Induced Unbalanced Linguistic Ordered Weighted Average

Research ArticleInduced Unbalanced Linguistic Ordered Weighted Average andIts Application in Multiperson Decision Making

Lucas Marin1 Aida Valls1 David Isern1 Antonio Moreno1 and Joseacute M Merigoacute2

1 Departament drsquoEnginyeria Informatica i Matematiques Universitat Rovira i Virgili Avinguda Paısos Catalans 2643007 Tarragona Catalonia Spain

2MBS Decision Solutions Manchester Business School Decision and Cognitive Sciences Research Centre (DCSRC)The University of Manchester Booth Street West Manchester M15 4PB UK

Correspondence should be addressed to Lucas Marin lucasmarinurvcat

Received 23 February 2014 Accepted 10 June 2014 Published 17 July 2014

Academic Editor Francisco J Cabrerizo

Copyright copy 2014 Lucas Marin 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

Linguistic variables are very useful to evaluate alternatives in decision making problems because they provide a vocabulary innatural language rather than numbers Some aggregation operators for linguistic variables force the use of a symmetric anduniformly distributed set of terms The need to relax these conditions has recently been posited This paper presents the inducedunbalanced linguistic ordered weighted average (IULOWA) operator This operator can deal with a set of unbalanced linguisticterms that are represented using fuzzy sets We propose a new order-inducing criterion based on the specificity and fuzziness of thelinguistic terms Different relevancies are given to the fuzzy values according to their uncertainty degree To illustrate the behaviourof the precision-based IULOWA operator we present an environmental assessment case study in which amultipersonmulticriteriadecision making model is applied

1 Introduction

Aggregation operators aim to reduce a set of values into a sin-gle one that summarizes the inputs in a certain way [1 2]Aggregation is a key point in decision making and infor-mation fusion In this paper we propose a new aggregationoperator from the family of OWA operators for use withlinguistic dataThe ordered weighted averaging (OWA) oper-ator is characterised by ordering the values of the argumentsbefore they are aggregated according to a certain combinationpolicy [3] with which the decision maker can determinethe compensatory behaviour of the aggregation from highsimultaneity (andness) to complete replaceability (orness)

An interesting extension of the OWA operator is theinduced OWA (IOWA) operator [4] Its main difference isthat it uses a reordering process that is based on order-induc-ing variablesThus it is able to consider an additional reorder-ing criterion that does not depend on the values of the argu-ments Since it appeared it has been studied bymany authorsthat have developed several extensions [5ndash10] for groupdecision making problems

Although theOWAand IOWAoperators have been tradi-tionally applied to numerical data we can findmany applica-tions where they have been used with linguistic variables [11]Most of the studies in this area have assumed a uniform andsymmetrical distribution of the linguistic terms that definethe linguistic variable [12 13] (see Figure 1(a)) However thereare some situations that cannot be modelled with symmetriclinguistic variables [14ndash16] For example some decisionmaking problems such as personnel examination or projectinvestment selection often require a linguistic scale thatassigns a different precision to each label This issue is illus-trated in the term set (b) in Figure 1 where the deviationbetween the indices of two adjoining labels is much largerbetween very low and medium than between medium andhigh Some unbalanced and linguistic aggregation operatorshave recently appeared [2 16ndash19]

This paper takes another step forward by suggestingnew unbalanced linguistic aggregation operators for decisionmaking in particular we define the induced unbalancedlinguistic ordered weighted averaging (IULOWA) operator Inorder to make the operator applicable in many situations

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 642165 19 pageshttpdxdoiorg1011552014642165

2 The Scientific World Journal

Very low Low Medium High Very high

(a)

Very low Medium High Very high Perfect

(b)

Figure 1 Examples of balanced (a) and unbalanced (b) linguistic term sets with five labels

we allow the set of linguistic labels to be unbalanced Labelscan be represented by asymmetric fuzzy sets and they can benonuniformly distributed in the domain Each label has anassociated fuzzy set on the variablersquos reference scale of mea-surementThe fuzzy sets of the labels define a fuzzy partitionFirst the ULOWA (unbalanced linguistic ordered weightingaveraging) operator is presented Unlike its predecessor theLOWA operator [20] ULOWA is based on the extensionprinciple and so it carries out operations on the fuzzy setsassociated with the labels thus defining a new procedure foraggregating a pair of labels according to their membershipfunctions and the set of available linguistic terms After thiswe propose a generalization which we call IULOWA whichenables us to work with inducing variables

This paper shows that IULOWA fulfills the monotonic-ity identity idempotency and boundary conditions usuallyrequired in aggregation operators Like the other OWA oper-ators IULOWA provides a family of aggregation operatorsthat is parameterized between the linguistic minimum andmaximum and that includes a wide range of particular casessuch as the unbalanced linguistic average (ULA) the unbal-anced linguistic OWA (ULOWA) the unbalanced linguisticweighted average (ULWA) and many others Note that inthe literature there are approaches that employ abbreviationssimilar to the ones used in this paper although they are con-ceptually different In particular it is worth mentioning thatthe work by Xu on uncertain linguistic variables produces theinduced uncertain linguistic OWA (IULOWA) operator [21]However that approach has important differences withrespect to the IULOWA operator presented in this paperThepresent study is focused on the use of unbalanced linguisticinformation while Xursquos work [21] is based on uncertain lin-guistic variables that deal with imprecise information whenrepresenting the linguistic labels Section 6 compares bothapproaches and highlights their similarities and differences

Another important contribution of this paper is the pro-posal to use some of the information related to the definitionof the labels as an order-inducing criterion in the IULOWAoperator Although the order of the arguments can be decidedby taking into account the domain requirements it is some-times desirable to take into consideration the amount ofinformation contained in the terms themselves In this paperwe propose a method to calculate the set of weights of thearguments taking into account the degree of uncertainty ofthe labels This permits us to order the arguments by givingpriority to more specific values because these representmore precise information The method uses two well-knownmeasures of fuzzy sets namely fuzziness and specificity

We demonstrate the behaviour of the precision-basedIULOWA operator with a case study in a real applicationSpecifically we analyse the results obtained from the eval-uation of the environmental impact produced when sewage

sludge coming from wastewater treatment plants is used asfertilizer on agricultural soils In this application a two-stageaggregation is needed because we have a set of experts thatevaluate a set of options using the same set of criteria (ievariables) For this reason we further extend the IULOWAoperator by usingmultiperson techniques in the analysis [22]and in doing so we define the multi-person-IULOWA (MP-IULOWA) operator By including the opinions of severalexperts we obtain more reliable results because we can basethe decision on the knowledge of a group of people ratherthan on the opinion of a single individual Moreover the useof unbalanced and induced information enables us to dealwith complex environments where some of the informationismore representative and therefore needs to be prioritised inorder to correctly assess the aggregation

The rest of the paper is organised as follows Section 2 pro-vides some preliminaries introducing the OWA and IOWAoperators the linguistic OWA operator and the managementof unbalanced linguistic labels which are the basis of the newoperator Section 3 defines the new induced unbalancedLOWA operator studies its properties and presents somespecific operators that can be derived from the general for-mulation Section 4 describes how the different degrees ofuncertainty in the unbalanced terms can be used to induce anorder among them It also explains how the set of weights forthe operator can be determined using the uncertainty mea-sures Section 5 shows the application of the new IULOWAoperator to a multiperson multicriteria problem (MP-IULOWA) Section 6 compares the operator defined in thispaper with the homonym IULOWAoperator proposed by Xu[21] Finally Section 7 gives themain conclusions of the paperand suggests some lines for future research

2 Preliminaries

This section describes the set of concepts required before theintroduction of the IULOWA operator First we presentnumerical aggregation with the ordered weighted average(OWA) together with its order-induced extension theinduced OWA operator We then present the LOWA (lin-guistic OWA) operator as the basis of the IULOWA operatorproposed in this paper Finally we introduce recent studies onthe management of unbalanced sets of terms

21 InducedOrderedWeightedAverage (IOWA)Operator TheOWA operator is formally defined as follows [3]

An OWA operator of dimension 119898 is a mapping OWA

119877119898

rarr 119877 defined by an associated weighting vector 119882 of

The Scientific World Journal 3

dimension 119898 so that sum119908119894= 1 and 119908

119894isin [0 1] according to

the following formula

OWA119882

(1198861 1198862 119886

119898) = 119882

119879

sdot 119861 =

119898

sum

119895=1

119908119895119887119895 (1)

where 119887119895is the 119895th largest 119886

119894

One of the main problems of the OWA operator is itsdependency upon the form of the weighting vectorThere aretwomain approaches [23 24] to define this vector the orness-based approach and the analytical-based approach The firstfamily of methods tries to optimize certain features (eg thevariance the maximum dispersion and entropy) under agiven orness level The second type of methods defines theweighting method using natural language [25 26] Thesemethods permit the use of absolute quantifiers such asmuchmore than 10 and relative quantifiers such as half all and thereexists

Yager [25] proposed amethod to obtain theOWAweight-ing vector by using regular increasingmonotone (RIM) quan-tifiers A RIM quantifier defines a fuzzy subset 119876 of the realline with (0) = 1119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 With aRIMquantifier119876 the OWAweighting vector can be obtainedas follows

119908119894= 119876(

119894

119899

) minus 119876(

119894 minus 1

119899

) (2)

For instance the quantifier all is represented by the fuzzysubsets 119876(1) = 1 and 119876(119909) = 0 for 119909 = 1

Once the weights have been established the aggregationpolicy is fully determined because the order of vector 119861 in theOWAoperator is based only on the value of the arguments 119886

119895

However as shown by Yager and Filev [4] by allowing otherorderings for the arguments we can obtain a more generalaggregation operator the IOWA (induced OWA) This gen-eralization takes into account the ordering that an additionalvariable (119906) induces in the set of values to be aggregated

The IOWA operator is defined as follows [4]

IOWA119882

(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882119879

sdot 119861119906=

119898

sum

119895=1

119908119895119887119895

(3)

where 119882 = (1199081 119908

119898) is the usual weighting vector that

defines the aggregation policy of theOWAoperator with119908119894isin

[0 1] sum119908119894= 1 The ordered argument vector 119861

119906is obtained

by taking 119887119895as the 119886

119894value of the pair ⟨119906

119894 119886119894⟩ which has the

119895th largest 119906119894value Yager refers to 119906 as the order-inducing

variable and 119886 as the argument variableIt is important to note that the only requirement for the 119906

variable is that itmust be drawn from a space inwhich there issome linear orderingThis allows different kinds of criteria tobe used for the order-inducing variables An important aspectof the IOWAoperator is the fact that the order induced by thevariable 119906 can produce ties in some arguments In this casethe relative order of two arguments 119886

119894and 119886119895with 119906

119894= 119906119895is

relevant because theymay correspond to different values that

is 119886119894

= 119886119895 Many papers adopt the solution of replacing 119886

119894

and 119886119895with their arithmetic average (119886

119894+ 119886119895)2 Another

mechanism for solving ties consists of including a secondaryordering criterion [27] as we will propose in this paper

The IOWA operator has the properties of monotonicityidempotency symmetry homogeneity shift-invariance andduality [28]

The semantics of the OWA operator is a generalization ofthe idea of averaging or summarizing the arguments How-ever IOWA permits other kinds of aggregation of the argu-ment variables which can be modelled by choosing theappropriate order-inducing variable Since the introductionof the IOWA operator several authors have proposed differ-ent ways of inducing the order For example Pasi and Yager[29] used IOWA to define the majority opinion in groupdecision making by inducing the order of the arguments onthe basis of the similarity amongone value and its neighboursThe combination of this ordering criterion with linguisticquantifiers allows calculating the fulfillment of the proposi-tion ldquothe satisfaction value of most of the criteriardquo rather thanldquomost of the criteria have to be satisfiedrdquo (which would be theresult of classical OWA) So IOWA can give different aggre-gation semanticsMerigo and Casanovas have developed sev-eral applications of IOWA with uncertain information [30]and with distance measures [22] Wei et al [31] and Xia andXu [32] have studied several extensions by using intuitionisticfuzzy sets and fuzzy numbers

Themain advantage of the IOWA operator over the OWAoperator is that it can deal with complex reordering processeswhere the highest value is not the first one in the reorderingTherefore the induced variables solve an important drawbackof the OWA operator which is exclusively based on a weight-ing policy For example a journal may determine an optimalaverage number of pages per paper Thus by using theIOWA operator we can ensure that extremely long papers arenot the first in the reordering process because they are notoptimal in this analysis Other interesting examples can befound when analysing several key variables of the humanbody including temperature calories and weight

The main classes of models with induced aggregationare classified [28] as standard auxiliary ordering where theinducing variable is an attribute associated with the input thatis not considered in the actual aggregation process but that isinformative about the object itself nearest-neighbour ruleswhere the order-inducing variable represents the similarity ordistance among the aggregating elements best-yesterdaymodels applied in models where it is necessary to predict theorder based on previous observations aggregation of complexobjects in which it is necessary to operate with compoundobjects such as aggregating matrices where the order is notdirectly defined and needs to be estimated with some addi-tional measure group decision making an area in which it hasbeen proposed that the consensus can be better achieved withinducing variables based on the support of each individualscore andmultiple inducing variables where a priority orderis established among more than one inducing variable

22 Linguistic OWA (LOWA) Operator In this paper we willstudy the aggregation of a set of linguistic termsAll the values


belong to the same linguistic scale of measurement 119878 = 119904119894

119894 isin 0 119879 This set 119878 is defined as a finite and totallyordered term set on a reference domain 119883 = [0 1] with anodd cardinal where one of the labels corresponds to theneutral value and the remaining terms are placed around it [715 20] The cardinality of the set must be small enough so asnot to impose useless precision and rich enough in order toallow an appropriate discrimination levelThe usual cardinal-ity values are 7 or 9

There are different approximations for working with lin-guistic labels Four paradigms are distinguished [13] opera-tors based on the linear ordering of the labels operators basedon the extension principle operators based on the 2-tuplemodel and operators defined directly on the symbols (iecomputing with words) The aggregation operators that fol-low the extension principle on linguistic data can be definedas

119878119899119865lowast

997888997888rarr 119865 (119877)

app(sdot)997888997888997888997888rarr 119878

(4)

where 119878119899 symbolizes the 119899 Cartesian product of 119878 119865lowast is an

aggregation operator based on the extension principle 119865(119877)

is the set of fuzzy sets over the set of real numbers119877 and app

119865(119877) rarr 119878 is a linguistic approximation function that returnsa label from 119878 whose meaning is the closest to the obtainedunlabelled fuzzy number

Following this principle the LOWA operator [20] is thebasis of the operator that we will present in this paper LOWAis an adaptation of theOWAoperator for dealingwith linguis-tic labels It operates directly on the labels that are aggregatedin pairs by using a convex combination operation based onlyon the position of the labels in the scale 119878 Because of itssimplicity it has been used in many domains [33ndash35]

