119862119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

ge ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

(38)


set 119862119896119897 is given by

119863119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

lt ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

= 119869

minus 119862119896119897

(39)



119860119896with respect to 119860




by

119862119896119897

= ([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

= sum119895isin119862119896119897

([1199081119895 1199082119895 1199083119895 1199084119895] [120583119871

119895 120583119880

119895] [120592119871

119895 120592119880

119895])

(40)




119862119898times119898

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213


11986221

minus 11986223



1198621198981

1198621198982

1198621198983

sdot sdot sdot 119862119898119898

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(41)



119896is




119863119868119896119897

=max119895isin119863119896119897

10038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871

119896119895 120583119880

119896119895] [120592119871

119896119895 120592119880

119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

10038161003816100381610038161003816

max119895isin119869

100381610038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871119897119895 120583119880119897119895] [120592119871119897119895 120592119880119897119895])

100381610038161003816100381610038161003816 (42)



119896119897(119896 119897 =


which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898






1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)



119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)




1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897


119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)



119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)


119896119897= 1 then

119860119896is preferred to 119860





119905119896119897


119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example




3) and


namelyDM1 DM2 andDM




1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)




11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601




1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








(119909) for the


119881120583() = int

119878119909()

0

119871(120572) + 119877

(120572)

2119891 (120572) 119889120572

119881120592() = int

1

119867119883()

119871(120573) + 119877

(120573)

2119892 (120573) 119889120573

(13)


119909()] with 119891(0) = 0 and

int119878119909()

0119891(120572)119889120572 = 119878



119892(120573) = 1 and int1

119867119883()

119892(120573)119889120573 = 1 minus 119867119909()

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1 minus 120573)(1 minus 119867

119909())

Consider

119881120583() = int

119878119909()

0

1

2[119886 +

120572 (119887 minus 119886)

119878119909()

+ 119889 +120572 (119888 minus 119889)

119878119909()

]

sdot2120572

119878119909()

119889120572 =1

119878119909()

[int119878119909()

0

(119886 + 119889) 120572 119889120572]

+1

119878119909

2 ()[int119878119909()

0

(119887 + 119888 minus 119886 minus 119889) 1205722119889120572]

=119886 + 2119887 + 2119888 + 119889

6119878119909()

119881120592() = int

1

119867119909()

1

2[(1 minus 120573) 119887 + (120573 minus 119867

119909()) 119886

1 minus 119867119909()

+(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)

+ 120573 (119886 + 119889) minus 119867119909() (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 + 119887)


minus 120573) 119889120573 =1

(1 minus 119867119909())2

sdot int1

119867119909()


+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=1

(1 minus 119867119909())2int1

119867119909()


+ (1 minus 119867119909()) (119886 + 119889)] (1 minus 120573) 119889120573

=119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())

(14)




119860120583() = int

119878119909()

0

(119877(120572) minus 119871

(120572)) 119891 (120572) 119889120572

119860120592() = int

1

119867119883()

(119877(120573) minus 119871

(120573)) 119892 (120573) 119889120573

(15)

For 119891(120572) = 2120572119878119909() 119892(120573) = 2(1minus120573)(1minus119867

119909()) Consider

119860120583() = int

119878119909()

0

[(119889 +120572 (119888 minus 119889)

119878119909()

)

minus (119886 +120572 (119887 minus 119886)

119878119909()

)]2120572

119878119909()

119889120572

=2

119878119909()

[int119878119909()

0

(119889 minus 119886) 120572 119889120572]

+2

119878119909

2 ()[int119878119909()

0

(119888 minus 119889 + 119886 minus 119887) 1205722119889120572]


3119878119909()

119860120592() = int

1

119867119909()

[(1 minus 120573) 119888 + (120573 minus 119867

119909() 119889)

1 minus 119867119909()


119909()) 119886

1 minus 119867119909()

]2 (1 minus 120573)

1 minus 119867119909()

119889120573


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


minus 120573) 119889120573 =2

(1 minus 119867119909())2

sdot int1

119867119909()




3(1 minus 119867

119909())

(16)


119871

120583119880

] [120592119871

120592119880

]) is

defined as

119881() = 119896119881120583() + (1 minus 119896)119881120592 ()

