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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 160610 11 pageshttpdxdoiorg1011552013160610
Research ArticleSome Induced Correlated Aggregating Operators with IntervalGrey Uncertain Linguistic Information and Their Application toMultiple Attribute Group Decision Making
Zu-Jun Ma Nian Zhang and Ying Dai
School of Economic and Management Southwest Jiaotong University Chengdu Sichuan 610031 China
Correspondence should be addressed to Nian Zhang chinazhangnian163com
Received 3 November 2012 Accepted 30 December 2012
Academic Editor Youqing Wang
Copyright copy 2013 Zu-Jun Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We propose the interval grey uncertain linguistic correlated ordered arithmetic averaging (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered arithmetic averaging (I-IGULCOA) operator based on the correlationproperties of the Choquet integral and the interval grey uncertain linguistic variables to investigate the multiple attribute groupdecision making (MAGDM) problems in which both the attribute weights and the expert weights are correlative Firstly therelative concepts of interval grey uncertain linguistic variables are defined and the operation rules between the two interval greyuncertain linguistic variables are establishedThen two new aggregation operators the interval grey uncertain linguistic correlatedordered arithmetic averaging (IGULCOA) operator and the induced interval grey uncertain linguistic correlated ordered arithmeticaveraging (I-IGULCOA) operator are developed and some desirable properties of the I-IGULCOA operator are studied suchas commutativity idempotency monotonicity and boundness Furthermore the IGULCOA and I-IGULCOA operators basedapproach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights are correlativeand the attribute values take the form of the interval grey uncertain linguistic variables Finally an illustrative example is given toverify the developed approach and to demonstrate its practicality and effectiveness
1 Introduction
Recently multiple attribute group decision making(MAGDM) has been extensively applied to various areassuch as society economics management and military It iswell known that the object things are complex and uncertainand human thinking is ambiguous Hence the majority ofmultiple attribute group decision making is also uncertainand fuzzy and fuzziness is the major factor in the processof decision making However in dealing with the problemof incomplete information caused by poor informationdecision making also demonstrates its greyness The ldquofuzzyrdquomeans those uncertain factors in the evaluation informationwhich are caused by the fuzziness of human thinking whilethe ldquogreyrdquo means that objective uncertainty caused by theinsufficient and incomplete information Therefore theldquofuzzyrdquo and the ldquogreyrdquo are different concepts many scholars
have studied the grey fuzzy multiple attribute decisionmaking which demonstrates not only its fuzziness but alsoits greyness
Since Zadeh introduced the concept of the fuzzy set [1]and Deng firstly presented the grey system theory [2] whichwere well applied inmultiple attribute group decisionmaking[3ndash13] the research on the grey fuzzy group decision makingproblem has been widely investigated and applied to a varietyof fields Chen [14] introduced the concept of the grey fuzzy indetail in his book Bu and Zhang [15] presented an approachto transform the grey fuzzy number into the interval numberand then utilized the ranking method of interval numberto rank the order of alternatives Basing on the grey fuzzymultiple attribute decision making in which both the fuzzypart and the grey part are real numbers Jin and Lou [1617] used the decision making model which utilized thehamming distance to measure the alternatives and utilized
2 Mathematical Problems in Engineering
the difference between the fuzzy positive ideal solution andthe negative ideal solution to rank the order In order tosolve the grey fuzzy decision making Dang and Sifeng [18]developed the maximum entropy formulism to determineattribute weight and ranked the order of alternatives basedon the linear combination of fuzzy information and greyinformation Meng et al [19] proposed the grey degree andfuzzy degree with the interval numbers and then basedon this the mathematical model of interval-valued greyfuzzy comprehensive evaluation was established At last itsapplication to the selection of the preferred project was givenIn many real-life decision making problems the linguisticvariable is easier to express fuzzy information and closer toactual condition the research on linguistic decision makinghas witnessed rich achievements [3 11 20ndash27] Liu and Jin[4] defined the concept of the interval grey linguistic variablewhere the fuzzy part and the grey part took the form ofthe uncertain linguistic variable and the interval numberrespectively studied the operation rules and developed themultiple attribute decision making method based on theinterval grey linguistic variable Liu and Zhang [5] proposedthe interval grey linguistic variables weighted geometricaggregation (IGLWGA) operator and the interval grey lin-guistic variables ordered weighted geometric aggregation(IGLOWGA) operator and the interval grey linguistic vari-ables hybrid weighted geometric aggregation (IGLHWGA)operator and then suggested a method for solving multipleattribute group decision making based on those operatorsZhang and Wei [28] introduced the interval grey linguisticvariables ordered weighted aggregation (IGLOWA) operatorand then used the Choquet integral to develop the intervalgrey linguistic correlated ordered arithmetic aggregation(IGLCOA)operator and the interval grey linguistic correlatedordered geometric aggregation (IGLCOGA) operator Thoseoperators not only consider the importance of the elementsbut also can reflect the correlations among the elementsThen they developed an approach to multiple attributedecisionmaking problems with correlative weights where theattribute values are given in terms of interval grey linguisticvariables information based on those operators
The existing grey fuzzy multiple attribute group decisionmaking only considers the situation where all the elements inthe grey fuzzy set are independent However in many prac-tical situations the elements in the grey fuzzy set are usuallycorrelativeTherefore we need to find some new ways to dealwith the situations in which the decision data in questionare correlative and the weights are correlative The Choquetintegral [29] is a very useful way of measuring the expectedutility of an uncertain event and can be utilized to depict thecorrelations of the decision data under consideration Yager[30 31] introduced the idea of order-induced aggregation tothe Choquet aggregation operator and defined an inducedChoquet ordered weighted averaging (C-OWA) operatorwhich allowed the ordering of the arguments to be basedupon some other associated variables instead of orderingthe arguments based on their values Tan and Chen [32]developed the induced Choquet ordered averaging (I-COA)operator and applied it to aggregate fuzzy preference relationsin group decision making Xu [25] utilized the Choquet
integral to propose the interval-valued intuitionistic fuzzycorrelated averaging (IVIFCA) operator and the interval-valued intuitionistic fuzzy correlated geometric (IVIFCG)operator to aggregate interval-valued intuitionistic fuzzyinformation and applied them to a practical decision makingproblem involving the prioritization of information technol-ogy improvement projects Wei and Zhao [33] developed theinduced intuitionistic fuzzy correlated averaging (I-IFCA)operator and induced intuitionistic fuzzy correlated geomet-ric (I-IFCG) operator and developed to solve the MAGDMproblems in which both the attribute weights and the expertweights are usually correlative and attribute values take theform of intuitionistic fuzzy values
Motivated by the correlation properties of the Choquetintegral and the uncertain linguistic variables in this paperwe propose the interval grey uncertain linguistic correlatedordered arithmetic averaging (IGULCOA) operator and theinduced interval grey uncertain linguistic correlated orderedarithmetic averaging (I-IGULCOA) operator with intervalgrey uncertain linguistic variables information The promi-nent characteristic of those operators is that they cannot onlyconsider the importance of the elements or their orderedpositions but also reflect the correlations among the elementsor their ordered positions And we introduce those inducedcorrelated aggregating operators to deal with group decisionmaking problems The aim of this paper is to investigate theMAGDM problems in which both the attribute weights andthe expert weights are correlative and the attribute values takethe form of interval grey uncertain linguistic variables Inorder to do so the remainder of this paper is set out as followsIn the next section we introduce some basic concepts relatedto interval grey uncertain linguistic variables and some oper-ational laws of interval grey uncertain linguistic variablesIn Section 3 we have developed two interval grey uncertainlinguistic correlated aggregation operators the IGULCOAoperator and the I-IGULCOA operator In Section 4 we havedeveloped an approach to multiple attribute group decisionmaking in which both the attribute weights and the expertweights are correlative and the attribute values take theform of interval grey uncertain linguistic variables based onthe IGULCOA operator and the I-IGULCOA operator withinterval grey uncertain linguistic variables information InSection 5 an illustrative example is pointed out In Section 6we conclude the paper and give some remarks
2 Preliminaries
In this section we briefly review some basic concepts to beused throughout the paper
The linguistic approach is an approximate techniquewhich represents qualitative aspects as linguistic values bymeans of linguistic variables [34 35]
Suppose that 119878 = 119904119894| 119894 = minus119905 minus1 0 1 119905 is a
finite and totally ordered discrete term set whose cardinalityvalue is odd [35] Any label 119904
119894represents a possible value for
a linguistic variable (as shown in Figure 1) and it has thefollowing characteristics (1) the set is ordered as 119904
120572gt 119904120573 if
120572 gt 120573 and (2) there is the negative operator neg(119904120572) = 119904minus120572
Mathematical Problems in Engineering 3
FairSlightlygood Good
Verygood
Extremelygood
Extremelypoor
Verypoor Poor
Slightlypoor
119904minus4 119904minus3 119904minus2 119904minus1 1199040 1199041 1199042 1199043 1199044
Figure 1 The linguistic labels
We call this linguistic label set 119878 the additive linguistic scaleFor example a set of nine terms 119878 could be defined as follows
119878 = 119904minus4= extremely poor 119904
minus3= very poor 119904
minus2= poor
119904minus1= slightly poor 119904
0= fair 119904
1= slightly good
1199042= good 119904
3= very good s
4= extremely good
(1)
in which 119904119894lt 119904119895if 119894 lt 119895
To preserve all the given information we extend thediscrete term set 119878 to a continuous term set 119878 = 119904
120572| 120572 isin
[minus119905 119905] If 119904120572lt 119878 then we call 119904
120572an original linguistic term
otherwise we call 119904120572a virtual linguistic term In general the
decision maker uses the original linguistic terms to evaluatealternatives and the virtual linguistic terms can only appearin operation [10 36]
Let 119904 = [119904120572 119904120573] where 119904
120572 119904120573isin 119878 119904
120572and 119904120573are the lower
and the upper limits respectively we then call 119904 an uncertainlinguistic variable Let 119878 be the set of all the uncertainlinguistic variables [10]
Consider any three uncertain linguistic variables 119904 =
[119904120572 119904120573] 1199041= [1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] then we define theoperations 119904
1oplus 1199042and 120582119904 as follows
(1) 1199041oplus 1199042= [1199041205721
1199041205731
] oplus [1199041205722
1199041205732
] = [1199041205721
oplus 1199041205722
1199041205731
oplus 1199041205732
] =
[1199041205721+1205722
1199041205731+1205732
](2) 120582119904 = 120582[119904
120572 119904120573] = [119904120582120572 119904120582120573] where 120582 isin [0 1]
(3) 1199041oplus 1199042= 1199042oplus 1199041
(4) 120582(1199041oplus 1199042) = 120582119904
1oplus 1205821199042 where 120582 isin [0 1]
(5) (1205821+ 1205822)119904 = 120582
1119904 oplus 1205822119904 where 120582
1 1205822isin [0 1]
Definition 1 (see [14]) Let 119860(119909) be the fuzzy subset in thespace 119883 = 119909 if the membership degree 120583
119860(119909) of 119909 to 119860(119909)
is the grey in the interval [0 1] and its grey is 120584119860(119909) then
119860(119909) is called the grey fuzzy set in space119883 (GF set for short)denoted by 119860
otimes(119909) as follows
119860otimes(119909) = (119909 120583119860 (119909) 120584119860 (119909)) | 119909 isin 119883 (2)
The set pair mode is 119860otimes(119909) = (119860(119909) 119860
otimes(119909)) where 119860(119909) =
(119909 120583119860(119909)) | 119909 isin 119883 is called the fuzzy part of 119860
otimes(119909) and
119860otimes(119909) = (119909 120584
119860(119909)) | 119909 isin 119883 is called the grey part of 119860
otimes(119909)
So the grey fuzzy set is regarded as the generalization of thefuzzy set and the grey set
119904
119904120573
119904120572
119892119860119892119860119880119871
Figure 2 The interval grey linguistic variable
Definition 2 Let 119860otimes(119909)=(119860(119909) 119860
otimes(119909)) be the grey fuzzy num-
ber if its fuzzy part is an uncertain linguistic variable 119904 =
[119904120572 119904120573] where 119904
120572 119904120573isin 119878 where 119878 is a finite and totally ordered
discrete term set and its grey part 119860otimes(119909) is a closed interval
[119892119871
119860 119892119880
119860] then 119860
otimes(119909) is called the interval grey uncertain
linguistic variable as shown in Figure 2
Suppose that 119860otimes(119909) = (119904
1 [119892119871
119860 119892119880
119860]) 119861otimes(119909) = (119904
2 [119892119871
119861 119892119880
119861])
are two interval grey uncertain linguistic variables where 1199041=
[1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] The continuous ordered weightedaveraging ((C-OWA) for short) operator which is developedby Yager [37] can be usefully applied to aggregate the greypart the greyness of the grey part would be transformed intoa real number and then the fuzzy part integrates with thegrey part that is to say the size of the interval uncertaingrey linguistic variables can be gotten through comparingthe size of [119904
1205721
1199041205731
] times 119891120588([(1 minus 119892
119880
119860) (1 minus 119892
119871
