Research Article An Extended TODIM Method for Group ...

Research ArticleAn Extended TODIM Method for Group Decision Making withthe Interval Intuitionistic Fuzzy Sets

Yanwei Li Yuqing Shan and Peide Liu

College of Economics and Management Civil Aviation University of China Tianjin 300300 China

Correspondence should be addressed to Yuqing Shan shanyuqing27163com

Received 3 December 2014 Revised 6 March 2015 Accepted 24 March 2015

Academic Editor Elena Benvenuti

Copyright copy 2015 Yanwei Li et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For amultiple-attribute groupdecision-making problemwith interval intuitionistic fuzzy sets amethod based on extendedTODIMis proposed First the concepts of interval intuitionistic fuzzy set and its algorithms are defined and then the entropy method todetermine the weights is put forwardThen based on theHamming distance and the Euclidean distance of the interval intuitionisticfuzzy set both of which have been defined function mapping is given for the attribute Finally to solve multiple-attribute groupdecision-making problems using interval intuitionistic fuzzy sets a method based on extended TODIM is put forward and a casethat deals with the site selection of airport terminals is given to prove the method

1 Introduction

Zadeh [1] put forward the concept of fuzzy sets in 1965shortly afterward the theory of fuzzy sets gradually devel-oped In 1986 Atanassov [2] proposed the theory of intuition-istic fuzzy set (IFS) However the common fuzzy set canbe seen as the special form of IFS In real life an accuratedefinition for the specific membership and the specific non-membership degree of IFS is relatively difficult [3] In 1989Atanassov and Gargov [4] extended the IFS into the intervalintuitionistic fuzzy set (IIFS) In the following years the prop-erties of the IIFSs were further expanded the changes mainlyincluded related algorithms the correlation and decom-position theorem topological properties correlation coeffi-cient and the relationship between other fuzzy sets [5ndash9]Xu [10] also put forward several kinds of weighted averagingoperators for the IIFSs in 2007 Also there are some applica-tions and extensions for these operators [11ndash13]

Multiple-attribute decision making (MADM) belongs tomulticriteria decision making (MCDM) which has charac-teristics of discrete types and limited alternativesThe processof decisionmaking is to gather the opinions of all the decisionmakers for several alternatives MADM means that there isnot only one attribute and how to integrate the attributesof various alternatives is very important Therefore many

researchers have devoted themselves to the study of MADMand rich achievements have been obtained Specific methodshave been used to solve the problems of multiple-attributedecision making such as the method of choice [14 15] theordering method of compromise [16] the method of greycorrelation analysis [17 18] and the TOPSIS method [19 20]There are also many other methods [21ndash25] and multiple-attribute group decision making has a wide array of theoryand practice basis [26 27]

Park et al [28] pointed out that problems in which theattribute weights were unknown and interval intuitionisticfuzzy decision was being applied were difficult At presentgeneral methods used to solve the problem of multiple-attribute decision making with IFS are based on the determi-nation attribute weights [29 30] In the recent years underthe condition in which attribute weights are determinedstudies have made great progress solving the problem ofmultiattribute group decision making using interval numberinformation For example Bryson and Mobolurin [31] pro-posed a method of linear programming which was based onthe deviation degree Xu [32] presented themethod of relativemembership degree Su et al [33] put forward an extendedVIKOR method for dynamic multiattribute decision makingusing interval numbers And Wang et al did some researchthat included the condition of uncertainty [34 35]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 672140 9 pageshttpdxdoiorg1011552015672140

2 Mathematical Problems in Engineering

However the methods used to solve the problems thatconcerned MADM with IIFS have been extended Somemethods involving MADM with IIFS can be understood asfollows Hu and Xu [36] proposed the TOPSIS method Qiet al [37] gave an approach to include automatic conver-gence A dynamicmultiple-attribute grey incidence decision-making method was put forward by Liu et al [38] Chen [39]developed a qualities-based method for handling problemsand he applied a confidence interval to the membershipdegree and the nonmembership degree

Gomes and Lima [40 41] proposed the TODIM (anacronym in Portuguese of interactive and multicriteria deci-sion making) method which was based on prospect theory[42] The TODIM method is used to establish the expectedfunction whichmeasures the degree to which one alternativeis superior to others by calculating accurate number infor-mation The TODIM method was successfully applied to theMADMfor the first time byGomes andRangel [43] and it hasproved to be an effective method for dealing with MADMKrohling and De Souza [44] applied the TODIM methodto the attribute values which are specifically expressed asthe IIFS In addition the TODIM method has provided agreat help in solving real problems TODIM methods wereexpanded by Fan et al [45] In the future TODIM methodscan deal with three types of information including thereal number the interval number and the triangular fuzzynumber [46]

The traditional TODIM method mainly handles realnumbers where the weights are known and only one decisionmaker appears in the MADM problems The IIFS is used toexpress the opinions of every expert for every alternativebecause its description is more accurate than other math-ematical linguistics In this paper multiple-attribute groupdecision making is studied by using the extended TODIMmethod which is represented by IIFSs The TODIM methodaims to optimize the ranking of alternatives In additionthe entropy method for the determination of weights is putforward We can derive the expert weights according to thescore function and the attribute weights can be derived bythe idea of maximum entropy

The structure of this paper is arranged as follows InSection 2 the definition and algorithms of the IIFS areproposed In Section 3 the group decision-makingmatrix forthe IIFSs is described In Section 4 the method to determinethe weights is given in detail In Section 5 the extendedTODIMmethod for interval intuitionistic fuzzy informationis put forward and the steps of how to solve the problem ofmultiple-attribute group decision making are listed one byone In the end a case concerning the site selection of airportterminals is shown to demonstrate the proposed method andits results

2 Definitions of IIFSs

Because the concept of IIFS is derived from intuitionisticfuzzy sets we first define IFS On the basis of that IIFS willbe defined The basic algorithms of the IIFS are given as wellas a description of its operators

Definition 1 (intuitionistic fuzzy set (IFS) [2]) Let 119860 be anIFS it can be represented as follows

119860 = ⟨119909 119906 (119909) V (119909) | 119909 isin 119877⟩ (1)

In this definition 119906(119909) represents the membership of theelement 119909 in the IFS and ](119909) represents the nonmembershipdegree of the element 119909 in the IFS Let 120587(119909) represent thehesitancy degree of 119860 that is 120587(119909) = 1 minus 119906(119909) minus V(119909)

This expression should satisfy the following conditions(1) 119877 is a nonempty set(2) 0 le 119906(119909) le 1 0 le V(119909) le 1 and 0 le 120587(119909) le 1

Definition 2 (interval intuitionistic fuzzy set (IIFS) [4]) Let be an IIFS it can be represented as follows

= ⟨[119906120572

(119909) 119906120573

(119909)] [V120572 (119909) V120573 (119909)] | 119909 isin 119877⟩ (2)

In this definition 119906120572(119909) represents the lower limit of themembership degree of the element 119909 119906120573(119909) represents theupper limit of the membership degree of the element 119909 V120572(119909)represents the lower limit of the nonmembership degree ofthe element 119909 and V120573(119909) represents the upper limit of thenonmembership degree of the element119909 In addition let (119909)be the hesitancy degree of that is (119909) = (120583(119909) cup ](119909))119888This is equal to [1 minus 119906120573(119909) minus V120573(119909) 1 minus 119906120572(119909) minus V120572(119909)]

This expression should satisfy the following conditions(1) 119877 is a nonempty set(2) 119906120572(119909) isin [0 1] 119906120573(119909) isin [0 1] and V120572(119909) isin [0 1]

V120573(119909) isin [0 1](3) [119906120572(119909) 119906120573(119909)] sub [0 1] [V120572(119909) V120573(119909)] sub [0 1] and0 le 119906120572

(119909) + V120572(119909) le 1

Definition 3 (the basic algorithms of IIFS [4]) Given the fol-lowing assumptions let ℎ

1and ℎ2be two interval intuitionis-

tic fuzzy numbers

ℎ1= ([119906

120572

1(119909) 119906

120573

1(119909)] [V120572

1(119909) V120573

1(119909)])

ℎ2= ([119906

120572

2(119909) 119906

120573

2(119909)] [V120572

2(119909) V120573

2(119909)])

(3)

Then we have these basic algorithms

ℎ1+ ℎ2= ([119906

120572

1(119909) + 119906

120572

2(119909) minus 119906

120572

1(119909) lowast 119906

120572

2(119909)

119906120573

1(119909) + 119906

120573

2(119909) minus 119906

120573

1(119909) lowast 119906

120573

2(119909)]

[V1205721(119909) lowast V120572

2(119909) V120573

1(119909) lowast V120573

2(119909)])

ℎ1lowast ℎ2= ([119906

120572

1(119909) lowast 119906

120572

2(119909) 119906

120573

1(119909) lowast 119906

120573

2(119909)]

[V1205721(119909) + V120572

2(119909) minus V120572

1(119909) lowast V120572

2(119909)

V1205731(119909) + V120573

2(119909) minus V120573

1(119909) lowast V120573

2(119909)])

120582ℎ1= ([1 minus (1 minus 119906

120572

1(119909))120582

1 minus (1 minus 119906120573

1(119909))120582

]

[V1205721(119909)120582

V1205731(119909)120582

])

(4)