= 119896 [119886 + 2119887 + 2119888 + 119889

6119878119909()] + (1 minus 119896)

sdot [119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())]

= [119886 + 2 (119887 + 119888) + 119889

6]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(17)


For 119896 = 05

119881() =119886 + 2 (119887 + 119888) + 119889

12(1 + 119878

119909() minus 119867

119909()) (18)

If [120583119871 120583119880

] = [1 1] and [120592

119871

120592119880



119881() =119886 + 2119887 + 2119888 + 119889

12 (19)


([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880

]) is defined as

119860() = 119896119860120583() + (1 minus 119896)119860120592 ()


3119878119909()] + (1 minus 119896)


3(1 minus 119867

119909())]


3]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(20)


For 119896 = 05


6(1 + 119878

119909() minus 119867

119909()) (21)



1= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

])

2= ([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(22)

(1) if 119881(1) lt 119881(

2) then

1lt 2

(2) if 119881(1) gt 119881(

2) then

1gt 2

(3) if 119881(1) = 119881(


1) and 119860(

2)

(i) if 119860(1) lt 119860(

2) then

1lt 2

(ii) if 119860(1) gt 119860(

2) then

1gt 2

(iii) if 119860(1) = 119860(

2) then

1= 2




= ([02 03 04 05] [04 06] [02 03])

= ([04 05 06 07] [03 05] [02 03])

(23)



119868 (119878119909()) = 00875

119868 (119878119909()) = 00825

(24)


119881() = 004375

119881 () = 004125

(25)

and therefore gt

Example 15 Consider

= ([03 04 05 06] [1 1] [0 0])

= ([02 03 06 07] [1 1] [0 0])

= ([01 04 05 08] [1 1] [0 0])

(26)


119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

997904rArr = =

(27)


119881() = 0225

119881 () = 0225

119881 () = 0225

119860 () = 00833

119860 () = 01833

119860 () = 015

997904rArr gt gt

(28)

Example 16 Consider

= ([05 06 07 075] [1 1] [0 0])

= ([045 065 07 075] [1 1] [0 0])

(29)

Then

119868 (119878119909()) = 06375

119868 (119878119909()) = 06375

119868 (119867119909()) = 06375

119868 (119867119909()) = 06375

997904rArr =

(30)

By proposed ranking

119881() = 03208

119881 () = 0325

(31)





Let 1198601 1198602 1198603 119860


let 1198621 1198622 1198623 119862




119894119895= ([119886119894119895 119887119894119895 119888119894119895 119889119894119895] [120583minus

119894119895

120583+

119894119895

] [120592minus

119894119895

120592+

119894119895

]) (32)

Let = [119896

119895]119896times119899


119896

119895= ([119908

119896

1119895 119908119896

2119895 119908119896

3119895 119908119896

4119895] [120583119871

119896

119895

120583119880

119896

119895

] [120592119871

119896

119895

120592119880

119896

119895

]) (33)




119895=

1

119896[1

119895oplus 2


119896

119895] (34)

here 119896



119894119895=

1

119896[1

119894119895oplus 2


119896

119894119895] (35)

where 119896




119894119895]119898times119899


119894119895=

119894119895

radicsum119898

119894=12

119894119895

(36)


119877 = [119894119895]119898times119899

(37)

where 119894119895

= 119894119895




119896and 119860

119897 where 119896 119897 =


is



119862119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

ge ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

(38)


set 119862119896119897 is given by

119863119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

lt ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

= 119869

minus 119862119896119897

(39)



119860119896with respect to 119860




by

119862119896119897

= ([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

= sum119895isin119862119896119897

([1199081119895 1199082119895 1199083119895 1199084119895] [120583119871

119895 120583119880

119895] [120592119871

119895 120592119880

119895])

(40)




119862119898times119898

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213


11986221

minus 11986223



1198621198981

1198621198982

1198621198983

sdot sdot sdot 119862119898119898

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(41)



119896is




119863119868119896119897

=max119895isin119863119896119897

10038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871

119896119895 120583119880

119896119895] [120592119871

119896119895 120592119880

119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

10038161003816100381610038161003816

max119895isin119869

100381610038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871119897119895 120583119880119897119895] [120592119871119897119895 120592119880119897119895])

100381610038161003816100381610038161003816 (42)



