research article uncertainty in interval type-2 fuzzy...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 452780 16 pageshttpdxdoiorg1011552013452780

Research ArticleUncertainty in Interval Type-2 Fuzzy Systems

Sadegh Aminifar and Arjuna Marzuki

School of Electrical and Electronics Engineering Universiti Sains Malaysia Engineering Campus 14300 Nibong TebalPulau Pinang Malaysia

Correspondence should be addressed to Sadegh Aminifar sadeghaminifargmailcom

Received 8 June 2013 Accepted 1 August 2013

Academic Editor Alexander P Seyranian

Copyright copy 2013 S Aminifar and A Marzuki This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper studies uncertainty and its effect on system response displacementThe paper also describes how IT2MFs (interval type-2 membership functions) differentiate from T1MFs (type-1 membership functions) by adding uncertaintyThe effect of uncertaintyis modeled clearly by introducing a technique that describes how uncertainty causes membership degree reduction and changingthe fuzzy word meanings in fuzzy logic controllers (FLCs) Several criteria are discussed for the measurement of the imbalance rateof internal uncertainty and its effect on system behavior Uncertainty removal is introduced to observe the effect of uncertainty onthe outputThe theorem of uncertainty avoidance is presented for describing the role of uncertainty in interval type-2 fuzzy systems(IT2FSs) Another objective of this paper is to derive a novel uncertainty measure for IT2MFs with lower complexity and clearerpresentation Finally for proving the affectivity of novel interpretation of uncertainty in IT2FSs several investigations are done

1 Introduction

Type-n fuzzy sets were discussed generally and comprehen-sively in [1ndash12] The standardized version of interval type-2 (IT2) fuzzy systems (IT2FSs) [13] was referred to in [14ndash16] The standardized IT2FS was completed in 2007 TheIT2FS provides a suitable environment for computing withwords (CW) [17] However complexities high volumes ofcalculations and ambiguities in applying uncertainty hinderextensive applications This paper investigates the sources ofuncertainty and the uncertainty handling behavior of IT2FSs

Interval type-2 fuzzy sets have widely been accepted asmore capable of modeling higher orders of uncertainty thantype-1 fuzzy sets [4 9ndash11 18 19] This property has been thedriving force behind much of the advancement of intervaltype-2 fuzzy set theories and applications [10] One of themost important causes of such forces is finding a suitableinterpretation that is theoretically meaningful and practicallytractable Uncertainty plays an important role in fuzzy logictheory and applications The amount of uncertainty inherentin a fuzzy set has been quantified using different methodsKlir and Parviz presented the generalized information theoryas a foundation for a research project that unifies the theoriesdealing with uncertainty in [20] In this paper interval type-2fuzzy sets (IT2FS) are utilized in the field of generalized

information theory and general theory of uncertainty Thecontribution presented in this paper paves the way for activeresearch on the relationship between the amount of injecteduncertainty and response displacement after injection ofuncertainty

The authors presented an ensemble of fuzzy systemsin [21] for capturing uncertainty In that paper one type-1 fuzzy logic controller chooses the output position in anarea produced by another type-1 fuzzy system This paperattempts to discover the new features of IT2FSs specificallythe significant relationship between uncertainty and defuzzi-fication of the center of gravity (COG) to show the effect ofuncertainty on system output based on extracted conceptsThis study investigates that the output of interval type-2 (IT2)fuzzy logic system (FLS) in comparison with type-1 (T1)FLS moves to points with less uncertainty Uncertainty canenter into T1FSs and affect system response The effect ofuncertainty is modeled clearly by introducing a techniquethat describes how uncertainty causes membership degreereduction and changing the fuzzy word meanings Severalcriteria are discussed for the measurement of the imbalancerate of internal uncertainty and its effect on system behaviorThis paper shows the imbalance rate of uncertainty thatcan be seen as a criterion for displacing COG The effect

2 Mathematical Problems in Engineering

of uncertainty on displacing COG is presented as a noveluncertainty measure in this paper