119860)]) and [119904
1205722
1199041205732
] times
119891120588([(1minus119892
119880
119861) (1minus119892
119871
119861)]) Assume the ordering value119876(119860
otimes(119909)) =
[1199041205721
1199041205731
] times 119891120588([(1 minus 119892
119880
119860) (1 minus 119892
119871
119860)]) and 119876(119861
otimes(119909)) = [119904
1205722
1199041205732
] times
119891120588([(1 minus 119892
119880
119861) (1 minus 119892
119871
119861)]) which can be obtained based on the
continuous ordered weighted averaging (C-OWA) operatorsuch as 119891
119901([119886 119887]) = int
1
0(119889120588(119910)119889119910)(119887 minus 119910(119887 minus 119886))119889119910 In order
to compare uncertain linguistic variables we use the degreeof possibility
The function 120588 is denoted as basic unit-interval mono-tonic (BUM) functions If (120575 ge 0) then 120588(119910) = 119910120575
Definition 3 (see [10]) Let 1199041= [1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] betwo uncertain linguistic variables and let len(119904
1) = 120573
1minus 1205721
4 Mathematical Problems in Engineering
and len(1199042) = 1205732minus 1205722 then the degree of possibility of 119904
1ge 1199042
is defined as follows
119901 (1199041ge 1199042)
=Max 0 len (119904
1) + len (119904
2) minusMax (120573
2minus 1205721 0)
len (1199041) + len (119904
2)
(3)
Particularly if both the uncertain linguistic variables 1199041and
1199042express precise information (ie if len(119904
1) + len(119904
2)) then
we define the degree of possibility of 1199041ge 1199042as follows
119901 (1199041ge 1199042) =
1 if 1199041gt 1199042
1
2if 1199041= 1199042
0 if 1199041lt 1199042
(4)
The operation rules of ranking are defined as follows
(1) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 1 then we have 119860
otimes(119909) gt
119861otimes(119909)
(2) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 0 then we have 119860
otimes(119909) lt
119861otimes(119909)
(3) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 1
then we have 119860otimes(119909) gt 119861
otimes(119909)
(4) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 12
then we have 119860otimes(119909) = 119861
otimes(119909)
(5) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 0
then we have 119860otimes(119909) lt 119861
otimes(119909)
The operation rules of the interval grey uncertain linguis-tic variables are defined as follows
(1) 119860otimes(119909) oplus 119861
otimes(119909) = (119904
1oplus 1199042 [(1 minus (1 minus 119892
119871
119860) times (1 minus 119892
119871
119861)) (1 minus
(1 minus 119892119880
119860) times (1 minus 119892
119880
119861))])
(2) 120582119860otimes(119909) = (120582119904
1 [119892119871
119860 119892119880
119860])
Definition 4 An IGULWAA operator of dimension 119899 is afunction IGULWAA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909119895)
=([119904sum119899
119895=1120596119895120572119895
119904sum119899
119895=1120596119895120573119895
] [
[
1minus
119899
prod
119895=1
(1minus119892119871
119895) 1minus
119899
prod
119895=1
(1minus119892119880
119895)]
]
)
(5)
Definition 5 An IGULOWA operator of dimension 119899 is afunction IGULOWA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum
119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ( [119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(6)
where there is a permutation such that 119860otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) that is 119860otimes(119909120591(119895)
) the 119895th largest value in the set119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)
3 Some Interval Grey UncertainLinguistic Correlated Ordered ArithmeticAveraging Operators
In multiple attribute group decision making the consideredattributes usually have different levels of importance andthus need to be assigned different weights Some operatorshave been introduced to aggregate the interval grey uncertainlinguistic variables together with independent weighted ele-ments but they only consider the addition of the importanceof individual elements However in some practical situationsthe elements in the interval grey uncertain linguistic variableshave some correlations with each other and thus it isnecessary to consider this issue For real decision makingproblems there is always some degree of interdependentcharacteristics between attributes Usually there is interac-tion among attributes of decision makers However thisassumption is too strong to match decision behaviors in thereal world The independence axiom generally cannot besatisfied Thus it is necessary to consider this issue
Let 119898(119909119894) (119894 = 1 2 119899) be the weight of the element
119909119894isin 119883 (119894 = 1 2 119899) where 119898 is a fuzzy measure defined
as follows
Definition 6 (see [38]) A fuzzymeasure119898 on the set119883 is a setfunction119898 120579(119909) rarr [0 1] satisfying the following axioms
(1) 119898(120601) = 0 119898(119883) = 1
(2) 119860 sube 119861 implies that119898(119860) le 119898(119861) for all 119860 119861 sube 119883
(3) 119898(119860cup119861) = 119898(119860)+119898(119861)+120588119898(119860)119898(119861) for all119860 119861 sube
119883 and 119860 cap 119861 = 120601 where 120588 isin (minus1infin)
Mathematical Problems in Engineering 5
Particularly if 120588 = 0 then the condition (3) reduces to theaxiom of additive measure
119898(119860 cup 119861) = 119898 (119860) + 119898 (119861) forall119860 119861 sube 119883 119860 cap 119861 = 120601
(7)
If all the elements in119883 are independent then we have
119898(119860) = sum
119909119894isin119860
119898(119909119894) forall119860 sube 119883 (8)
Based on Definition 5 in what follows we use the well-known Choquet integral [29] to develop an operator foraggregating the interval grey uncertain linguistic variableswith correlative weights
Definition 7 Let 119898 be a fuzzy measure on 119883 and Let119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899) be 119899 interval
grey uncertain linguistic variables then we call
(C) int119860otimes(119909) 119889119898
= IGULCOA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
(119898 (119861 (119909120591(119895)
)) minus 119898 (119861 (119909120591(119895minus1)
)))119860otimes(119909120591(119895)
)
(9)
the interval grey uncertain linguistic correlated ordered arith-metic averaging (IGULCOA) operator where (C) int119860
otimes119889119898
denotes the Choquet integral (120591(1) 120591(2) 120591(119899)) is a per-mutation of (1 2 119899) such that 119860
otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) 119861(119909120591(119895)
) = 119909120591(119896)
| 119896 le 119895 119896 ge 1 and119861(119909120591(0)
) = 120601 whose aggregated value is also an interval greyuncertain linguistic variable ((IGULV) for short)
Below we discuss two special cases of the IGULCOAoperator
(1) If 120588=0 then119898(119861 cup 119862) = 119898(119861)+119898(119862) and119898(119909120591(119895)
) =
119898(119861(119909120591(119895)
)) minus 119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 In this case theIGULCOAoperator (9) reduces to the interval grey uncertainlinguistic weighted arithmetic averaging (IGULWAA) opera-tor
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
119898(119909119895)119860otimes(119909119895)
= ([119904sum119899
119895=1119898(119909119895)120572119895
119904sum119899
119895=1119898(119909119895)120573119895
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
)
(10)
In particular if 119898(119909119895) = 1119899 for all 119895 = 1 2 119899 then
the IGULWAA operator (10) reduces to the interval grey
uncertain linguistic arithmetic averaging (IGULAA) opera-tor
IGULAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
[119904sum119899
119895=1(120572119895119899) 119904sum119899
119895=1(120573119895119899)]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
(11)
(2) If 119898(119861) = sum|119861|
119895=1120596119895 for all 119861 sube 119883 where |119861| is the
number of the elements in the set 119861 then 120596119895= 119898(119861(119909
120591(119895))) minus
119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 where 120596 = (1205961 1205962 120596
119899)119879
120596119895ge 0 119895 = 1 2 119899 and sum
119899
119895=1120596119895= 1 In this case the
IGULCOA operator (9) reduces to the interval grey uncer-tain linguistic ordered weighted arithmetic averaging (IGU-LOWA) operator
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ([119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(12)
In particular if 119898(119861) = |119861|119899 for all 119861 sube 119883 then boththe IGULCOA operator (9) and the IGULOWA operator (12)reduce to the IGULAA operator
Definition 8 Let 119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899)
be a collection of interval grey uncertain linguistic variablesand 120583 be a fuzzy measure on 119883 an induced interval greyuncertain linguistic correlated ordered arithmetic averaging(I-IGULCOA) operator is defined as follows
I-IGLCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
(13)
6 Mathematical Problems in Engineering
where (120590(1) 120590(2) 120590(119899)) is a permutation of (1 2 119899)for all 119895 = 1 2 119899 that is ⟨119906
119895 119860otimes(119909119895)⟩ is the 2-tuple with
119906120590(119895)
the 119895th largest values in the set 1199061 1199062 119906
119899 and 119906
119895in
⟨119906119895 119860otimes(119909119895)⟩ is referred to as the order-inducing variable and
119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) as the interval grey uncertain
linguistic value 119861(119909120590(119895)
) = 119909120590(119896)
| 119896 le 119895 119896 ge 1 for 119896 ge 1and 119861(119909
120590(0)) = 120601
In the above definition the reordering of the set ofvalues to aggregate (119860
otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) is induced by
the reordering of the set of values 1199061 1199062 119906
119899 associ-
ated with them which is based on their magnitude Themain difference between the IGULCOA operator and theI-IGULCOA operator resides in the reordering step of theargument variable In the case of IGULCOA operator thisreordering is based on the magnitude of the interval greyuncertain linguistic values to be aggregated whereas in thecase of I-IGULCOA operator an order-inducing variable isused as the criterion to induce that reordering Obviously inDefinition 8 if 119906
119895= 119860otimes(119909119895) for all 119895 then the I-IGULCOA
operator is reduced to the IGULCOA operatorSimilar to the other induced operators the I-IGULCOA
operator has the following properties
Theorem 9 (commutativity) Consider
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(14)
where (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(15)
Since (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩) we
have 119860otimes(119909119895) = 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) and then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(16)
Theorem 10 (idempotency) If 119860otimes(119909119895)= ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) =
119860otimes(119909) = ([119904
120572 119904120573] [119892119871 119892119880]) (119895 = 1 2 119899) for all 119895 then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= 119860otimes(119909) = ([119904120572 119904120573] [119892
119871 119892119880])
(17)
Proof Since 119860otimes(119909119895) = 119860otimes(119909) for all 119895 we have
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909)
= 119860otimes(119909)
(18)
Theorem 11 (monotonicity) If 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2
119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(19)
Mathematical Problems in Engineering 7
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061 119860otimes
1015840
(1199091)⟩ ⟨119906
2 119860otimes
1015840
(1199092)⟩
⟨119906119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(20)
Since 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) it follows that
119860otimes(119909120590(119895)
) le 119860otimes
1015840
(119909120590(119895)
) (119895 = 1 2 119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(21)
Theorem 12 (boundness) Consider
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(22)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
(23)
Since Min119895119860otimes(119909119895) le119860otimes(119909119895) le Max
119895119860otimes(119909119895) (119895 = 1 2 119899)
then119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
= Min119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
= Max119895
119860otimes(119909119895)
(24)
Hence
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(25)
4 An Approach to Multiple AttributeGroup Decision Making Method Based onthe IGULCOA Operator andthe I-IGULCOA Operator
In this section we will develop an approach to multipleattribute group decision making with interval grey uncertainlinguistic variables information and correlated weight asfollows
Let 119860 = 1198861 1198862 119886
119898 be a discrete set of alternatives
and let 119883 = 1199091 1199092 119909
119899 be the set of attributes119898 a fuzzy
measure on 119883 where 0 le 119898(1199091 119909
119895) le 1 119898(119883) =
1 and 119898(120601) = 0 Let 119863 = 1198891 1198892 119889
119905 be the set
of decision makers 120583 a fuzzy measure on 119863 where 0 le
120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0 Suppose that
119896= (119860otimes
(119896)
119894
(119909119895))119898times119899
is the interval grey uncertain linguistic
decision matrix where 119860otimes
(119896)
119894
(119909119895) indicates the attribute value
that the alternative 119886119894satisfies the attribute 119909
119895given by the
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
the difference between the fuzzy positive ideal solution andthe negative ideal solution to rank the order In order tosolve the grey fuzzy decision making Dang and Sifeng [18]developed the maximum entropy formulism to determineattribute weight and ranked the order of alternatives basedon the linear combination of fuzzy information and greyinformation Meng et al [19] proposed the grey degree andfuzzy degree with the interval numbers and then basedon this the mathematical model of interval-valued greyfuzzy comprehensive evaluation was established At last itsapplication to the selection of the preferred project was givenIn many real-life decision making problems the linguisticvariable is easier to express fuzzy information and closer toactual condition the research on linguistic decision makinghas witnessed rich achievements [3 11 20ndash27] Liu and Jin[4] defined the concept of the interval grey linguistic variablewhere the fuzzy part and the grey part took the form ofthe uncertain linguistic variable and the interval numberrespectively studied the operation rules and developed themultiple attribute decision making method based on theinterval grey linguistic variable Liu and Zhang [5] proposedthe interval grey linguistic variables weighted geometricaggregation (IGLWGA) operator and the interval grey lin-guistic variables ordered weighted geometric aggregation(IGLOWGA) operator and the interval grey linguistic vari-ables hybrid weighted geometric aggregation (IGLHWGA)operator and then suggested a method for solving multipleattribute group decision making based on those operatorsZhang and Wei [28] introduced the interval grey linguisticvariables ordered weighted aggregation (IGLOWA) operatorand then used the Choquet integral to develop the intervalgrey linguistic correlated ordered arithmetic aggregation(IGLCOA)operator and the interval grey linguistic correlatedordered geometric aggregation (IGLCOGA) operator Thoseoperators not only consider the importance of the elementsbut also can reflect the correlations among the elementsThen they developed an approach to multiple attributedecisionmaking problems with correlative weights where theattribute values are given in terms of interval grey linguisticvariables information based on those operators