119896119897(119896 119897 =


which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898






1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)



119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)




1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897


119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)



119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)


119896119897= 1 then

119860119896is preferred to 119860





119905119896119897


119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example




3) and


namelyDM1 DM2 andDM




1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)




11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601




1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


=2

(1 minus 119867119909())2int1

119867119909()

[(1 minus 120573) (119888 minus 119887)


minus 120573) 119889120573 =2

(1 minus 119867119909())2

sdot int1

119867119909()




3(1 minus 119867

119909())

(16)


119871

120583119880

] [120592119871

120592119880

]) is

defined as

119881() = 119896119881120583() + (1 minus 119896)119881120592 ()

= 119896 [119886 + 2119887 + 2119888 + 119889

6119878119909()] + (1 minus 119896)

sdot [119886 + 2119887 + 2119888 + 119889

6(1 minus 119867

119909())]

= [119886 + 2 (119887 + 119888) + 119889

6]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(17)


For 119896 = 05

119881() =119886 + 2 (119887 + 119888) + 119889

12(1 + 119878

119909() minus 119867

119909()) (18)

If [120583119871 120583119880

] = [1 1] and [120592

119871

120592119880



119881() =119886 + 2119887 + 2119888 + 119889

12 (19)


([119886 119887 119888 119889] [120583119871

120583119880

] [120592119871

120592119880

]) is defined as

119860() = 119896119860120583() + (1 minus 119896)119860120592 ()


3119878119909()] + (1 minus 119896)


3(1 minus 119867

119909())]


3]

sdot (119896119878119909() + (1 minus 119896) (1 minus 119867

119909()))

(20)


For 119896 = 05


6(1 + 119878

119909() minus 119867

119909()) (21)



1= ([1198861 1198871 1198881 1198891] [120583119871

1

120583119880

1

] [120592119871

1

120592119880

1

])

2= ([1198862 1198872 1198882 1198892] [120583119871

2

120583119880

2

] [120592119871

2

120592119880

2

])

(22)

(1) if 119881(1) lt 119881(

2) then

1lt 2

(2) if 119881(1) gt 119881(

2) then

1gt 2

(3) if 119881(1) = 119881(


1) and 119860(

2)

(i) if 119860(1) lt 119860(

2) then

1lt 2

(ii) if 119860(1) gt 119860(

2) then

1gt 2

(iii) if 119860(1) = 119860(

2) then

1= 2




= ([02 03 04 05] [04 06] [02 03])

= ([04 05 06 07] [03 05] [02 03])

(23)



119868 (119878119909()) = 00875

119868 (119878119909()) = 00825

(24)


119881() = 004375

119881 () = 004125

(25)

and therefore gt

Example 15 Consider

= ([03 04 05 06] [1 1] [0 0])

= ([02 03 06 07] [1 1] [0 0])

= ([01 04 05 08] [1 1] [0 0])

(26)


119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

997904rArr = =

(27)


119881() = 0225

119881 () = 0225

119881 () = 0225

119860 () = 00833

119860 () = 01833

119860 () = 015

997904rArr gt gt

(28)

Example 16 Consider

= ([05 06 07 075] [1 1] [0 0])

= ([045 065 07 075] [1 1] [0 0])

(29)

Then

119868 (119878119909()) = 06375

119868 (119878119909()) = 06375

119868 (119867119909()) = 06375

119868 (119867119909()) = 06375

997904rArr =

(30)

By proposed ranking

119881() = 03208

119881 () = 0325

(31)





Let 1198601 1198602 1198603 119860


let 1198621 1198622 1198623 119862




119894119895= ([119886119894119895 119887119894119895 119888119894119895 119889119894119895] [120583minus

119894119895

120583+

119894119895

] [120592minus

119894119895

120592+

119894119895

]) (32)

Let = [119896

119895]119896times119899


119896

119895= ([119908

119896

1119895 119908119896

2119895 119908119896

3119895 119908119896

4119895] [120583119871

119896

119895

120583119880

119896

119895

] [120592119871

119896

119895

120592119880

119896

119895

]) (33)




119895=

1

119896[1

119895oplus 2


119896

119895] (34)

here 119896



119894119895=

1

119896[1

119894119895oplus 2


119896

119894119895] (35)

where 119896




119894119895]119898times119899


119894119895=

119894119895

radicsum119898

119894=12

119894119895

(36)