The concept of uncertainty in fuzzy systems is interpretedin a new manner as illustrated in Figure 1 This researchconsiders the presence of uncertainty injectors and removersdifferent from conventional fuzzy systems (Figure 1) T1 fuzzysystems do not have these two blocks compared with theproposed fuzzy system The main difference between T1fuzzy systems and IT2FSs [13] is found in the fuzzifier anddefuzzifier blocks

Uncertainty removal is introduced to observe the effectof uncertainty on the output In Section 2 (Uncertainty) theconceptual descriptions regarding fuzzification block basedon the interpretations derived from this paper are discussedWe explain the application of these concepts in a systemwherein each variable corresponds to a number with oneinterval in each moment Our method attempts to find a T1fuzzy membership function for each IT2MF based on theeffect of uncertainty on certain data We also avoid usingiterative procedures

The direct role of uncertainty is introduced as the maindifference between T1FSs and T2FSs The justification ofapplication of IT2FLSs in fuzzy systems is highly depen-dent on uncertainty issue This paper aims to recognizedifferent types of uncertainty by introducing new definitionsand expressions thus creating informed and goal-orientedmaneuvers during the design process and increasing theapplication of interval T2 fuzzy controllers

Another objective of this paper is about new uncertaintymeasure for interval type-2 fuzzy membership functions(IT2MFs) Uncertainty measure is a necessity because to usefuzzy sets (FSs) as granules in general theory of uncertainty(GTU) which is introduced by Zadeh [22] it is necessaryto quantify the uncertainty associated with fuzzy sets (FS)because based on [23] ldquoonce uncertainty (and information)measures become well justified they can very effectively beutilized for managing uncertainty and associated informa-tion For example they can be utilized for extrapolatingevidence assessing the strength of relationship between givengroups of variables assessing the influences of given inputvariables on given output variables measuring the loss ofinformation when a system is simplified and the likerdquo

Klir and Folger [24] Klir and Parviz [20] and Harmanec[25] have developed three fundamental principles to guidethe use of uncertainty measures under different circum-stances

(1) ldquoThe principle of minimum uncertainty which statesthat solutionswith the least loss of information shouldbe selected can be used in simplification and conflictresolution problemsrdquo

(2) ldquoThe principle of maximum uncertainty which statesthat a conclusion should maximize the relevantuncertainty within constraints given by the verifiedpremises is widely used within classical probabilityframework [26 27]rdquo

(3) ldquoThe principle of uncertainty invariance which statesthat the amount of uncertainty should be preserved in

each transformation of uncertainty from one math-ematical framework to another is widely studied inthe context of probability-possibility transformation[28 29]rdquo

This paper tries to discuss the effect of uncertainty ofIT2FS based on the first principal of uncertainty

The remainder of this paper is organized as followsSection 2 discusses uncertainty and IT2MF efficacy andpresents improvements in this field Section 3 discusses theimbalance rate of internal uncertainty and its effect onsystem behavior Section 4 explains new uncertainty mea-sure Section 5 formulizes the uncertainty effect on systemoutput Sections 6 and 7 display the simulation analyses andconclusions respectively

2 Uncertainty

Zadeh points out in [22] that ldquouncertainty is an attribute ofinformationrdquo and introduced the general theory of uncer-tainty (GTU) ldquobecause existing approaches to representationof uncertain information are inadequate for dealing withproblems in which uncertain information is perception-based and is expressed in a natural languagerdquo He also statesthat ldquoIn GTU uncertainty is linked to information throughthe concept of granular structure a concept which plays a keyrole in human interaction with the real world [30ndash32]rdquo

Fuzzy systems are mainly applied for calculations thatuse lexical variables (ie CW) [33] Verbal interpretationsfor different operations are the key elements in CW Forfuzzification general T2 fuzzy systems are more effectivein modeling defined quantities by linguistic constraintsIT2FSs which have less complexity have discovered a fieldof application that is higher than T2 fuzzy sets This paperfocuses on IT2FLSs Figure 2 shows an example of an IT2MFThe membership degree of (119909 = 119886) takes the form of aninterval or a line segment [120583119897119886 120583119906119886] (Figure 2) The lengthof this line segment indicates the uncertainty bandwidth(119906