Mathematical Problems in Engineering 3

Definition 4 (IIFWA [10]) The interval intuitionistic fuzzyweighted averaging (IIFWA) operator is a kind of aggre-gating operator which evolved from the intuitionistic fuzzyweighted averaging (IFWA) operator This operator has beenproven and its specific form is

IIFWA119908(ℎ1 ℎ2 ℎ

119899)

= ([1 minus

119899

prod

119894=1

(1 minus 119906120572

119894(119909))119908119894 1 minus

119899

prod

119894=1

(1 minus 119906120573

119894(119909))119908119894

]

[

119899

prod

119894=1

V120572119894(119909)119908119894

119899

prod

119894=1

V120573119894(119909)119908119894])

(5)

where 119908119894ge 0 119894 isin 0 1 2 119899 and sum119899

119894=1119908119894= 1 and 119908

119894is the

weight of ℎ119894

Definition 5 (the distances of IIFSs) Alternatives will beevaluated according to the distance of IIFS Based on theHamming distance and the Euclidean distance of IFS we canextend to the normalizedHamming distance and the normal-ized Euclidean distance of IIFS Its specific form is as follows

Definition 51 The normalized Hamming distance of intervalintuitionistic fuzzy is as follows

119889119867

(ℎ1 ℎ2) =

1

4(1003816100381610038161003816119906120572

1(119909) minus 119906

120572

2(119909)1003816100381610038161003816 +100381610038161003816100381610038161003816119906120573

1(119909) minus 119906

120573

2(119909)100381610038161003816100381610038161003816

+1003816100381610038161003816V120572

1(119909) minus V120572

2(119909)1003816100381610038161003816 +100381610038161003816100381610038161003816V1205731(119909) minus V120573

2(119909)100381610038161003816100381610038161003816)

(6)

Definition 52 The normalized Euclidean distance of intervalintuitionistic fuzzy is as follows

119889119864

(ℎ1 ℎ2)

=1

2((119906120572

1(119909) minus 119906

120572

2(119909))2

+ (119906120573

1(119909) minus 119906

120573

2(119909))2

+ (V1205721(119909) minus V120572

1(119909))2

+ (V1205731(119909) minus V120573

2(119909))2

)

12

(7)

Definition 6 (the score function of IIFS [10]) Let ℎ be aninterval intuitionistic fuzzy number ℎ = ([119906

120572

(119909) 119906120573

(119909)]

[V120572(119909) V120573(119909)]) and let 119878(ℎ) be the score function of ℎ

119878 (ℎ) =1

2((119906120572

(119909) + 119906120573

(119909)) minus (V120572 (119909) + V120573 (119909))) (8)

When 119878(ℎ1) lt 119878(ℎ

2) there is ℎ

1lt ℎ2

3 Description of the Group Decision-MakingMatrix with IIFSs Information

For a multiple-attribute group decision-making problem wesuppose that there is an alternative set 119860 its form is 119860 =

(1 2 3 119899) and the attributes of the alternative set can

be expressed as 119862 = (1 2 3 119898) Let 119863 be the decision-maker set that is 119863 = (1 2 3 ℎ) The decision-maker119896 evaluates the attributes 119895 of alternative 119894 by the form ofIIFSThis process of assignment can be expressed as a matrixof 119896119894119895(119899 times 119898) = ([119906

120572

119894119895119896(119909) 119906120573

119894119895119896(119909)] [V120572

119894119895119896(119909) V120573119894119895119896(119909)]) and the

evaluation matrix of all decision makers can be expressed as(1

119894119895 2

119894119895

ℎ

119894119895)119879

Definition 7 (the aggregating of the experts matrix) TheIIFWA operator is adopted to aggregate the expert matrix Inthis case we suppose that the weight of expert 119902

119894is known

After a series of calculations we will derive the aggregationof the experts matrix as (119898 times 119899) the element of (119898 times 119899)is 119903119894119895 which means that experts evaluate every attribute 119895 of

every alternative 119894 That is to say when all of the expertsrsquoevaluations of all of the alternatives are put together theyequal one expertrsquos evaluation of all of the alternatives So weconsider this matrix an overall evaluation matrix

119903119894119895= ([1 minus

ℎ

prod

119896=1

(1 minus 119906120572

119894119895119896(119909))119902119896

1 minus

ℎ

prod

119896=1

(1 minus 119906120573

119894119895119896(119909))

119902119896

]

[

ℎ

prod

119896=1

V120572119894119895119896(119909)119902119896

ℎ

prod

119896=1

V120573119894119895119896(119909)119902119896])

(9)

where 119902119896ge 0 119894 isin 0 1 2 ℎ and sumℎ

119896=1119902119896= 1

4 Entropy Weight Method for Group DecisionMaking with IIFS

The concept of entropy is derived from thermodynamics CE Shannon has recently applied it to information theoryThebasic principle is that entropy can be used as a measure of theuseful information that the data providesThe entropy weightmethod has beenwidely used in the decision-making processIn this paper the entropy coefficient method is applied todetermine the weights of experts and attributes in groupdecision making using IIFS

41 Entropy Weight Method for Determination ofExpert Weights

Definition 8 (the entropy of expert) Let 119864119896be the entropy of

expert 119896

119864119896= minus

1

ln 119899

119894=119899

sum

119894=1

(119904119894119896

sum119899

119894=1119904119894119896

lowast ln100381610038161003816100381610038161003816100381610038161003816

119904119894119896

sum119899

119894=1119904119894119896

100381610038161003816100381610038161003816100381610038161003816

) (10)

where 119904119894119896can be calculated by formula (12) and its form is as

follows

119904119894119896=1

2

119898

sum

119895=1

((119906120572

119894119895119896(119909) + 119906

120573

119894119895119896(119909)) minus (V120572

119894119895119896(119909) + V120573

119894119895119896(119909)))

(11)


Formula (11) is based on the score function which is pro-posed by Xu [10] It is used to calculate the overall evaluatingvalues of expert 119896 to the alternative 119894

Definition 9 (the expert weights) Using the properties ofentropy we find that if the degree of disorder in a systemis high the entropy value will correspondingly be larger Ingroup decisionmaking if experts have similar opinions aboutdifferent alternatives the entropy value will be larger That isthe greater the entropy value of expert 119864

119896is the smaller the

differences the results will show Let 119902119896be the entropy weight

of expert 119896 its form will be as follows

119902119896=

1 minus 119864119896

ℎ minus sumℎ

119896=1(119864119896)

(12)

where 0 le 119902119896le 1 and sumℎ

119896=1119902119896= 1

42 Entropy Weight Method for Determination ofAttribute Weights

Definition 10 (the entropy of attributes) Let 119890119895be the entropy

of attribute 119895

119890119895= minus

1

ln 119899

119899

sum

119894=1

(

ℎ119894119895

sum119898

119895=1ℎ119894119895

lowast ln(ℎ119894119895

sum119898

119895=1ℎ119894119895

)) (13)

where ℎ119895can be calculated by formula (16) and its form is as

follows [37]

ℎ119894119895= (1 minus 119889 (([

120572

119894119895(119909)

120573

119894119895(119909)] [V120572

119894119895(119909) V120573

119894119895(119909)])

([05 05] [05 05])))

(14)

Formula (14) has to satisfy the following conditions

(1) the relative importance of evaluation attributes andalternatives is independent

(2)

119889 (119903119894119895 ([05 05] [05 05]))

=1

2((119906120572

119894119895(119909) minus 05)

2

+ (119906120573

119894119895(119909) minus 05)

2

+ (V120572119894119895(119909) minus 05)

2

+ (V120573119894119895(119909) minus 05)

2

)

12

(15)

(3) if there is 119889( ([05 05] [05 05])) ge 119889(120573 ([05 05][05 05])) for any two IIFSs then there will be ℎ() leℎ(120573)

Formula (14) is based on the Euclidean distance of IIFSThe interval intuitionistic fuzzy number ([05 05] [05 05])has no hesitancy degree but its entropy attains the maximumvalue That is to say the positive and negative evidence ofthis number are accounted equally and it is impossible touse fuzzy information to describe or make a reasonablejudgment

Definition 11 (the attribute weight) According to the theoryof entropy when the entropy of attribute 119895 is greater thanthe value of other attributes the value of attribute 119895 betweenevery alternative and the optimal strategy will have a smallerdifference In order to facilitate a comprehensive evaluationthe weight of attribute 119895 can be determined by 119890

119895 Let 119908

119895be

the weight of attribute 119895 it is formed as follows

119908119895=

1 minus 119890119895

119898 minus sum119898

119895=1(119890119895)

(16)

where 0 le 119908119895le 1 and sum119898

119895=1119908119895= 1

5 The Extended TODIM Method for GroupDecision Making with IIFS

Based on the above discussion which included the concept ofIIFSs as well as its basic algorithms and the gatheringmethodof the interval intuitionistic fuzzy matrix a new method ofprocessing multiattribute group decision making is givenwhich is called the extended TODIMmethodThe traditionalTODIM method measures the comprehensive degree of analternative better than others and is extended in this paperto handle IIFS Assuming the conditions that (a) weights ofthe experts and attributes are unknown and (b) there is morethan one decision maker we outline the steps of how to usethe TODIMmethod in the next segment

Step 1 There are 119870 experts to evaluate every attribute of119899 alternatives When evaluating alternatives IIFSs are usedto express the evaluating values After that an assessmentmatrix 119896