119877 = [119894119895]119898times119899

(37)

where 119894119895

= 119894119895




119896and 119860

119897 where 119896 119897 =


is



119862119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

ge ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

(38)


set 119862119896119897 is given by

119863119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

lt ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

= 119869

minus 119862119896119897

(39)



119860119896with respect to 119860




by

119862119896119897

= ([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

= sum119895isin119862119896119897

([1199081119895 1199082119895 1199083119895 1199084119895] [120583119871

119895 120583119880

119895] [120592119871

119895 120592119880

119895])

(40)




119862119898times119898

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213


11986221

minus 11986223



1198621198981

1198621198982

1198621198983

sdot sdot sdot 119862119898119898

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(41)



119896is




119863119868119896119897

=max119895isin119863119896119897

10038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871

119896119895 120583119880

119896119895] [120592119871

119896119895 120592119880

119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

10038161003816100381610038161003816

max119895isin119869

100381610038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871119897119895 120583119880119897119895] [120592119871119897119895 120592119880119897119895])

100381610038161003816100381610038161003816 (42)



119896119897(119896 119897 =


which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898






1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)



119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)




1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897


119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)



119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)


119896119897= 1 then

119860119896is preferred to 119860





119905119896119897


119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example




3) and


namelyDM1 DM2 andDM




1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)




11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601




1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



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119868 (119878119909()) = 00875

119868 (119878119909()) = 00825

(24)


119881() = 004375

119881 () = 004125

(25)

and therefore gt

Example 15 Consider

= ([03 04 05 06] [1 1] [0 0])

= ([02 03 06 07] [1 1] [0 0])

= ([01 04 05 08] [1 1] [0 0])

(26)


119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119878119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

119868 (119867119909()) = 045

997904rArr = =

(27)


119881() = 0225

119881 () = 0225

119881 () = 0225

119860 () = 00833

119860 () = 01833

119860 () = 015

997904rArr gt gt

(28)

Example 16 Consider

= ([05 06 07 075] [1 1] [0 0])

= ([045 065 07 075] [1 1] [0 0])

(29)

Then

119868 (119878119909()) = 06375

119868 (119878119909()) = 06375

119868 (119867119909()) = 06375

119868 (119867119909()) = 06375

997904rArr =

(30)

By proposed ranking

119881() = 03208

119881 () = 0325

(31)





Let 1198601 1198602 1198603 119860


let 1198621 1198622 1198623 119862




119894119895= ([119886119894119895 119887119894119895 119888119894119895 119889119894119895] [120583minus

119894119895

120583+

119894119895

] [120592minus

119894119895

120592+

119894119895

]) (32)

Let = [119896

119895]119896times119899


119896

119895= ([119908

119896

1119895 119908119896

2119895 119908119896

3119895 119908119896

4119895] [120583119871

119896

119895

120583119880

119896

119895

] [120592119871

119896

119895

120592119880

119896

119895

]) (33)




119895=

1

119896[1

119895oplus 2


119896

119895] (34)

here 119896



119894119895=

1

119896[1

119894119895oplus 2


119896

119894119895] (35)

where 119896




119894119895]119898times119899


119894119895=

119894119895

radicsum119898

119894=12

119894119895

(36)


119877 = [119894119895]119898times119899

(37)

where 119894119895

= 119894119895




119896and 119860

119897 where 119896 119897 =


is



119862119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

ge ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

(38)


set 119862119896119897 is given by

119863119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

lt ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

= 119869

minus 119862119896119897

(39)



119860119896with respect to 119860




by

119862119896119897

= ([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

= sum119895isin119862119896119897

([1199081119895 1199082119895 1199083119895 1199084119895] [120583119871

119895 120583119880

119895] [120592119871

119895 120592119880

119895])

(40)




119862119898times119898

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213


11986221

minus 11986223



1198621198981

1198621198982

1198621198983

sdot sdot sdot 119862119898119898

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(41)



119896is




119863119868119896119897

=max119895isin119863119896119897

10038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871

119896119895 120583119880

119896119895] [120592119871

119896119895 120592119880

119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

10038161003816100381610038161003816

max119895isin119869

100381610038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871119897119895 120583119880119897119895] [120592119871119897119895 120592119880119897119895])