The existing grey fuzzy multiple attribute group decisionmaking only considers the situation where all the elements inthe grey fuzzy set are independent However in many prac-tical situations the elements in the grey fuzzy set are usuallycorrelativeTherefore we need to find some new ways to dealwith the situations in which the decision data in questionare correlative and the weights are correlative The Choquetintegral [29] is a very useful way of measuring the expectedutility of an uncertain event and can be utilized to depict thecorrelations of the decision data under consideration Yager[30 31] introduced the idea of order-induced aggregation tothe Choquet aggregation operator and defined an inducedChoquet ordered weighted averaging (C-OWA) operatorwhich allowed the ordering of the arguments to be basedupon some other associated variables instead of orderingthe arguments based on their values Tan and Chen [32]developed the induced Choquet ordered averaging (I-COA)operator and applied it to aggregate fuzzy preference relationsin group decision making Xu [25] utilized the Choquet
integral to propose the interval-valued intuitionistic fuzzycorrelated averaging (IVIFCA) operator and the interval-valued intuitionistic fuzzy correlated geometric (IVIFCG)operator to aggregate interval-valued intuitionistic fuzzyinformation and applied them to a practical decision makingproblem involving the prioritization of information technol-ogy improvement projects Wei and Zhao [33] developed theinduced intuitionistic fuzzy correlated averaging (I-IFCA)operator and induced intuitionistic fuzzy correlated geomet-ric (I-IFCG) operator and developed to solve the MAGDMproblems in which both the attribute weights and the expertweights are usually correlative and attribute values take theform of intuitionistic fuzzy values
Motivated by the correlation properties of the Choquetintegral and the uncertain linguistic variables in this paperwe propose the interval grey uncertain linguistic correlatedordered arithmetic averaging (IGULCOA) operator and theinduced interval grey uncertain linguistic correlated orderedarithmetic averaging (I-IGULCOA) operator with intervalgrey uncertain linguistic variables information The promi-nent characteristic of those operators is that they cannot onlyconsider the importance of the elements or their orderedpositions but also reflect the correlations among the elementsor their ordered positions And we introduce those inducedcorrelated aggregating operators to deal with group decisionmaking problems The aim of this paper is to investigate theMAGDM problems in which both the attribute weights andthe expert weights are correlative and the attribute values takethe form of interval grey uncertain linguistic variables Inorder to do so the remainder of this paper is set out as followsIn the next section we introduce some basic concepts relatedto interval grey uncertain linguistic variables and some oper-ational laws of interval grey uncertain linguistic variablesIn Section 3 we have developed two interval grey uncertainlinguistic correlated aggregation operators the IGULCOAoperator and the I-IGULCOA operator In Section 4 we havedeveloped an approach to multiple attribute group decisionmaking in which both the attribute weights and the expertweights are correlative and the attribute values take theform of interval grey uncertain linguistic variables based onthe IGULCOA operator and the I-IGULCOA operator withinterval grey uncertain linguistic variables information InSection 5 an illustrative example is pointed out In Section 6we conclude the paper and give some remarks
2 Preliminaries
In this section we briefly review some basic concepts to beused throughout the paper
The linguistic approach is an approximate techniquewhich represents qualitative aspects as linguistic values bymeans of linguistic variables [34 35]
Suppose that 119878 = 119904119894| 119894 = minus119905 minus1 0 1 119905 is a
finite and totally ordered discrete term set whose cardinalityvalue is odd [35] Any label 119904
119894represents a possible value for
a linguistic variable (as shown in Figure 1) and it has thefollowing characteristics (1) the set is ordered as 119904
120572gt 119904120573 if
120572 gt 120573 and (2) there is the negative operator neg(119904120572) = 119904minus120572
Mathematical Problems in Engineering 3
FairSlightlygood Good
Verygood
Extremelygood
Extremelypoor
Verypoor Poor
Slightlypoor
119904minus4 119904minus3 119904minus2 119904minus1 1199040 1199041 1199042 1199043 1199044
Figure 1 The linguistic labels
We call this linguistic label set 119878 the additive linguistic scaleFor example a set of nine terms 119878 could be defined as follows
119878 = 119904minus4= extremely poor 119904
minus3= very poor 119904
minus2= poor
119904minus1= slightly poor 119904
0= fair 119904
1= slightly good
1199042= good 119904
3= very good s
4= extremely good
(1)
in which 119904119894lt 119904119895if 119894 lt 119895
To preserve all the given information we extend thediscrete term set 119878 to a continuous term set 119878 = 119904
120572| 120572 isin
[minus119905 119905] If 119904120572lt 119878 then we call 119904
120572an original linguistic term
otherwise we call 119904120572a virtual linguistic term In general the
decision maker uses the original linguistic terms to evaluatealternatives and the virtual linguistic terms can only appearin operation [10 36]
Let 119904 = [119904120572 119904120573] where 119904
120572 119904120573isin 119878 119904
120572and 119904120573are the lower
and the upper limits respectively we then call 119904 an uncertainlinguistic variable Let 119878 be the set of all the uncertainlinguistic variables [10]
Consider any three uncertain linguistic variables 119904 =
[119904120572 119904120573] 1199041= [1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] then we define theoperations 119904
1oplus 1199042and 120582119904 as follows
(1) 1199041oplus 1199042= [1199041205721
1199041205731
] oplus [1199041205722
1199041205732
] = [1199041205721
oplus 1199041205722
1199041205731
oplus 1199041205732
] =
[1199041205721+1205722
1199041205731+1205732
](2) 120582119904 = 120582[119904
120572 119904120573] = [119904120582120572 119904120582120573] where 120582 isin [0 1]
(3) 1199041oplus 1199042= 1199042oplus 1199041
(4) 120582(1199041oplus 1199042) = 120582119904
1oplus 1205821199042 where 120582 isin [0 1]
(5) (1205821+ 1205822)119904 = 120582
1119904 oplus 1205822119904 where 120582
1 1205822isin [0 1]
Definition 1 (see [14]) Let 119860(119909) be the fuzzy subset in thespace 119883 = 119909 if the membership degree 120583
119860(119909) of 119909 to 119860(119909)
is the grey in the interval [0 1] and its grey is 120584119860(119909) then
119860(119909) is called the grey fuzzy set in space119883 (GF set for short)denoted by 119860
otimes(119909) as follows
119860otimes(119909) = (119909 120583119860 (119909) 120584119860 (119909)) | 119909 isin 119883 (2)
The set pair mode is 119860otimes(119909) = (119860(119909) 119860
otimes(119909)) where 119860(119909) =
(119909 120583119860(119909)) | 119909 isin 119883 is called the fuzzy part of 119860
otimes(119909) and
119860otimes(119909) = (119909 120584
119860(119909)) | 119909 isin 119883 is called the grey part of 119860
otimes(119909)
So the grey fuzzy set is regarded as the generalization of thefuzzy set and the grey set
119904
119904120573
119904120572
119892119860119892119860119880119871
Figure 2 The interval grey linguistic variable
Definition 2 Let 119860otimes(119909)=(119860(119909) 119860
otimes(119909)) be the grey fuzzy num-
ber if its fuzzy part is an uncertain linguistic variable 119904 =
[119904120572 119904120573] where 119904
120572 119904120573isin 119878 where 119878 is a finite and totally ordered
discrete term set and its grey part 119860otimes(119909) is a closed interval
[119892119871
119860 119892119880
119860] then 119860
otimes(119909) is called the interval grey uncertain
linguistic variable as shown in Figure 2
Suppose that 119860otimes(119909) = (119904
1 [119892119871
119860 119892119880
119860]) 119861otimes(119909) = (119904
2 [119892119871
119861 119892119880
119861])
are two interval grey uncertain linguistic variables where 1199041=
[1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] The continuous ordered weightedaveraging ((C-OWA) for short) operator which is developedby Yager [37] can be usefully applied to aggregate the greypart the greyness of the grey part would be transformed intoa real number and then the fuzzy part integrates with thegrey part that is to say the size of the interval uncertaingrey linguistic variables can be gotten through comparingthe size of [119904
1205721
1199041205731
] times 119891120588([(1 minus 119892
119880
119860) (1 minus 119892
119871
119860)]) and [119904
1205722
1199041205732
] times
119891120588([(1minus119892
119880
119861) (1minus119892
119871
119861)]) Assume the ordering value119876(119860
otimes(119909)) =
[1199041205721
1199041205731
] times 119891120588([(1 minus 119892
119880
119860) (1 minus 119892
119871
119860)]) and 119876(119861
otimes(119909)) = [119904
1205722
1199041205732
] times
119891120588([(1 minus 119892
119880
119861) (1 minus 119892
119871
119861)]) which can be obtained based on the
continuous ordered weighted averaging (C-OWA) operatorsuch as 119891
119901([119886 119887]) = int
1
0(119889120588(119910)119889119910)(119887 minus 119910(119887 minus 119886))119889119910 In order
to compare uncertain linguistic variables we use the degreeof possibility
The function 120588 is denoted as basic unit-interval mono-tonic (BUM) functions If (120575 ge 0) then 120588(119910) = 119910120575
Definition 3 (see [10]) Let 1199041= [1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] betwo uncertain linguistic variables and let len(119904
1) = 120573
1minus 1205721
4 Mathematical Problems in Engineering
and len(1199042) = 1205732minus 1205722 then the degree of possibility of 119904
1ge 1199042
is defined as follows
119901 (1199041ge 1199042)
=Max 0 len (119904
1) + len (119904
2) minusMax (120573
2minus 1205721 0)
len (1199041) + len (119904
2)
(3)
Particularly if both the uncertain linguistic variables 1199041and
1199042express precise information (ie if len(119904
1) + len(119904
2)) then
we define the degree of possibility of 1199041ge 1199042as follows
119901 (1199041ge 1199042) =
1 if 1199041gt 1199042
1
2if 1199041= 1199042
0 if 1199041lt 1199042
(4)
The operation rules of ranking are defined as follows
(1) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 1 then we have 119860
otimes(119909) gt
119861otimes(119909)
(2) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 0 then we have 119860
otimes(119909) lt
119861otimes(119909)
(3) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 1
then we have 119860otimes(119909) gt 119861
otimes(119909)
(4) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 12
then we have 119860otimes(119909) = 119861
otimes(119909)
(5) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 0
then we have 119860otimes(119909) lt 119861
otimes(119909)
The operation rules of the interval grey uncertain linguis-tic variables are defined as follows
(1) 119860otimes(119909) oplus 119861
otimes(119909) = (119904
1oplus 1199042 [(1 minus (1 minus 119892
119871
119860) times (1 minus 119892
119871
119861)) (1 minus
(1 minus 119892119880
119860) times (1 minus 119892
119880
119861))])
(2) 120582119860otimes(119909) = (120582119904
1 [119892119871
119860 119892119880
119860])
Definition 4 An IGULWAA operator of dimension 119899 is afunction IGULWAA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909119895)
=([119904sum119899
119895=1120596119895120572119895
119904sum119899
119895=1120596119895120573119895
] [
[
1minus
119899
prod
119895=1
(1minus119892119871
119895) 1minus
119899
prod
119895=1
(1minus119892119880
119895)]
]
)
(5)
Definition 5 An IGULOWA operator of dimension 119899 is afunction IGULOWA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum
119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ( [119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(6)
where there is a permutation such that 119860otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) that is 119860otimes(119909120591(119895)
) the 119895th largest value in the set119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)
3 Some Interval Grey UncertainLinguistic Correlated Ordered ArithmeticAveraging Operators
In multiple attribute group decision making the consideredattributes usually have different levels of importance andthus need to be assigned different weights Some operatorshave been introduced to aggregate the interval grey uncertainlinguistic variables together with independent weighted ele-ments but they only consider the addition of the importanceof individual elements However in some practical situationsthe elements in the interval grey uncertain linguistic variableshave some correlations with each other and thus it isnecessary to consider this issue For real decision makingproblems there is always some degree of interdependentcharacteristics between attributes Usually there is interac-tion among attributes of decision makers However thisassumption is too strong to match decision behaviors in thereal world The independence axiom generally cannot besatisfied Thus it is necessary to consider this issue
Let 119898(119909119894) (119894 = 1 2 119899) be the weight of the element
119909119894isin 119883 (119894 = 1 2 119899) where 119898 is a fuzzy measure defined
as follows
Definition 6 (see [38]) A fuzzymeasure119898 on the set119883 is a setfunction119898 120579(119909) rarr [0 1] satisfying the following axioms
(1) 119898(120601) = 0 119898(119883) = 1
(2) 119860 sube 119861 implies that119898(119860) le 119898(119861) for all 119860 119861 sube 119883
(3) 119898(119860cup119861) = 119898(119860)+119898(119861)+120588119898(119860)119898(119861) for all119860 119861 sube