119894119895of every 119896th expert will be derived each row

represents an attribute and the columns represent alterna-tives All of the assessment matrices can be expressed as(1

119894119895 2

119894119895

ℎ

119894119895)119879

Step 2 Calculate the expert weights The weight measure 119902119896

can be determined according to formulas (10) through (12)

Step 3 Aggregate all of the assessment matrices 119896119894119895 On the

premise that the obtained weight values of every expert 119902119896

can be expressed by (119894 = 1 2 ℎ) formula (9) is applied toassemble the advice of ℎ experts and the aggregating matrix(119898 times 119899) can be derived

Step 4 Calculate the attribute weights On the basis of Step3 the weight measure 119908

119895can be determined according to

formulas (13) through (16)

Step 5 According to the Hamming distance (Definition 51)and the Euclidean distance (Definition 52) of IIFSs thedistance between every element in the aggregatingmatrix andthe interval intuitionistic fuzzy number ([0 0] [1 1]) can becalculated respectively and matrix 119884 can be obtained afterthe distance is gathered into expression (18) which is shownbelow Let 119910

119894119895be the element of matrix119884 where 119903

119894119895represents

the element of the aggregating matrix (119898 times 119899)

119910119894119895= 119889 (119903

119894119895 ([0 0] [1 1])) (17)


The interval intuitionistic fuzzy number ([0 0] [1 1]) isthe minimum so we can define it as the negative ideal pointWhen the distance between an alternative and this point islarger the result will be more accurate

Step 6 When matrix 119884 that has been derived is broughtinto expression (18) matrix 119868 will be derived The function119868(119860119894 119860119895) is used to represent the degree to which alternative

119894 is better than alternative 119895 and is the sum of the subfunc-tion Φ

119888(119860119894 119860119895) where 119888 applies from 1 to 119898 Subfunction

Φ119888(119860119894 119860119895) indicates the degree to which alternative 119894 is better

than alternative 119895 when a particular attribute 119888 is givenIn expression (18) there is a constant parameter 120587 which

is used to represent the sensitive coefficient of risk aversionWhen the parameter 120587 has different values the values ofsubfunctionΦ

119888(119860119894 119860119895) will change correspondingly

Φ119888(119860119894 119860119895) =

radic119908119888(119910119894119888minus 119910119895119888)

sum119898

119888=1119908119888

119910119894119888gt 119910119895119888

minus1

120587

radic(sum119898

119888=1119908119888) (119910119895119888minus 119910119894119888)

119908119888

119910119894119888lt 119910119895119888

0 119910119894119888= 119910119895119888

(18)

119868 (119860119894 119860119895) =

119898

sum

119888=1

Φ(119860119894 119860119895) 119894 119895 isin 119860 (19)

Step 7 According to expression (20) the overall appraisalvalues of alternatives can be worked out and these valuesshould be sorted If the value 120585(119860

119894) is larger than other values

then alternative 119894will bemore optimal Function 120585(119860119894)will be

obtained by matrix 119868 and 120585(119860119894) represents the degree of pri-

ority bywhich alternative 119894 is preferred over the othersThere-fore these alternatives can be sorted according to the resultsof function 120585(119860

119894) That is to say if the result of function 120585(119860

119894)

is larger this alternative will be more optimal than others

120585 (119860119894)

=

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

max1le119894le119899

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

(20)

Step 8 Rank the values of function 120585(119860119894) and analyze the

results

6 Analysis of Examples The Site Selection ofAirport Terminals

An airfield plans to construct different airport terminalsin surrounding cities After preliminary screening severalcities remain for further analysis and research First a com-prehensive evaluation index system should be establishedstarting from the following three aspects number of potentialcustomers degree of traffic connections and the existingcompetitive capacity

The number of potential customers can refer not only tothe volume of airline passenger transportation and the rateof growth in recent years but also to the local economicconditions such as local GDP industrial patterns and otherenterprises The degree of traffic connections mainly takesother modes of transport into account considers whetherthe others are convenient and fast and have a high coveragerate throughout the city and also considers the conveniencedegree of each traffic connection from the city to thesurrounding areas The existing competitive capacity mainlymeans the competitive pressures coming fromhigh-speed railand airport terminals of other airfields the degree of localgovernment supporting the air transportation industry andthe advantages of relevant policies

Suppose there are five alternative cities 119860119894(119894 = 1 2

3 4 5) Three experts 119863119896(119896 = 1 2 3) in the field of aviation

are invited to evaluate the five cities using IIFSs Accordingto the above evaluation index system about remote airportterminals a similar description of attributes can be made asfollows attribute 119862

1signifies the amount of potential cus-

tomers attribute1198622signifies the degree of traffic connections

and attribute 1198623signifies the existing competitive capacity

For the above three indices IIFSs are adopted to express theevaluated values When assigning the evaluating values weassume that the attribute dimensions have been eliminatedThus this data can be seen as normalized

Specific steps are as follows

Step 1 Determination of basic data the three matrices belowrepresent the three expertsrsquo evaluation of the five cities forevery attribute

1198751

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([060 079] [010 015]) ([040 050] [030 040]) ([050 060] [030 035])

1198602

([070 075] [010 020]) ([060 065] [020 030]) ([070 080] [005 010])

1198603

1198604

1198605

([050 060] [020 025])

([030 035] [020 025])

([070 075] [015 020])

([070 075] [010 020])

([050 060] [010 015])

([030 040] [020 025])

([030 040] [030 040])

([060 070] [020 025])

([065 070] [020 030])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


1198752

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([050 065] [010 020]) ([030 040] [020 030]) ([040 050] [035 045])

1198602

([080 090] [005 010]) ([050 060] [020 030]) ([060 070] [020 030])

1198603

1198604

1198605

([060 070] [020 025])

([050 060] [020 030])

([070 080] [010 020])

([050 060] [020 025])

([070 080] [010 015])

([020 030] [020 025])

([030 040] [020 030])

([070 075] [020 025])

([050 055] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198753

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([040 050] [020 025]) ([050 060] [020 025]) ([030 040] [050 060])

1198602

([060 065] [020 030]) ([030 035] [010 020]) ([060 065] [030 035])

1198603

1198604

1198605

([040 045] [010 015])

([020 030] [040 050])

([090 100] [030 040])

([080 090] [000 010])

([060 065] [020 025])

([020 030] [040 045])

([040 045] [020 030])

([055 015] [040 050])

([060 070] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)

Step 2 Calculating the values of expert weight according toformulas (10) through (12) the results are shown in Table 1and the following

119864119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0981 0965 0883

1003816100381610038161003816100381610038161003816100381610038161003816

(22)

Finally the expert weights can be worked out and theresults are

119902119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0114 0203 0683

1003816100381610038161003816100381610038161003816100381610038161003816

(23)

Step 3 The above three matrices of the five cities can applyDefinition 6 and in this manner they can be integrated intoa matrix The results of the matrix are shown in Table 2

Step 4 Calculate the values of the attribute weights On thebasis of Step 3 the weight measure 119908

119895can be determined

according to formulas (13) through (16) and the results areas shown in Table 3 and the following

119890119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

09978 09974 09989

1003816100381610038161003816100381610038161003816100381610038161003816

(24)

Finally the attribute weights can be worked out and theresults are as follows

119882119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

0371 0435 0194

1003816100381610038161003816100381610038161003816100381610038161003816

(25)

Step 5 Calculate the distance between the IIFSs in matrix 119877with ([0 0] [1 1])

(1) Calculating by the Hamming distance the results areas follows

119884119867

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010656 0631 0456

11986020758 0621 0694

11986030672 0868 0571

11986040482 0731 0508

11986050829 0457 0702

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)

(2) Calculating by the Euclidean distance the results areas follows

119884119864

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010675 0645 0462

11986020762 0655 0696

11986030696 0872 0597

11986040507 0736 0526

11986050837 0499 0706

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(27)

Step 6 The degree by which a given alternative is superior toall of the others can be calculated using the extended TODIMmethod First work out the values of function Φ

119888(119860119894 119860119895)

and then the values of function 119868may be calculatedThe valueof parameter 120587 is given as 09 because of the principle that


Table 1

119904119894119896

1198631

1198632

1198633

1198601

0850 0575 03501198602

1625 1475 08501198603

0900 0850 12751198604

0950 1425 01001198605

1100 0925 0850Total 5425 5250 3425

Table 2

1198621

1198622

1198623

1198601

0448 0561 0161 0225 0453 0554 0209 0274 0347 0448 0439 05321198602

0664 0739 0139 0229 0387 0451 0125 0227 0613 0682 0225 02941198603

0459 0531 0125 0176 0748 0853 0000 0130 0370 0435 0209 03101198604

0284 0380 0321 0417 0613 0683 0161 0213 0591 0411 0321 04011198605

0858 1000 0222 0321 0212 0312 0321 0374 0588 0674 0200 0255

decision makers usually tend to avoid risk and pursue profitThe results are as follows

(1) Calculating by the Hamming distance the results are

1198681=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605

11986010000 minus1666 minus2077 minus1091 minus1848

11986020268 0000 minus0835 minus0346 minus0810

11986030543 minus1214 0000 0344 minus1499

1198604minus0952 minus2056 minus2398 0000 minus2192

1198605minus0478 minus0836 minus1147 minus0790 0000

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)

(2) Calculating by the Euclidean distance the results are

1198682=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030568 minus1057 0000 0427 minus1405