100381610038161003816100381610038161003816 (42)



119896119897(119896 119897 =


which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898






1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)



119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)




1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897


119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)



119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)


119896119897= 1 then

119860119896is preferred to 119860





119905119896119897


119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example




3) and


namelyDM1 DM2 andDM




1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)




11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601




1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119895=

1

119896[1

119895oplus 2


119896

119895] (34)

here 119896



119894119895=

1

119896[1

119894119895oplus 2


119896

119894119895] (35)

where 119896




119894119895]119898times119899


119894119895=

119894119895

radicsum119898

119894=12

119894119895

(36)


119877 = [119894119895]119898times119899

(37)

where 119894119895

= 119894119895




119896and 119860

119897 where 119896 119897 =


is



119862119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

ge ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

(38)


set 119862119896119897 is given by

119863119896119897

=

119895

([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895])

lt ([1199031119897119895

1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

= 119869

minus 119862119896119897

(39)



119860119896with respect to 119860




by

119862119896119897

= ([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

= sum119895isin119862119896119897

([1199081119895 1199082119895 1199083119895 1199084119895] [120583119871

119895 120583119880

119895] [120592119871

119895 120592119880

119895])

(40)




119862119898times119898

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213


11986221

minus 11986223



1198621198981

1198621198982

1198621198983

sdot sdot sdot 119862119898119898

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(41)



119896is




119863119868119896119897

=max119895isin119863119896119897

10038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871

119896119895 120583119880

119896119895] [120592119871

119896119895 120592119880

119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871

119897119895 120583119880

119897119895] [120592119871

119897119895 120592119880

119897119895])

10038161003816100381610038161003816

max119895isin119869

100381610038161003816100381610038161003816119881 ([1199031119896119895

1199032119896119895

1199033119896119895

1199034119896119895

] [120583119871119896119895 120583119880119896119895] [120592119871119896119895 120592119880119896119895]) minus ([119903

1119897119895 1199032119897119895

1199033119897119895

1199034119897119895

] [120583119871119897119895 120583119880119897119895] [120592119871119897119895 120592119880119897119895])

100381610038161003816100381610038161003816 (42)



119896119897(119896 119897 =


which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898






1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)



119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)




1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897


119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)



119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)


119896119897= 1 then

119860119896is preferred to 119860





119905119896119897


119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example




3) and


namelyDM1 DM2 andDM




1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)




11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601




1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119896119897(119896 119897 =


which is given by

119863119898times119898

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986311986812

11986311986813

sdot sdot sdot 1198631198681119898

11986311986821

minus 11986311986823

sdot sdot sdot 1198631198682119898


1198631198681198981

1198631198681198982

1198631198681198983

119863119868119898times119898minus1

minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(43)

with 0 le 119863119868119896119897

le 1 for 119896 119897 = 1 2 3 119898






1198621015840

= ([1198621198681015840

1119896119897 1198621198681015840

2119896119897 1198621198681015840

3119896119897 1198621198681015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

([1198621198681119896119897

1198621198682119896119897

1198621198683119896119897

1198621198684119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

119898 (119898 minus 1)(44)



119891119896119897

=

1 if 119862 ge 1198621015840

0 if 119862 lt 1198621015840

(45)




1198631198681015840=

119898

sum119896=1

119896 =119897

119898

sum119897=1

119897 =119896

119873119868119896119897

119898(119898 minus 1) (46)

The elements 119892119896119897


119892119896119897

= 1 if 119863119868119896119897

le 1198631198681015840

119892119896119897

= 0 if 119863119868119896119897

gt 1198631198681015840

(47)



119896119897of 119879 are defined as

119905119896119897

= 119891119896119897

sdot 119892119896119897 (48)


119896119897= 1 then

119860119896is preferred to 119860





119905119896119897


119905119896119897

= 0 forall119894 for 119894 = 1 2 119898 119894 = 119896 119894 = 119897(49)

5 Numerical Example




3) and


namelyDM1 DM2 andDM




1198821198881119888211988831198884

= ([03 04 05 06] [03 05] [01 02])

([03 04 05 06] [04 05] [03 04])

([02 04 05 06] [04 06] [02 04])

([04 05 07 08] [03 04] [02 04])