= 120583119906 to 120583119897) on that point This paper emphasizes thehypothesis in [34] in which IT2MF serves as the expansion ofT1 fuzzy MF with an equal amount of uncertainty expansionat both ends A principal MF that is located in the middleof the upper and lower bands of uncertainty is definedfor each IT2MF where the membership degree of eachmember 120583119901 is the average of the upper and lowermembershipdegrees (120583119901 = (120583119906 + 120583119871)2) [34] For each T1MF (whichis referred to as ldquodeterministic membership degreesrdquo thatis one-to-one correspondence between each variable and itsmembership degree) an unlimited number of IT2MFs arecreated because of the presence of uncertainty on either sideof each point of the member function These IT2MFs arecalled uncertaintified MFs of the T1MF

Four sources with different nature were mentioned foruncertainty in [35]These sources introduce the equal IT2MFfor uncertainty modeling For instance the uncertainty thatis transmitted to the data of a local factor such as noise iscompletely different from the uncertainty that is an insep-arable part of a word in mind Therefore the uncertaintyshould be modeled separately or the description trend of

Mathematical Problems in Engineering 3

Inference engine

Type II

Inputs

Outputs

Uncertainty remover

IT2 defuzzifierIT2 fuzzifier

Uncertaintifier

(Uncertainty injector)

Fuzzifier

T1

DefuzzifierT1

Figure 1 The role of uncertainty in an IT2FS or IT2 FLC

120583

Principal function(PF)

Upper bound of uncertainty(UMF)

Lower bound of uncertainty(LMF)

120583ux

120583lx

x

Uncertainty band (usim)

Figure 2 Principal function and one of its uncertaintified functions

this uncertainty should be determined Several methods ofextracting the MF of an IT2FS were explained in [36ndash38]Nevertheless none of these sources have shown clearly theeffect of these types of uncertainty on control processesandor their effect on output In the literature related toIT2FSs application the cause of the differences betweensystem outputs in the presence or absence of uncertaintyhas not been associated with the lack of certainty in asystemor data by using documented and significant formulasDiscussions in [39] indicate the minor role of uncertainty inthis issue

If we divide the uncertainty sources introduced in [35]into two groups based on the classifications performedon uncertainty sources [40ndash42] one group will be relatedto uncertainty which naturally exists in every word Thisuncertainty is independent from the field wherein adverbsof quantity such as ldquolittlerdquo ldquofairlyrdquo ldquoveryrdquo and ldquotoordquo aredetermined The quantity and end points of these endpointshave been determined by expertsMendel proposed amethodin [38] to determine the start and end points Naturaluncertainty is nondecreasing that is uncertaintymay changefrom person to person and from one period to the next [38]but its nature does not change the general form of the MF

Modeling the function of interval T2 membership functions(IT2MFs) for this type of uncertainty is very rational CertainIT2MF words which are used more in IT2FS were studiedin [18] These IT2MF words accurately model uncertaintyproperties within the nature of a word IT2MF words can besummarized by three modes small sounding words (low)medium sounding words (medium) and large soundingwords (high) (Figure 3) [19] Gaussian MF is introduced forldquohighrdquo to generalize the discussion (Figure 3)

In a specific problem choosing the same type of MF forall inputs and outputs is preferred

Regardless of whether the triangular trapezoidal orGaussian MFs are selected at points where the input rangehas quantitatively higher or lower membership degrees theuncertainty band will be narrower and wider at points thathave medium membership degrees respectively This modelis consistent with our attitudes toward words For instanceat a low mode the membership degree is high when theuncertainty of placing very small data in a field becomes low(Points zero and one in low shape in Figure 3) When databecomes larger the membership degree can be stated moredecisively to be not small (Points 119888 and 119889 in low shape inFigure 3)However at Point 119887 themembership degree cannotbe determined at a low mode furthermore disagreementsamong different people and the uncertainty of one person aregreater in points close to Point 119887 than the rest of the points(Figure 3)