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)

Step 7 According to expression (20) the overall evaluationvalue of each alternative can be worked out and the resultsare as follows

120585 (119860119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605

on condition that Hamming distance 0156 1 0983 0 0740

on condition that Euclidean distance 0114 1 0984 0 0736

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(30)

Step 8 The values of function 120585(119860119894) should be ranked by

ascending order after Step 7

No matter what kinds of distance formulas were adaptedin this process such as the Hamming distance formula or theEuclidean distance formula the final results of the sorting areconsistent And it is 119860

2≻ 1198603≻ 1198605≻ 1198601≻ 1198604 Therefore

1198602is better than other alternatives that is to say that when

the airfield constructs an airport terminals in the surroundingcities the second city will be the best choice However thefinal results of the sorting can also be used to judge therationality of the position of the existing airport terminal Forexample assume that city119860

3and city119860

1already have airport

terminals In this case if another airport terminal needs to beconstructed city 119860

5should also be taken into consideration

7 Conclusion

Amethod of multiple-attribute group decision making basedon TODIM is proposed to evaluate the information of theIIFSs The differences between this paper and other studiesthat discuss the TODIM method of group decision makingin the case of unknown weights [29 30 43 44 47] are madeclear The entropy evaluation method is applied to determineattribute weights and expert weights The attribute value isexpressed as the IIFSs as distinct from previous expressionssuch as the two-dimension linguistic variable and the trianglefuzzy value [11 18 45 47]

First we show that expert weights can be obtained usingthe original expert matrices Second we show how thematrices of IIFSs including different expert opinions can be


Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185

integratedThen we discuss how the attribute weights can beworked out on top of the aggregated expert matrix Finallyusing the Hamming distance and the Euclidean distance ofthe IIFSs we accomplish the transformation of the matrixThe traditional TODIMmethod is improved so that it can beused to handle the IIFS information allowing for a generalranking of the alternatives Finally a case concerning the siteselection of airport terminals is described to show that thesteps of the above method are operable and easy

However according to the Hamming distance and theEuclidean distance of the IIFSs the transformation of thematrix is simple and there must be a more accurate mappingfunction that can be used to express the relationship Anotherdeficiency of this paper is the simplicity of the method usedto determine weights Some studies have proposed othermethods to solve the unknown weights problems whichmay be applied to the extended TODIM method for groupdecision making with the IIFSs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors also would like to express sincere appreciation tothe anonymous reviewers and Editors for their very helpfulcomments that improved the paper This paper is supportedby the National Social Science Fund of China (no 12BJY112)and the Fundamental Research Funds for the Central Univer-sities of China (no ZXH2010B002)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] K-T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] D-F Li ldquoCloseness coefficient based nonlinear programmingmethod for interval-valued intuitionistic fuzzy multiattributedecision making with incomplete preference informationrdquoApplied Soft Computing Journal vol 11 no 4 pp 3402ndash34182011

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

[5] K T Atanassov ldquoOperators over interval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 64 no 2 pp 159ndash1741994

[6] H Bustince and P Burillo ldquoCorrelation of interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Sets and Systems vol 74 no 2pp 237ndash244 1995

[7] T K Mondal and S K Samanta ldquoTopology of interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Sets and Systems vol 119 no 3pp 483ndash494 2001

[8] W-L Hung and J-W Wu ldquoCorrelation of intuitionistic fuzzysets by centroid methodrdquo Information Sciences vol 144 no 1ndash4pp 219ndash225 2002

[9] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[10] Z-S Xu ldquoMethods for aggregating interval-valued intuitionis-tic fuzzy information and their application to decisionmakingrdquoControl and Decision vol 22 no 2 pp 215ndash219 2007

[11] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoNewoperators on triangular intuitionistic fuzzy numbers and theirapplications in system fault analysisrdquo Information Sciences vol251 pp 79ndash95 2013

[12] P-D Liu Y-C Chu Y-W Li and Y-B Chen ldquoSome gener-alized neutrosophic number Hamacher aggregation operatorsand their application to Group Decision Makingrdquo InternationalJournal of Fuzzy Systems vol 16 no 2 pp 242ndash255 2014

[13] J-Q Wang P Zhou K-J Li H-Y Zhang and X-H ChenldquoMulti-criteria decision-making method based on normalintuitionistic fuzzy-induced generalized aggregation operatorrdquoTOP vol 22 no 3 pp 1103ndash1122 2014

[14] B Roy and B Bertier ldquoLa metode ELECTRE IIrdquo in SixiemeConference Internationale de Rechearche Operationelle DublinIreland 1972

[15] P-D Liu and X Zhang ldquoResearch on the supplier selection of asupply chain based on entropyweight and improved ELECTRE-III methodrdquo International Journal of Production Research vol49 no 3 pp 637ndash646 2011

[16] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[17] G-W Wei and Y Wei ldquoModel of grey relational analysis forinterval multiple attribute decision making with preferenceinformation on alternativesrdquo Chinese Journal of ManagementScience vol 16 pp 158ndash162 2008

[18] P D Liu and M H Wang ldquoAn extended VIKOR method formultiple attribute group decision making based on generalizedinterval-valued trapezoidal fuzzy numbersrdquo Scientific Researchand Essays vol 6 no 4 pp 766ndash776 2011

[19] P Liu ldquoMulti-attribute decision-makingmethod research basedOn interval vague set and topsis methodrdquo Technological andEconomic Development of Economy vol 15 no 3 pp 453ndash4632009

[20] Z-L Yue ldquoAn extended TOPSIS for determining weights ofdecision makers with interval numbersrdquo Knowledge-Based Sys-tems vol 24 no 1 pp 146ndash153 2011

[21] X Zhao and G-W Wei ldquoSome intuitionistic fuzzy Einsteinhybrid aggregation operators and their application to multipleattribute decision makingrdquo Knowledge-Based Systems vol 37pp 472ndash479 2013


[22] P-D Liu and Y Liu ldquoAn approach to multiple attributegroup decisionmaking based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo International Journalof Computational Intelligence Systems vol 7 no 2 pp 291ndash3042014

[23] P-D Liu and Y-M Wang ldquoMultiple attribute decision-mak-ing method based on single-valued neutrosophic normalizedweighted Bonferroni meanrdquo Neural Computing and Applica-tions vol 25 no 7-8 pp 2001ndash2010 2014

[24] P-D Liu and Y-M Wang ldquoMultiple attribute group decisionmaking methods based on intuitionistic linguistic power gen-eralized aggregation operatorsrdquo Applied Soft Computing vol 17pp 90ndash104 2014

[25] J-Q Wang J-T Wu J Wang H-Y Zhang and X-H ChenldquoMulti-criteria decision-making methods based on the Haus-dorff distance of hesitant fuzzy linguistic numbersrdquo Soft Com-puting 2015

[26] W Tao and G Zhang ldquoTrusted interaction approach fordynamic service selection using multi-criteria decision makingtechniquerdquo Knowledge-Based Systems vol 32 pp 116ndash122 2012

[27] W-Y Chiu G-H Tzeng and H-L Li ldquoA new hybrid MCDMmodel combining DANP with VIKOR to improve e-store busi-nessrdquo Knowledge-Based Systems vol 37 pp 48ndash61 2013

[28] D G Park Y C Kwun J H Park and I Y Park ldquoCorrelationcoefficient of interval-valued intuitionistic fuzzy sets and itsapplication to multiple attribute group decision making prob-lemsrdquoMathematical and Computer Modelling vol 50 no 9-10pp 1279ndash1293 2009

[29] D-F Li ldquoLinear programming method for MADM with inter-val-valued intuitionistic fuzzy setsrdquo Expert Systems with Appli-cations vol 37 no 8 pp 5939ndash5945 2010

[30] Z-S Xu and X-Q Cai ldquoNonlinear optimization models formultiple attribute group decision making with intuitionisticfuzzy informationrdquo International Journal of Intelligent Systemsvol 25 no 6 pp 489ndash513 2010

[31] N Bryson and A Mobolurin ldquoAn action learning evaluationprocedure for multiple criteria decision making problemsrdquoEuropean Journal ofOperational Research vol 96 no 2 pp 379ndash386 1997

[32] Z Xu ldquoOn method for uncertain multiple attribute decisionmaking problems with uncertain multiplicative preferenceinformation on alternativesrdquo Fuzzy Optimization and DecisionMaking vol 4 no 2 pp 131ndash139 2005

[33] Z-X Su LWang and G-P Xia ldquoExtended VIKORmethod fordynamic multi-attribute decision making with interval num-bersrdquo Control and Decision vol 25 no 6 pp 836ndash840 2010

[34] J-Q Wang and H-Y Zhang ldquoMulticriteria decision-makingapproach based on atanassovrsquos intuitionistic fuzzy sets withincomplete certain information on weightsrdquo IEEE Transactionson Fuzzy Systems vol 21 no 3 pp 510ndash515 2013

[35] J-Q Wang R-R Nie H-Y Zhang and X-H Chen ldquoIntu-itionistic fuzzy multi-criteria decision-making method basedon evidential reasoningrdquo Applied Soft Computing Journal vol13 no 4 pp 1823ndash1831 2013

[36] H Hu and Z-S Xu ldquoTOPSIS method for multiple attributedecision making with interval-valued intuitionistic fuzzy infor-mationrdquo Fuzzy Systems and Mathematics vol 21 no 5 pp 108ndash112 2007