If 119860 = 1198861 119886

119898 is the set of labels to aggregate the

LOWA operator is defined as

LOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898

= 1199081otimes 1198871oplus (1 minus 119908

1)

otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(5)

where 119882 = (1199081 119908

119898) is an 119898-dimensional weighting

vector so that 119908119894isin [0 1] and sum119908

119894= 1 120573

ℎ= 119908ℎsum119898

2119908119896 ℎ =

2 119898 and 119861 is the associated ordered label vector (eachelement 119887

119894isin 119861 is the 119894th largest label in 119860) 119862119898 is the convex

combination operator of 119898 labels if 119898 = 2 it is defined as

1198622

119908119894 119887119894 119894 = 1 2 = 119908

1otimes 119904119895oplus (1 minus 119908

1) otimes 119904119894= 119904119896

119904119895 119904119894isin 119878 (119895 ge 119894)

(6)

so that

119896 = min 119879 119894 + round (1199081sdot (119895 minus 119894)) (7)

where 1198871= 119904119895and b2= si If119908119895 = 1 and119908

119896= 0 for all k = j then

119862119898

119908119894 119887119894 119894 = 1 119898 = 119887

119895 and round is the common

mathematical function which translates a real number to itsnearest integer value

It is worth noting that the LOWA operator is commuta-tive monotonic bounded and idempotent These propertieswill be used later to analyse our proposed IULOWA operator

23 Management of Unbalanced Linguistic Labels The sema-ntics of each linguistic label is usually given by a trapezoidalor triangular membership function 120583 119883 rarr [0 1] that isrepresented with a tuple P = (p

1 p2 p3 p4) where p

1le p2le

p3le p4are the points in the reference domainX which define

the trapezoid Some special cases can be defined If p1= p2

and p3= p4 then P corresponds to a crisp interval If p

2= p3

the fuzzy set P is triangular otherwise it is a trapezoid If p1=

p2= p3= p4 then P is called a crisp real number In most of

the applications that use linguistic information the fuzzy setsassociated with each label are equal symmetric and uni-formly distributed throughout the domain (eg the term set(a) in Figure 2)This kind of representation makes it easier tomanage and understand the meaning of each label and themathematical operations carried out over them For instancethe LOWA aggregation operator takes advantage of thissimplicity to consider only the order in which the labelsappear in the domain which means that it can operate in asymbolic way [20] (see Section 22)

Howevermany daily situations involve term sets inwhichthe fuzzy sets are not symmetric or are not distributeduniformly across the domain [16 36] For instance when astudent is evaluated there is usually only one negative term(failed) but several positive terms (pass good very good andexcellent) If a control system is analysing the value of a sen-sor there is normally a range of terms that can be used withinthe standard interval of values but above a certain thresholdthere is only one term (alarm) In general the designersof a fuzzy system may be more interested in defining acertain interval of the domain more precisely than otherparts leading to the use of more labels in that interval Thedevelopment of unbalanced linguistic label sets and operatorsto aggregate these kinds of labels has been highlighted as animportant area of research [13]

There have been few studies proposing the use of unbal-anced sets of terms Herrera et al proposed [14 36] someaggregation operators for linguistic data represented in fuzzyunbalanced linguistic term sets defined using the 2-tuplemodel [12] Terms are represented by a pair (s 120572) where s isthe linguistic label and 120572 is a number that represents thetranslation of symbols into a real scale This model is able todeal with unbalanced linguistic variables by means of hierar-chical linguistic contexts A linguistic hierarchy [37] is a set oflevels where each level is a balanced linguistic term set with agranularity different from that of the remaining levels in thehierarchy Each level belonging to a linguistic hierarchy isdenoted as 119897(119905 119899(119905)) where t is the number that indicates thelevel of the hierarchy and n(t) indicates the granularity ofthe linguistic term set of that level To build a linguistichierarchy the authors propose the fact that the linguistic set ofterms for level t + 1 is obtained from its predecessor using theexpression 119871(119905 119899(119905)) rarr 119871(119905 + 1 2 sdot 119899(119905) minus 1)


0 100908070605040302010

1

0 100908070605040302010

1

0 100908070605040302010

1

(a)

0 100908070605040302010

1

(b)

Figure 2 Unbalanced term set with 5 linguistic labels (b) obtainedfrom a linguistic hierarchy of 3 levels (a)

In order to work with terms from different levels of thehierarchy it is necessary to use transformation functions totranslate linguistic terms from one level to another Con-sidering that LH=119880

119905119897(119905 119899(119905)) is a linguistic hierarchy whose

linguistic term sets are denoted as Sn(t) = 119904119899(119905)

0 119904

119899(119905)

119899(119905)minus1 a

transformation function from a linguistic label in level t to alabel in level 1199051015840 is defined [37] as

TF1199051199051015840 (119904119899(119905)

119894 120572119899(119905)

) = Δ(

Δminus1

(119904119899(119905)

119894 120572119899(119905)

) sdot (119899 (1199051015840

) minus 1)

119899 (119905) minus 1

)

(8)

where Δ(120573) = (119904round(120573) 120573-round(120573)) and Δminus1(si 120572) = 119894 + 120572

Herrera-Viedma et al [36] define the LOWAun andILOWAun operators on the basis of these transformationfunctions to make an aggregation (and an induced aggrega-tion) of unbalanced linguistic terms

To give an illustrative example Figure 2(a) shows a 3-levellinguistic hierarchyThe parts in red are used to construct theunbalanced term set depicted in Figure 2(b)

As can be seen in this example the set of labels used ineach of the levels of the hierarchy is balanced and uniformlydistributed but by taking some pieces of each level and

S(5)minus4 S(5)minus2 S(5)minus1 S(5)minus04 S(5)0 S(5)04 S(5)1 S(5)2 S(5)4

Extre

mely

poor

Very

poo

r

Poor

Slig

htly

poo

rFa

irSl

ight

ly g

ood

Goo

d

Very

goo

d

Extre

mely

good

Figure 3 A set of nine linguistic labels (from [16])

putting them together it is possible to model an unbalancedterm set The main drawbacks to this approach are that itis quite complex to define an appropriate set of labels andthe number of levels (and labels) to consider can be quitelarge In the example shown above there are 17 labels in the 3levels of the hierarchy whereas the unbalanced term set tobe modelled only has 5 labels Moreover the definition of thefuzzy sets in each of the levels is very strict so it is not easyto model an arbitrary set of unbalanced terms In contrast aswill be mentioned in the rest of the paper in our proposal thelabels can be represented either with trapezoidal or withtriangular fuzzy sets with the sole requirement that theydefine a fuzzy partition

Another prominent proposal for modelling unbalancedterm sets is given by Xu [16] He argues that when defining anunbalanced term set the absolute value of the deviationbetween the indices of two adjoining linguistic labels shouldincrease as the indices of the linguistic labels steadily increase(the term in the centre has index 0 so there are terms withpositive and negative indices) Following this idea he pro-poses defining a term set with 2119905 minus 1 labels in the followingway

119878119905

= 119904119905

120573| 120573 = 1 minus 119905

2

3

(2 minus 119905)

2

4

(3 minus 119905)

0

2

4

(119905 minus 3)

2

3

(119905 minus 2) 119905 minus 1

(9)

For example Figure 3 shows an unbalanced term set withnine labels

It can be seen that this way of defining the unbalancedterm sets is very rigid It is only possible to model those situ-ations in which the labels in the middle are very precise andthe labels in the extremes have a wider range Moreover thelabels are symmetrically located with respect to the centre ofthe domain so it is not possible to have more positive labelsthan negative labels This strict definition of the term setsallows Xu to define simple functions that permit terms in oneset to be transformed into terms in another set These trans-formations are meant to be used when different experts haveused different term sets to evaluate a set of alternatives Eachlabel is basically represented by a point in the domain ratherthan by a fuzzy set

One of the main aims of our work was to devise a methodfor working with any unbalanced set of terms without anyrestriction on the definition of the fuzzy set associated witheach term (as long as a fuzzy partition is obtained) Xursquos pro-posal [16] does not provide this flexibility since the term setsare very precisely defined and they have to be symmetrically


located with respect to the centre of the domain Moreover itonly considers situations in which precise labels are requiredin the centre and imprecise labels in the extremes The studyby Herrera et al [14 36] allows some unbalanced sets to bemodelled provided that the labels are composed by takingpieces from each level of the hierarchy In contrast ourapproach permits the direct definition of the unbalancedterm set that fits better with the needs of the applicationchoosing any fuzzy set for each label The price to be paid forthis flexibility is that the aggregation operator must operateon fuzzy sets

A related proposal by Xu [21] with which our approach iscompared on Section 6 is based on the idea of representinguncertainty by using intervals of values (eg an expert couldsay that the quality of an attribute of an object is ldquobetweengood and very goodrdquo) Xu proposed to consider a fixed andtotally ordered discrete term set for example 119878 = 119904

minus4 119904minus3

1199043 1199044 An uncertain linguistic variable s is defined as an

interval [119904119886 119904119887] where a is lower than or equal to bThe wider

interval is the more uncertain is the evaluation it providesHe defines two basic operations on these intervals

(i) 1199041

+ 1199042

= [1199041198861

1199041198871

] + [1199041198862

1199041198872

] = [max119904minus4

min1199041198861+1198862

1199044 max119904

minus4 min119904

1198871+1198872

1199044]

(ii) ks = k [sa sb] = [ska skb] with k between 0 and 1

Note that all the operations are performed on the indexesof the labels so it is assumed that all terms are homogeneouslydistributed through the domain With these two operationsXu defines the induced uncertain linguistic OWA aggregator(IULOWA) as follows

Xu-IULOWA119908

= (⟨1199061 1199041⟩ ⟨1199062 1199042⟩ ⟨119906

119899 119904119899⟩)

= 11990811199041198871

+ 11990821199041198872

+ sdot sdot sdot + 119908119899119904119887119899

(10)

where119882 = (1199081 1199082 119908

119899) is the usual weighting vector 119906 is

the order-inducing variable and 119904119887119895

is the 119904 value of the ⟨119906 119904⟩

pair with the jth largest 119906 If two pairs have the same 119906 valuetheir two 119904 values are replaced by their average

As will be shown in the comparison in Section 6 theflexibility of our approach allows it to be used to simulatethe uncertainty represented by the intervals in Xursquos proposalby replacing these intervals with appropriately defined unbal-anced fuzzy sets

3 The Induced ULOWA Operator

The first part of this section focuses on defining theunbalanced linguistic ordered weight averaging operator(ULOWA) which is later extended to create its order-inducedversion IULOWA

31 Unbalanced LOWA (ULOWA) Operator The ULOWAoperator was defined [17] as an extension of the LOWA oper-ator [20] designed to deal with unbalanced linguistic termsThe ULOWA operator takes the same form as the LOWAwhich is as follows

If 119860 = 1198861 119886


ULOWA operator is defined as

ULOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(11)

Thedifferencewith LOWA lies in the convex combinationof two linguistic terms Whereas LOWA uses a symbol-basedapproach ULOWAoperates on fuzzy sets that is it takes intoaccount the membership functions of the terms In this waywe are able to take into account the information given by thefuzzy sets during the aggregation process obtaining a moreprecise result So whenm= 2 the convex combination of thetwo terms b

1= sj and b2 = si with sj si isin S (jge i) is calculated

taking into account the membership functions of the labels sjand si

1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896

where 119904119896isin 119878 with 119896 = arg max

119894le119901le119895

Sim (119878119901 120575)

(12)

In this expression 120575 is a crisp number defined as 120575=(xk xk xk xk) with 119909

119896= 119909lowast

119904119894

+ 1199081(119909lowast

119904119895

minus 119909lowast

119904119894

) where 119909lowast

119904119894

is thex-component of the centre of gravity (COG) of the fuzzy setassociated with the label si = (p

1 p2 p3 p4)

COG (119904119860) =

119910lowast

119860=

1

6

(

1199013minus 1199012

1199014minus 1199011

+ 2) if 1199011

= 1199014

1

2

if 1199011= 1199014

119909lowast

119860=

119910lowast

119860(1199013+ 1199012) + (119901

4+ 1199011) (1 minus 119910

lowast

119860)

2

(13)

In this way the aggregation result sk is the linguistic termconsisting of sj and si with the greatest similarity to the inter-mediate point 120575 It is worth noting that the result is one of theterms available in the scale of reference S The similaritybetween two fuzzy sets proposed after a thorough study ofthe literature [17] is calculated as follows

Sim (119875 119876) =4

radic

4

prod

119894=1

(2 minus1003816100381610038161003816119901119894minus 119902119894

1003816100381610038161003816) minus 1 (14)

Figure 4 depicts the aggregation procedure of the twoextreme labels of an unbalanced 7-term set (VL and P) fordifferent policies The figure shows the COGs of both labelsand their intermediate crisp number 120575 that is used to find theresult of the aggregation when varying the vector W Threeusual aggregation policies have been used After the similarityfunction is applied using (14) ULOWA obtains the neutraltermM in the case ofmean H in the case of at least half andL in the case of as many as possible


0

1

ylowast

xlowastVL xlowastP120575

VL L PHM AH VH

(a) Policy mean (119908 = (05 05))

0

1

ylowast


VL L PHM AH VH

(b) Policy at least half (119908 = (07 03))

0

1

ylowast


VL L PHM AH VH

(c) Policy as many as possible (119908 = (03 07))

Figure 4 Examples of ULOWA aggregation of two labels (VL and P)

32 Definition of IULOWA The induced unbalanced LOWA(IULOWA) is an aggregation operator for linguistic valuesthat are defined on an unbalanced vocabulary S As it is basedon IOWA the operator is able to manage complex decisionproblems by using order-inducing variables

Definition 1 The induced unbalanced linguistic orderedweighted average based on the ordering criterion u iscalculated as

IULOWA119908(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882 sdot 119861119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1) otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(15)

where B is the induced ordered vector that is 119861 = (1198861015840

120590(1)

1198861015840

120590(2) 119886

1015840

120590(119895)) where 119886

1015840

120590(119895)corresponds to the value aj having

the jth largest ui 119882 = (1199081 119908

119898) is the usual weighting

vector that defines the aggregation policy of the OWA opera-tor with 119908

119894isin [0 1] sum119908

119894= 1 The final convex combination

of two linguistic terms is the same as in the ULOWAoperatordefined in (11)

33 Properties of the IULOWA As stated in [3] an aggre-gation operator should have the following properties mono-tonicity commutativity idempotency and boundedness

Property 1 The IULOWA operator is increasingly mono-tonous with respect to the argument values if the associatedorder-inducing values remain unchanged