119883 and 119860 cap 119861 = 120601 where 120588 isin (minus1infin)
Mathematical Problems in Engineering 5
Particularly if 120588 = 0 then the condition (3) reduces to theaxiom of additive measure
119898(119860 cup 119861) = 119898 (119860) + 119898 (119861) forall119860 119861 sube 119883 119860 cap 119861 = 120601
(7)
If all the elements in119883 are independent then we have
119898(119860) = sum
119909119894isin119860
119898(119909119894) forall119860 sube 119883 (8)
Based on Definition 5 in what follows we use the well-known Choquet integral [29] to develop an operator foraggregating the interval grey uncertain linguistic variableswith correlative weights
Definition 7 Let 119898 be a fuzzy measure on 119883 and Let119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899) be 119899 interval
grey uncertain linguistic variables then we call
(C) int119860otimes(119909) 119889119898
= IGULCOA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
(119898 (119861 (119909120591(119895)
)) minus 119898 (119861 (119909120591(119895minus1)
)))119860otimes(119909120591(119895)
)
(9)
the interval grey uncertain linguistic correlated ordered arith-metic averaging (IGULCOA) operator where (C) int119860
otimes119889119898
denotes the Choquet integral (120591(1) 120591(2) 120591(119899)) is a per-mutation of (1 2 119899) such that 119860
otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) 119861(119909120591(119895)
) = 119909120591(119896)
| 119896 le 119895 119896 ge 1 and119861(119909120591(0)
) = 120601 whose aggregated value is also an interval greyuncertain linguistic variable ((IGULV) for short)
Below we discuss two special cases of the IGULCOAoperator
(1) If 120588=0 then119898(119861 cup 119862) = 119898(119861)+119898(119862) and119898(119909120591(119895)
) =
119898(119861(119909120591(119895)
)) minus 119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 In this case theIGULCOAoperator (9) reduces to the interval grey uncertainlinguistic weighted arithmetic averaging (IGULWAA) opera-tor
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
119898(119909119895)119860otimes(119909119895)
= ([119904sum119899
119895=1119898(119909119895)120572119895
119904sum119899
119895=1119898(119909119895)120573119895
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
)
(10)
In particular if 119898(119909119895) = 1119899 for all 119895 = 1 2 119899 then
the IGULWAA operator (10) reduces to the interval grey
uncertain linguistic arithmetic averaging (IGULAA) opera-tor
IGULAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
[119904sum119899
119895=1(120572119895119899) 119904sum119899
119895=1(120573119895119899)]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
(11)
(2) If 119898(119861) = sum|119861|
119895=1120596119895 for all 119861 sube 119883 where |119861| is the
number of the elements in the set 119861 then 120596119895= 119898(119861(119909
120591(119895))) minus
119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 where 120596 = (1205961 1205962 120596
119899)119879
120596119895ge 0 119895 = 1 2 119899 and sum
119899
119895=1120596119895= 1 In this case the
IGULCOA operator (9) reduces to the interval grey uncer-tain linguistic ordered weighted arithmetic averaging (IGU-LOWA) operator
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ([119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(12)
In particular if 119898(119861) = |119861|119899 for all 119861 sube 119883 then boththe IGULCOA operator (9) and the IGULOWA operator (12)reduce to the IGULAA operator
Definition 8 Let 119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899)
be a collection of interval grey uncertain linguistic variablesand 120583 be a fuzzy measure on 119883 an induced interval greyuncertain linguistic correlated ordered arithmetic averaging(I-IGULCOA) operator is defined as follows
I-IGLCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
(13)
6 Mathematical Problems in Engineering
where (120590(1) 120590(2) 120590(119899)) is a permutation of (1 2 119899)for all 119895 = 1 2 119899 that is ⟨119906
119895 119860otimes(119909119895)⟩ is the 2-tuple with
119906120590(119895)
the 119895th largest values in the set 1199061 1199062 119906
119899 and 119906
119895in
⟨119906119895 119860otimes(119909119895)⟩ is referred to as the order-inducing variable and
119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) as the interval grey uncertain
linguistic value 119861(119909120590(119895)
) = 119909120590(119896)
| 119896 le 119895 119896 ge 1 for 119896 ge 1and 119861(119909
120590(0)) = 120601
In the above definition the reordering of the set ofvalues to aggregate (119860
otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) is induced by
the reordering of the set of values 1199061 1199062 119906
119899 associ-
ated with them which is based on their magnitude Themain difference between the IGULCOA operator and theI-IGULCOA operator resides in the reordering step of theargument variable In the case of IGULCOA operator thisreordering is based on the magnitude of the interval greyuncertain linguistic values to be aggregated whereas in thecase of I-IGULCOA operator an order-inducing variable isused as the criterion to induce that reordering Obviously inDefinition 8 if 119906
119895= 119860otimes(119909119895) for all 119895 then the I-IGULCOA
operator is reduced to the IGULCOA operatorSimilar to the other induced operators the I-IGULCOA
operator has the following properties
Theorem 9 (commutativity) Consider
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(14)
where (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(15)
Since (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩) we
have 119860otimes(119909119895) = 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) and then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(16)
Theorem 10 (idempotency) If 119860otimes(119909119895)= ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) =
119860otimes(119909) = ([119904
120572 119904120573] [119892119871 119892119880]) (119895 = 1 2 119899) for all 119895 then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= 119860otimes(119909) = ([119904120572 119904120573] [119892
119871 119892119880])
(17)
Proof Since 119860otimes(119909119895) = 119860otimes(119909) for all 119895 we have
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909)
= 119860otimes(119909)
(18)
Theorem 11 (monotonicity) If 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2
119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(19)
Mathematical Problems in Engineering 7
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061 119860otimes
1015840
(1199091)⟩ ⟨119906
2 119860otimes
1015840
(1199092)⟩
⟨119906119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(20)
Since 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) it follows that
119860otimes(119909120590(119895)
) le 119860otimes
1015840
(119909120590(119895)
) (119895 = 1 2 119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(21)
Theorem 12 (boundness) Consider
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(22)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
(23)
Since Min119895119860otimes(119909119895) le119860otimes(119909119895) le Max
119895119860otimes(119909119895) (119895 = 1 2 119899)
then119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
= Min119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
= Max119895
119860otimes(119909119895)
(24)
Hence
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(25)
4 An Approach to Multiple AttributeGroup Decision Making Method Based onthe IGULCOA Operator andthe I-IGULCOA Operator
In this section we will develop an approach to multipleattribute group decision making with interval grey uncertainlinguistic variables information and correlated weight asfollows
Let 119860 = 1198861 1198862 119886
119898 be a discrete set of alternatives
and let 119883 = 1199091 1199092 119909
119899 be the set of attributes119898 a fuzzy
measure on 119883 where 0 le 119898(1199091 119909
119895) le 1 119898(119883) =
1 and 119898(120601) = 0 Let 119863 = 1198891 1198892 119889
119905 be the set
of decision makers 120583 a fuzzy measure on 119863 where 0 le
120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0 Suppose that
119896= (119860otimes
(119896)
119894
(119909119895))119898times119899
is the interval grey uncertain linguistic
decision matrix where 119860otimes
(119896)
119894
(119909119895) indicates the attribute value
that the alternative 119886119894satisfies the attribute 119909
119895given by the
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
FairSlightlygood Good
Verygood
Extremelygood
Extremelypoor
Verypoor Poor
Slightlypoor
119904minus4 119904minus3 119904minus2 119904minus1 1199040 1199041 1199042 1199043 1199044
Figure 1 The linguistic labels
We call this linguistic label set 119878 the additive linguistic scaleFor example a set of nine terms 119878 could be defined as follows
119878 = 119904minus4= extremely poor 119904
minus3= very poor 119904
minus2= poor
119904minus1= slightly poor 119904
0= fair 119904
1= slightly good
1199042= good 119904
3= very good s
4= extremely good
(1)
in which 119904119894lt 119904119895if 119894 lt 119895
To preserve all the given information we extend thediscrete term set 119878 to a continuous term set 119878 = 119904
120572| 120572 isin
[minus119905 119905] If 119904120572lt 119878 then we call 119904
120572an original linguistic term
otherwise we call 119904120572a virtual linguistic term In general the
decision maker uses the original linguistic terms to evaluatealternatives and the virtual linguistic terms can only appearin operation [10 36]
Let 119904 = [119904120572 119904120573] where 119904
120572 119904120573isin 119878 119904
120572and 119904120573are the lower
and the upper limits respectively we then call 119904 an uncertainlinguistic variable Let 119878 be the set of all the uncertainlinguistic variables [10]
Consider any three uncertain linguistic variables 119904 =
[119904120572 119904120573] 1199041= [1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] then we define theoperations 119904
1oplus 1199042and 120582119904 as follows
(1) 1199041oplus 1199042= [1199041205721
1199041205731
] oplus [1199041205722
1199041205732
] = [1199041205721
oplus 1199041205722
1199041205731
oplus 1199041205732
] =
[1199041205721+1205722
1199041205731+1205732
](2) 120582119904 = 120582[119904
120572 119904120573] = [119904120582120572 119904120582120573] where 120582 isin [0 1]
(3) 1199041oplus 1199042= 1199042oplus 1199041
(4) 120582(1199041oplus 1199042) = 120582119904
1oplus 1205821199042 where 120582 isin [0 1]
(5) (1205821+ 1205822)119904 = 120582
1119904 oplus 1205822119904 where 120582
1 1205822isin [0 1]
Definition 1 (see [14]) Let 119860(119909) be the fuzzy subset in thespace 119883 = 119909 if the membership degree 120583
119860(119909) of 119909 to 119860(119909)
is the grey in the interval [0 1] and its grey is 120584119860(119909) then
119860(119909) is called the grey fuzzy set in space119883 (GF set for short)denoted by 119860
otimes(119909) as follows
119860otimes(119909) = (119909 120583119860 (119909) 120584119860 (119909)) | 119909 isin 119883 (2)
The set pair mode is 119860otimes(119909) = (119860(119909) 119860
otimes(119909)) where 119860(119909) =
(119909 120583119860(119909)) | 119909 isin 119883 is called the fuzzy part of 119860
otimes(119909) and
119860otimes(119909) = (119909 120584
119860(119909)) | 119909 isin 119883 is called the grey part of 119860
otimes(119909)
So the grey fuzzy set is regarded as the generalization of thefuzzy set and the grey set
119904
119904120573
119904120572
119892119860119892119860119880119871
Figure 2 The interval grey linguistic variable
Definition 2 Let 119860otimes(119909)=(119860(119909) 119860
otimes(119909)) be the grey fuzzy num-
ber if its fuzzy part is an uncertain linguistic variable 119904 =
[119904120572 119904120573] where 119904
120572 119904120573isin 119878 where 119878 is a finite and totally ordered
discrete term set and its grey part 119860otimes(119909) is a closed interval
[119892119871
119860 119892119880
119860] then 119860
otimes(119909) is called the interval grey uncertain
linguistic variable as shown in Figure 2
Suppose that 119860otimes(119909) = (119904
1 [119892119871
119860 119892119880
119860]) 119861otimes(119909) = (119904
2 [119892119871
119861 119892119880
119861])
are two interval grey uncertain linguistic variables where 1199041=
[1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] The continuous ordered weightedaveraging ((C-OWA) for short) operator which is developedby Yager [37] can be usefully applied to aggregate the greypart the greyness of the grey part would be transformed intoa real number and then the fuzzy part integrates with thegrey part that is to say the size of the interval uncertaingrey linguistic variables can be gotten through comparingthe size of [119904
1205721
1199041205731
] times 119891120588([(1 minus 119892
119880
119860) (1 minus 119892
119871
119860)]) and [119904
1205722
1199041205732
] times
119891120588([(1minus119892
119880
119861) (1minus119892
119871
119861)]) Assume the ordering value119876(119860
otimes(119909)) =
[1199041205721
1199041205731
] times 119891120588([(1 minus 119892
119880
119860) (1 minus 119892
119871
119860)]) and 119876(119861
otimes(119909)) = [119904
1205722
1199041205732
] times
119891120588([(1 minus 119892
119880
119861) (1 minus 119892
119871
119861)]) which can be obtained based on the
continuous ordered weighted averaging (C-OWA) operatorsuch as 119891
119901([119886 119887]) = int
1
0(119889120588(119910)119889119910)(119887 minus 119910(119887 minus 119886))119889119910 In order
to compare uncertain linguistic variables we use the degreeof possibility
The function 120588 is denoted as basic unit-interval mono-tonic (BUM) functions If (120575 ge 0) then 120588(119910) = 119910120575
Definition 3 (see [10]) Let 1199041= [1199041205721
1199041205731
] and 1199042= [1199041205722
1199041205732
] betwo uncertain linguistic variables and let len(119904
1) = 120573
1minus 1205721
4 Mathematical Problems in Engineering
and len(1199042) = 1205732minus 1205722 then the degree of possibility of 119904
1ge 1199042
is defined as follows
119901 (1199041ge 1199042)
=Max 0 len (119904
1) + len (119904
2) minusMax (120573
2minus 1205721 0)
len (1199041) + len (119904
2)
(3)
Particularly if both the uncertain linguistic variables 1199041and
1199042express precise information (ie if len(119904
1) + len(119904
2)) then
we define the degree of possibility of 1199041ge 1199042as follows
119901 (1199041ge 1199042) =
1 if 1199041gt 1199042
1
2if 1199041= 1199042
0 if 1199041lt 1199042
(4)
The operation rules of ranking are defined as follows
(1) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 1 then we have 119860
otimes(119909) gt
119861otimes(119909)
(2) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 0 then we have 119860