[37] X-W Qi C-Y Liang Q-W Cao and Y Ding ldquoAutomatic con-vergent approach in interval-valued intuitionistic fuzzy multi-attribute group decision makingrdquo Systems Engineering andElectronics vol 33 no 1 pp 110ndash115 2011

[38] Y Liu J Forrest S-F Liu H-H Zhao and L-R Jian ldquoDynamicmultiple attribute grey incidence decision making mothedbased on interval valued intuitionisitc fuzzy numberrdquo Controland Decision vol 28 no 9 pp 1303ndash1321 2013

[39] T-Y Chen ldquoData construction process and qualiflex-basedmethod for multiple-criteria group decision making with inter-val-valued intuitionistic fuzzy setsrdquo International Journal ofInformation Technology and Decision Making vol 12 no 3 pp425ndash467 2013

[40] L F A M Gomes and M M P P Lima ldquoTODIM basicsand application to multicriteria ranking of projects with envi-ronmental impactsrdquo Foundations of Computing and DecisionSciences vol 16 no 4 pp 113ndash127 1992

[41] L F A M Gomes and M M P P Lima ldquoFrom modelingindividual preferences to multicriteria ranking of discrete alter-natives a look at prospect theory and the additive differencemodelrdquo Foundations of Computing and Decision Sciences vol17 no 3 pp 171ndash184 1992

[42] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 p 263 1979

[43] L F A M Gomes and L A D Rangel ldquoAn application of theTODIM method to the multicriteria rental evaluation of resi-dential propertiesrdquo European Journal of Operational Researchvol 193 no 1 pp 204ndash211 2009

[44] R A Krohling and T T M De Souza ldquoCombining prospecttheory and fuzzy numbers to multi-criteria decision makingrdquoExpert Systems with Applications vol 39 no 13 pp 11487ndash114932012

[45] Z-P Fan X Zhang F-D Chen and Y Liu ldquoExtended TODIMmethod for hybrid multiple attribute decision making prob-lemsrdquo Knowledge-Based Systems vol 42 pp 40ndash48 2013

[46] X-W Qi C-Y Liang E-Q Zhang and Y Ding ldquoApproachto interval-valued intuitionistic fuzzy multiple attributes groupdecisionmaking based onmaximum entropyrdquo System Engineer-ing Theory and Practice vol 31 no 10 pp 1940ndash1948 2011

[47] P Liu and F Teng ldquoAn extended TODIM method for multipleattribute group decision-making based on 2-dimension uncer-tain linguistic Variablerdquo Complexity 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


However the methods used to solve the problems thatconcerned MADM with IIFS have been extended Somemethods involving MADM with IIFS can be understood asfollows Hu and Xu [36] proposed the TOPSIS method Qiet al [37] gave an approach to include automatic conver-gence A dynamicmultiple-attribute grey incidence decision-making method was put forward by Liu et al [38] Chen [39]developed a qualities-based method for handling problemsand he applied a confidence interval to the membershipdegree and the nonmembership degree

Gomes and Lima [40 41] proposed the TODIM (anacronym in Portuguese of interactive and multicriteria deci-sion making) method which was based on prospect theory[42] The TODIM method is used to establish the expectedfunction whichmeasures the degree to which one alternativeis superior to others by calculating accurate number infor-mation The TODIM method was successfully applied to theMADMfor the first time byGomes andRangel [43] and it hasproved to be an effective method for dealing with MADMKrohling and De Souza [44] applied the TODIM methodto the attribute values which are specifically expressed asthe IIFS In addition the TODIM method has provided agreat help in solving real problems TODIM methods wereexpanded by Fan et al [45] In the future TODIM methodscan deal with three types of information including thereal number the interval number and the triangular fuzzynumber [46]

The traditional TODIM method mainly handles realnumbers where the weights are known and only one decisionmaker appears in the MADM problems The IIFS is used toexpress the opinions of every expert for every alternativebecause its description is more accurate than other math-ematical linguistics In this paper multiple-attribute groupdecision making is studied by using the extended TODIMmethod which is represented by IIFSs The TODIM methodaims to optimize the ranking of alternatives In additionthe entropy method for the determination of weights is putforward We can derive the expert weights according to thescore function and the attribute weights can be derived bythe idea of maximum entropy

The structure of this paper is arranged as follows InSection 2 the definition and algorithms of the IIFS areproposed In Section 3 the group decision-makingmatrix forthe IIFSs is described In Section 4 the method to determinethe weights is given in detail In Section 5 the extendedTODIMmethod for interval intuitionistic fuzzy informationis put forward and the steps of how to solve the problem ofmultiple-attribute group decision making are listed one byone In the end a case concerning the site selection of airportterminals is shown to demonstrate the proposed method andits results

2 Definitions of IIFSs

Because the concept of IIFS is derived from intuitionisticfuzzy sets we first define IFS On the basis of that IIFS willbe defined The basic algorithms of the IIFS are given as wellas a description of its operators

Definition 1 (intuitionistic fuzzy set (IFS) [2]) Let 119860 be anIFS it can be represented as follows

119860 = ⟨119909 119906 (119909) V (119909) | 119909 isin 119877⟩ (1)

In this definition 119906(119909) represents the membership of theelement 119909 in the IFS and ](119909) represents the nonmembershipdegree of the element 119909 in the IFS Let 120587(119909) represent thehesitancy degree of 119860 that is 120587(119909) = 1 minus 119906(119909) minus V(119909)

This expression should satisfy the following conditions(1) 119877 is a nonempty set(2) 0 le 119906(119909) le 1 0 le V(119909) le 1 and 0 le 120587(119909) le 1

Definition 2 (interval intuitionistic fuzzy set (IIFS) [4]) Let be an IIFS it can be represented as follows

= ⟨[119906120572

(119909) 119906120573

(119909)] [V120572 (119909) V120573 (119909)] | 119909 isin 119877⟩ (2)

In this definition 119906120572(119909) represents the lower limit of themembership degree of the element 119909 119906120573(119909) represents theupper limit of the membership degree of the element 119909 V120572(119909)represents the lower limit of the nonmembership degree ofthe element 119909 and V120573(119909) represents the upper limit of thenonmembership degree of the element119909 In addition let (119909)be the hesitancy degree of that is (119909) = (120583(119909) cup ](119909))119888This is equal to [1 minus 119906120573(119909) minus V120573(119909) 1 minus 119906120572(119909) minus V120572(119909)]

This expression should satisfy the following conditions(1) 119877 is a nonempty set(2) 119906120572(119909) isin [0 1] 119906120573(119909) isin [0 1] and V120572(119909) isin [0 1]

V120573(119909) isin [0 1](3) [119906120572(119909) 119906120573(119909)] sub [0 1] [V120572(119909) V120573(119909)] sub [0 1] and0 le 119906120572

(119909) + V120572(119909) le 1

Definition 3 (the basic algorithms of IIFS [4]) Given the fol-lowing assumptions let ℎ

1and ℎ2be two interval intuitionis-

tic fuzzy numbers

ℎ1= ([119906

120572

1(119909) 119906

120573

1(119909)] [V120572

1(119909) V120573

1(119909)])

ℎ2= ([119906

120572

2(119909) 119906

120573

2(119909)] [V120572

2(119909) V120573

2(119909)])

(3)

Then we have these basic algorithms

ℎ1+ ℎ2= ([119906

120572

1(119909) + 119906

120572

2(119909) minus 119906

120572

1(119909) lowast 119906

120572

2(119909)

119906120573

1(119909) + 119906

120573

2(119909) minus 119906

120573

1(119909) lowast 119906

120573

2(119909)]

[V1205721(119909) lowast V120572

2(119909) V120573

1(119909) lowast V120573

2(119909)])

ℎ1lowast ℎ2= ([119906

120572

1(119909) lowast 119906

120572

2(119909) 119906

120573

1(119909) lowast 119906

120573

2(119909)]

[V1205721(119909) + V120572

2(119909) minus V120572

1(119909) lowast V120572

2(119909)

V1205731(119909) + V120573

2(119909) minus V120573

1(119909) lowast V120573

2(119909)])

120582ℎ1= ([1 minus (1 minus 119906

120572

1(119909))120582

1 minus (1 minus 119906120573

1(119909))120582

]

[V1205721(119909)120582

V1205731(119909)120582

])

(4)




119899)

= ([1 minus

119899

prod

119894=1

(1 minus 119906120572

119894(119909))119908119894 1 minus

119899

prod

119894=1

(1 minus 119906120573

119894(119909))119908119894

]

[

119899

prod

119894=1

V120572119894(119909)119908119894

119899

prod

119894=1

V120573119894(119909)119908119894])

(5)


119894=1119908119894= 1 and 119908

119894is the

weight of ℎ119894



119889119867

(ℎ1 ℎ2) =

1

4(1003816100381610038161003816119906120572

1(119909) minus 119906

120572

2(119909)1003816100381610038161003816 +100381610038161003816100381610038161003816119906120573

1(119909) minus 119906

120573

2(119909)100381610038161003816100381610038161003816

+1003816100381610038161003816V120572

1(119909) minus V120572

2(119909)1003816100381610038161003816 +100381610038161003816100381610038161003816V1205731(119909) minus V120573

2(119909)100381610038161003816100381610038161003816)

(6)


119889119864

(ℎ1 ℎ2)