Let us consider two order-induced vectors 119875 = ⟨1199061 1199011⟩

⟨119906119898 119901119898⟩ and 119876 = ⟨119906

1015840

1 1199021⟩ ⟨119906

1015840

119898 119902119898⟩ so that

forall119895 119906119895

= 1199061015840

119895and forall119895 119901

119895ge 119902119895 then IULOWA

119908(119875) ge

IULOWA119908(119876)

That means as will be detailed in Section 4 if we replaceeach term with another one that has the same specificity andfuzziness but a greater preference in the scale S the resultwill also be an equal or better term in the preference scale Infact this case reduces the proof to the ULOWA operator [17]because the inducing variable does not change the order

Proof Let IULOWA119908(119875) = IULOWA

119908(⟨1199061 1199011⟩

⟨119906119898 119901119898⟩) and IULOWA

119908(119876) = IULOWA

119908(⟨1199061015840

1 1199021⟩

⟨1199061015840

119898 119902119898⟩) If forall119895 119906

119895= 1199061015840

119895and forall119895 119901

119895ge 119902119895 any induced per-

mutation of the elements satisfies the condition forall119895 1199011015840

120590(119895)ge

1199021015840

120590(119895) and IULOWA

119908(1199011015840

120590(1) 119901

1015840

120590(119898)) ge IULOWA

119908(1199021015840

120590(1)

1199021015840

120590(119898)) due to the monotonicity of the ULOWA operator

Then IULOWA119908(119875) ge IULOWA

119908(119876)

Property 2 The IULOWA operator is commutativeIULOWA

119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= IULOWA

119908(⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) where (⟨119906

1015840

1 1198861015840

1⟩ ⟨119906

1015840

119898 1198861015840

119898⟩) is any

permutation of the elements in (⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)

Proof The IULOWA operator reorders the arguments acco-rding to the order-inducing variable Thus if 119860 = (⟨119906

1 1198861⟩

⟨119906119898 119886119898⟩) is any permutation of 119860

1015840

= (⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) the order induced for A and 119860

1015840 will be the sameTherefore IULOWA

119908(119860) = IULOWA

119908(1198601015840

)

Property 3 The IULOWAoperator is idempotent in the sensethat IULOWA

119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= a if forall119895 119886

119895= 119886

Proof The proof does not depend on the inducing variablebecause in this case the values to be aggregated are the samefor all the arguments Then in line with the definition of theIULOWA operator we have a final step (12) that consists ofIULOWA

119908(119886119898minus1

119886119898) = 119904119896 where 119896 = 119908

1otimes 119904119895oplus (1minus119908

1) otimes 119904119894=

argmax119894le119901le119895

Sim(119904119901 120575) In this case 119894 = 119895 so 119904

119896= 119904119894= 119904119895=

119886 Recursively we obtain IULOWA119908(1198861 119886

119898) = 119886


Property 4 The IULOWA operator is bounded That is forany weighting vectorW

min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)

Proof Given that 119904119895 119904119894isin 119878 (119895 ge 119894) we have defined the convex

combination of these two terms as 1198622

119908119894 119887119894 119894 = 1 2 =


1) otimes 119904119894= 119904119896 According to (12) we have 119896 =

argmax119894le119901le119895

Sim(119904119901 120575) That is the resulting label from the

combination of two labels is1198622(119904119894 119904119895) = 119904119896with 119894 le 119896 le 119895This

means that we cannot obtain a result out of the limits givenby the labels that are aggregated at each step

34 Families of IULOWA Operators The IULOWA operatorpermits the definition of a wide range of families of unbal-anced linguistic aggregation operators following themethod-ology used in the OWA literature [6 21] Note that eachspecific case is useful in certain situations depending on theobjectives of the analysis For example when aggregating mlabels we can study the following cases

(i) If119908119895= 1119898 for all j we get the unbalanced linguistic

average (ULA)(ii) The induced unbalanced linguistic maximum is

obtained if 1199081

= 1 and 119908119895

= 0 for all j = 1 whichgives as result the value ai with maximum ui becauseu1=maxui after the reordering stage

(iii) The induced unbalanced linguistic minimum isobtained if 119908

119898= 1 and 119908

119895= 0 for all j = m which

gives as result the value ai with minimum ui becauseum =minui after the reordering stage

(iv) The unbalanced linguistic weighted average (ULWA)appears if ui gt ui+1 for all i

(v) The unbalanced LOWA operator is obtained if the jthlargest label sj according to the scale S is also orderedat position j according to the inducing variable U forall j

(vi) Step-IULOWA it occurs if there is a position 1le k le

m so that 119908119896= 1 and 119908

119895= 0 for all j = k

(vii) Median-IULOWA if m is odd we assign 119908119901

= 1 and119908119895= 0 for all others with p the position of the [(119898 +

1)2]th largest ui Ifm is even we assign for example119908119901

= 119908119902= 05 and 119908

119895= 0 for all others with p and q

being the positions of the (m2)th and [(1198982) + 1]thlargest ui

(viii) Olympic-IULOWA it occurs if 119908119901

= 119908119902

= 0 with119906119901

rarr max119906119894 and uq rarr min119906

119894 and for all others

119908119895= 1(119898 minus 2)

(ix) Window-IULOWA it occurs if 119908119895= 1119889 for 119896 le 119895 le

119896+119889minus1 and119908119895= 0 for 119895 gt 119896+119889 and 119895 lt 119896 Note that

k and dmust be positive integers so that 119896+119889minus1 le 119898(x) Centred-IULOWA it occurs if the aggregation is

symmetric strongly decaying and inclusive It issymmetric if 119908

119895= 119908119898minus119895+1

It is strongly decaying

when 119894 lt 119895 le (119898 + 1)2 then 119908119894lt 119908119895and when igt

jge (m+ 1)2 then119908119894lt 119908119895 It is inclusive if119908

119895gt 0 for

all j(xi) Slide-IULOWA three versions of this operator can be

defined on the basis of the degrees of andness (120572) andorness (120573) where 120572 120573 isin [0 1] and 120572 + 120573 le 1 asfollows

(1) Generalised slide-IULOWA occurs when 1199081

=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898

(2) Orlike slide-IULOWA occurs if 120572 = 0(3) Andlike slide-IULOWA occurs if 120573 = 0

4 Order-Inducing Variables

In this section we analyse the feature of the IULOWA oper-ator that distinguishes it from the ULOWA operator whichis the order-inducing variable used in the reordering processof the linguistic labels With this type of operator we are ableto deal with complex reordering processes in which thehighest linguistic value in S is not the optimal value for thedecision maker

41 Order Induction As pointed out in Section 21 the order-inducing variable can be obtained using different proceduresThe decision maker can express hisher personal orderingdirectly on the values of the domain of reference but it is alsointeresting to have automatic processes to generate the order-inducing criterion In this latter case the order is linked tocertain features of the set of arguments such as the distanceamong the values the past history of values or the confidencein the values

In this paper we propose a new way of inducing the orderthat is related to the additional information given by the shapeof unbalanced terms As mentioned in the introductionunbalanced terms permit the definition of linguistic variableswith different granularity anddistribution for the positive andthe negative values

Considering the distribution of the terms VL L M AHH VH P in Figure 4 let us assume that we are going to usethem to evaluate the performance of a certain object Peopleusually do not assign extreme values unless they are reallysure about the performance of the object thus we havedefined two very precise fuzzy sets for VL and P (the mostnegative andmost positive terms) Being interested in findingobjects with a good performance there are three terms toindicate different degrees of positivity while only one indi-cates a low performance (L) Therefore L is much moreuncertain than the others Similarly one can consider that thelabels AH (almost high) and VH (very high) are qualifyingthe term H (high) indicating ldquoa little more than highrdquo or ldquoabit less than highrdquo respectively Thus they are more precisethan high These specific semantics of the different labels canonly be captured using an unbalanced set of terms

The difference on the certainty of the terms should betaken into account during the aggregation process as eachlabel is providing a different amount of information about


the evaluated alternative In fact if we consider that bothtriangular and trapezoidal fuzzy sets can be associated withthe labels (as in Figure 4) then the uncertainty of the labels isnot only related to their support intervals in the referencedomain but also related to their kernel (ie the set of pointswith value 1)

Taking into account the different features of the definitionof the linguistic variables pointed out before we proposeusing a measure of the uncertainty of the linguistic labels asthe order-inducing criterion for the aggregation Thus thearguments will be ordered by decreasing uncertainty In thisway the contribution of precise labels is prioritized while theeffect of uncertain labels is reduced

In the literature [7 38ndash40] two types of uncertainty infuzzy sets are recognized (1) specificity related to the mea-surement of imprecision which is based on the cardinality ofthe set and (2) fuzziness or entropy which measures thevagueness of the set as a result of having imprecise bound-aries

With regard to the measure of specificity [8] let X be a setand let [0 1]

119883 be the class of fuzzy sets on X A measure ofspecificity is a function Sp [0 1]119883 rarr [0 1] so that

(1) Sp(Oslash) = 0(2) Sp(120583) = 1 if and only if 120583 is a singleton(3) if 120583 and 120574 are normal fuzzy sets in 119883 and 120583 sub 120574 then

Sp(120583) ge Sp(120574)

The following specificity measure for a fuzzy set Adefined on X is defined as a generalization of other previousformulations [8]

Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)

In this expression T is a T-norm int120572sup0

is a Choquet inte-gral 120572sup is the superior 120572-cut N is a negation operator andM is a fuzzy measure

A special case of (17) is given in (18) by considering theT-norm min the standard negation 119873(119909) = 1 minus 119909 and theLebesgue-Stieltjes fuzzy measure 119872([119886 119887]) = 119887 minus 119886 Takingthese parameters and a normalized fuzzy set (with 120572sup = 1)the specificity of a fuzzy set defined in the [119886 119887] interval canbe calculated as

Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)

With respect to the measure of fuzziness [41] let X be aset and let [0 1]119909 be the class of fuzzy sets on X A measure offuzziness is a function Fz [0 1]

119909

rarr [0 1] such that

(1) Fz(119860) = 0 if 119860 is a crisp set(2) Fz(119860) = 1 if forall119909 isin 119883119860(119909) = 12(3) Fz(119860) le Fz(119861) if119860 is less fuzzy than 119861 that is119860(119909) le

119861(119909) le 12 or 119860(119909) ge 119861(119909) ge 12 for every 119909 isin 119883

Fuzziness may be seen as the lack of distinction betweenthe fuzzy set A and its complement AC A general definition

01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)

Figure 5 Two fuzzy sets with the same specificity and differentfuzziness

of this type of fuzziness measure is based on an aggregationoperator h and a distance function d as follows

Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)

For the case of continuous domains using the standardnegation operation and the Hamming distance (19) corre-sponds to

Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)

Specificity and fuzziness refer to two different characteris-tics of fuzzy sets Specificity (or its counterpart nonspecificity[42])measures the degree of truth of the sentence ldquocontainingjust one elementrdquo Fuzziness measures the difference from acrisp set For decision making purposes it seems desirable tohave labels that correspond to single elements rather than tolarge sets of values which may hamper the selection of theappropriate alternative For this reason we propose usingspecificity as the order-inducing variable in the aggregation oflinguistic terms that qualify a set of alternatives in a decisionmaking process

If there are ties between terms with the same specificity asecond ordering criterionmay be their fuzziness An increas-ing ordering of fuzzinesswill be used aswe prefer those termswith less uncertainty If this second criterion also leads tosome ties a decreasing ordering on the preference scale 119878

associated with the terms can be used In Figure 5 we showtwo fuzzy sets with the same specificity (Sp(119860) = Sp(119861) =

09) according to (18)

Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)


Table 1 Uncertainty measures for the terms in Figure 6

Term Definition Specificity Fuzziness IndexA (00 00 00 01) 095 005 0B (00 01 02 03) 080 010 1C (02 03 03 04) 090 010 2D (03 04 04 06) 085 015 3E (04 06 06 08) 080 020 4F (06 08 08 09) 085 015 5G (08 09 10 10) 085 005 6

A GFEDC

0 100908070605040302010

1B

Figure 6 Linguistic variable with 7 terms (test 1)

but different fuzziness (Fz(A) = 01 and Fz(B) = 005) accord-ing to (20)

Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| = 1 minus int

1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)

In this example the set 119860 is fuzzier than 119861 so 119861 ispreferred

Definition 2 Precision-based IULOWA given a set of unbal-anced linguistic arguments 119886

1 119886

119898 we calculate their

induced aggregation according to their uncertainty by usingthe IULOWA equation (15) where B is the induced orderingvector so that 119861 = (119887

1 1198872 119887

119898) satisfies these conditions

(i) forall1198961 le 119896 lt 119898 Sp(119887119896) ge Sp(119887

119896+ 1)

(ii) forall1198961 le 119896 lt 119898 if Sp(119887119896) = Sp(119887

119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)

(iii) forall1198961 le 119896 lt 119898 if Sp(119887119896) = Sp(119887

119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887

119896gt 119887119896+ 1 according to the linguistic

scale 119878

Notice that if the terms correspond to crisp numbersIULOWA is reduced to the OWA operator

The following example will show how the terms depictedin Figure 6 would be sorted according to the previous order-ing rules Table 1 shows the information regarding each of theterms needed to conduct the sorting procedure Specificity iscalculated with (18) whereas fuzziness is obtained with (20)

Taking into account the specificity the labels are orderedas A gt C gt (D F G) gt (BE) Note that there are two tiesthe first one between D F and G (with Sp = 085) and thesecond one between B and E (Sp = 08) Using the fuzzinessmeasure to solve the ties we put G (Fz = 005) before D and F(Fz = 015) in the first tie because we give priority to less fuzzyterms In the second tie B (Fz = 010) precedes E (Fz = 020)As we can see by measuring fuzziness we are still unable todecide the order between D and F so we use the index ofthe terms to decide their position putting F (index = 5) beforeD (index = 3) Thus the induced order according to theprocedure proposed in this paper (Definition 2) is A gt C gt

G gt F gt D gt B gt E

42 Weight Generation As mentioned above the OWA wei-ghts 119908

119894are used to define different conjunctiondisjunction

aggregation models [43 44] As proposed in the literature[1 5 45] the inclusion of an additional variable in the OWAaggregator may also involve the transformation of the set ofweights

In this section we propose modifying the set of weightsassociated with the arguments by taking into considerationthe uncertainty of the values that are aggregated The ratio-nale is that the more specific values should have a higherweight whereas the less specific terms (that are less reliable)should have a lower weight

Using the family of fuzzy quantifiers proposed by Yager[3] the set of weights is obtained with the expression

119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)

and 120590 is the permutation according tothe order-inducing procedure established before Q(p) indi-cates the degree of compatibility of p with the conceptdenoted by Q For example if Q represents a linguisticquantifier such as ldquomost of rdquo and 119876(095) = 1 then it can besaid that a value of 95 is completely compatible with the ideaconveyed by this linguistic quantifier