otimes(119909) lt
119861otimes(119909)
(3) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 1
then we have 119860otimes(119909) gt 119861
otimes(119909)
(4) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 12
then we have 119860otimes(119909) = 119861
otimes(119909)
(5) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 0
then we have 119860otimes(119909) lt 119861
otimes(119909)
The operation rules of the interval grey uncertain linguis-tic variables are defined as follows
(1) 119860otimes(119909) oplus 119861
otimes(119909) = (119904
1oplus 1199042 [(1 minus (1 minus 119892
119871
119860) times (1 minus 119892
119871
119861)) (1 minus
(1 minus 119892119880
119860) times (1 minus 119892
119880
119861))])
(2) 120582119860otimes(119909) = (120582119904
1 [119892119871
119860 119892119880
119860])
Definition 4 An IGULWAA operator of dimension 119899 is afunction IGULWAA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909119895)
=([119904sum119899
119895=1120596119895120572119895
119904sum119899
119895=1120596119895120573119895
] [
[
1minus
119899
prod
119895=1
(1minus119892119871
119895) 1minus
119899
prod
119895=1
(1minus119892119880
119895)]
]
)
(5)
Definition 5 An IGULOWA operator of dimension 119899 is afunction IGULOWA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum
119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ( [119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(6)
where there is a permutation such that 119860otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) that is 119860otimes(119909120591(119895)
) the 119895th largest value in the set119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)
3 Some Interval Grey UncertainLinguistic Correlated Ordered ArithmeticAveraging Operators
In multiple attribute group decision making the consideredattributes usually have different levels of importance andthus need to be assigned different weights Some operatorshave been introduced to aggregate the interval grey uncertainlinguistic variables together with independent weighted ele-ments but they only consider the addition of the importanceof individual elements However in some practical situationsthe elements in the interval grey uncertain linguistic variableshave some correlations with each other and thus it isnecessary to consider this issue For real decision makingproblems there is always some degree of interdependentcharacteristics between attributes Usually there is interac-tion among attributes of decision makers However thisassumption is too strong to match decision behaviors in thereal world The independence axiom generally cannot besatisfied Thus it is necessary to consider this issue
Let 119898(119909119894) (119894 = 1 2 119899) be the weight of the element
119909119894isin 119883 (119894 = 1 2 119899) where 119898 is a fuzzy measure defined
as follows
Definition 6 (see [38]) A fuzzymeasure119898 on the set119883 is a setfunction119898 120579(119909) rarr [0 1] satisfying the following axioms
(1) 119898(120601) = 0 119898(119883) = 1
(2) 119860 sube 119861 implies that119898(119860) le 119898(119861) for all 119860 119861 sube 119883
(3) 119898(119860cup119861) = 119898(119860)+119898(119861)+120588119898(119860)119898(119861) for all119860 119861 sube
119883 and 119860 cap 119861 = 120601 where 120588 isin (minus1infin)
Mathematical Problems in Engineering 5
Particularly if 120588 = 0 then the condition (3) reduces to theaxiom of additive measure
119898(119860 cup 119861) = 119898 (119860) + 119898 (119861) forall119860 119861 sube 119883 119860 cap 119861 = 120601
(7)
If all the elements in119883 are independent then we have
119898(119860) = sum
119909119894isin119860
119898(119909119894) forall119860 sube 119883 (8)
Based on Definition 5 in what follows we use the well-known Choquet integral [29] to develop an operator foraggregating the interval grey uncertain linguistic variableswith correlative weights
Definition 7 Let 119898 be a fuzzy measure on 119883 and Let119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899) be 119899 interval
grey uncertain linguistic variables then we call
(C) int119860otimes(119909) 119889119898
= IGULCOA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
(119898 (119861 (119909120591(119895)
)) minus 119898 (119861 (119909120591(119895minus1)
)))119860otimes(119909120591(119895)
)
(9)
the interval grey uncertain linguistic correlated ordered arith-metic averaging (IGULCOA) operator where (C) int119860
otimes119889119898
denotes the Choquet integral (120591(1) 120591(2) 120591(119899)) is a per-mutation of (1 2 119899) such that 119860
otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) 119861(119909120591(119895)
) = 119909120591(119896)
| 119896 le 119895 119896 ge 1 and119861(119909120591(0)
) = 120601 whose aggregated value is also an interval greyuncertain linguistic variable ((IGULV) for short)
Below we discuss two special cases of the IGULCOAoperator
(1) If 120588=0 then119898(119861 cup 119862) = 119898(119861)+119898(119862) and119898(119909120591(119895)
) =
119898(119861(119909120591(119895)
)) minus 119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 In this case theIGULCOAoperator (9) reduces to the interval grey uncertainlinguistic weighted arithmetic averaging (IGULWAA) opera-tor
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
119898(119909119895)119860otimes(119909119895)
= ([119904sum119899
119895=1119898(119909119895)120572119895
119904sum119899
119895=1119898(119909119895)120573119895
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
)
(10)
In particular if 119898(119909119895) = 1119899 for all 119895 = 1 2 119899 then
the IGULWAA operator (10) reduces to the interval grey
uncertain linguistic arithmetic averaging (IGULAA) opera-tor
IGULAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
[119904sum119899
119895=1(120572119895119899) 119904sum119899
119895=1(120573119895119899)]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
(11)
(2) If 119898(119861) = sum|119861|
119895=1120596119895 for all 119861 sube 119883 where |119861| is the
number of the elements in the set 119861 then 120596119895= 119898(119861(119909
120591(119895))) minus
119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 where 120596 = (1205961 1205962 120596
119899)119879
120596119895ge 0 119895 = 1 2 119899 and sum
119899
119895=1120596119895= 1 In this case the
IGULCOA operator (9) reduces to the interval grey uncer-tain linguistic ordered weighted arithmetic averaging (IGU-LOWA) operator
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ([119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(12)
In particular if 119898(119861) = |119861|119899 for all 119861 sube 119883 then boththe IGULCOA operator (9) and the IGULOWA operator (12)reduce to the IGULAA operator
Definition 8 Let 119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899)
be a collection of interval grey uncertain linguistic variablesand 120583 be a fuzzy measure on 119883 an induced interval greyuncertain linguistic correlated ordered arithmetic averaging(I-IGULCOA) operator is defined as follows
I-IGLCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
(13)
6 Mathematical Problems in Engineering
where (120590(1) 120590(2) 120590(119899)) is a permutation of (1 2 119899)for all 119895 = 1 2 119899 that is ⟨119906
119895 119860otimes(119909119895)⟩ is the 2-tuple with
119906120590(119895)
the 119895th largest values in the set 1199061 1199062 119906
119899 and 119906
119895in
⟨119906119895 119860otimes(119909119895)⟩ is referred to as the order-inducing variable and
119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) as the interval grey uncertain
linguistic value 119861(119909120590(119895)
) = 119909120590(119896)
| 119896 le 119895 119896 ge 1 for 119896 ge 1and 119861(119909
120590(0)) = 120601
In the above definition the reordering of the set ofvalues to aggregate (119860
otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) is induced by
the reordering of the set of values 1199061 1199062 119906
119899 associ-
ated with them which is based on their magnitude Themain difference between the IGULCOA operator and theI-IGULCOA operator resides in the reordering step of theargument variable In the case of IGULCOA operator thisreordering is based on the magnitude of the interval greyuncertain linguistic values to be aggregated whereas in thecase of I-IGULCOA operator an order-inducing variable isused as the criterion to induce that reordering Obviously inDefinition 8 if 119906
119895= 119860otimes(119909119895) for all 119895 then the I-IGULCOA
operator is reduced to the IGULCOA operatorSimilar to the other induced operators the I-IGULCOA
operator has the following properties
Theorem 9 (commutativity) Consider
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(14)
where (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(15)
Since (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩) we
have 119860otimes(119909119895) = 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) and then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(16)
Theorem 10 (idempotency) If 119860otimes(119909119895)= ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) =
119860otimes(119909) = ([119904
120572 119904120573] [119892119871 119892119880]) (119895 = 1 2 119899) for all 119895 then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= 119860otimes(119909) = ([119904120572 119904120573] [119892
119871 119892119880])
(17)
Proof Since 119860otimes(119909119895) = 119860otimes(119909) for all 119895 we have
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909)
= 119860otimes(119909)
(18)
Theorem 11 (monotonicity) If 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2
119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(19)
Mathematical Problems in Engineering 7
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061 119860otimes
1015840
(1199091)⟩ ⟨119906
2 119860otimes
1015840
(1199092)⟩
⟨119906119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(20)
Since 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) it follows that
119860otimes(119909120590(119895)
) le 119860otimes
1015840
(119909120590(119895)
) (119895 = 1 2 119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(21)
Theorem 12 (boundness) Consider
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(22)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
(23)
Since Min119895119860otimes(119909119895) le119860otimes(119909119895) le Max
119895119860otimes(119909119895) (119895 = 1 2 119899)
then119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
= Min119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
= Max119895
119860otimes(119909119895)
(24)
Hence
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(25)
4 An Approach to Multiple AttributeGroup Decision Making Method Based onthe IGULCOA Operator andthe I-IGULCOA Operator
In this section we will develop an approach to multipleattribute group decision making with interval grey uncertainlinguistic variables information and correlated weight asfollows
Let 119860 = 1198861 1198862 119886
119898 be a discrete set of alternatives
and let 119883 = 1199091 1199092 119909
119899 be the set of attributes119898 a fuzzy
measure on 119883 where 0 le 119898(1199091 119909
119895) le 1 119898(119883) =
1 and 119898(120601) = 0 Let 119863 = 1198891 1198892 119889
119905 be the set
of decision makers 120583 a fuzzy measure on 119863 where 0 le
120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0 Suppose that
119896= (119860otimes
(119896)
119894
(119909119895))119898times119899
is the interval grey uncertain linguistic
decision matrix where 119860otimes
(119896)
119894
(119909119895) indicates the attribute value
that the alternative 119886119894satisfies the attribute 119909
119895given by the
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
and len(1199042) = 1205732minus 1205722 then the degree of possibility of 119904
1ge 1199042
is defined as follows
119901 (1199041ge 1199042)
=Max 0 len (119904
1) + len (119904
2) minusMax (120573
2minus 1205721 0)
len (1199041) + len (119904
2)
(3)
Particularly if both the uncertain linguistic variables 1199041and
1199042express precise information (ie if len(119904
1) + len(119904
2)) then
we define the degree of possibility of 1199041ge 1199042as follows
119901 (1199041ge 1199042) =
1 if 1199041gt 1199042
1
2if 1199041= 1199042
0 if 1199041lt 1199042
(4)
The operation rules of ranking are defined as follows
(1) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 1 then we have 119860
otimes(119909) gt
119861otimes(119909)
(2) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 0 then we have 119860
otimes(119909) lt
119861otimes(119909)
(3) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 1
then we have 119860otimes(119909) gt 119861
otimes(119909)
(4) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 12
then we have 119860otimes(119909) = 119861
otimes(119909)
(5) If 119875(119876(119860otimes(119909)) gt 119876(119861
otimes(119909))) = 12 and 119875(119904
1gt 1199042) = 0
then we have 119860otimes(119909) lt 119861
otimes(119909)
The operation rules of the interval grey uncertain linguis-tic variables are defined as follows
(1) 119860otimes(119909) oplus 119861
otimes(119909) = (119904
1oplus 1199042 [(1 minus (1 minus 119892
119871
119860) times (1 minus 119892
119871
119861)) (1 minus
(1 minus 119892119880
119860) times (1 minus 119892
119880
119861))])
(2) 120582119860otimes(119909) = (120582119904
1 [119892119871
119860 119892119880
119860])
Definition 4 An IGULWAA operator of dimension 119899 is afunction IGULWAA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909119895)
=([119904sum119899
119895=1120596119895120572119895
119904sum119899
119895=1120596119895120573119895
] [
[
1minus
119899
prod
119895=1
(1minus119892119871
119895) 1minus
119899
prod
119895=1
(1minus119892119880
119895)]
]
)
(5)
Definition 5 An IGULOWA operator of dimension 119899 is afunction IGULOWA Ω
119899rarr Ω which has associated
a set of weights or weighting vector 119908 = (1205961 1205962 120596
119899)
with 120596119895isin [0 1] sum
119899
119895=1120596119895= 1 and is defined to aggregate
a list of values 119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899) where 119860
otimes(119909119895) =
([119904120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) According to the following expression
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ( [119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(6)
where there is a permutation such that 119860otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) that is 119860otimes(119909120591(119895)
) the 119895th largest value in the set119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)
3 Some Interval Grey UncertainLinguistic Correlated Ordered ArithmeticAveraging Operators