Another type of uncertainty is related to data Two outof 4 items discussed in [35] are included in this categoryData uncertainty is not limited to these items and otheritems were mentioned in [41 42] Most parts of these types ofuncertainty can be reduced if a system is better understoodSome types of uncertainty are inevitable because of scientificreasons and the nature of the problem The efficiencies andabilities of IT2MF have not often been explained clearly Theusual applied forms in [18] and the three basicMFs (Figure 3)against modeling data uncertainty are not as efficient asmodeling the uncertainty in the nature of a wordThese typesof uncertainty are not always in line with the changes of


b

120583

a0 c d

Low

(a)

120583

a0 b c d

Medium

(b)

a0 b c

120583High

(c)

Figure 3 Basic IT2MF shapes

Table 1 Sample data that denotes uncertainty as an interval

Interval [27 33] [07 33] [minus13 33] [minus33 33]Average 3 2 1 0Interval [63 77] [54 66] [45 55] [36 44]Average 7 6 5 4Interval [11 11] [9825 0175] [865 923] [7475 8525]Average 11 10 9 8

the uncertainty band in the nature of a word in terms ofdecreasing or increasing uncertainty band

If the uncertainty in the nature of a word is applied to aT1MF we will achieve the same common IT2MFs Howeverif the same approach is implemented on data uncertaintywe will not achieve IT2MFs with common forms In thisexample two types of uncertainties are applied to a basic T1fuzzy function in two steps

Example Sample data that have been measured for apoint obtain an interval for numbers rather than a certainnumber because of the iteration of measurement or noiseTable 1 shows these intervals and their middle points Theuncertainty bounds of these intervals are the maximumpossible bounds in related systems for correspondent data(second row Table 1) This master interval (the uncertaintyof incoming online data does not exceed these intervals)can be achieved clearly from an iterative measurement of aparameter point or from different system conditions whichis applied in a single-input system or a system analyzer Forthese recent conditions the data obtained from two precisetemperature sensors can be considered as an interval ofuncertainty the sensors are installed on both ends of anautoclave room or a steam room The temperature is notnecessarily the same at both ends of the room at anymomentThe average bounds of the temperature are applied as a singleinput to the control system of the steam room temperatureThis type of uncertainty is created to remove the complexmodeling of temperature changes inside an autoclave room

In this example we assume that the general field ofthe data related is the verbal expression of ldquolowrdquo which isdefined as T1 fuzzy MF (Figure 4(b)) If the low bound ofall points in the intervals of the first row of Table 1 (alsoshown in Figure 4(a)) is determined by the verbal expression

ldquolowrdquo with the MF shown in Figure 4(b) (according tothe graphical description of Figure 4(b)) an IT2MF willbe created After deleting the saturated part an IT2MF isachieved (Figure 4(c)) that is the IT2MF is obtained fromdirect inclusion of the data uncertainty in Table 1 to the T1MFof Figure 4(b)

Each set of optional data which represents one set offigures in each interval of uncertainty in Table 1 has a first-degree fuzzy MF embedded in the IT2MF of Figure 4(c)

In the following step we intend to inject data uncertaintyto an IT2MF that defines only the natural uncertainty insidethe verbal expression (Figure 5(a)) We repeat all the above-mentioned steps for every single T1 embedded membershipdegree function in Figure 4(c) which is limited to the bound-aries shown in Figure 5(b) and takes the form of Figure 5(c)

According to the definition provided in this section theMFs of Figures 4(c) and 5(c) are the IT2MFs for ldquolowrdquo (alsocalled uncertaintified) of Figure 4(b) per data uncertaintyand per natural uncertainty respectively

The effect of these different MFs on system behavior isinvestigated in Section 6 (Investigation 6)