The properties of the quantifier function must be takeninto account in order to generate a coherent set of weights fortheOWAoperator Taking the usual quantifierQ(r) = ra [3] if119886 isin [0 1] then the weighting function is concave whichensures that the larger the specificity the higher theweight119908

119896

of the corresponding argument [45] It is worth noting thatwith 119886 isin [0 1] the aggregation policy is disjunctive whichmeans that uncertain evaluations can be replaced with themost specific available values

Table 2 shows an example of weights obtained withouttaking into account the specificities We have made severaltests with different values of the parameter a ranging from01 (where we mostly base the result on the first argument) to1 (which corresponds to an arithmetic average of the argu-ments as the weights are equal for all the values)

To evaluate the impact of the specificity measure in theset of weights two tests have been performed The first oneis based on the linguistic variable with 7 terms represented inFigure 6 We generate the weights for the values (A C F B


Table 2 Weights obtained without specificity

119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)

Table 3 Weights obtained in test 1

119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)

and B)with specificities (095 09 085 08 and 08) respec-tively (see Table 1) The results are shown in Table 3

In this test the specificities of the terms that are aggre-gated are very similar For this reason the weights in Table 3are quite similar to those in Table 2 where specificity was notconsidered This shows that when the specificity of the termsis similar the weights are not heavily modified

For the second test we have used another set of termswithdifferent degrees of specificity shown in Figure 7 In this casewe aggregate the values (E B B C and C) with specificities(095 08 08 05 and 05) respectively

In this second test the last two terms have a specificity(05) much lower than the first three terms (095 and 08)Theresults given in Table 4 show that this difference affects theweights as expected giving more weight to the less uncertainterms We can see a notable increase in the overall weight ofthe first three terms and a decrease in the weight of the lasttwo terms

5 IULOWA Multiperson MulticriteriaCase Study

In this section we use the operator defined in this paper toaddress a real environmental evaluation problem In particu-lar we study the impact of disposing sewage sludge in agricul-tural soils Environmental impact assessment is defined by theInternational Association for Impact Assessment (IAIA) asldquothe process of identifying predicting evaluating and mit-igating the biophysical social and other relevant effects of

0 100908070605040302010

1A EDCB

Figure 7 Linguistic variable with 5 terms (Test 2)

development proposals prior to major decisions being takenand commitments maderdquo In the last decades the increase ofsewage sludge production as a residue of wastewater treat-ment plants (WWTP) has become an environmental problemin several countries To maintain sustainability countries areencouraged to promote the value of sewage sludge as auseful by-product One of the most widespread practices hasbeen to apply sewage sludge to agricultural soils as fertilizerAlthough this option is generally accepted because it reducesfertilizer costs it may have ecological and human impacts Inthe SOSTAQUA Spanish research project these impacts havebeen studied and evaluated using many different criteriaCriteria were structured along three basic axes economicaspects environmental suitability and human health risks[46ndash48] For sludge managers the decision on how to dis-tribute the available sludge (from different WWTPs) amongtheir clients (farmerswith different agricultural fields) is quitecomplex due to the large amount of information that has to beconsidered and due to the expert knowledge that is requiredtomake a correct evaluation For this reason it is important tohave tools that evaluate the degree of suitability of using agiven sewage sludge on different types of soils in order to findthe best possible combination

In this paper we focus on the problem of obtaining anoverall suitability index that evaluates the impact of certaintypes of sludge on soil This overall suitability is obtained byaggregating the five criteria presented in Table 5 The evalua-tion of these criteria is not straightforward and can be madeusing different methodologies [46 49] Moreover some ofthe information considered in the evaluation model is sub-jectively defined by a domain expert so we can have differentopinions from different people

51 The Multiperson Multicriteria Aggregation Process It isquite common to find problematic decisions when a set ofalternatives have been evaluated by different experts on a setof criteria In this scenario an aggregation process with twosteps is carried out First the expertsrsquo evaluations of eachcriterion are fused in order to find a collective result for eachcriterion Afterwards collective criteria are aggregated inorder to find the overall evaluation for each alternative Thistwo-stage process is illustrated in Figure 8

Let 119874 = 1198741 1198742 119874

119899 be a finite set of options (or

alternatives) to be considered in the group decision makingproblem Let 119862 = 119862

1 1198622 119862

119898 be a set of criteria (or

attributes) Let 119864 = 1198641 1198642 119864

119902 be a finite set of experts

(or decision makers or stakeholders) who participate in the


Table 5 Environmental criteria

Criterion name Description Information used

Biodiversitysuitability

Biodiversity is an indicator of the health of ecosystems Biodiversity can beadversely affected by metal and organic compound contaminationdepending on the characteristics of the soil

Metal concentration in the sludgeOrganic compounds in the sludgeSludge treatment typeOrganic matter in the soilSoil textureSoil carbonate level

Nitrates suitabilityContamination of the soil by nutrients should be minimized Applyingsludge containing nitrates to a soil may affect its recommended level ofnitrates

Organic matter in the sludgeSludge treatment typeNitrates available in the sludgeSoil textureNitrates available in the soil

Organic mattersuitability

Soil organic matter regulates several processes (eg as OMmineralizesslowly nutrients are released at a slower pace reducing the potential risk ofnitrogen leaching to groundwater)

Organic matter in the sludgeOrganic matter in the soilSludge treatment type

pH suitabilityMetal contamination in soils is related to its pH For this reason basic soilsare preferred for sewage sludge treatment Acid soils should receive sludgewith a high pH

Sludge pHSoil pH

Soil contaminationsuitability

Soil contamination refers to the presence of heavy metals and organiccompounds in a soil The presence of contaminants in sewage sludge mayresult in risks to humans and ecosystems The contaminantrsquos movementbetween environmental compartments may lead to soil contamination

Metal concentration in the sludgeOrganic compounds in the sludgeSludge treatment typeOrganic matter in the soilSoil textureSoil carbonates levelSoil pH

Expert A

Expert B

Expert C

MP-IULOWAaggregation

IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes

Attributes Evaluations

Figure 8 Diagram of the multiperson multicriteria aggregation process

decision making process so that each expert Ek provides hisher own payoff matrix (119886

119894119895)119899times119898

The process can be defined asfollows

Step 1 For each option Oi and each criterion Cj take the qvalues of the experts E and calculate the weighting vectorWto be used in the IULOWA operator according to the order-inducing variable U (ie the specificity and fuzziness of thelabels) following the method proposed in Section 4 Thenapply IULOWA to aggregate the q values of the experts Eusing the weighting vector W following Definition 2 Theresult is the collective payoff matrix (119886

119894119895)119899times119898

Step 2 For each option Oi and its collective scores obtainedin Step 1 calculate the weighting vector W to be used in theIULOWA operator considering the order-inducing variableU (ie the specificity and fuzziness of the linguistic labels ofthe ith row of the matrix) and the method proposed inSection 4 Then calculate the overall aggregated results withthe IULOWA operator using Definition 2

Step 3 Adopt decisions according to the results found in theprevious steps Select the alternative that provides the bestresult Otherwise establish an ordering or a ranking of thealternatives from the most- to the least-preferred alternativeto enable the consideration of more than one selection


Table 6 Definition and values of specificity and fuzziness of thelinguistic terms

Linguistic value Definition Specificity FuzzinessDangerous (D) (00 00 00 01) 0950 0050Risky (R) (00 01 01 02) 0900 0099Poor (PO) (01 02 035 05) 0725 0125Acceptable (A) (035 05 05 065) 0850 0150Good (G) (05 065 065 0875) 0812 0187Excellent (E) (065 0875 09 10) 0812 0162Perfect (PF) (09 10 10 10) 0950 0050

This double-aggregation process is applied to a givenoption Oj described with m criteria by q experts and can beexpressed as a function MP-IULOWA 119878119898 times 119878

119902

rarr 119878 so that

MP-IULOWA ((⟨1199061

1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))

= IULOWA (IULOWA119895=1119898

times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)

Note that in the literature there is a wide range of meth-ods for group decisionmaking including those that use expertsystems voting systems and game theory Observe that gametheory is focused on competitive decision making where theindividuals make a decision considering the potential actionsof their opponents In this context it is worth noting that thework of Yager [50] considered the use of OWA operator ingame theory This paper has not received much attention inthe scientific community but it may bring strong implicationsfor the development of new approaches for group decisionmakingwith game theory For example all the extensions andgeneralizations of the OWA operator also have the potentialto be implemented under this framework including theIULOWA operator presented in this paper By using theIULOWA operator deeper assumptions should be madebecause the first step would be to assume that the availableinformation is given in the formof linguistic variables poten-tially defined on unbalanced fuzzy sets Although this pointis not considered in this paper note that this issue may bringsome potential developments in future research

52 Solving the Case Study In this section we apply the MP-IULOWA operator to an example with 3 types of sludge (S1S2 and S3) and 4 agricultural fields (F1F2F3 andF4) whichleads to a total of 12 different combinations or cases Let usassume that three experts (E1 E2 and E3) have evaluatedthose cases with the five criteria explained in Table 5 andusing the unbalanced linguistic variable depicted in Figure 9

The linguistic vocabulary gives 7 degrees of suitabilityranging from a dangerous to a perfect situationThis linguis-tic scale presents an unbalanced set of terms with differentspecificity and fuzziness (see Table 6)Themost specific termsare those that correspond to themost extreme scores (danger-ous and perfect) followed by the term ldquoriskyrdquoThis specificity

D PFEAPOR

0 100908070605040302010

1G

Figure 9 Evaluation scale for the criteria (D ldquodangerousrdquo Rldquoriskyrdquo PO ldquopoorrdquo A ldquoacceptablerdquo G ldquogoodrdquo E ldquoexcellentrdquo and PFldquoperfectrdquo)

is needed because those labels refer to very critical and precisesituations A not so specific neutral term is available for useif there is a combination of values that is neither positive nornegative from the point of view of environmental suitabilityThe other terms permit the identification of different suitabil-ity levels without the need to be too precise

Notice that in this vocabulary the specificity of the termsis a useful indicator to induce the aggregationweights becausethe most specific values correspond to those terms thatare detecting the most interesting situations from the deci-sionmakerrsquos point of view In fact themost specific terms givemore information than the rest It is also necessary to takeinto account the fact that when the specificity of the evalu-ations is the same fuzziness is used to solve those ties Thedefinition specificity and fuzziness of the terms depicted inFigure 9 are shown in Table 6 In this example it is assumedthat the scientists that provide the evaluations have similarexpertise and it is not necessary to assign different confi-dences to each of them

Table 7 corresponds to the three expertsrsquo evaluationsof the twelve cases taking into account the environmentalcriteria explained above

After obtaining the evaluations of the three experts it isnecessary to aggregate all of this information into a singlematrix to represent the group opinion regarding the alterna-tives for the five criteria As we want to give more confidenceto values with high precision we will use the proposed two-stage IULOWA aggregation process (see Figure 8)

First when we aggregate the three expertsrsquo evaluationsto obtain a single evaluation for each attribute of eachalternative we will apply (18) and (20) to give more confi-dence to the labels with more specificity and less fuzzinessTable 8 shows the matrix obtained after the aggregation ofthe three expertsrsquo opinions using the IULOWA operatorinduced by the specificity and by the quantifier Q(r) = r05which corresponds to high ornessThis policy establishes thatthe evaluations given by the more uncertain values will bealmost ignored and the overall result will be mostly based onthe most specific evaluation given by one of the experts (119908

1

around 055) For example for alternative 1 and the ldquobiodi-versityrdquo criterion the ranking of the values given by the threeexperts is risky ⪰ acceptable ⪰ excellent The result given byIULOWA is poor and is mainly based on the combination ofthe two first labels (119908

1+1199082asymp 085)Thus as the most precise

expert has indicated only a ldquoriskyrdquo level of suitability and


Table 7 Evaluations of experts 1198641 1198642 and 1198643

Case Biodiversity Nutrients suitability Organic matter suitability PH suitability Absence of soil contamination1198641 1198642 1198643 1198641 1198642 1198643 1198641 1198642 1198643 1198641 1198642 1198643 1198641 1198642 1198643

1 A R E D PO PO R R R A E G R R D2 G G G A G A PF G PF PF PF PF PO PO D3 PO G A D D D A PO PO R R A R R R4 A R PO A PO A G E G PF G PF PO G PO5 R A D A E A A R A E G PF G A PF6 PO G PO E G G G G E G G A PO D PO7 G PO A G G A G PO G PF PF PF A G E8 E E E G A G E E E E E PF G G G9 R D R PO G PO A A PO D D D A A A10 A A R G G PF G G G R D R PO PO PO11 G E A G G G E E G G G G G G A12 PO PO PO A A PO A A A PF PF PF PO PO PO

Table 8 Collective data matrix including the overall suitability value

Case Biodiversity Nutrients suitability Organic matter suitability PH suitability Absence of soil contamination Overall suitability1 Poor Risky Risky Acceptable Risky Risky2 Good Acceptable Excellent Perfect Risky Good3 Acceptable Dangerous Acceptable Risky Risky Risky4 Poor Acceptable Excellent Excellent Acceptable Good5 Risky Acceptable Poor Excellent Excellent Acceptable6 Acceptable Excellent Excellent Acceptable Risky Acceptable7 Acceptable Acceptable Acceptable Perfect Acceptable Good8 Excellent Acceptable Excellent Excellent Good Good9 Risky Acceptable Acceptable Dangerous Acceptable Poor10 Poor Excellent Good Risky Poor Acceptable11 Acceptable Good Excellent Good Acceptable Good12 Poor Acceptable Acceptable Perfect Poor Good

taking into account the precise medium evaluation given byldquoacceptablerdquo the result is ldquopoorrdquo

In the resulting matrix which contains the collectiveevaluation IULOWA is applied again to each row in order toobtain a final overall evaluation for each alternative Wefollow the same aggregation policy which weights the con-tribution of the values according to their precision The lastcolumn of Table 8 shows the overall suitability of each of thetwelve alternatives considered

As indicated above evaluating the environmental impactof treating soil with sewage sludge is quite a delicate task andthe experts will want to give more importance to the caseswhere they have detected extreme values such as ldquodangerousrdquoor ldquoperfectrdquo Using the IULOWA weighting mechanism andapplying a disjunctive aggregation policy we find that themost specific label contributes around 45 to the final result(1199081= 045) The remaining 55 is divided among the other

labels in particular the one in the second position afterthe ranking according to the uncertainty For example incase 8 the ranking is acceptable ⪰ excellent ⪰ excellent ⪰

excellent ⪰ good so the final result is mainly a combination ofldquoacceptablerdquo and ldquoexcellentrdquo which gives the result ldquogoodrdquo