In multiple attribute group decision making the consideredattributes usually have different levels of importance andthus need to be assigned different weights Some operatorshave been introduced to aggregate the interval grey uncertainlinguistic variables together with independent weighted ele-ments but they only consider the addition of the importanceof individual elements However in some practical situationsthe elements in the interval grey uncertain linguistic variableshave some correlations with each other and thus it isnecessary to consider this issue For real decision makingproblems there is always some degree of interdependentcharacteristics between attributes Usually there is interac-tion among attributes of decision makers However thisassumption is too strong to match decision behaviors in thereal world The independence axiom generally cannot besatisfied Thus it is necessary to consider this issue
Let 119898(119909119894) (119894 = 1 2 119899) be the weight of the element
119909119894isin 119883 (119894 = 1 2 119899) where 119898 is a fuzzy measure defined
as follows
Definition 6 (see [38]) A fuzzymeasure119898 on the set119883 is a setfunction119898 120579(119909) rarr [0 1] satisfying the following axioms
(1) 119898(120601) = 0 119898(119883) = 1
(2) 119860 sube 119861 implies that119898(119860) le 119898(119861) for all 119860 119861 sube 119883
(3) 119898(119860cup119861) = 119898(119860)+119898(119861)+120588119898(119860)119898(119861) for all119860 119861 sube
119883 and 119860 cap 119861 = 120601 where 120588 isin (minus1infin)
Mathematical Problems in Engineering 5
Particularly if 120588 = 0 then the condition (3) reduces to theaxiom of additive measure
119898(119860 cup 119861) = 119898 (119860) + 119898 (119861) forall119860 119861 sube 119883 119860 cap 119861 = 120601
(7)
If all the elements in119883 are independent then we have
119898(119860) = sum
119909119894isin119860
119898(119909119894) forall119860 sube 119883 (8)
Based on Definition 5 in what follows we use the well-known Choquet integral [29] to develop an operator foraggregating the interval grey uncertain linguistic variableswith correlative weights
Definition 7 Let 119898 be a fuzzy measure on 119883 and Let119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899) be 119899 interval
grey uncertain linguistic variables then we call
(C) int119860otimes(119909) 119889119898
= IGULCOA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
(119898 (119861 (119909120591(119895)
)) minus 119898 (119861 (119909120591(119895minus1)
)))119860otimes(119909120591(119895)
)
(9)
the interval grey uncertain linguistic correlated ordered arith-metic averaging (IGULCOA) operator where (C) int119860
otimes119889119898
denotes the Choquet integral (120591(1) 120591(2) 120591(119899)) is a per-mutation of (1 2 119899) such that 119860
otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) 119861(119909120591(119895)
) = 119909120591(119896)
| 119896 le 119895 119896 ge 1 and119861(119909120591(0)
) = 120601 whose aggregated value is also an interval greyuncertain linguistic variable ((IGULV) for short)
Below we discuss two special cases of the IGULCOAoperator
(1) If 120588=0 then119898(119861 cup 119862) = 119898(119861)+119898(119862) and119898(119909120591(119895)
) =
119898(119861(119909120591(119895)
)) minus 119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 In this case theIGULCOAoperator (9) reduces to the interval grey uncertainlinguistic weighted arithmetic averaging (IGULWAA) opera-tor
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
119898(119909119895)119860otimes(119909119895)
= ([119904sum119899
119895=1119898(119909119895)120572119895
119904sum119899
119895=1119898(119909119895)120573119895
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
)
(10)
In particular if 119898(119909119895) = 1119899 for all 119895 = 1 2 119899 then
the IGULWAA operator (10) reduces to the interval grey
uncertain linguistic arithmetic averaging (IGULAA) opera-tor
IGULAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
[119904sum119899
119895=1(120572119895119899) 119904sum119899
119895=1(120573119895119899)]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
(11)
(2) If 119898(119861) = sum|119861|
119895=1120596119895 for all 119861 sube 119883 where |119861| is the
number of the elements in the set 119861 then 120596119895= 119898(119861(119909
120591(119895))) minus
119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 where 120596 = (1205961 1205962 120596
119899)119879
120596119895ge 0 119895 = 1 2 119899 and sum
119899
119895=1120596119895= 1 In this case the
IGULCOA operator (9) reduces to the interval grey uncer-tain linguistic ordered weighted arithmetic averaging (IGU-LOWA) operator
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ([119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(12)
In particular if 119898(119861) = |119861|119899 for all 119861 sube 119883 then boththe IGULCOA operator (9) and the IGULOWA operator (12)reduce to the IGULAA operator
Definition 8 Let 119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899)
be a collection of interval grey uncertain linguistic variablesand 120583 be a fuzzy measure on 119883 an induced interval greyuncertain linguistic correlated ordered arithmetic averaging(I-IGULCOA) operator is defined as follows
I-IGLCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
(13)
6 Mathematical Problems in Engineering
where (120590(1) 120590(2) 120590(119899)) is a permutation of (1 2 119899)for all 119895 = 1 2 119899 that is ⟨119906
119895 119860otimes(119909119895)⟩ is the 2-tuple with
119906120590(119895)
the 119895th largest values in the set 1199061 1199062 119906
119899 and 119906
119895in
⟨119906119895 119860otimes(119909119895)⟩ is referred to as the order-inducing variable and
119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) as the interval grey uncertain
linguistic value 119861(119909120590(119895)
) = 119909120590(119896)
| 119896 le 119895 119896 ge 1 for 119896 ge 1and 119861(119909
120590(0)) = 120601
In the above definition the reordering of the set ofvalues to aggregate (119860
otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) is induced by
the reordering of the set of values 1199061 1199062 119906
119899 associ-
ated with them which is based on their magnitude Themain difference between the IGULCOA operator and theI-IGULCOA operator resides in the reordering step of theargument variable In the case of IGULCOA operator thisreordering is based on the magnitude of the interval greyuncertain linguistic values to be aggregated whereas in thecase of I-IGULCOA operator an order-inducing variable isused as the criterion to induce that reordering Obviously inDefinition 8 if 119906
119895= 119860otimes(119909119895) for all 119895 then the I-IGULCOA
operator is reduced to the IGULCOA operatorSimilar to the other induced operators the I-IGULCOA
operator has the following properties
Theorem 9 (commutativity) Consider
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(14)
where (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(15)
Since (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩) we
have 119860otimes(119909119895) = 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) and then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(16)
Theorem 10 (idempotency) If 119860otimes(119909119895)= ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) =
119860otimes(119909) = ([119904
120572 119904120573] [119892119871 119892119880]) (119895 = 1 2 119899) for all 119895 then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= 119860otimes(119909) = ([119904120572 119904120573] [119892
119871 119892119880])
(17)
Proof Since 119860otimes(119909119895) = 119860otimes(119909) for all 119895 we have
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909)
= 119860otimes(119909)
(18)
Theorem 11 (monotonicity) If 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2
119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(19)
Mathematical Problems in Engineering 7
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061 119860otimes
1015840
(1199091)⟩ ⟨119906
2 119860otimes
1015840
(1199092)⟩
⟨119906119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(20)
Since 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) it follows that
119860otimes(119909120590(119895)
) le 119860otimes
1015840
(119909120590(119895)
) (119895 = 1 2 119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(21)
Theorem 12 (boundness) Consider
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(22)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
(23)
Since Min119895119860otimes(119909119895) le119860otimes(119909119895) le Max
119895119860otimes(119909119895) (119895 = 1 2 119899)
then119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
= Min119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
= Max119895
119860otimes(119909119895)
(24)
Hence
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(25)
4 An Approach to Multiple AttributeGroup Decision Making Method Based onthe IGULCOA Operator andthe I-IGULCOA Operator
In this section we will develop an approach to multipleattribute group decision making with interval grey uncertainlinguistic variables information and correlated weight asfollows
Let 119860 = 1198861 1198862 119886
119898 be a discrete set of alternatives
and let 119883 = 1199091 1199092 119909
119899 be the set of attributes119898 a fuzzy
measure on 119883 where 0 le 119898(1199091 119909
119895) le 1 119898(119883) =
1 and 119898(120601) = 0 Let 119863 = 1198891 1198892 119889
119905 be the set
of decision makers 120583 a fuzzy measure on 119863 where 0 le
120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0 Suppose that
119896= (119860otimes
(119896)
119894
(119909119895))119898times119899
is the interval grey uncertain linguistic
decision matrix where 119860otimes
(119896)
119894
(119909119895) indicates the attribute value
that the alternative 119886119894satisfies the attribute 119909
119895given by the
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Particularly if 120588 = 0 then the condition (3) reduces to theaxiom of additive measure
119898(119860 cup 119861) = 119898 (119860) + 119898 (119861) forall119860 119861 sube 119883 119860 cap 119861 = 120601
(7)
If all the elements in119883 are independent then we have
119898(119860) = sum
119909119894isin119860
119898(119909119894) forall119860 sube 119883 (8)
Based on Definition 5 in what follows we use the well-known Choquet integral [29] to develop an operator foraggregating the interval grey uncertain linguistic variableswith correlative weights
Definition 7 Let 119898 be a fuzzy measure on 119883 and Let119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899) be 119899 interval
grey uncertain linguistic variables then we call
(C) int119860otimes(119909) 119889119898
= IGULCOA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
(119898 (119861 (119909120591(119895)
)) minus 119898 (119861 (119909120591(119895minus1)
)))119860otimes(119909120591(119895)
)
(9)
the interval grey uncertain linguistic correlated ordered arith-metic averaging (IGULCOA) operator where (C) int119860
otimes119889119898
denotes the Choquet integral (120591(1) 120591(2) 120591(119899)) is a per-mutation of (1 2 119899) such that 119860
otimes(119909120591(1)
) ge 119860otimes(119909120591(2)
) ge
sdot sdot sdot ge 119860otimes(119909120591(119899)
) 119861(119909120591(119895)
) = 119909120591(119896)
| 119896 le 119895 119896 ge 1 and119861(119909120591(0)
) = 120601 whose aggregated value is also an interval greyuncertain linguistic variable ((IGULV) for short)
Below we discuss two special cases of the IGULCOAoperator
(1) If 120588=0 then119898(119861 cup 119862) = 119898(119861)+119898(119862) and119898(119909120591(119895)
) =
119898(119861(119909120591(119895)
)) minus 119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 In this case theIGULCOAoperator (9) reduces to the interval grey uncertainlinguistic weighted arithmetic averaging (IGULWAA) opera-tor
IGULWAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
119899
⨁
119895=1
119898(119909119895)119860otimes(119909119895)
= ([119904sum119899
119895=1119898(119909119895)120572119895
119904sum119899
119895=1119898(119909119895)120573119895
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
)
(10)
In particular if 119898(119909119895) = 1119899 for all 119895 = 1 2 119899 then
the IGULWAA operator (10) reduces to the interval grey
uncertain linguistic arithmetic averaging (IGULAA) opera-tor
IGULAA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899))
=
[119904sum119899
119895=1(120572119895119899) 119904sum119899
119895=1(120573119895119899)]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
119895) 1 minus
119899
prod
119895=1
(1 minus 119892119880
119895)]
]
(11)
(2) If 119898(119861) = sum|119861|
119895=1120596119895 for all 119861 sube 119883 where |119861| is the
number of the elements in the set 119861 then 120596119895= 119898(119861(119909
120591(119895))) minus
119898(119861(119909120591(119895minus1)
)) 119895 = 1 2 119899 where 120596 = (1205961 1205962 120596
119899)119879
120596119895ge 0 119895 = 1 2 119899 and sum
119899
119895=1120596119895= 1 In this case the
IGULCOA operator (9) reduces to the interval grey uncer-tain linguistic ordered weighted arithmetic averaging (IGU-LOWA) operator
IGULOWA (119860otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) =
119899
⨁
119895=1
120596119895119860otimes(119909120591(119895)
)
= ([119904sum119899
119895=1120596119895120572120591(119895)
119904sum119899
119895=1120596119895120573120591(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120591(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120591(119895))]
]
)
(12)
In particular if 119898(119861) = |119861|119899 for all 119861 sube 119883 then boththe IGULCOA operator (9) and the IGULOWA operator (12)reduce to the IGULAA operator
Definition 8 Let 119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) (119895 = 1 2 119899)
be a collection of interval grey uncertain linguistic variablesand 120583 be a fuzzy measure on 119883 an induced interval greyuncertain linguistic correlated ordered arithmetic averaging(I-IGULCOA) operator is defined as follows
I-IGLCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
(13)
6 Mathematical Problems in Engineering
where (120590(1) 120590(2) 120590(119899)) is a permutation of (1 2 119899)for all 119895 = 1 2 119899 that is ⟨119906
119895 119860otimes(119909119895)⟩ is the 2-tuple with
119906120590(119895)
the 119895th largest values in the set 1199061 1199062 119906
119899 and 119906
119895in
⟨119906119895 119860otimes(119909119895)⟩ is referred to as the order-inducing variable and
119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) as the interval grey uncertain
linguistic value 119861(119909120590(119895)
) = 119909120590(119896)
| 119896 le 119895 119896 ge 1 for 119896 ge 1and 119861(119909
120590(0)) = 120601
In the above definition the reordering of the set ofvalues to aggregate (119860
otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) is induced by
the reordering of the set of values 1199061 1199062 119906
119899 associ-
ated with them which is based on their magnitude Themain difference between the IGULCOA operator and theI-IGULCOA operator resides in the reordering step of theargument variable In the case of IGULCOA operator thisreordering is based on the magnitude of the interval greyuncertain linguistic values to be aggregated whereas in thecase of I-IGULCOA operator an order-inducing variable isused as the criterion to induce that reordering Obviously inDefinition 8 if 119906
119895= 119860otimes(119909119895) for all 119895 then the I-IGULCOA
operator is reduced to the IGULCOA operatorSimilar to the other induced operators the I-IGULCOA
operator has the following properties
Theorem 9 (commutativity) Consider
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(14)
where (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(15)
Since (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩) we
have 119860otimes(119909119895) = 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) and then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(16)
Theorem 10 (idempotency) If 119860otimes(119909119895)= ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) =
119860otimes(119909) = ([119904
120572 119904120573] [119892119871 119892119880]) (119895 = 1 2 119899) for all 119895 then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= 119860otimes(119909) = ([119904120572 119904120573] [119892
119871 119892119880])
(17)
Proof Since 119860otimes(119909119895) = 119860otimes(119909) for all 119895 we have
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909)
= 119860otimes(119909)
(18)
Theorem 11 (monotonicity) If 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2
119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(19)
Mathematical Problems in Engineering 7
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061 119860otimes
1015840
(1199091)⟩ ⟨119906
2 119860otimes
1015840
(1199092)⟩
⟨119906119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(20)
Since 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) it follows that
119860otimes(119909120590(119895)
) le 119860otimes
1015840
(119909120590(119895)
) (119895 = 1 2 119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(21)
Theorem 12 (boundness) Consider
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(22)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
(23)
Since Min119895119860otimes(119909119895) le119860otimes(119909119895) le Max
119895119860otimes(119909119895) (119895 = 1 2 119899)
then119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
= Min119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
= Max119895
119860otimes(119909119895)
(24)
Hence
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(25)
4 An Approach to Multiple AttributeGroup Decision Making Method Based onthe IGULCOA Operator andthe I-IGULCOA Operator
In this section we will develop an approach to multipleattribute group decision making with interval grey uncertainlinguistic variables information and correlated weight asfollows
Let 119860 = 1198861 1198862 119886
119898 be a discrete set of alternatives
and let 119883 = 1199091 1199092 119909
119899 be the set of attributes119898 a fuzzy
measure on 119883 where 0 le 119898(1199091 119909
119895) le 1 119898(119883) =
1 and 119898(120601) = 0 Let 119863 = 1198891 1198892 119889
119905 be the set
of decision makers 120583 a fuzzy measure on 119863 where 0 le
120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0 Suppose that
119896= (119860otimes
(119896)
119894
(119909119895))119898times119899
is the interval grey uncertain linguistic
decision matrix where 119860otimes
(119896)
119894
(119909119895) indicates the attribute value
that the alternative 119886119894satisfies the attribute 119909
119895given by the
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
where (120590(1) 120590(2) 120590(119899)) is a permutation of (1 2 119899)for all 119895 = 1 2 119899 that is ⟨119906
119895 119860otimes(119909119895)⟩ is the 2-tuple with
119906120590(119895)
the 119895th largest values in the set 1199061 1199062 119906
119899 and 119906
119895in
⟨119906119895 119860otimes(119909119895)⟩ is referred to as the order-inducing variable and
119860otimes(119909119895) = ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) as the interval grey uncertain
linguistic value 119861(119909120590(119895)
) = 119909120590(119896)
| 119896 le 119895 119896 ge 1 for 119896 ge 1and 119861(119909
120590(0)) = 120601
In the above definition the reordering of the set ofvalues to aggregate (119860
otimes(1199091) 119860otimes(1199092) 119860
otimes(119909119899)) is induced by
the reordering of the set of values 1199061 1199062 119906
119899 associ-
ated with them which is based on their magnitude Themain difference between the IGULCOA operator and theI-IGULCOA operator resides in the reordering step of theargument variable In the case of IGULCOA operator thisreordering is based on the magnitude of the interval greyuncertain linguistic values to be aggregated whereas in thecase of I-IGULCOA operator an order-inducing variable isused as the criterion to induce that reordering Obviously inDefinition 8 if 119906
119895= 119860otimes(119909119895) for all 119895 then the I-IGULCOA
operator is reduced to the IGULCOA operatorSimilar to the other induced operators the I-IGULCOA
operator has the following properties
Theorem 9 (commutativity) Consider
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(14)
where (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(15)
Since (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨1199061015840
2 119860otimes
1015840
(1199092)⟩ ⟨119906
1015840
119899 119860otimes
1015840
(119909119899)⟩) is any
permutation of (⟨1199061 119860otimes(1199091)⟩ ⟨1199062 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩) we
have 119860otimes(119909119895) = 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) and then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(16)
Theorem 10 (idempotency) If 119860otimes(119909119895)= ([119904
120572119895
119904120573119895
] [119892119871
119895 119892119880
119895]) =
119860otimes(119909) = ([119904
120572 119904120573] [119892119871 119892119880]) (119895 = 1 2 119899) for all 119895 then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
= 119860otimes(119909) = ([119904120572 119904120573] [119892
119871 119892119880])
(17)
Proof Since 119860otimes(119909119895) = 119860otimes(119909) for all 119895 we have
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909)
= 119860otimes(119909)
(18)
Theorem 11 (monotonicity) If 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2
119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨11990610158401 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(19)
Mathematical Problems in Engineering 7
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061 119860otimes
1015840
(1199091)⟩ ⟨119906
2 119860otimes
1015840
(1199092)⟩
⟨119906119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(20)
Since 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) it follows that
119860otimes(119909120590(119895)
) le 119860otimes
1015840
(119909120590(119895)
) (119895 = 1 2 119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(21)
Theorem 12 (boundness) Consider
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(22)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
(23)
Since Min119895119860otimes(119909119895) le119860otimes(119909119895) le Max
119895119860otimes(119909119895) (119895 = 1 2 119899)
then119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
= Min119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
= Max119895
119860otimes(119909119895)
(24)
Hence
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(25)
4 An Approach to Multiple AttributeGroup Decision Making Method Based onthe IGULCOA Operator andthe I-IGULCOA Operator
In this section we will develop an approach to multipleattribute group decision making with interval grey uncertainlinguistic variables information and correlated weight asfollows
Let 119860 = 1198861 1198862 119886
119898 be a discrete set of alternatives
and let 119883 = 1199091 1199092 119909
119899 be the set of attributes119898 a fuzzy
measure on 119883 where 0 le 119898(1199091 119909
119895) le 1 119898(119883) =
1 and 119898(120601) = 0 Let 119863 = 1198891 1198892 119889
119905 be the set
of decision makers 120583 a fuzzy measure on 119863 where 0 le
120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0 Suppose that
119896= (119860otimes
(119896)
119894
(119909119895))119898times119899
is the interval grey uncertain linguistic
decision matrix where 119860otimes
(119896)
119894
(119909119895) indicates the attribute value
that the alternative 119886119894satisfies the attribute 119909
119895given by the
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
I-IGULCOA (⟨1199061 119860otimes
1015840
(1199091)⟩ ⟨119906
2 119860otimes
1015840
(1199092)⟩
⟨119906119899 119860otimes
1015840
(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
1015840
(119909120590(119895)
)
(20)
Since 119860otimes(119909119895) le 119860otimes
1015840
(119909119895) (119895 = 1 2 119899) it follows that
119860otimes(119909120590(119895)
) le 119860otimes
1015840
(119909120590(119895)
) (119895 = 1 2 119899) then
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
le I-IGULCOA (⟨1199061015840
1 119860otimes
1015840
(1199091)⟩ ⟨119906
1015840
2 119860otimes
1015840
(1199092)⟩
⟨1199061015840
119899 119860otimes
1015840
(119909119899)⟩)
(21)
Theorem 12 (boundness) Consider
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(22)
Proof Let
I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩ ⟨119906
119899 119860otimes(119909119899)⟩)
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
(23)
Since Min119895119860otimes(119909119895) le119860otimes(119909119895) le Max
119895119860otimes(119909119895) (119895 = 1 2 119899)
then119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes(119909120590(119895)
)
le
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Min119895
119860otimes(119909119895)
= Min119895
119860otimes(119909119895)
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))Max119895
119860otimes(119909119895)
= Max119895
119860otimes(119909119895)
(24)
Hence
Min119895
119860otimes(119909119895)
le I-IGULCOA (⟨1199061 119860otimes(1199091)⟩ ⟨119906
2 119860otimes(1199092)⟩
⟨119906119899 119860otimes(119909119899)⟩)
le Max119895
119860otimes(119909119895)
(25)
4 An Approach to Multiple AttributeGroup Decision Making Method Based onthe IGULCOA Operator andthe I-IGULCOA Operator
In this section we will develop an approach to multipleattribute group decision making with interval grey uncertainlinguistic variables information and correlated weight asfollows
Let 119860 = 1198861 1198862 119886
119898 be a discrete set of alternatives
and let 119883 = 1199091 1199092 119909
119899 be the set of attributes119898 a fuzzy
measure on 119883 where 0 le 119898(1199091 119909
119895) le 1 119898(119883) =
1 and 119898(120601) = 0 Let 119863 = 1198891 1198892 119889
119905 be the set
of decision makers 120583 a fuzzy measure on 119863 where 0 le
120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0 Suppose that
119896= (119860otimes
(119896)
119894
(119909119895))119898times119899
is the interval grey uncertain linguistic
decision matrix where 119860otimes
(119896)
119894
(119909119895) indicates the attribute value
that the alternative 119886119894satisfies the attribute 119909
119895given by the
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
decision maker 119889119896 and 119906
(119896)
119895indicates the inducing values
that the decision maker 119889119896satisfies the attribute 119909
119895 119894 =
1 2 119898 119895 = 1 2 119899 and 119896 = 1 2 119905In the following we apply the IGULCOA and I-
IGULCOA operators to multiple attribute group decisionmaking based on interval grey uncertain linguistic infor-mation and correlated weight The method involves thefollowing steps
Step 1 Utilize the decision information given in matrix 119896
and the I-IGULCOA operator which has correlative weightsvector
119903(119896)
119894
= I-IGLCOA(⟨1199061 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
(⟨119906119899 119860otimes
(119896)
119894
(119909119899)⟩))
=
119899
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
= ([119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120572120590(119895)
119904sum119899
119895=1(119898(119861(119909
120590(119895)))minus119898(119861(119909
120590(119895minus1))))120573120590(119895)
]
[
[
1 minus
119899
prod
119895=1
(1 minus 119892119871
120590(119895)) 1 minus
119899
prod
119895=1
(1 minus 119892119880
120590(119895))]
]
)
119894 = 1 2 119898 119895 = 1 2 119899 119896 = 1 2 119905
(26)
to aggregate all the decision matrices 119896(119896 = 1 2 119905) into
a collective decision matrix = (119903(119896)
119894)119898times119905
where 119898 is a fuzzymeasure on 119883 0 le 119898(119909
1 119909
119895) le 1 119898(119883) = 1 and
119898(120601) = 0 (119895 = 1 2 119899)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903
(119905)
119894)
=
119905
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
119894
119894 = 1 2 119898
(27)
to derive the collective overall preference values 119903119894(119894 =
1 2 119898) of the alternative 119860119894 where 120583 is a fuzzy measure
on119863 0 le 120583(1198891 119889
119896) le 1 120583(119863) = 1 and 120583(120601) = 0
Step 3 Let the basic unit-interval monotonic (BUM)function be 120588(119910) = 119910
120575 We get the size of the collective
overall pre-ference values 119894(119894 = 1 2 119898) by using
119876 (119894)
= [119904120572119894
119904120573119894
] times 119891120588([(1 minus 119892
119880
119894) (1 minus 119892
119871
119894)])
= [119904120572119894
119904120573119894
] times int
1
0
119889120588 (119910)
119889119910[(1 minus 119892
119871
119894) minus 119910 (119892
119880
119894minus 119892119871
119894)] 119889119910
(28)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 119898) we first compare each 119876(
119894) with all
119876(119894) (119894 = 1 2 119898) by using (9) For simplicity we let
119901119894119895= 119901(119876(
119894) ge 119876(
119895)) then we develop a complementary
matrix as 119875 = (119901119894119895)119899times119899
where 119901119894119895ge 0 119901
119894119895+ 119901119895119894= 1 119901
119894119894= 12
119894 119895 = 1 2 119899 Summing all elements in each line of matrix119875 we have119901
119894= sum119899
119895=1119901119894119895 119894 = 1 2 119899 Consider
119901 (119876 (119894) ge 119876 (
119895))
=
Max 0 len (119904119894) + len (119904
119895) minusMax (120573
119895minus 120572119894 0)
len (119904119894) + len (119904
119895)
(29)