=1

2((119906120572

1(119909) minus 119906

120572

2(119909))2

+ (119906120573

1(119909) minus 119906

120573

2(119909))2

+ (V1205721(119909) minus V120572

1(119909))2

+ (V1205731(119909) minus V120573

2(119909))2

)

12

(7)


120572

(119909) 119906120573

(119909)]


119878 (ℎ) =1

2((119906120572

(119909) + 119906120573

(119909)) minus (V120572 (119909) + V120573 (119909))) (8)

When 119878(ℎ1) lt 119878(ℎ

2) there is ℎ

1lt ℎ2





120572

119894119895119896(119909) 119906120573

119894119895119896(119909)] [V120572

119894119895119896(119909) V120573119894119895119896(119909)]) and the


119894119895 2

119894119895

ℎ

119894119895)119879


119894is known



119903119894119895= ([1 minus

ℎ

prod

119896=1

(1 minus 119906120572

119894119895119896(119909))119902119896

1 minus

ℎ

prod

119896=1

(1 minus 119906120573

119894119895119896(119909))

119902119896

]

[

ℎ

prod

119896=1

V120572119894119895119896(119909)119902119896

ℎ

prod

119896=1

V120573119894119895119896(119909)119902119896])

(9)


119896=1119902119896= 1





expert 119896

119864119896= minus

1

ln 119899

119894=119899

sum

119894=1

(119904119894119896

sum119899

119894=1119904119894119896

lowast ln100381610038161003816100381610038161003816100381610038161003816

119904119894119896

sum119899

119894=1119904119894119896

100381610038161003816100381610038161003816100381610038161003816

) (10)


follows

119904119894119896=1

2

119898

sum

119895=1

((119906120572

119894119895119896(119909) + 119906

120573

119894119895119896(119909)) minus (V120572

119894119895119896(119909) + V120573

119894119895119896(119909)))

(11)







119902119896=

1 minus 119864119896

ℎ minus sumℎ

119896=1(119864119896)

(12)


119896=1119902119896= 1



of attribute 119895

119890119895= minus

1

ln 119899

119899

sum

119894=1

(

ℎ119894119895

sum119898

119895=1ℎ119894119895

lowast ln(ℎ119894119895

sum119898

119895=1ℎ119894119895

)) (13)


follows [37]

ℎ119894119895= (1 minus 119889 (([

120572

119894119895(119909)

120573

119894119895(119909)] [V120572

119894119895(119909) V120573

119894119895(119909)])

([05 05] [05 05])))

(14)



(2)

119889 (119903119894119895 ([05 05] [05 05]))

=1

2((119906120572

119894119895(119909) minus 05)

2

+ (119906120573

119894119895(119909) minus 05)

2

+ (V120572119894119895(119909) minus 05)

2

+ (V120573119894119895(119909) minus 05)

2

)

12

(15)




119895 Let 119908

119895be


119908119895=

1 minus 119890119895

119898 minus sum119898

119895=1(119890119895)

(16)


119895=1119908119895= 1






119894119895 2

119894119895

ℎ

119894119895)119879













119910119894119895= 119889 (119903

119894119895 ([0 0] [1 1])) (17)










Φ119888(119860119894 119860119895) =

radic119908119888(119910119894119888minus 119910119895119888)

sum119898

119888=1119908119888

119910119894119888gt 119910119895119888

minus1

120587

radic(sum119898

119888=1119908119888) (119910119895119888minus 119910119894119888)

119908119888

119910119894119888lt 119910119895119888

0 119910119894119888= 119910119895119888

(18)

119868 (119860119894 119860119895) =

119898

sum

119888=1

Φ(119860119894 119860119895) 119894 119895 isin 119860 (19)







119894)


120585 (119860119894)

=

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

max1le119894le119899

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

(20)


results













1198751

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([060 079] [010 015]) ([040 050] [030 040]) ([050 060] [030 035])

1198602

([070 075] [010 020]) ([060 065] [020 030]) ([070 080] [005 010])

1198603

1198604

1198605

([050 060] [020 025])

([030 035] [020 025])

([070 075] [015 020])

([070 075] [010 020])

([050 060] [010 015])

([030 040] [020 025])

([030 040] [030 040])

([060 070] [020 025])

([065 070] [020 030])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


1198752

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([050 065] [010 020]) ([030 040] [020 030]) ([040 050] [035 045])

1198602

([080 090] [005 010]) ([050 060] [020 030]) ([060 070] [020 030])

1198603

1198604

1198605

([060 070] [020 025])

([050 060] [020 030])

([070 080] [010 020])

([050 060] [020 025])

([070 080] [010 015])

([020 030] [020 025])

([030 040] [020 030])

([070 075] [020 025])

([050 055] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198753

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([040 050] [020 025]) ([050 060] [020 025]) ([030 040] [050 060])

1198602

([060 065] [020 030]) ([030 035] [010 020]) ([060 065] [030 035])

1198603

1198604

1198605

([040 045] [010 015])

([020 030] [040 050])

([090 100] [030 040])

([080 090] [000 010])

([060 065] [020 025])

([020 030] [040 045])

([040 045] [020 030])

([055 015] [040 050])

([060 070] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)


119864119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0981 0965 0883

1003816100381610038161003816100381610038161003816100381610038161003816

(22)


119902119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0114 0203 0683

1003816100381610038161003816100381610038161003816100381610038161003816

(23)





119890119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

09978 09974 09989

1003816100381610038161003816100381610038161003816100381610038161003816

(24)


119882119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

0371 0435 0194

1003816100381610038161003816100381610038161003816100381610038161003816

(25)



119884119867

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010656 0631 0456

11986020758 0621 0694

11986030672 0868 0571

11986040482 0731 0508

11986050829 0457 0702

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


119884119864

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010675 0645 0462

11986020762 0655 0696

11986030696 0872 0597

11986040507 0736 0526

11986050837 0499 0706

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(27)


119888(119860119894 119860119895)



Table 1

119904119894119896

1198631

1198632

1198633

1198601

0850 0575 03501198602

1625 1475 08501198603

0900 0850 12751198604

0950 1425 01001198605

1100 0925 0850Total 5425 5250 3425

Table 2

1198621

1198622

1198623

1198601

0448 0561 0161 0225 0453 0554 0209 0274 0347 0448 0439 05321198602

0664 0739 0139 0229 0387 0451 0125 0227 0613 0682 0225 02941198603

0459 0531 0125 0176 0748 0853 0000 0130 0370 0435 0209 03101198604

0284 0380 0321 0417 0613 0683 0161 0213 0591 0411 0321 04011198605

0858 1000 0222 0321 0212 0312 0321 0374 0588 0674 0200 0255



1198681=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030543 minus1214 0000 0344 minus1499



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)


1198682=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030568 minus1057 0000 0427 minus1405



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)


120585 (119860119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(30)




2≻ 1198603≻ 1198605≻ 1198601≻ 1198604 Therefore



3and city119860




7 Conclusion




Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899)

= ([1 minus

119899

prod

119894=1

(1 minus 119906120572

119894(119909))119908119894 1 minus

119899

prod

119894=1

(1 minus 119906120573

119894(119909))119908119894

]

[

119899

prod

119894=1

V120572119894(119909)119908119894

119899

prod

119894=1

V120573119894(119909)119908119894])

(5)


119894=1119908119894= 1 and 119908

119894is the

weight of ℎ119894



119889119867

(ℎ1 ℎ2) =

1

4(1003816100381610038161003816119906120572

1(119909) minus 119906

120572

2(119909)1003816100381610038161003816 +100381610038161003816100381610038161003816119906120573

1(119909) minus 119906

120573

2(119909)100381610038161003816100381610038161003816

+1003816100381610038161003816V120572

1(119909) minus V120572

2(119909)1003816100381610038161003816 +100381610038161003816100381610038161003816V1205731(119909) minus V120573

2(119909)100381610038161003816100381610038161003816)

(6)


119889119864

(ℎ1 ℎ2)

=1

2((119906120572

1(119909) minus 119906

120572

2(119909))2

+ (119906120573

1(119909) minus 119906

120573

2(119909))2

+ (V1205721(119909) minus V120572

1(119909))2

+ (V1205731(119909) minus V120573

2(119909))2

)

12

(7)


120572

(119909) 119906120573

(119909)]


119878 (ℎ) =1

2((119906120572

(119909) + 119906120573

(119909)) minus (V120572 (119909) + V120573 (119909))) (8)

When 119878(ℎ1) lt 119878(ℎ

2) there is ℎ

1lt ℎ2





120572

119894119895119896(119909) 119906120573

119894119895119896(119909)] [V120572

119894119895119896(119909) V120573119894119895119896(119909)]) and the


119894119895 2

119894119895

ℎ

119894119895)119879


119894is known



119903119894119895= ([1 minus

ℎ

prod

119896=1

(1 minus 119906120572

119894119895119896(119909))119902119896

1 minus

ℎ

prod

119896=1

(1 minus 119906120573

119894119895119896(119909))

119902119896

]

[

ℎ

prod

119896=1

V120572119894119895119896(119909)119902119896

ℎ

prod

119896=1

V120573119894119895119896(119909)119902119896])

(9)