(50)




11986212

= 1 2

11986213

= 1 2 3 4

11986221

= 3 4

11986223

= 1 2 3 4

11986231

= 120601




1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1198602

1198603

1198621

([02 03 04 05] [03 06] [01 03]) ([01 03 04 05] [05 06] [02 04]) ([03 05 06 07] [03 04] [02 03])1198622

([04 06 07 08] [04 07] [01 02]) ([02 04 05 08] [04 05] [03 04]) ([01 03 04 05] [04 06] [01 02])1198623

([03 05 06 08] [03 04] [02 03]) ([04 06 08 10] [04 06] [02 04]) ([02 03 04 05] [04 05] [03 04])1198624

([01 03 04 06] [06 07] [02 03]) ([03 05 06 07] [03 05] [01 04]) ([01 02 03 05] [02 05] [01 04])



1198602

1198603

1198621

([06 07 08 09] [06 07] [01 02]) ([05 06 07 08] [05 06] [03 04]) ([07 08 09 10] [05 08] [01 02])1198622

([01 02 04 05] [03 05] [02 04]) ([04 05 06 08] [04 06] [02 03]) ([03 04 05 06] [05 06] [02 04])1198623

([03 04 05 06] [05 06] [03 04]) ([01 02 03 05] [04 06] [02 04]) ([02 03 04 05] [07 08] [01 02])1198624

([01 03 04 05] [05 07] [02 03]) ([03 05 06 08] [03 04] [01 02]) ([02 03 04 05] [04 07] [02 03])

11986232

= 120601

11986312

= 3 4

11986313

= 120601

11986321

= 1 2

11986323

= 120601

11986331

= 1 2 3 4

11986332

= 1 2 3 4

(51)


For instance

11986212

= ([11990811 11990821 11990831 11990841] [120583119871

1 120583119877

1] [120592119871

1 120592119877

1])

oplus ([11990812 11990822 11990832 11990842] [120583119871

2 120583119877

2] [120592119871

2 120592119877

2])

= ([03 04 05 06] [03 05] [01 02])

oplus ([03 04 05 06] [04 05] [03 04])

= ([06 08 10 12] [03 05] [03 04])

(52)

Similarly we get

11986213

= ([12 17 22 26] [03 04] [03 04])

11986221

= ([06 09 12 14] [03 04] [02 04])

11986223

= ([12 17 22 26] [03 04] [03 04])

11986231

= 120601

11986232

= 120601

(53)


1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(54)



11986311986812

= 1

11986311986813

= 0

11986311986821

= 0

11986311986823

= 0

11986311986831

= 1

11986311986832

= 1

(55)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 0

0 minus 0

1 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([36 51 66 78] [03 04] [03 04])

6

= ([06 085 11 13] [03 04] [03 04])

(57)




1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([05 06 07 09] [03 06] [01 02]) ([03 04 05 06] [05 06] [01 04])1198622

([05 06 07 09] [02 04] [01 02]) ([04 05 07 08] [04 05] [03 04]) ([02 03 04 05] [04 05] [02 05])1198623

([04 05 06 07] [07 08] [01 02]) ([02 03 04 05] [06 08] [01 02]) ([02 04 05 06] [05 06] [02 04])1198624

([05 06 07 08] [05 06] [02 04]) ([06 07 08 09] [03 04] [01 02]) ([07 08 09 10] [04 05] [03 05])



1198602

1198603

1198621

([04 05 06 07] [04 05] [03 04]) ([036 05 06 073] [03 06] [03 04]) ([043 056 066 076] [03 04] [05 06])1198622

([033 046 06 073] [02 04] [02 04]) ([033 046 06 08] [04 05] [03 04]) ([02 033 043 053] [04 05] [02 05])1198623

([033 046 056 07] [03 04] [03 04]) ([023 036 05 066] [04 06] [02 04]) ([02 033 043 053] [04 05] [03 04])1198624

([023 036 05 063] [05 06] [02 04]) ([04 056 066 08] [03 04] [01 04]) ([033 043 053 066] [02 05] [03 05])


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(58)



1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (59)


119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(60)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 0 1

1 minus 1

0 0 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(61)


1198602gt 1198601gt 1198603 (62)




2) (3)