Notice that with this criterion a precise evaluation isconsidered more important because experts are more con-fident when they give a specific evaluation that when theychoose a more general one This rationale is clearly exempli-fied in cases 7 and 9 where an extreme evaluation (positiveldquoperfectrdquo for case 7 and negative ldquodangerousrdquo for case 9) hasdirect consequences on the final overall suitability evaluationIn case 7 despite having an ldquoacceptablerdquo evaluation for mostof the attributes the fact of having a single but very specificldquoperfectrdquo evaluation makes the overall evaluation ldquogoodrdquo Asimilar event occurs in case 9 where having a ldquodangerousrdquoevaluation reduces the final evaluation to ldquopoorrdquo althoughmost of the attributes have an ldquoacceptablerdquo suitability

6 Comparison with Xu-IULOWA Aggregator

In this section we compare the performance of the proposedIULOWA aggregator with another well-known induced lin-guistic operator with the same name defined by Xu [21] Themethod proposed by Xu intends to aggregate the informationprovided by a set of uncertain linguistic variables Each ofthese variables is defined with an interval of two linguistic


0

1

0125 10

0875

075

0625

0375

025 05

sminus3 sminus2 sminus1 s0 s1 s2 s3 s4sminus4

(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)

Figure 10 Linguistic terms used in the comparison (a) Basic set of 9 labels equally distributed (b) terms aggregated in case 1 (c) termsaggregated in case 2 (d) terms aggregated in case 3

terms from a fixed finite set of predefined terms Figure 10(a)shows an example set of 9 labels numbered from 119904

minus4to 1199044

whose meaning may be taken to be implicitly represented bya set of symmetric and uniformly distributed triangular fuzzysets Moreover each item to be aggregated has an associatedvalue that is used to induce the order in which the items haveto be aggregated however Xu does not propose any specificinduction order In Section 2 (10) we have provided thedefinition of the Xu-IULOWA operator which only dependson the indexes of the aggregated elements (no operationsare performed explicitly on fuzzy sets) The result of theaggregation is also an uncertain linguistic variable that isan interval [sa sb] where a and b do not have to be valuesfrom the original set of terms For example the result of anaggregation could be [119904

minus13 11990424

]In this section we show how our proposal is flexible

enough to be able to ldquosimulaterdquo the uncertainty represented bythe intervals in Xursquos workThe flexibility of our method relieson the possibility of associating any fuzzy set to a label evenif it is unbalanced or the labels are not uniformly distributedthroughout the domain of discourse

Let us consider three case studies In each of them theaim is to aggregate 4 uncertain linguistic values as definedby Xu (ie 4 intervals of terms defined on a set of 9 labels)using Xursquosmethod and our novel IULOWAoperator In orderto apply our method first it is necessary to translate eachinterval to a fuzzy set Our idea is to simulate each intervalwith a trapezoid using the triangular fuzzy sets shown inFigure 10(a) for instance the interval (119904

minus2 1199043) would be

simulated with the trapezoid (0125 025 0875 and 10)Table 9 shows the 4 values to be aggregated in each of the 3cases and the trapezoids associated with each interval These

trapezoids are also shown in Figures 10(b) 10(c) and 10(d)In order to induce the aggregation order in our methodit is necessary to know the specificity fuzziness and indexassociated with each of the fuzzy sets to be aggregatedThe values are shown in Table 9 (the index refers to therelative order of the lowest extreme of the intervals) Havingthese three values the order in which the items have to beaggregated is determined (the last column of Table 9) Firstwe take the more specific sets If two sets have the samespecificity the less fuzzy set is preferred If two sets have thesame specificity and fuzziness the one with lower index takesprecedence

The next step of our method is to calculate the weightingvector which depends on the aggregation policy (the param-eter a described in Section 42) and on the specificity of thevalues to be aggregated (themore specific values have a higherweight than the less specific values) Table 10 shows the valuesof the weighting vector for 3 different policies (119886 = 01119886 = 05 and 119886 = 1) as described on Section 42

After all this process we can apply our IULOWAaggrega-tor and Xursquos IULOWA operator using in both cases the sameinduced order and the same weighting vector The results (alabel in our approach an interval in Xursquos case) are also shownin Table 10 It is worth noting that in our approach after theaggregation of each pair of labels we take as result the closestlabel of the original 9 term set

The main difference between both proposals is that theresult of our method is by definition a single label whereasthe result of Xursquos approach is an interval Note that each of thenine original labels may be taken to represent in some waya certain interval (eg label 119904

1corresponds basically to the

values between 05625 and 06875) Table 10 shows how in


Table 9 Fuzzy terms and features used during the comparison

Case ULOWA pair Membership function (1199011 1199012 1199013 1199014) Specificity Fuzziness Index Induced order

1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1

Table 10 Aggregation of terms

Case Terms to aggregate Policy Weights Xu-IULOWA Our proposal

1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043

all cases our result is indeed very close to the middle of theuncertain interval obtained by Xu

The three cases can be mentioned in more detail Incase 1 (Figure 10(b)) there are two specific labels (minus1 0) and(0 1) and two slightly more general ones (1 3) and (minus3 minus1)defined in a symmetric way with respect to the centre of thedomain When 119886 = 1 the two more specific intervals have aslightly higher weight than the others but the positive onescompensate for the negative ones and the result is perfectlysymmetric When 119886 = 05 almost 74 of the weight relies onthe two initial intervals although the result is not perfectlysymmetric with respect to 0 the first one (minus1 0) has asignificantly higher weight When 119886 = 01 almost all theweight (878) is taken by the first label (minus1 0) so the resultis very close to this interval

In case 2 (Figure 10(c)) there are two specific positivelabels (1 2) and (1 3) and two labels that are more general (anegative one (minus3 0) and a positive one (minus1 4))The differencein the specificity of the labels is clearly reflected on theweightsassigned to each label even when 119886 = 1 as the first label ismore than twice the weight of the last one (433 and 144respectively) This difference is even bigger when 119886 = 05where the two initial labels already take 792 of the totalweight and it takes its maximum expression when 119886 = 01

and the first label takes almost 90 of the whole weight Thetwo initial labels move the result towards the (1 2) intervalalthough the third label displaces a little bit this interval

towards the negative side (especially when 119886 = 1 and thisthird interval still has a weight of 228)

Finally in case 3 (Figure 10(d)) there are two verypositive and specific intervals (3 4) and (2 4) and two verynegative and uncertain intervals (minus4 1) and (minus3 2) The largedifference of specificity between the first two intervals and theother two is clearly shown in their associated weights whichgive an overall weight to the first two values of 727 (119886 = 1)852 (119886 = 05) and 968 (119886 = 01) Thus the lower is theparameter a themore positive is the result of the aggregation

In summary the comparison between our novel methodand the one proposed by Xu shows how our IULOWAmethodology is flexible enough to model the uncertaintyrepresented by Xursquos intervals giving an overall result thatdespite being a single label closely reflects the middle of theresulting uncertain intervals of Xu-IULOWA

7 Conclusions and Future Work

The fusion of partial evaluations to obtain an overall score foreach alternative is usually done with aggregation functionsThis paper has presented a new aggregation operator calledIULOWA which enables complex reordering processes to becarried out by using order-inducing variables In particularthe IOWA operator has been extended to deal with linguisticvariables that use unbalanced fuzzy sets Unbalanced setsof terms allow managing values with different degrees of


uncertainty thus permitting the design of sets of linguisticterms for variables that due to their nature require differentdegrees of precision in different parts of the domain

First we have proposed a new procedure to aggregateterms with different degrees of precision This method isbased on the extension principle and it uses operations onthe fuzzy sets associated with the linguistic terms that areaggregated The procedure is recursive following the well-known LOWA operator

Second we have carefully analysed the use of inducedvariables in unbalanced sets of linguistic termsThe paper hasproposed a procedure to use the measurement of uncertaintyas an order-inducing criterion in IULOWA In this approachthe decision is based on the less uncertain valuesThe conceptof minimum uncertainty is interpreted as maximum speci-ficity and minimum fuzziness two well-known measures infuzzy theory Ties are solved by taking the linguistic scale ofevaluation as the preference degreeThe paper also shows thatit is useful to modify the weighting policy according to thelevel of uncertainty to make a coherent aggregation of thevalues

It can be clearly seen that we have defined a generaloperator that includes the ULOWA operator when all theterms have the same specificity and fuzzinessThis can also bereduced to the LOWA operator if the terms are balanced Infact the IULOWA operator provides a wide range of familiesof unbalanced linguistic aggregation operators following themethodology used in the OWA literature

On the basis of the IULOWA operator a multipersonmulticriteria scenario has been presented proposing a solu-tion to the decision making problem in two steps (1) usingIULOWA to obtain a collective value for each criterion ofeach alternative and (2) using IULOWA to combine theaggregated values of the different criteria into a single overallevaluation The final overall linguistic value will identify thebest alternative alternatives This model has been used in areal environmental assessment problem using a set of criteriadefined in the Spanish research project SOSTAQUA Theresults obtained show that when specific values give moreinformation than the more uncertain ones the IULOWAoperator and the weighting policy proposed in this paper givegood and consistent results

Finally a thorough comparison with an aggregationoperator for uncertain linguistic values defined by Xu [21] hasbeen performedThe versatility of our approach allowing thedefinition of unbalanced and not uniformly distributed fuzzysets to represent the meaning of linguistic labels permitsmodelling the uncertain intervals considered by Xu andobtaining results that lie closely within the centre of theintervals obtained with that method

In future research we plan to develop further aggregationoperators for unbalanced linguistic variables including thepossibility of introducing the importance of the variablesThis combination of weighting criteria has been already stud-ied in the numerical case so different methodologies can beconsidered in future research [51] We are also interested inusing this approach for group decision making We intend tostudy measures of consensus to evaluate the degree ofagreement between the experts and to estimate the proximity

between the individual opinion and the collective one [52]The similarity measure between fuzzy sets that we apply tocombine pairs of labels during the aggregation process couldalso be used for this purpose As mentioned in the paper therelationship between this novel method of group decisionmaking and game theory is also worth studying in detailFinally this study is part of the Spanish DAMASK researchproject in which we are developing a web-based online rec-ommender system In particular we will deploy a version ofthis system that is aimed at helping the user to find holidaydestinations and restaurants according to hisher personalpreferences [53ndash55]

Conflict of Interests

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

Acknowledgments

This study has been supported by Universitat Rovira i Virgili(a predoctoral grant for L Marin) the Spanish Ministry ofScience and Innovation (DAMASK project Data MiningAlgorithms with Semantic Knowledge TIN2009-11005 andthe SHADE project Semantic and Hierarchical Attributes inDecision Making TIN2012-34369) and the Spanish Govern-ment (PlanE Spanish Economy and Employment Stimula-tion Plan) The authors would also like to acknowledge thesupport provided by the SOSTAQUA project founded byCDTI within the framework of the Ingenio 2010 - CENITProgram

References

[1] V Torra and Y Narukawa Modeling Decisions InformationFusion and Aggregation Operators Springer Berlin Germany2007

[2] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Berlin Germany 2007

[3] R R Yager ldquoOn ordered weighted averaging aggregation oper-ators in multicriteria decision makingrdquo IEEE Transactions onSystems Man and Cybernetics vol 18 no 1 pp 183ndash190 1988

[4] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEE Transactions on Systems Man and CyberneticsB Cybernetics vol 29 no 2 pp 141ndash150 1999

[5] F J Cabrerizo I J Perez and E Herrera-Viedma ldquoManagingthe consensus in group decisionmaking in an unbalanced fuzzylinguistic context with incomplete informationrdquo Knowledge-Based Systems vol 23 no 2 pp 169ndash181 2010

[6] J M Merigo and M Casanovas ldquoThe uncertain induced quasi-arithmetic OWA operatorrdquo International Journal of IntelligentSystems vol 26 no 1 pp 1ndash24 2011

[7] P Bonissone and K Decker ldquoSelecting uncertainty calculi andgranularity an experiment in trading-off precision and com-plexityrdquo in Proceedings of the 1st Annual Conference on Uncer-tainty in Artificial Intelligence (UAI rsquo85) L N Kanal and J FLemmer Eds pp 217ndash248 New York NY USA 1985

[8] R R Yager ldquoMeasures of specificity over continuous spacesunder similarity relationsrdquo Fuzzy Sets and Systems vol 159 no17 pp 2193ndash2210 2008


[9] Z Xu ldquoOn generalized induced linguistic aggregation opera-torsrdquo International Journal of General Systems vol 35 no 1 pp17ndash28 2006

[10] Z S 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

[11] Z S Xu andQ L Da ldquoAn overview of operators for aggregatinginformationrdquo International Journal of Intelligent Systems vol 18no 9 pp 953ndash969 2003

[12] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[13] Z S Xu ldquoLinguistic aggregation operators an overviewrdquo inFuzzy Sets and Their Extensions Representation Aggregationand Models H Bustince F Herrera and J Montero Eds pp163ndash181 Springer Berlin Germany 2008

[14] F Herrera E Herrera-Viedma and L Martınez ldquoA fuzzy lin-guistic methodology to deal with unbalanced linguistic termsetsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp 354ndash370 2008

[15] L A Zadeh ldquoThe concept of a linguistic variable and its appli-cation to approximate reasoning-Irdquo vol 8 no 3 pp 199ndash2491975

[16] Z S 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

[17] D Isern L Marin A Valls and A Moreno ldquoThe unbalancedlinguistic ordered weighted averaging operatorrdquo in Proceedingsof the IEEE International Conference on Fuzzy Systems (FUZZrsquo10) pp 1ndash8 Barcelona Spain July 2010

[18] Z Xu ldquoMulti-period multi-attribute group decision-makingunder linguistic assessmentsrdquo International Journal of GeneralSystems vol 38 no 8 pp 823ndash850 2009

[19] X Yu Z S Xu and X Zhang ldquoUniformization ofmultigranularlinguistic labels and their application to group decision mak-ingrdquo Journal of Systems Science and Systems Engineering vol 19no 3 pp 257ndash276 2010

[20] 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

[21] Z Xu ldquoInduced uncertain linguistic OWA operators applied togroup decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006

[22] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquo Computersand Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[23] X Liu and S Han ldquoOrness and parameterized RIM quantifieraggregation with OWA operators a summaryrdquo InternationalJournal of Approximate Reasoning vol 48 no 1 pp 77ndash97 2008

[24] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005

[25] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996

[26] J Kacprzyk ldquoGroup decision making with a fuzzy linguisticmajorityrdquo Fuzzy Sets and Systems vol 18 no 2 pp 105ndash118 1986

[27] J M Merigo and M Casanovas ldquoThe uncertain generalizedOWAoperator and its application to financial decisionmakingrdquoInternational Journal of Information Technology and DecisionMaking vol 10 no 2 pp 221ndash230 2011