Then we rank 119876(119894) (119894 = 1 2 119898) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 119898)
Step 5 The ranking of the alternatives can be gained and thebest one can be found out
5 Illustrative Example
Let us suppose there is an investment company which wantsto invest a sum of money in the best option There is a panelwith four possible alternatives to invest the money
(1) 1198861is a car company
(2) 1198862is a food company
(3) 1198863is a computer company
(4) 1198864is an arms company
The investment company must take a decision according tothe following four attributes
(1) 1199091is the risk analysis
(2) 1199092is the growth analysis
(3) 1199093is the social-political impact analysis
(4) 1199094is the environmental impact analysis
The four possible alternatives 1198861 1198862 1198863 1198864 are to be eval-
uated by the three decision makers 1198891 1198892 1198893 under the
above four attributes and construct respectively the inducing
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 1 Inducing variables 119880
Experts Attribute(1199091)
Attribute(1199092)
Attribute(1199093)
Attribute(1199094)
1198631 17 15 22 12
1198632 15 22 25 13
1198633 16 21 25 28
variables 119880 which are shown in Table 1 And the decisionmatrices
119896= (119860otimes
(119896)
119894
(119909119895))4times4
are as follows
1=
[[[
[
([1199044 1199045] [02 05]) ([119904
3 1199044] [03 04])
([1199043 1199045] [04 05]) ([119904
4 1199045] [04 05])
([1199043 1199046] [02 04]) ([119904
3 1199044] [02 04])
([1199044 1199046] [03 06]) ([119904
2 1199045] [02 05])
([1199044 1199045] [03 05]) ([119904
2 1199045] [02 04])
([1199042 1199044] [02 04]) ([119904
3 1199044] [03 05])
([1199043 1199044] [03 04]) ([119904
4 1199045] [02 05])
([1199042 1199044] [02 05]) ([119904
2 1199044] [03 04])
]]]
]
2=[[[
[
([1199043 1199045] [01 03]) ([119904
2 1199044] [02 03])
([1199044 1199045] [03 05]) ([119904
2 1199045] [03 04])
([1199043 1199044] [02 04]) ([119904
3 1199044] [02 03])
([1199044 1199045] [03 04]) ([119904
3 1199045] [04 05])
([1199042 1199043] [02 03]) ([119904
3 1199046] [04 05])
([1199043 1199044] [02 04]) ([119904
2 1199043] [02 04])
([1199041 1199043] [03 04]) ([119904
2 1199043] [03 04])
([1199041 1199044] [03 05]) ([119904
3 1199044] [02 05])
]]]
]
3=[[[
[
([1199044 1199045] [02 04]) ([119904
2 1199044] [03 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [03 04])
([1199043 1199044] [02 03]) ([119904
3 1199045] [03 04])
([1199042 1199045] [02 03]) ([119904
2 1199045] [01 03])
([1199043 1199044] [04 05]) ([119904
3 1199044] [02 03])
([1199041 1199044] [01 04]) ([119904
2 1199045] [01 03])
([1199041 1199043] [01 05]) ([119904
3 1199044] [02 03])
([1199043 1199044] [03 04]) ([119904
4 1199045] [01 04])
]]]
]
(30)
The experts evaluate the enterprises 119886119894(119894 = 1 2 3 4) in
relation to the factors 119909119895(119895 = 1 2 3 4) Let
119898(1199091) = 03 119898 (119909
2) = 035
119898 (1199093) = 030 119898 (119909
4) = 022
119898 (1199091 1199092) = 07 119898 (119909
1 1199093) = 06
119898 (1199091 1199094) = 055 119898 (119909
2 1199093) = 05
119898 (1199092 1199094) = 045 119898 (119909
3 1199094) = 04
119898 (1199091 1199092 1199093) = 082 119898 (119909
1 1199092 1199094) = 087
119898 (1199091 1199093 1199094) = 075 119898 (119909
2 1199093 1199094) = 06
119898 (1199091 1199092 1199093 1199094) = 1
(31)
The fuzzymeasure of weighting vector of decisionmakers119863119896(119896 = 1 2 3) and sets of decision makes119863 as follows
120583 (120601) = 0 120583 (1198631) = 025
120583 (1198632) = 035 120583 (119863
3) = 03
120583 (1198631 1198632) = 07 120583 (119863
1 1198633) = 065
120583 (1198632 1198633) = 05 120583 (119863
1 1198632 1198633) = 1
(32)
Step 1 Utilize the decision information given in matrix119896(119896 = 1 2 3) inducing variables 119880 and the I-IGULCOA
operator which has correlative weights vector
119903(119896)
119894= I-IGULCOA(⟨119906
1 119860otimes
(119896)
119894
(1199091)⟩ ⟨119906
2 119860otimes
(119896)
119894
(1199092)⟩
⟨1199063 119860otimes
(119896)
119894
(1199093)⟩ ⟨119906
4 119860otimes
(119896)
119894
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(119896)
119894
(119909120590(119895)
)
(33)
where 119906(1)3
ge 119906(1)
1ge 119906(1)
2ge 119906(1)
4 then 119860
otimes
(1)
1
(119909120590(1)
) = 119860otimes
(1)
1
(1199093)
119860otimes
(1)
1
(119909120590(2)
) = 119860otimes
(1)
1
(1199091) 119860otimes
(1)
1
(119909120590(3)
) = 119860otimes
(1)
1
(1199092) 119860otimes
(1)
1
(119909120590(4)
) =
119860otimes
(1)
1
(1199094) We have
119898(119861 (119909120591(1)
)) = 119898 (1199093) = 03
119898 (119861 (119909120591(2)
)) = 119898 (1199091 1199093) = 06
119898 (119861 (119909120591(3)
)) = 119898 (1199091 1199092 1199093) = 082
119898 (119861 (119909120591(4)
)) = 119898 (1199091 1199092 1199093 1199094) = 1
119903(1)
1= I-IGULCOA(⟨119906
1 119860otimes
(1)
1
(1199091)⟩ ⟨119906
2 119860otimes
(1)
1
(1199092)⟩
⟨1199063 119860otimes
(1)
1
(1199093)⟩ ⟨119906
4 119860otimes
(1)
1
(1199094)⟩)
=
4
⨁
119895=1
(119898 (119861 (119909120590(119895)
)) minus 119898 (119861 (119909120590(119895minus1)
)))119860otimes
(1)
1
(119909120590(119895)
)
= ([119904342
119904478
] [0686 0910])
(34)
Similarly we have
119903(1)
1= ([119904342
119904478
] [0686 0910])
119903(1)
2= ([119904292
119904452
] [0798 0925])
119903(1)
3= ([119904318
119904478
] [0642 0892])
119903(1)
4= ([119904260
119904482
] [0686 0940])
119903(2)
1= ([119904250
119904438
] [0654 0829])
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
119903(2)
2= ([119904294
119904434
] [0686 0892])
119903(2)
3= ([119904222
119904352
] [0686 0849])
119903(2)
4= ([119904272
119904452
] [0765 0925])
119903(3)
1= ([119904320
119904440
] [0731 0853])
119903(3)
2= ([119904262
119904442
] [0603 0849])
119903(3)
3= ([119904264
119904402
] [0597 0853])
119903(3)
4= ([119904262
119904482
] [0546 0824])
(35)
Step 2 Utilize the decision information given inmatrix andthe IGULCOA operator
119894= IGULCOA (119903
(1)
119894 119903(2)
119894 119903(3)
119894)
=
3
⨁
119896=1
(120583 (119861 (119889120591(119896)
)) minus 120583 (119861 (119889120591(119896minus1)
))) 119903(119896)
120591(119894)
(36)
We obtain the collective overall preference values 119894of the
decision makers119863119896(119896 = 1 2 3) Consider
1= ([1199043091
1199044509
] [0971 0998])
2= ([1199042834
1199044454
] [0975 0999])
3= ([1199042709
1199044149
] [0955 0998])
4= ([1199042648
1199044715
] [0967 0999])
(37)
Step 3 Let the basic unit-interval monotonic (BUM) func-tion be 120588(119910) = 119910
2 We get the size of the collective overallpreference values 119876(
119894) (119894 = 1 2 3 4) by using
119876 (1) = [119904
0062 1199040091
]
119876 (2) = [119904
0049 1199040076
]
119876 (3) = [119904
0084 1199040129
]
119876 (4) = [119904
0060 1199040106
]
(38)
Step 4 To rank these collective overall preference values119876(119894) (119894 = 1 2 3 4) by using (29) and then construct a
complementary matrix
119875 =[[[
[
0500 0752 0095 0415
0248 0500 0000 0223
0905 1000 0500 0755
0585 0777 0245 0500
]]]
]
(39)
Summing all elements in each line of matrix 119875 we have
1199011= 0230 119901
2= 0164
1199013= 0347 119901
4= 0259
(40)
Then we rank 119876(119894) (119894 = 1 2 3 4) in a descending order
in accordance with the values of 119901119894(119894 = 1 2 3 4) 119876(
3) ≻
119876(4) ≻ 119876(
1) ≻ 119876(
2) so we get the
3≻ 4≻ 1≻ 2
Step 5 Rank all the alternatives 119886119894(119894 = 1 2 3 4) in accor-
dance with 119894(119894 = 1 2 3 4) 119886
3≻ 1198864≻ 1198861≻ 1198862 Thus the best
alternative is 1198863
6 Conclusion
In this paper we use the Choquet integral to propose theinterval grey uncertain linguistic correlated ordered arith-metic aggregation (IGULCOA) operator and the inducedinterval grey uncertain linguistic correlated ordered geo-metric aggregation (I-IGULCOA) operator which are usedto discuss the correlative interval grey uncertain linguisticvariables Furthermore we also analyze the relation betweenit and some known operators and develop an approach tomultiple attribute group decisionmaking with the correlativeattributes weights and the correlative expert weights inwhich the attribute values are given in terms of the inducedinterval grey uncertain linguistic variables based on the inter-val grey uncertain linguistic correlated ordered arithmeticaggregation (IGULCOA) operator and the induced inter-val grey uncertain linguistic correlated ordered geometricaggregation (I-IGULCOA) operator Finally an illustrativeexample has been given to show the developed method Theapplications of the operator in many actual fields such asdecisionmaking pattern recognition and clustering analysisare open questions for future research
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 90924012 71090402 71103149)Program for New Century Excellent Talents in University(no NCET-10-0706) Specialized Research Fund for the Doc-toral Program of Higher Education (no 20090184110029)Humanities and Social Science Research Foundation of theMinistry of Education (no 08JC630067) Sichuan YouthScience and Technology Foundation (no 09ZQ026-021)Fund for Cultivating Academic and Technological Leadersin Sichuan Province (no 2011-441) Fundamental ResearchFunds for the Central Universities (no SWJTU11CX152)
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[2] D Ju-Long ldquoControl problems of grey systemsrdquo Systems ampControl Letters vol 1 no 5 pp 288ndash294 1982
[3] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986
[4] P Liu and F Jin ldquoThe multi-attribute group decision makingmethod based on the interval grey linguistic variablesrdquo AfricanJournal of Business Management vol 4 no 17 pp 3708ndash37152010
[5] P D Liu and X Zhang ldquoMulti-attribute group decision makingmethod based on interval grey linguistic variables weighted
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
geometric aggregation operatorrdquo Control and Decision vol 26no 5 pp 743ndash747 2011
[6] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012
[7] Z Yue ldquoA method for group decision-making based on deter-mining weights of decision makers using TOPSISrdquo AppliedMathematical Modelling vol 35 no 4 pp 1926ndash1936 2011
[8] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011
[9] G Wei X Zhao R Lin and H Wang ldquoGeneralized triangu-lar fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012
[10] Z Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004
[11] Z Xu ldquoA note on linguistic hybrid arithmetic averaging opera-tor in multiple attribute group decision making with linguisticinformationrdquo Group Decision and Negotiation vol 15 no 6 pp593ndash604 2006
[12] Z Xu ldquoAn approach based on the uncertain LOWG andinduced uncertain LOWG operators to group decision makingwith uncertain multiplicative linguistic preference relationsrdquoDecision Support Systems vol 41 no 2 pp 488ndash499 2006
[13] J M Merigo and G Wei ldquoProbabilistic aggregation opera-tors and their application in uncertain multi-person decision-makingrdquo Technological and Economic Development of Economyvol 17 no 2 pp 335ndash351 2011
[14] D W Chen Grey Fuzzy Set Introduction Heilongjiang Scienceand Technology Press Harbin China 1994
[15] G Z Bu and Y W Zhang ldquoGrey fuzzy comprehensive eval-uation based on the theory of grey fuzzy relationrdquo SystemsEngineering vol 4 no 4 pp 141ndash144 2002
[16] N Jin and S C Lou ldquoStudy on multi-attribute decision makingmodel based on the theory of grey fuzzy relationrdquo IntelligenceCommand Control and Simulation Techniques vol 7 pp 44ndash472003
[17] N Jin and S Lou ldquoA grey fuzzymulti-attribute decisionmakingmethodrdquo Fire Control amp Command Control vol 4 pp 26ndash282004
[18] L Dang and L Sifeng ldquoAnalytic method to a kind of grey fuzzydecision making based on entropyrdquo Engineering Science vol 10pp 48ndash51 2004
[19] KMeng Y Li CWang andWYang ldquoInterval-value grey fuzzycomprehensive evaluation based on the preference of the riskand its applicationrdquo Fire Control and Command Control vol2007 no 4 pp 109ndash111 2007
[20] E Cables M S Garcıa-Cascales and M T Lamata ldquoTheLTOPSIS an alternative to TOPSIS decision-making approachfor linguistic variablesrdquo Expert Systems with Applications vol39 no 2 pp 2119ndash2126 2011
[21] M Delgado J L Verdegay andM A Vila ldquoLinguistic decision-makingmodelsrdquo International Journal of Intelligent Systems vol7 no 5 pp 479ndash492 1992
[22] F Herrera E Herrera-Viedma and J L Verdegay ldquoDirectapproach processes in group decision making using linguisticOWA operatorsrdquo Fuzzy Sets and Systems vol 79 no 2 pp 175ndash190 1996
[23] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000
[24] J M Merigo M Casanovas and L Martınez ldquoLinguisticaggregation operators for linguistic decision making based onthe Dempster-Shafer theory of evidencerdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 18no 3 pp 287ndash301 2010
[25] Z Xu ldquoChoquet integrals of weighted intuitionistic fuzzyinformationrdquo Information Sciences vol 180 no 5 pp 726ndash7362010
[26] S-M Zhou F Chiclana R I John and J M Garibaldi ldquoType-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiersrdquoFuzzy Sets and Systems vol 159 no 24 pp 3281ndash3296 2008
[27] Y H Chen T C Wang and C Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011
[28] N Zhang and G W Wei ldquoMethod for aggregating correlatedinterval grey linguistic variables and its application to decisionmakingrdquo Technological and Economic Development of Economyvol 19 no 1 pp 60ndash77 2013
[29] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1953
[30] R R Yager ldquoInduced aggregation operatorsrdquo Fuzzy Sets andSystems vol 137 no 1 pp 59ndash69 2003
[31] R Yager ldquoOWA aggregation over a continuous interval argu-ment with applications to decision makingrdquo IEEE Transactionson Systems Man and Cybernetics B vol 34 no 5 pp 1952ndash19632004
[32] C Tan andX Chen ldquoInducedChoquet ordered averaging oper-ator and its application to group decisionmakingrdquo InternationalJournal of Intelligent Systems vol 25 no 1 pp 59ndash82 2010
[33] G W Wei and X Zhao ldquoSome induced correlated aggregat-ing operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012
[34] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningmdashIrdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975
[35] Z Xu ldquoDeviation measures of linguistic preference relations ingroup decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005
[36] Z Xu ldquoAn interactive approach to multiple attribute groupdecision making with multigranular uncertain linguistic infor-mationrdquo Group Decision and Negotiation vol 18 no 2 pp 119ndash145 2009
[37] R R Yager ldquoChoquet aggregation using order inducing var-iablesrdquo International Journal of Uncertainty Fuzziness andKnowledge-Based Systems vol 12 no 1 pp 69ndash88 2004
[38] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum Pub-lishing New York NY USA 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of