4) The crisp



1198821198881119888211988831198884

= ([01 02 03 04] [03 04] [02 03])

([005 01 015 02] [03 05] [02 04])

([01 03 04 05] [04 06] [03 04])

([03 04 05 06] [04 05] [03 04])

(63)



11986212

= 1 2 3 4

11986213

= 1 2 3 4

11986221

= 120601

11986223

= 2

11986231

= 120601

11986232

= 1 3 4

11986312

= 120601

11986313

= 120601

11986321

= 1 2 3 4

11986323

= 1 3 4

11986331

= 1 2 3 4

11986332

= 2

(64)


Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table5Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([035044053062][0405][0304])

([031044053064][0306][0304])

([034045053061][0304][0506])

1198622

([029041054066][0204][0204])

([028041052069][0405][0304])

([025042054067][0405][0205])

1198623

([031043052066][0304][0304])

([024038053070][0406][0204])

([025042054067][0405][0304])

1198624

([025039

054069][0506][0204])

([032044053064][0304][0104])

([032042052065][0205][0305])


Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table6Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([010017026037][0305][0304])

([009017026038][0305][0304])

([010018

026036][0304][0506])

1198622

([008016

027039][0204][0304])

([008016

026041][0405][0304])

([007016

027040

][0405][0305])

1198623

([006017026039][0304][0304])

([004015

026042][0406][0204])

([005016

027040

][0405][0304])

1198624

([01019037

055][0304][0204])

([012022037051][0304][0204])

([012021036052

][0204][0305])




1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1198602

1198603

1198621

([01 02 03 04] [05 06] [02 03]) ([04 05 06 07] [03 05] [02 04]) ([02 04 05 08] [05 07] [01 02])1198622

([02 03 04 05] [04 07] [02 03]) ([01 03 05 06] [02 04] [04 05]) ([02 03 04 05] [01 04] [05 06])1198623

([03 04 05 06] [03 06] [03 04]) ([02 04 06 07] [04 07] [02 03]) ([01 02 04 05] [06 08] [01 02])1198624

([01 03 05 06] [04 05] [02 04]) ([03 05 06 08] [00 03] [05 07]) ([01 02 04 06] [02 04] [03 06])



1198623times3

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 11986212

11986213

11986221

minus 11986223

11986231

11986232

minus

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(65)


1198633times3

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 120601 120601

1 minus 1

1 00292 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1198621015840

= ([1198621015840

1119896119897 1198621015840

2119896119897 1198621015840

3119896119897 1198621015840

4119896119897] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

=

119898

sum119896=1

119898

sum119897=1

([1198621119896119897

1198622119896119897

1198623119896119897

1198624119896119897

] [120583119871

119896119897 120583119880

119896119897] [120592119871

119896119897 120592119880

119896119897])

3 (3 minus 1)

=([165 3 405 51] [03 04] [03 04])

6

= ([0275 05 0675 085] [03 04] [03 04])

(67)


1198631198681015840=

3

sum119896=1

3

sum119897=1

119863119868119896119897

3 (3 minus 1) (68)


119865 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119866 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)


119879 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus 1 1

0 minus 0

0 1 minus

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(70)


1198601gt 1198603gt 1198602 (71)




4which


6 Conclusion



Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table8Normalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018036054073][0506][0203])

([035044053062][0305][0204])

([019038047076

][0507][01

02])

1198622

([027040054068][0407][0203])

([011035059071][0204][0405])

([027040054068][0104][0506])

1198623

([032043053064][0306][0304])

([019039

058068][0407][0203])

([014029058073][0608][0102])

1198624

([011035059071][0405][0204])

([025043051069][0003][0507])

([013026052079][0204][0306])


Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table9Weightedno

rmalized

decisio

nmatrix

Criteria

Alternatives

1198601

1198602

1198603

1198621

([018007016029][0304][0203])

([003008016024][0304][0204])

([001007014030][0304][0203])

1198622

([013004008013

][0305][0204])

([0005003008014][0204][0405])

([001004008013

][01

04][0506])

1198623

([003012

021032][0306][0304])

([001011019

038][0406][0304])

([001008023036][0406][0304])

1198624

([003014

029042][0405][0304])

([007017025041][0003][0507])

([003010

026047][0204][0306])


Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Competing Interests


References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in













Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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