119896=1119902119896= 1





expert 119896

119864119896= minus

1

ln 119899

119894=119899

sum

119894=1

(119904119894119896

sum119899

119894=1119904119894119896

lowast ln100381610038161003816100381610038161003816100381610038161003816

119904119894119896

sum119899

119894=1119904119894119896

100381610038161003816100381610038161003816100381610038161003816

) (10)


follows

119904119894119896=1

2

119898

sum

119895=1

((119906120572

119894119895119896(119909) + 119906

120573

119894119895119896(119909)) minus (V120572

119894119895119896(119909) + V120573

119894119895119896(119909)))

(11)







119902119896=

1 minus 119864119896

ℎ minus sumℎ

119896=1(119864119896)

(12)


119896=1119902119896= 1



of attribute 119895

119890119895= minus

1

ln 119899

119899

sum

119894=1

(

ℎ119894119895

sum119898

119895=1ℎ119894119895

lowast ln(ℎ119894119895

sum119898

119895=1ℎ119894119895

)) (13)


follows [37]

ℎ119894119895= (1 minus 119889 (([

120572

119894119895(119909)

120573

119894119895(119909)] [V120572

119894119895(119909) V120573

119894119895(119909)])

([05 05] [05 05])))

(14)



(2)

119889 (119903119894119895 ([05 05] [05 05]))

=1

2((119906120572

119894119895(119909) minus 05)

2

+ (119906120573

119894119895(119909) minus 05)

2

+ (V120572119894119895(119909) minus 05)

2

+ (V120573119894119895(119909) minus 05)

2

)

12

(15)




119895 Let 119908

119895be


119908119895=

1 minus 119890119895

119898 minus sum119898

119895=1(119890119895)

(16)


119895=1119908119895= 1






119894119895 2

119894119895

ℎ

119894119895)119879













119910119894119895= 119889 (119903

119894119895 ([0 0] [1 1])) (17)










Φ119888(119860119894 119860119895) =

radic119908119888(119910119894119888minus 119910119895119888)

sum119898

119888=1119908119888

119910119894119888gt 119910119895119888

minus1

120587

radic(sum119898

119888=1119908119888) (119910119895119888minus 119910119894119888)

119908119888

119910119894119888lt 119910119895119888

0 119910119894119888= 119910119895119888

(18)

119868 (119860119894 119860119895) =

119898

sum

119888=1

Φ(119860119894 119860119895) 119894 119895 isin 119860 (19)







119894)


120585 (119860119894)

=

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

max1le119894le119899

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

(20)


results













1198751

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([060 079] [010 015]) ([040 050] [030 040]) ([050 060] [030 035])

1198602

([070 075] [010 020]) ([060 065] [020 030]) ([070 080] [005 010])

1198603

1198604

1198605

([050 060] [020 025])

([030 035] [020 025])

([070 075] [015 020])

([070 075] [010 020])

([050 060] [010 015])

([030 040] [020 025])

([030 040] [030 040])

([060 070] [020 025])

([065 070] [020 030])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


1198752

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([050 065] [010 020]) ([030 040] [020 030]) ([040 050] [035 045])

1198602

([080 090] [005 010]) ([050 060] [020 030]) ([060 070] [020 030])

1198603

1198604

1198605

([060 070] [020 025])

([050 060] [020 030])

([070 080] [010 020])

([050 060] [020 025])

([070 080] [010 015])

([020 030] [020 025])

([030 040] [020 030])

([070 075] [020 025])

([050 055] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198753

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([040 050] [020 025]) ([050 060] [020 025]) ([030 040] [050 060])

1198602

([060 065] [020 030]) ([030 035] [010 020]) ([060 065] [030 035])

1198603

1198604

1198605

([040 045] [010 015])

([020 030] [040 050])

([090 100] [030 040])

([080 090] [000 010])

([060 065] [020 025])

([020 030] [040 045])

([040 045] [020 030])

([055 015] [040 050])

([060 070] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)


119864119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0981 0965 0883

1003816100381610038161003816100381610038161003816100381610038161003816

(22)


119902119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0114 0203 0683

1003816100381610038161003816100381610038161003816100381610038161003816

(23)





119890119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

09978 09974 09989

1003816100381610038161003816100381610038161003816100381610038161003816

(24)


119882119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

0371 0435 0194

1003816100381610038161003816100381610038161003816100381610038161003816

(25)



119884119867

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010656 0631 0456

11986020758 0621 0694

11986030672 0868 0571

11986040482 0731 0508

11986050829 0457 0702

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


119884119864

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010675 0645 0462

11986020762 0655 0696

11986030696 0872 0597

11986040507 0736 0526

11986050837 0499 0706

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(27)


119888(119860119894 119860119895)



Table 1

119904119894119896

1198631

1198632

1198633

1198601

0850 0575 03501198602

1625 1475 08501198603

0900 0850 12751198604

0950 1425 01001198605

1100 0925 0850Total 5425 5250 3425

Table 2

1198621

1198622

1198623

1198601

0448 0561 0161 0225 0453 0554 0209 0274 0347 0448 0439 05321198602

0664 0739 0139 0229 0387 0451 0125 0227 0613 0682 0225 02941198603

0459 0531 0125 0176 0748 0853 0000 0130 0370 0435 0209 03101198604

0284 0380 0321 0417 0613 0683 0161 0213 0591 0411 0321 04011198605

0858 1000 0222 0321 0212 0312 0321 0374 0588 0674 0200 0255



1198681=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030543 minus1214 0000 0344 minus1499



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)


1198682=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030568 minus1057 0000 0427 minus1405



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)


120585 (119860119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(30)




2≻ 1198603≻ 1198605≻ 1198601≻ 1198604 Therefore



3and city119860




7 Conclusion




Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119902119896=

1 minus 119864119896

ℎ minus sumℎ

119896=1(119864119896)

(12)


119896=1119902119896= 1



of attribute 119895

119890119895= minus

1

ln 119899

119899

sum

119894=1

(

ℎ119894119895

sum119898

119895=1ℎ119894119895

lowast ln(ℎ119894119895

sum119898

119895=1ℎ119894119895

)) (13)


follows [37]

ℎ119894119895= (1 minus 119889 (([

120572

119894119895(119909)

120573

119894119895(119909)] [V120572

119894119895(119909) V120573

119894119895(119909)])

([05 05] [05 05])))

(14)



(2)

119889 (119903119894119895 ([05 05] [05 05]))

=1

2((119906120572

119894119895(119909) minus 05)

2

+ (119906120573

119894119895(119909) minus 05)

2

+ (V120572119894119895(119909) minus 05)

2

+ (V120573119894119895(119909) minus 05)

2

)

12

(15)




119895 Let 119908

119895be


119908119895=

1 minus 119890119895

119898 minus sum119898

119895=1(119890119895)

(16)


119895=1119908119895= 1






119894119895 2

119894119895

ℎ

119894119895)119879













119910119894119895= 119889 (119903

119894119895 ([0 0] [1 1])) (17)










Φ119888(119860119894 119860119895) =

radic119908119888(119910119894119888minus 119910119895119888)

sum119898

119888=1119908119888

119910119894119888gt 119910119895119888

minus1

120587

radic(sum119898

119888=1119908119888) (119910119895119888minus 119910119894119888)

119908119888

119910119894119888lt 119910119895119888

0 119910119894119888= 119910119895119888

(18)

119868 (119860119894 119860119895) =

119898

sum

119888=1

Φ(119860119894 119860119895) 119894 119895 isin 119860 (19)







119894)


120585 (119860119894)

=

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

max1le119894le119899

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

(20)


results













1198751

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([060 079] [010 015]) ([040 050] [030 040]) ([050 060] [030 035])

1198602

([070 075] [010 020]) ([060 065] [020 030]) ([070 080] [005 010])

1198603

1198604

1198605

([050 060] [020 025])

([030 035] [020 025])

([070 075] [015 020])

([070 075] [010 020])

([050 060] [010 015])

([030 040] [020 025])

([030 040] [030 040])

([060 070] [020 025])

([065 070] [020 030])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


1198752

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([050 065] [010 020]) ([030 040] [020 030]) ([040 050] [035 045])

1198602

([080 090] [005 010]) ([050 060] [020 030]) ([060 070] [020 030])

1198603

1198604

1198605

([060 070] [020 025])

([050 060] [020 030])

([070 080] [010 020])

([050 060] [020 025])

([070 080] [010 015])

([020 030] [020 025])

([030 040] [020 030])

([070 075] [020 025])

([050 055] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198753

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([040 050] [020 025]) ([050 060] [020 025]) ([030 040] [050 060])

1198602

([060 065] [020 030]) ([030 035] [010 020]) ([060 065] [030 035])

1198603

1198604

1198605

([040 045] [010 015])

([020 030] [040 050])

([090 100] [030 040])

([080 090] [000 010])

([060 065] [020 025])

([020 030] [040 045])

([040 045] [020 030])

([055 015] [040 050])

([060 070] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)


119864119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0981 0965 0883

1003816100381610038161003816100381610038161003816100381610038161003816

(22)


119902119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0114 0203 0683

1003816100381610038161003816100381610038161003816100381610038161003816

(23)





119890119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

09978 09974 09989

1003816100381610038161003816100381610038161003816100381610038161003816

(24)


119882119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

0371 0435 0194

1003816100381610038161003816100381610038161003816100381610038161003816

(25)



119884119867

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010656 0631 0456

11986020758 0621 0694

11986030672 0868 0571

11986040482 0731 0508

11986050829 0457 0702

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


119884119864

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010675 0645 0462

11986020762 0655 0696

11986030696 0872 0597

11986040507 0736 0526

11986050837 0499 0706

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(27)