[28] G Beliakov and S James ldquoInduced ordered weighted averagingoperatorsrdquo in Recent Developments in the Ordered WeightedAveraging Operators Theory and Practice R R Yager JKacprzyk and G Beliakov Eds vol 265 pp 29ndash47 SpringerBerlin Germany 2011

[29] G Pasi and R R Yager ldquoModeling the concept of majorityopinion in group decision makingrdquo Information Sciences vol176 no 4 pp 390ndash414 2006

[30] J M Merigo and M Casanovas ldquoDecision making with dis-tance measures and linguistic aggregation operatorsrdquo Interna-tional Journal of Fuzzy Systems vol 12 no 3 pp 190ndash198 2010

[31] G Wei X Zhao and R Lin ldquoSome induced aggregatingoperators with fuzzy number intuitionistic fuzzy informationand their applications to group decision makingrdquo InternationalJournal of Computational Intelligence Systems vol 3 no 1 pp84ndash95 2010

[32] M Xia and Z S Xu ldquoEntropycross entropy-based groupdecisionmaking under intuitionistic fuzzy environmentrdquo Infor-mation Fusion vol 13 no 1 pp 31ndash47 2012

[33] B Arfi ldquoLinguistic fuzzy-logic decision-making processrdquo inLinguistic Fuzzy Logic Methods in Social Sciences pp 43ndash62Springer Berlin Germany 2010

[34] R Wang and S J Chuu ldquoGroup decision-making using a fuzzylinguistic approach for evaluating the flexibility in a manufac-turing systemrdquo European Journal of Operational Research vol154 no 3 pp 563ndash572 2004

[35] Z Pei Y Xu D Ruan and K Qin ldquoExtracting complexlinguistic data summaries from personnel database via simplelinguistic aggregationsrdquo Information Sciences vol 179 no 14 pp2325ndash2332 2009

[36] E Herrera-Viedma F J Cabrerizo I J Perez M J Cobo SAlonso and F Herrera ldquoApplying linguistic OWA operatorsin consensus models under unbalanced linguistic informationrdquoin Recent Developments in the Ordered Weighted AveragingOperators Theory and Practice R Yager J Kacprzyk andG Beliakov Eds vol 265 of Studies in Fuzziness and SoftComputing pp 167ndash186 Springer Berlin Germany 2011

[37] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001

[38] L Garmendia R R Yager E Trillas and A Salvador ldquoA t-norm based specificity for fuzzy sets on compact domainsrdquoInternational Journal of General Systems vol 35 no 6 pp 687ndash698 2006

[39] G J Klir ldquoDevelopments in uncertainty-based informationrdquo inAdvances in Computers C Y Marshall Ed pp 255ndash332Elsevier New York NY USA 1993

[40] R R Yager ldquoOrdinal measures of specificityrdquo InternationalJournal of General Systems vol 17 no 1 pp 57ndash72 1990

[41] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andComputation vol 20 pp 301ndash312 1972

[42] G J Klir and B Yuan ldquoOn nonspecificity of fuzzy sets withcontinuous membership functionsrdquo in Proceedings of the 15thIEEE International Conference on Systems Man and Cybernetics(SMC rsquo95) pp 25ndash29 Vancouver Canada October 1995


[43] R R Yager ldquoIncluding importances inOWAaggregations usingfuzzy systems modelingrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 2 pp 286ndash294 1998

[44] R R Yager ldquoOn the dispersion measure of OWA operatorsrdquoInformation Sciences vol 179 no 22 pp 3908ndash3919 2009

[45] F Chiclana E Herrera-Viedma F Herrera and S AlonsoldquoSome induced ordered weighted averaging operators and theiruse for solving group decision-making problems based on fuzzypreference relationsrdquo European Journal of Operational Researchvol 182 no 1 pp 383ndash399 2007

[46] A Valls J PijuanM Schuhmacher A Passuello M Nadal andJ Sierra ldquoPreference assessment for the management of sewagesludge application on agricultural soilsrdquo International Journal ofMulticriteria Decision Making vol 1 pp 4ndash24 2010

[47] J Pijuan A Valls A Passuello and M Schuhmacher ldquoEvalu-ating the impact of sewage sludge application on agriculturalsoilsrdquo in 15th Congreso espanol sobre tecnologıas y logica fuzzy(ESTYLF rsquo10) pp 369ndash374 Huelva Spain 2010

[48] T Kaya andCKahraman ldquoAn integrated fuzzyAHP-ELECTREmethodology for environmental impact assessmentrdquo ExpertSystems with Applications vol 38 no 7 pp 8553ndash8562 2011

[49] A Passuello M Schuhmacher M Mari O Cadiach and MNadal ldquoA spatial multicriteria decision analysis to managesewage sludge application on agriculatural soilsrdquo in Environ-mental Modeling for Susteinable Regional Development SystemApproaches and Advanced Methods V Olej I Obrsalova and JKrupka Eds pp 221ndash241 2011

[50] R R Yager ldquoA game theoretic approach to decision makingunder uncertaintyrdquo International Journal of Intelligent Systemsin Accounting Finance and Management vol 8 no 2 pp 131ndash143 1999

[51] V Torra ldquoThe WOWA operator a reviewrdquo in Recent Develop-ments in the OrderedWeighted Averaging Operators Theory andPractice pp 17ndash28 Springer Berlin Germany 2011

[52] F Herrera E Herrera-Viedma and J L Verdegay ldquoA model ofconsensus in group decision making under linguistic assess-mentsrdquo Fuzzy Sets and Systems vol 78 no 1 pp 73ndash87 1996

[53] LMarin AMoreno andD Isern ldquoAutomatic preference learn-ing on numeric and multi-valued categorical attributesrdquoKnowledge-Based Systems vol 56 no 1 pp 201ndash215 2014

[54] L Marin D Isern and A Moreno ldquoDynamic adaptation ofnumerical attributes in a user profilerdquo Applied Intelligence vol39 no 2 pp 421ndash437 2013

[55] AMoreno AValls D Isern LMarin and J Borras ldquoSigTurE-Destination ontology-based personalized recommendation ofTourism and Leisure Activitiesrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 633ndash651 2013

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks


Advances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

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Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


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Industrial EngineeringJournal of


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



Very low Low Medium High Very high

(a)

Very low Medium High Very high Perfect

(b)

Figure 1 Examples of balanced (a) and unbalanced (b) linguistic term sets with five labels

we allow the set of linguistic labels to be unbalanced Labelscan be represented by asymmetric fuzzy sets and they can benonuniformly distributed in the domain Each label has anassociated fuzzy set on the variablersquos reference scale of mea-surementThe fuzzy sets of the labels define a fuzzy partitionFirst the ULOWA (unbalanced linguistic ordered weightingaveraging) operator is presented Unlike its predecessor theLOWA operator [20] ULOWA is based on the extensionprinciple and so it carries out operations on the fuzzy setsassociated with the labels thus defining a new procedure foraggregating a pair of labels according to their membershipfunctions and the set of available linguistic terms After thiswe propose a generalization which we call IULOWA whichenables us to work with inducing variables

This paper shows that IULOWA fulfills the monotonic-ity identity idempotency and boundary conditions usuallyrequired in aggregation operators Like the other OWA oper-ators IULOWA provides a family of aggregation operatorsthat is parameterized between the linguistic minimum andmaximum and that includes a wide range of particular casessuch as the unbalanced linguistic average (ULA) the unbal-anced linguistic OWA (ULOWA) the unbalanced linguisticweighted average (ULWA) and many others Note that inthe literature there are approaches that employ abbreviationssimilar to the ones used in this paper although they are con-ceptually different In particular it is worth mentioning thatthe work by Xu on uncertain linguistic variables produces theinduced uncertain linguistic OWA (IULOWA) operator [21]However that approach has important differences withrespect to the IULOWA operator presented in this paperThepresent study is focused on the use of unbalanced linguisticinformation while Xursquos work [21] is based on uncertain lin-guistic variables that deal with imprecise information whenrepresenting the linguistic labels Section 6 compares bothapproaches and highlights their similarities and differences

Another important contribution of this paper is the pro-posal to use some of the information related to the definitionof the labels as an order-inducing criterion in the IULOWAoperator Although the order of the arguments can be decidedby taking into account the domain requirements it is some-times desirable to take into consideration the amount ofinformation contained in the terms themselves In this paperwe propose a method to calculate the set of weights of thearguments taking into account the degree of uncertainty ofthe labels This permits us to order the arguments by givingpriority to more specific values because these representmore precise information The method uses two well-knownmeasures of fuzzy sets namely fuzziness and specificity

We demonstrate the behaviour of the precision-basedIULOWA operator with a case study in a real applicationSpecifically we analyse the results obtained from the eval-uation of the environmental impact produced when sewage

sludge coming from wastewater treatment plants is used asfertilizer on agricultural soils In this application a two-stageaggregation is needed because we have a set of experts thatevaluate a set of options using the same set of criteria (ievariables) For this reason we further extend the IULOWAoperator by usingmultiperson techniques in the analysis [22]and in doing so we define the multi-person-IULOWA (MP-IULOWA) operator By including the opinions of severalexperts we obtain more reliable results because we can basethe decision on the knowledge of a group of people ratherthan on the opinion of a single individual Moreover the useof unbalanced and induced information enables us to dealwith complex environments where some of the informationismore representative and therefore needs to be prioritised inorder to correctly assess the aggregation

The rest of the paper is organised as follows Section 2 pro-vides some preliminaries introducing the OWA and IOWAoperators the linguistic OWA operator and the managementof unbalanced linguistic labels which are the basis of the newoperator Section 3 defines the new induced unbalancedLOWA operator studies its properties and presents somespecific operators that can be derived from the general for-mulation Section 4 describes how the different degrees ofuncertainty in the unbalanced terms can be used to induce anorder among them It also explains how the set of weights forthe operator can be determined using the uncertainty mea-sures Section 5 shows the application of the new IULOWAoperator to a multiperson multicriteria problem (MP-IULOWA) Section 6 compares the operator defined in thispaper with the homonym IULOWAoperator proposed by Xu[21] Finally Section 7 gives themain conclusions of the paperand suggests some lines for future research

2 Preliminaries

This section describes the set of concepts required before theintroduction of the IULOWA operator First we presentnumerical aggregation with the ordered weighted average(OWA) together with its order-induced extension theinduced OWA operator We then present the LOWA (lin-guistic OWA) operator as the basis of the IULOWA operatorproposed in this paper Finally we introduce recent studies onthe management of unbalanced sets of terms

21 InducedOrderedWeightedAverage (IOWA)Operator TheOWA operator is formally defined as follows [3]

An OWA operator of dimension 119898 is a mapping OWA

119877119898

rarr 119877 defined by an associated weighting vector 119882 of





OWA119882

(1198861 1198862 119886

119898) = 119882

119879

sdot 119861 =

119898

sum

119895=1

119908119895119887119895 (1)


119894



119908119894= 119876(

119894

119899

) minus 119876(

119894 minus 1

119899

) (2)



119895



IOWA119882

(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882119879

sdot 119861119906=

119898

sum

119895=1

119908119895119887119895

(3)

where 119882 = (1199081 119908




119906is obtained

by taking 119887119895as the 119886


119894 119886119894⟩ which has the




119894and 119886119895with 119906

119894= 119906119895is


is 119886119894


119894


119894+ 119886119895)2 Another











119878119899119865lowast

997888997888rarr 119865 (119877)

app(sdot)997888997888997888997888rarr 119878

(4)






If 119860 = 1198861 119886



LOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(5)

where 119882 = (1199081 119908



119894= 1 120573

ℎ= 119908ℎsum119898

2119908119896 ℎ =




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896

119904119895 119904119894isin 119878 (119895 ge 119894)

(6)

so that




119862119898

119908119894 119887119894 119894 = 1 119898 = 119887





1 p2 p3 p4) where p

1le p2le




2= p3







0 100908070605040302010

1

0 100908070605040302010

1

0 100908070605040302010

1

(a)

0 100908070605040302010

1

(b)





0 119904

119899(119905)

119899(119905)minus1 a


TF1199051199051015840 (119904119899(119905)

119894 120572119899(119905)

) = Δ(

Δminus1

(119904119899(119905)

119894 120572119899(119905)

) sdot (119899 (1199051015840

) minus 1)

119899 (119905) minus 1

)

(8)






Extre

mely

poor

Very

poo

r

Poor

Slig

htly

poo

rFa

irSl

ight

ly g

ood

Goo

d

Very

goo

d

Extre

mely

good




119878119905

= 119904119905

120573| 120573 = 1 minus 119905

2

3

(2 minus 119905)

2

4

(3 minus 119905)

0

2

4

(119905 minus 3)

2

3

(119905 minus 2) 119905 minus 1

(9)







minus4 119904minus3




(i) 1199041

+ 1199042

= [1199041198861

1199041198871

] + [1199041198862

1199041198872


min1199041198861+1198862

1199044 max119904

minus4 min119904

1198871+1198872

1199044]



Xu-IULOWA119908

= (⟨1199061 1199041⟩ ⟨1199062 1199042⟩ ⟨119906

119899 119904119899⟩)

= 11990811199041198871

+ 11990821199041198872

+ sdot sdot sdot + 119908119899119904119887119899

(10)

where119882 = (1199081 1199082 119908









If 119860 = 1198861 119886



ULOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(11)




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896


119894le119901le119895

Sim (119878119901 120575)

(12)


119896= 119909lowast

119904119894

+ 1199081(119909lowast

119904119895

minus 119909lowast

119904119894


119904119894


1 p2 p3 p4)

COG (119904119860) =

119910lowast

119860=

1

6

(

1199013minus 1199012

1199014minus 1199011

+ 2) if 1199011

= 1199014

1

2

if 1199011= 1199014

119909lowast

119860=

119910lowast

119860(1199013+ 1199012) + (119901

4+ 1199011) (1 minus 119910

lowast

119860)

2

(13)


Sim (119875 119876) =4

radic

4

prod

119894=1

(2 minus1003816100381610038161003816119901119894minus 119902119894

1003816100381610038161003816) minus 1 (14)



0

1

ylowast


VL L PHM AH VH

(a) Policy mean (119908 = (05 05))

0

1

ylowast


VL L PHM AH VH


0

1

ylowast


VL L PHM AH VH





IULOWA119908(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882 sdot 119861119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1) otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(15)


120590(1)

1198861015840

120590(2) 119886

1015840

120590(119895)) where 119886

1015840





119894isin [0 1] sum119908






⟨119906119898 119901119898⟩ and 119876 = ⟨119906

1015840

1 1199021⟩ ⟨119906

1015840

119898 119902119898⟩ so that

forall119895 119906119895

= 1199061015840

119895and forall119895 119901

119895ge 119902119895 then IULOWA

119908(119875) ge

IULOWA119908(119876)