119888(119860119894 119860119895)



Table 1

119904119894119896

1198631

1198632

1198633

1198601

0850 0575 03501198602

1625 1475 08501198603

0900 0850 12751198604

0950 1425 01001198605

1100 0925 0850Total 5425 5250 3425

Table 2

1198621

1198622

1198623

1198601

0448 0561 0161 0225 0453 0554 0209 0274 0347 0448 0439 05321198602

0664 0739 0139 0229 0387 0451 0125 0227 0613 0682 0225 02941198603

0459 0531 0125 0176 0748 0853 0000 0130 0370 0435 0209 03101198604

0284 0380 0321 0417 0613 0683 0161 0213 0591 0411 0321 04011198605

0858 1000 0222 0321 0212 0312 0321 0374 0588 0674 0200 0255



1198681=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030543 minus1214 0000 0344 minus1499



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)


1198682=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030568 minus1057 0000 0427 minus1405



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)


120585 (119860119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(30)




2≻ 1198603≻ 1198605≻ 1198601≻ 1198604 Therefore



3and city119860




7 Conclusion




Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Φ119888(119860119894 119860119895) =

radic119908119888(119910119894119888minus 119910119895119888)

sum119898

119888=1119908119888

119910119894119888gt 119910119895119888

minus1

120587

radic(sum119898

119888=1119908119888) (119910119895119888minus 119910119894119888)

119908119888

119910119894119888lt 119910119895119888

0 119910119894119888= 119910119895119888

(18)

119868 (119860119894 119860119895) =

119898

sum

119888=1

Φ(119860119894 119860119895) 119894 119895 isin 119860 (19)







119894)


120585 (119860119894)

=

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

max1le119894le119899

sum119899

119895=1119868 (119860119894 119860119895) minusmin

1le119894le119899sum119899

119895=1119868 (119860119894 119860119895)

(20)


results













1198751

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([060 079] [010 015]) ([040 050] [030 040]) ([050 060] [030 035])

1198602

([070 075] [010 020]) ([060 065] [020 030]) ([070 080] [005 010])

1198603

1198604

1198605

([050 060] [020 025])

([030 035] [020 025])

([070 075] [015 020])

([070 075] [010 020])

([050 060] [010 015])

([030 040] [020 025])

([030 040] [030 040])

([060 070] [020 025])

([065 070] [020 030])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


1198752

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([050 065] [010 020]) ([030 040] [020 030]) ([040 050] [035 045])

1198602

([080 090] [005 010]) ([050 060] [020 030]) ([060 070] [020 030])

1198603

1198604

1198605

([060 070] [020 025])

([050 060] [020 030])

([070 080] [010 020])

([050 060] [020 025])

([070 080] [010 015])

([020 030] [020 025])

([030 040] [020 030])

([070 075] [020 025])

([050 055] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198753

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([040 050] [020 025]) ([050 060] [020 025]) ([030 040] [050 060])

1198602

([060 065] [020 030]) ([030 035] [010 020]) ([060 065] [030 035])

1198603

1198604

1198605

([040 045] [010 015])

([020 030] [040 050])

([090 100] [030 040])

([080 090] [000 010])

([060 065] [020 025])

([020 030] [040 045])

([040 045] [020 030])

([055 015] [040 050])

([060 070] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)


119864119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0981 0965 0883

1003816100381610038161003816100381610038161003816100381610038161003816

(22)


119902119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0114 0203 0683

1003816100381610038161003816100381610038161003816100381610038161003816

(23)





119890119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

09978 09974 09989

1003816100381610038161003816100381610038161003816100381610038161003816

(24)


119882119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

0371 0435 0194

1003816100381610038161003816100381610038161003816100381610038161003816

(25)



119884119867

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010656 0631 0456

11986020758 0621 0694

11986030672 0868 0571

11986040482 0731 0508

11986050829 0457 0702

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


119884119864

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010675 0645 0462

11986020762 0655 0696

11986030696 0872 0597

11986040507 0736 0526

11986050837 0499 0706

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(27)


119888(119860119894 119860119895)



Table 1

119904119894119896

1198631

1198632

1198633

1198601

0850 0575 03501198602

1625 1475 08501198603

0900 0850 12751198604

0950 1425 01001198605

1100 0925 0850Total 5425 5250 3425

Table 2

1198621

1198622

1198623

1198601

0448 0561 0161 0225 0453 0554 0209 0274 0347 0448 0439 05321198602

0664 0739 0139 0229 0387 0451 0125 0227 0613 0682 0225 02941198603

0459 0531 0125 0176 0748 0853 0000 0130 0370 0435 0209 03101198604

0284 0380 0321 0417 0613 0683 0161 0213 0591 0411 0321 04011198605

0858 1000 0222 0321 0212 0312 0321 0374 0588 0674 0200 0255



1198681=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030543 minus1214 0000 0344 minus1499



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)


1198682=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030568 minus1057 0000 0427 minus1405



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)


120585 (119860119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(30)




2≻ 1198603≻ 1198605≻ 1198601≻ 1198604 Therefore



3and city119860




7 Conclusion




Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198752

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([050 065] [010 020]) ([030 040] [020 030]) ([040 050] [035 045])

1198602

([080 090] [005 010]) ([050 060] [020 030]) ([060 070] [020 030])

1198603

1198604

1198605

([060 070] [020 025])

([050 060] [020 030])

([070 080] [010 020])

([050 060] [020 025])

([070 080] [010 015])

([020 030] [020 025])

([030 040] [020 030])

([070 075] [020 025])

([050 055] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198753

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

1198601

([040 050] [020 025]) ([050 060] [020 025]) ([030 040] [050 060])

1198602

([060 065] [020 030]) ([030 035] [010 020]) ([060 065] [030 035])

1198603

1198604

1198605

([040 045] [010 015])

([020 030] [040 050])

([090 100] [030 040])

([080 090] [000 010])

([060 065] [020 025])

([020 030] [040 045])

([040 045] [020 030])

([055 015] [040 050])

([060 070] [020 025])

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)


119864119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0981 0965 0883

1003816100381610038161003816100381610038161003816100381610038161003816

(22)


119902119896=

1003816100381610038161003816100381610038161003816100381610038161003816

1198631

1198632

1198633

0114 0203 0683

1003816100381610038161003816100381610038161003816100381610038161003816

(23)





119890119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

09978 09974 09989

1003816100381610038161003816100381610038161003816100381610038161003816

(24)


119882119895=

1003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

0371 0435 0194

1003816100381610038161003816100381610038161003816100381610038161003816

(25)



119884119867

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010656 0631 0456

11986020758 0621 0694

11986030672 0868 0571

11986040482 0731 0508

11986050829 0457 0702

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


119884119864

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198621

1198622

1198623

11986010675 0645 0462

11986020762 0655 0696

11986030696 0872 0597

11986040507 0736 0526

11986050837 0499 0706

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(27)


119888(119860119894 119860119895)



Table 1

119904119894119896

1198631

1198632

1198633

1198601

0850 0575 03501198602

1625 1475 08501198603

0900 0850 12751198604

0950 1425 01001198605

1100 0925 0850Total 5425 5250 3425

Table 2

1198621

1198622

1198623

1198601

0448 0561 0161 0225 0453 0554 0209 0274 0347 0448 0439 05321198602

0664 0739 0139 0229 0387 0451 0125 0227 0613 0682 0225 02941198603

0459 0531 0125 0176 0748 0853 0000 0130 0370 0435 0209 03101198604

0284 0380 0321 0417 0613 0683 0161 0213 0591 0411 0321 04011198605

0858 1000 0222 0321 0212 0312 0321 0374 0588 0674 0200 0255



1198681=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030543 minus1214 0000 0344 minus1499



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)


1198682=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030568 minus1057 0000 0427 minus1405



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)


120585 (119860119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(30)




2≻ 1198603≻ 1198605≻ 1198601≻ 1198604 Therefore



3and city119860




7 Conclusion




Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1

119904119894119896

1198631

1198632

1198633

1198601

0850 0575 03501198602

1625 1475 08501198603

0900 0850 12751198604

0950 1425 01001198605

1100 0925 0850Total 5425 5250 3425

Table 2

1198621

1198622

1198623

1198601

0448 0561 0161 0225 0453 0554 0209 0274 0347 0448 0439 05321198602

0664 0739 0139 0229 0387 0451 0125 0227 0613 0682 0225 02941198603

0459 0531 0125 0176 0748 0853 0000 0130 0370 0435 0209 03101198604

0284 0380 0321 0417 0613 0683 0161 0213 0591 0411 0321 04011198605

0858 1000 0222 0321 0212 0312 0321 0374 0588 0674 0200 0255



1198681=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030543 minus1214 0000 0344 minus1499



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(28)


1198682=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



11986030568 minus1057 0000 0427 minus1405



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)


120585 (119860119894) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198601

1198602

1198603

1198604

1198605



100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(30)




2≻ 1198603≻ 1198605≻ 1198601≻ 1198604 Therefore



3and city119860




7 Conclusion




Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 3

ℎ119894119895

1198621

1198622

1198623

1198601

0778 0812 09121198602

0732 0760 07981198603

0751 0622 08121198604

0842 0753 08801198605

0651 0796 0783Total 3754 3743 4185





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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