119908(⟨1199061 1199011⟩

⟨119906119898 119901119898⟩) and IULOWA

119908(119876) = IULOWA

119908(⟨1199061015840

1 1199021⟩

⟨1199061015840

119898 119902119898⟩) If forall119895 119906

119895= 1199061015840

119895and forall119895 119901



120590(119895)ge

1199021015840

120590(119895) and IULOWA

119908(1199011015840

120590(1) 119901

1015840

120590(119898)) ge IULOWA

119908(1199021015840

120590(1)

1199021015840



119908(119876)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= IULOWA

119908(⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) where (⟨119906

1015840

1 1198861015840

1⟩ ⟨119906

1015840

119898 1198861015840

119898⟩) is any


119898 119886119898⟩)


1 1198861⟩


1015840

= (⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840



119908(119860) = IULOWA

119908(1198601015840

)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= a if forall119895 119886

119895= 119886


119908(119886119898minus1

119886119898) = 119904119896 where 119896 = 119908


1) otimes 119904119894=

argmax119894le119901le119895


119896= 119904119894= 119904119895=


119898) = 119886



min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)



119908119894 119887119894 119894 = 1 2 =



argmax119894le119901le119895







obtained if 1199081

= 1 and 119908119895



119898= 1 and 119908






m so that 119908119896= 1 and 119908

119895= 0 for all j = k




= 119908119902= 05 and 119908




= 119908119902

= 0 with119906119901



119908119895= 1(119898 minus 2)





119895= 119908119898minus119895+1




119895gt 0 for




=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898















Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







OWA119882

(1198861 1198862 119886

119898) = 119882

119879

sdot 119861 =

119898

sum

119895=1

119908119895119887119895 (1)


119894



119908119894= 119876(

119894

119899

) minus 119876(

119894 minus 1

119899

) (2)



119895



IOWA119882

(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882119879

sdot 119861119906=

119898

sum

119895=1

119908119895119887119895

(3)

where 119882 = (1199081 119908




119906is obtained

by taking 119887119895as the 119886


119894 119886119894⟩ which has the




119894and 119886119895with 119906

119894= 119906119895is


is 119886119894


119894


119894+ 119886119895)2 Another











119878119899119865lowast

997888997888rarr 119865 (119877)

app(sdot)997888997888997888997888rarr 119878

(4)






If 119860 = 1198861 119886



LOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(5)

where 119882 = (1199081 119908



119894= 1 120573

ℎ= 119908ℎsum119898

2119908119896 ℎ =




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896

119904119895 119904119894isin 119878 (119895 ge 119894)

(6)

so that




119862119898

119908119894 119887119894 119894 = 1 119898 = 119887





1 p2 p3 p4) where p

1le p2le




2= p3







0 100908070605040302010

1

0 100908070605040302010

1

0 100908070605040302010

1

(a)

0 100908070605040302010

1

(b)





0 119904

119899(119905)

119899(119905)minus1 a


TF1199051199051015840 (119904119899(119905)

119894 120572119899(119905)

) = Δ(

Δminus1

(119904119899(119905)

119894 120572119899(119905)

) sdot (119899 (1199051015840

) minus 1)

119899 (119905) minus 1

)

(8)






Extre

mely

poor

Very

poo

r

Poor

Slig

htly

poo

rFa

irSl

ight

ly g

ood

Goo

d

Very

goo

d

Extre

mely

good




119878119905

= 119904119905

120573| 120573 = 1 minus 119905

2

3

(2 minus 119905)

2

4

(3 minus 119905)

0

2

4

(119905 minus 3)

2

3

(119905 minus 2) 119905 minus 1

(9)







minus4 119904minus3




(i) 1199041

+ 1199042

= [1199041198861

1199041198871

] + [1199041198862

1199041198872


min1199041198861+1198862

1199044 max119904

minus4 min119904

1198871+1198872

1199044]



Xu-IULOWA119908

= (⟨1199061 1199041⟩ ⟨1199062 1199042⟩ ⟨119906

119899 119904119899⟩)

= 11990811199041198871

+ 11990821199041198872

+ sdot sdot sdot + 119908119899119904119887119899

(10)

where119882 = (1199081 1199082 119908









If 119860 = 1198861 119886



ULOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(11)




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896


119894le119901le119895

Sim (119878119901 120575)

(12)


119896= 119909lowast

119904119894

+ 1199081(119909lowast

119904119895

minus 119909lowast

119904119894


119904119894


1 p2 p3 p4)

COG (119904119860) =

119910lowast

119860=

1

6

(

1199013minus 1199012

1199014minus 1199011

+ 2) if 1199011

= 1199014

1

2

if 1199011= 1199014

119909lowast

119860=

119910lowast

119860(1199013+ 1199012) + (119901

4+ 1199011) (1 minus 119910

lowast

119860)

2

(13)


Sim (119875 119876) =4

radic

4

prod

119894=1

(2 minus1003816100381610038161003816119901119894minus 119902119894

1003816100381610038161003816) minus 1 (14)



0

1

ylowast


VL L PHM AH VH

(a) Policy mean (119908 = (05 05))

0

1

ylowast


VL L PHM AH VH


0

1

ylowast


VL L PHM AH VH





IULOWA119908(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882 sdot 119861119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1) otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(15)


120590(1)

1198861015840

120590(2) 119886

1015840

120590(119895)) where 119886

1015840





119894isin [0 1] sum119908






⟨119906119898 119901119898⟩ and 119876 = ⟨119906

1015840

1 1199021⟩ ⟨119906

1015840

119898 119902119898⟩ so that

forall119895 119906119895

= 1199061015840

119895and forall119895 119901

119895ge 119902119895 then IULOWA

119908(119875) ge

IULOWA119908(119876)



119908(⟨1199061 1199011⟩

⟨119906119898 119901119898⟩) and IULOWA

119908(119876) = IULOWA

119908(⟨1199061015840

1 1199021⟩

⟨1199061015840

119898 119902119898⟩) If forall119895 119906

119895= 1199061015840

119895and forall119895 119901



120590(119895)ge

1199021015840

120590(119895) and IULOWA

119908(1199011015840

120590(1) 119901

1015840

120590(119898)) ge IULOWA

119908(1199021015840

120590(1)

1199021015840



119908(119876)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= IULOWA

119908(⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) where (⟨119906

1015840

1 1198861015840

1⟩ ⟨119906

1015840

119898 1198861015840

119898⟩) is any


119898 119886119898⟩)


1 1198861⟩


1015840

= (⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840



119908(119860) = IULOWA

119908(1198601015840

)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= a if forall119895 119886

119895= 119886


119908(119886119898minus1

119886119898) = 119904119896 where 119896 = 119908


1) otimes 119904119894=

argmax119894le119901le119895


119896= 119904119894= 119904119895=


119898) = 119886



min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)



119908119894 119887119894 119894 = 1 2 =



argmax119894le119901le119895







obtained if 1199081

= 1 and 119908119895



119898= 1 and 119908






m so that 119908119896= 1 and 119908

119895= 0 for all j = k




= 119908119902= 05 and 119908




= 119908119902

= 0 with119906119901



119908119895= 1(119898 minus 2)





119895= 119908119898minus119895+1




119895gt 0 for




=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898















Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







119878119899119865lowast

997888997888rarr 119865 (119877)

app(sdot)997888997888997888997888rarr 119878

(4)






If 119860 = 1198861 119886



LOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(5)

where 119882 = (1199081 119908



119894= 1 120573

ℎ= 119908ℎsum119898

2119908119896 ℎ =




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896

119904119895 119904119894isin 119878 (119895 ge 119894)

(6)

so that




119862119898

119908119894 119887119894 119894 = 1 119898 = 119887





1 p2 p3 p4) where p

1le p2le




2= p3







0 100908070605040302010

1

0 100908070605040302010

1

0 100908070605040302010

1

(a)

0 100908070605040302010

1

(b)





0 119904

119899(119905)

119899(119905)minus1 a


TF1199051199051015840 (119904119899(119905)

119894 120572119899(119905)

) = Δ(

Δminus1

(119904119899(119905)

119894 120572119899(119905)

) sdot (119899 (1199051015840

) minus 1)

119899 (119905) minus 1

)

(8)






Extre

mely

poor

Very

poo

r

Poor

Slig

htly

poo

rFa

irSl

ight

ly g

ood

Goo

d

Very

goo

d

Extre

mely

good




119878119905

= 119904119905

120573| 120573 = 1 minus 119905

2

3

(2 minus 119905)

2

4

(3 minus 119905)

0

2

4

(119905 minus 3)

2

3

(119905 minus 2) 119905 minus 1

(9)







minus4 119904minus3




(i) 1199041

+ 1199042

= [1199041198861

1199041198871

] + [1199041198862

1199041198872


min1199041198861+1198862

1199044 max119904

minus4 min119904

1198871+1198872

1199044]



Xu-IULOWA119908

= (⟨1199061 1199041⟩ ⟨1199062 1199042⟩ ⟨119906

119899 119904119899⟩)

= 11990811199041198871

+ 11990821199041198872

+ sdot sdot sdot + 119908119899119904119887119899

(10)

where119882 = (1199081 1199082 119908









If 119860 = 1198861 119886



ULOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(11)




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896


119894le119901le119895

Sim (119878119901 120575)

(12)


119896= 119909lowast

119904119894

+ 1199081(119909lowast

119904119895

minus 119909lowast

119904119894


119904119894


1 p2 p3 p4)

COG (119904119860) =

119910lowast

119860=

1

6

(

1199013minus 1199012

1199014minus 1199011

+ 2) if 1199011

= 1199014

1

2

if 1199011= 1199014

119909lowast

119860=

119910lowast

119860(1199013+ 1199012) + (119901

4+ 1199011) (1 minus 119910

lowast

119860)

2

(13)


Sim (119875 119876) =4

radic

4

prod

119894=1

(2 minus1003816100381610038161003816119901119894minus 119902119894

1003816100381610038161003816) minus 1 (14)



0

1

ylowast


VL L PHM AH VH

(a) Policy mean (119908 = (05 05))

0

1

ylowast


VL L PHM AH VH


0

1

ylowast


VL L PHM AH VH





IULOWA119908(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882 sdot 119861119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1) otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(15)


120590(1)

1198861015840

120590(2) 119886

1015840

120590(119895)) where 119886

1015840





119894isin [0 1] sum119908






⟨119906119898 119901119898⟩ and 119876 = ⟨119906

1015840

1 1199021⟩ ⟨119906

1015840

119898 119902119898⟩ so that

forall119895 119906119895

= 1199061015840

119895and forall119895 119901

119895ge 119902119895 then IULOWA

119908(119875) ge

IULOWA119908(119876)



119908(⟨1199061 1199011⟩

⟨119906119898 119901119898⟩) and IULOWA

119908(119876) = IULOWA

119908(⟨1199061015840

1 1199021⟩

⟨1199061015840

119898 119902119898⟩) If forall119895 119906

119895= 1199061015840

119895and forall119895 119901



120590(119895)ge

1199021015840

120590(119895) and IULOWA

119908(1199011015840

120590(1) 119901

1015840

120590(119898)) ge IULOWA

119908(1199021015840

120590(1)

1199021015840



119908(119876)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= IULOWA

119908(⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) where (⟨119906

1015840

1 1198861015840

1⟩ ⟨119906

1015840

119898 1198861015840

119898⟩) is any


119898 119886119898⟩)


1 1198861⟩


1015840

= (⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840



119908(119860) = IULOWA

119908(1198601015840

)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= a if forall119895 119886

119895= 119886


119908(119886119898minus1

119886119898) = 119904119896 where 119896 = 119908


1) otimes 119904119894=

argmax119894le119901le119895


119896= 119904119894= 119904119895=


119898) = 119886



min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)



119908119894 119887119894 119894 = 1 2 =



argmax119894le119901le119895







obtained if 1199081

= 1 and 119908119895



119898= 1 and 119908






m so that 119908119896= 1 and 119908

119895= 0 for all j = k




= 119908119902= 05 and 119908




= 119908119902

= 0 with119906119901



119908119895= 1(119898 minus 2)





119895= 119908119898minus119895+1




119895gt 0 for




=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898















Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 100908070605040302010

1

0 100908070605040302010

1

0 100908070605040302010

1

(a)

0 100908070605040302010

1

(b)





0 119904

119899(119905)

119899(119905)minus1 a


TF1199051199051015840 (119904119899(119905)

119894 120572119899(119905)

) = Δ(

Δminus1

(119904119899(119905)

119894 120572119899(119905)

) sdot (119899 (1199051015840

) minus 1)

119899 (119905) minus 1

)

(8)






Extre

mely

poor

Very

poo

r

Poor

Slig

htly

poo

rFa

irSl

ight

ly g

ood

Goo

d

Very

goo

d

Extre

mely

good




119878119905

= 119904119905

120573| 120573 = 1 minus 119905

2

3

(2 minus 119905)

2

4

(3 minus 119905)

0

2

4

(119905 minus 3)

2

3

(119905 minus 2) 119905 minus 1

(9)







minus4 119904minus3




(i) 1199041

+ 1199042

= [1199041198861

1199041198871

] + [1199041198862

1199041198872


min1199041198861+1198862

1199044 max119904

minus4 min119904

1198871+1198872

1199044]



Xu-IULOWA119908

= (⟨1199061 1199041⟩ ⟨1199062 1199042⟩ ⟨119906

119899 119904119899⟩)

= 11990811199041198871

+ 11990821199041198872

+ sdot sdot sdot + 119908119899119904119887119899

(10)

where119882 = (1199081 1199082 119908









If 119860 = 1198861 119886



ULOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(11)




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896


119894le119901le119895

Sim (119878119901 120575)

(12)


119896= 119909lowast

119904119894

+ 1199081(119909lowast

119904119895

minus 119909lowast

119904119894


119904119894


1 p2 p3 p4)

COG (119904119860) =

119910lowast

119860=

1

6

(

1199013minus 1199012

1199014minus 1199011

+ 2) if 1199011

= 1199014

1

2

if 1199011= 1199014

119909lowast

119860=

119910lowast

119860(1199013+ 1199012) + (119901

4+ 1199011) (1 minus 119910

lowast

119860)

2

(13)


Sim (119875 119876) =4

radic

4

prod

119894=1

(2 minus1003816100381610038161003816119901119894minus 119902119894

1003816100381610038161003816) minus 1 (14)



0

1

ylowast


VL L PHM AH VH

(a) Policy mean (119908 = (05 05))

0

1

ylowast


VL L PHM AH VH


0

1

ylowast


VL L PHM AH VH





IULOWA119908(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882 sdot 119861119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1) otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(15)


120590(1)

1198861015840

120590(2) 119886

1015840

120590(119895)) where 119886

1015840





119894isin [0 1] sum119908






⟨119906119898 119901119898⟩ and 119876 = ⟨119906

1015840

1 1199021⟩ ⟨119906

1015840

119898 119902119898⟩ so that

forall119895 119906119895

= 1199061015840

119895and forall119895 119901

119895ge 119902119895 then IULOWA

119908(119875) ge

IULOWA119908(119876)



119908(⟨1199061 1199011⟩

⟨119906119898 119901119898⟩) and IULOWA

119908(119876) = IULOWA

119908(⟨1199061015840

1 1199021⟩

⟨1199061015840

119898 119902119898⟩) If forall119895 119906

119895= 1199061015840

119895and forall119895 119901



120590(119895)ge

1199021015840

120590(119895) and IULOWA

119908(1199011015840

120590(1) 119901

1015840

120590(119898)) ge IULOWA

119908(1199021015840

120590(1)

1199021015840



119908(119876)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= IULOWA

119908(⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) where (⟨119906

1015840

1 1198861015840

1⟩ ⟨119906

1015840

119898 1198861015840

119898⟩) is any


119898 119886119898⟩)


1 1198861⟩


1015840

= (⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840



119908(119860) = IULOWA

119908(1198601015840

)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= a if forall119895 119886

119895= 119886


119908(119886119898minus1

119886119898) = 119904119896 where 119896 = 119908


1) otimes 119904119894=

argmax119894le119901le119895


119896= 119904119894= 119904119895=


119898) = 119886



min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)



119908119894 119887119894 119894 = 1 2 =



argmax119894le119901le119895







obtained if 1199081

= 1 and 119908119895



119898= 1 and 119908






m so that 119908119896= 1 and 119908

119895= 0 for all j = k




= 119908119902= 05 and 119908




= 119908119902

= 0 with119906119901



119908119895= 1(119898 minus 2)





119895= 119908119898minus119895+1




119895gt 0 for




=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898















Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






minus4 119904minus3




(i) 1199041

+ 1199042

= [1199041198861

1199041198871

] + [1199041198862

1199041198872


min1199041198861+1198862

1199044 max119904

minus4 min119904

1198871+1198872

1199044]



Xu-IULOWA119908

= (⟨1199061 1199041⟩ ⟨1199062 1199042⟩ ⟨119906

119899 119904119899⟩)

= 11990811199041198871

+ 11990821199041198872

+ sdot sdot sdot + 119908119899119904119887119899

(10)

where119882 = (1199081 1199082 119908









If 119860 = 1198861 119886



ULOWA (1198861 119886

119898) = 119882 sdot 119861

119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1)


120573ℎ 119887ℎ ℎ = 2 119898

(11)




1198622

119908119894 119887119894 119894 = 1 2 = 119908


1) otimes 119904119894= 119904119896


119894le119901le119895

Sim (119878119901 120575)

(12)


119896= 119909lowast

119904119894

+ 1199081(119909lowast

119904119895

minus 119909lowast

119904119894


119904119894


1 p2 p3 p4)

COG (119904119860) =

119910lowast

119860=

1

6

(

1199013minus 1199012

1199014minus 1199011

+ 2) if 1199011

= 1199014

1

2

if 1199011= 1199014

119909lowast

119860=

119910lowast

119860(1199013+ 1199012) + (119901

4+ 1199011) (1 minus 119910

lowast

119860)

2

(13)


Sim (119875 119876) =4

radic

4

prod

119894=1

(2 minus1003816100381610038161003816119901119894minus 119902119894

1003816100381610038161003816) minus 1 (14)



0

1

ylowast


VL L PHM AH VH

(a) Policy mean (119908 = (05 05))

0

1

ylowast


VL L PHM AH VH


0

1

ylowast


VL L PHM AH VH





IULOWA119908(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882 sdot 119861119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1) otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(15)


120590(1)

1198861015840

120590(2) 119886

1015840

120590(119895)) where 119886

1015840





119894isin [0 1] sum119908






⟨119906119898 119901119898⟩ and 119876 = ⟨119906

1015840

1 1199021⟩ ⟨119906

1015840

119898 119902119898⟩ so that

forall119895 119906119895

= 1199061015840

119895and forall119895 119901

119895ge 119902119895 then IULOWA

119908(119875) ge

IULOWA119908(119876)



119908(⟨1199061 1199011⟩

⟨119906119898 119901119898⟩) and IULOWA

119908(119876) = IULOWA

119908(⟨1199061015840

1 1199021⟩

⟨1199061015840

119898 119902119898⟩) If forall119895 119906

119895= 1199061015840

119895and forall119895 119901



120590(119895)ge

1199021015840

120590(119895) and IULOWA

119908(1199011015840

120590(1) 119901

1015840

120590(119898)) ge IULOWA

119908(1199021015840

120590(1)

1199021015840



119908(119876)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= IULOWA

119908(⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) where (⟨119906

1015840

1 1198861015840

1⟩ ⟨119906

1015840

119898 1198861015840

119898⟩) is any


119898 119886119898⟩)


1 1198861⟩


1015840

= (⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840



119908(119860) = IULOWA

119908(1198601015840

)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= a if forall119895 119886

119895= 119886


119908(119886119898minus1

119886119898) = 119904119896 where 119896 = 119908


1) otimes 119904119894=

argmax119894le119901le119895


119896= 119904119894= 119904119895=


119898) = 119886



min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)



119908119894 119887119894 119894 = 1 2 =



argmax119894le119901le119895







obtained if 1199081

= 1 and 119908119895



119898= 1 and 119908






m so that 119908119896= 1 and 119908

119895= 0 for all j = k




= 119908119902= 05 and 119908




= 119908119902

= 0 with119906119901



119908119895= 1(119898 minus 2)





119895= 119908119898minus119895+1




119895gt 0 for




=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898















Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

1

ylowast


VL L PHM AH VH

(a) Policy mean (119908 = (05 05))

0

1

ylowast


VL L PHM AH VH


0

1

ylowast


VL L PHM AH VH





IULOWA119908(⟨1199061 1198861⟩ ⟨1199062 1198862⟩ ⟨119906

119898 119886119898⟩)

= 119882 sdot 119861119879

= 119862119898

119908119896 119887119896 119896 = 1 119898


1) otimes 119862119898minus1

120573ℎ 119887ℎ ℎ = 2 119898

(15)


120590(1)

1198861015840

120590(2) 119886

1015840

120590(119895)) where 119886

1015840





119894isin [0 1] sum119908






⟨119906119898 119901119898⟩ and 119876 = ⟨119906

1015840

1 1199021⟩ ⟨119906

1015840

119898 119902119898⟩ so that

forall119895 119906119895

= 1199061015840

119895and forall119895 119901

119895ge 119902119895 then IULOWA

119908(119875) ge

IULOWA119908(119876)



119908(⟨1199061 1199011⟩

⟨119906119898 119901119898⟩) and IULOWA

119908(119876) = IULOWA

119908(⟨1199061015840

1 1199021⟩

⟨1199061015840

119898 119902119898⟩) If forall119895 119906

119895= 1199061015840

119895and forall119895 119901



120590(119895)ge

1199021015840

120590(119895) and IULOWA

119908(1199011015840

120590(1) 119901

1015840

120590(119898)) ge IULOWA

119908(1199021015840

120590(1)

1199021015840



119908(119876)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= IULOWA

119908(⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840

119898⟩) where (⟨119906

1015840

1 1198861015840

1⟩ ⟨119906

1015840

119898 1198861015840

119898⟩) is any


119898 119886119898⟩)


1 1198861⟩


1015840

= (⟨1199061015840

1 1198861015840

1⟩

⟨1199061015840

119898 1198861015840



119908(119860) = IULOWA

119908(1198601015840

)


119908(⟨1199061 1198861⟩ ⟨119906

119898 119886119898⟩)= a if forall119895 119886

119895= 119886


119908(119886119898minus1

119886119898) = 119904119896 where 119896 = 119908


1) otimes 119904119894=

argmax119894le119901le119895


119896= 119904119894= 119904119895=


119898) = 119886



min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)



119908119894 119887119894 119894 = 1 2 =



argmax119894le119901le119895







obtained if 1199081

= 1 and 119908119895



119898= 1 and 119908






m so that 119908119896= 1 and 119908

119895= 0 for all j = k




= 119908119902= 05 and 119908




= 119908119902

= 0 with119906119901



119908119895= 1(119898 minus 2)





119895= 119908119898minus119895+1




119895gt 0 for




=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898















Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





min (1198861 119886

119898) le IULOWA

119908(1199061 1198861 119906

119898 119886119898)

le max (1198861 119886

119898)

(16)



119908119894 119887119894 119894 = 1 2 =



argmax119894le119901le119895







obtained if 1199081

= 1 and 119908119895



119898= 1 and 119908






m so that 119908119896= 1 and 119908

119895= 0 for all j = k




= 119908119902= 05 and 119908




= 119908119902

= 0 with119906119901



119908119895= 1(119898 minus 2)





119895= 119908119898minus119895+1




119895gt 0 for




=

(1 minus (120572 + 120573))119898 + 120573119908119898

= (1 minus (120572 + 120573))119898 + 120572

when and 119908119895= (1 minus (120572 + 120573))119898















Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Sp(120583) ge Sp(120574)


Sp (119860) = 119879(120572sup 119873(int

120572sup

0

119872(119860120572) 119889120572)) (17)




Sp (119860) = 1 minus

area under 119860

119887 minus 119886

(18)


119909





01 03020

1

A = (01 02 02 03)

B = (0125 0175 0225 0275)



Fz (119860) = ℎ119909isin119860

(119889 (119860 (119883) 119860119862

(119909))) (19)


Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886

|2 sdot 119860 (119909) minus 1| (20)





Sp (119860) = 1 minus

area of 119860

119887 minus 119886

= 1 minus

(02 lowast 1) 2

1 minus 0

= 09

Sp (119861) = 1 minus

area of 119861

119887 minus 119886

= 1 minus

2 (0052) + 005

1 minus 0

= 09

(21)




A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






A GFEDC

0 100908070605040302010

1B



Fz (119860) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119860 (119909) minus 1|

= 1 minus 09 = 01

Fz (119861) = 1 minus

1

119887 minus 119886

int

119887

119886


1

0

|2 sdot 119861 (119909) minus 1|

= 1 minus 095 = 005

(22)



1 119886



1 1198872 119887



119896+ 1)


119896+ 1) then Fz(119887

119896) le

Fz(119887119896+ 1)


119896+ 1) and Fz(119887

119896) =

Fz(119887119896+ 1) then 119887


scale 119878










119908119896= 119876(

119878 (119896)

119878 (119899)

) minus 119876(

119878 (119896 minus 1)

119878 (119899)

) (23)

where 119878(119896) = sum119896

119897=1119906120590(119897)



119896






119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119886 Weights01 (0851 0061 0038 0028 0022)025 (0668 0127 0085 0066 0054)05 (0447 0185 0142 0120 0106)075 (0299 0204 0179 0164 0154)1 (0200 0200 0200 0200 0200)


119886 Weights01 (0860 0059 0036 0025 0020)025 (0686 0124 0080 0060 0050)05 (0470 0186 0136 0110 0098)075 (0322 0209 0174 0152 0143)1 (0221 0209 0198 0186 0186)


119886 Weights01 (0876 0056 0036 0017 0015)025 (0719 0119 0083 0042 0037)05 (0517 0187 0145 0079 0072)075 (0372 0216 0192 0112 0108)1 (0268 0225 0225 0141 0141)







0 100908070605040302010

1A EDCB





Let 119874 = 1198741 1198742 119874



1 1198622 119862


attributes) Let 119864 = 1198641 1198642 119864















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in















Sludge pHSoil pH




Expert A

Expert B

Expert C


IULOWAaggregation

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Alte

rnat

ives

Attributes

Attributes

Attributes




119894119895)119899times119898



119894119895)119899times119898







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







119902

rarr 119878 so that


1 1198861

1⟩ ⟨119906

1

119898 1198861

119898⟩)

(⟨119906119902

1 119886119902

1⟩ ⟨119906

119902

119898 119886119902

119898⟩))


times (⟨1199061

119895 1198861

119895⟩ ⟨119906

119902

119895 119886119902

119895⟩))

(24)




D PFEAPOR

0 100908070605040302010

1G







1



















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in


















0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

1

0125 10

0875

075

0625

0375

025 05


(a)

0

1

0125 10

0875

075

0625

0375

025 05

(b)

0

1

0125 10

0875

075

0625

0375

025 05

(c)

0

1

0125 10

0875

075

0625

0375

025 05

(d)



minus4to 1199044


minus13 11990424















1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1

[119904minus3

119904minus1

] (0 0125 0375 05) 0625 0125 0 4[1199040 1199041] (0375 05 0625 075) 075 0125 2 2

[119904minus1

1199040] (025 0375 05 0625) 075 0125 1 1

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 3

2

[1199041 1199043] (05 0625 0875 1) 0625 0125 3 2

[119904minus1

1199044] (025 0375 1 1) 03125 0062 1 4

[119904minus3

1199040] (0 0125 05 0625) 05 0125 0 3

[1199041 1199042] (05 0625 075 0875) 075 0125 2 1

3

[119904minus3

1199042] (0 0125 075 0875) 025 0125 1 4

[119904minus4

1199041] (0 0 0625 075) 03125 0062 0 3

[1199042 1199044] (0625 075 1 1) 06875 0062 2 2

[1199043 1199044] (075 0875 1 1) 08125 0062 3 1



1 ⟨(1 [119904minus1

1199040]) (2 [119904

0 1199041]) (3 [119904

1 1199043]) (4 [119904

minus3 119904minus1

])⟩

119886 = 1 (0273 0273 0227 0227) [119904minus072

119904072

] 1199040

119886 = 05 (0523 0216 0140 0121) [119904minus074

119904051

] 1199040

119886 = 01 (0878 0063 0033 0026) [119904minus092

119904013

] 1199040 almost 119904

minus1

2 ⟨(1 [1199041 1199042]) (2 [119904

1 1199043]) (3 [119904

minus3 1199040]) (4 [119904

minus1 1199044])⟩

119886 = 1 (0343 0285 0228 0144) [119904minus020

119904211

] 1199041

119886 = 05 (0585 0207 0133 0075) [119904031

119904209

] 1199041

119886 = 01 (0898 0056 0030 0016) [119904084

119904202

] 1199041 almost 119904

2

3 ⟨(1 [1199043 1199044]) (2 [119904

2 1199044]) (3 [119904

minus4 1199041]) (4 [119904

minus3 1199042])⟩

119886 = 1 (0394 0333 0151 0122) [119904087

119904330

] 1199042

119886 = 05 (0627 0225 0084 0064) [119904180

119904362

] 1199042 almost 119904

3

119886 = 01 (0911 0057 0018 0014) [119904273

119904391

] 1199043





















Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in














Acknowledgments


References

































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in
























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Induced Unbalanced Linguistic Ordered Weighted Average