3 Rate of Internal Uncertainty Imbalance andIts Effect on System Response

Discussions have been provided in [18] regarding uncertaintymeasurement By considering the concepts of these discus-sions new definitions that are compatible with the objectivesof this paper are presented in this section The differencein the internal uncertainty of words plays an important rolein finding the COG through the principles introduced inthe following sections of the present paper Thus a newformula and definition is provided for COG calculationBefore starting the discussion we define the function of theabsolute ratio of two parameters

119860119877 (119860 119861) = ((119860

119861if 119861 lt 119860) (

119861

119860if 119861 lt 119860)) (1)

Definition 1 For each IT2MF one defines a function of 119906

which shows the bandwidth of uncertainty (119906

= 120583119906 minus 120583119897) foreach point of horizontal axes of IT2MF as shown in Figure 6


0

11

120583l

Low principal

1

(b)

120583 = 1 minus y11

120583

120583u

11 times measured datasorted from small to big

Dat

a

(a)

11

110 1 2 3 4 5 6 7 8 9 10

Low IT2MF

1 New principal

(c)

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10

a bc

120583

11

Figure 4 (a) Uncertain space of measured data (b) Application of uncertain space to one TIMF (c) Obtained IT2MF

1

120583

110

120583u

Low IT2MFintrinsic uncertainty

9

120583l

(a)

0 1 2 3 4 5 6 7 8 9 10 11

120583Injection of

data uncertainty

1

07

(b)

0 1 2 3 4 5 6 7 8 9 10 11

120583Low IT2MF

intrinsic and datauncertainties

07

1

(c)

Figure 5 (a) IT2MF based on the intrinsic description of uncertainty (b) Application of uncertain space to one type-2 MF (c) ObtainedIT2MF

Definition 2 The absolute ratio of the surface areas on bothsides of a line to the length of the COG of the principlefunction enclosed between the upper and lower bounds ofuncertainty band is called the difference index in internaluncertainty also referred to as the absolute ratio of surface(ARS) If the uncertainty bandwidth by 119906

and 119906

(119910) is a

function that shows the uncertainty bandwidth of every

single point of the IT2MF of 119865 the amount of ARS is shownas follows

ARS (119865) = 119860119877(int119872

119886

119880

119889119910 int119887

119872

119880

119889119910) (2)

The role of ARS which is the role of the surface areabetween two uncertainty bands in internal uncertainty in


M

ba y

120583

120583u

120583l

Principal function

(a)

ba y

Left side of COGof principal

function


function

M

usim

usim= 120583u minus 120583l

Al

Ar

(b)

Figure 6 (a) Typical IT2MF and its principal function (b) Uncertain bandwidth function (119906) and its left and right areas

[18] is used in the remaining sections of this paper to provethe theorem proposed in this section ARS is also used as aclosed formula for finding COG as a factor to drift the COG

Theorem of Uncertainty Avoidance In proportion to the COGof the principal MF the COGs of uncertaintified MFs aredisplaced toward a more clarified domain A higher amountof uncertainty imbalance on both sides of the COG of theprinciple MF corresponds to more displacements that shifttoward certainty

For example as shown in Figure 7 if some uncertaintyis injected to one side of the COG of IT2MF the COGwill shift toward the other side An increasing uncertaintyon the left of Points 119887 and 119888 displaces the response (COG)toward the right compared with Point a (Figure 7) In Points119888 and 119889 creating a slight uncertainty toward the right slightlydisplaces the response (COG) toward the left The blueshadow around each point shows the uncertainty around thatpoint (Figure 7)

Verification We use the physical properties of the COG ofhomogeneous planes without employing large volumes ofcalculations even simple ones to prove the theorem AnIT2MF is formed by creating uncertainty at both sides of thefunction that is on the left side of the COG (Figure 8(b))TheMF of Figure 8(a) is the principal function of this T2MFAccording to the principles of the Karnik-Mendel (KM)calculations [34] the switching 119871 and 119877 points are locatedon the left and right sides of 119872 respectively 119871 and 119877 arethe COGs of T1 embedded fuzzy MF respectively (Figures8(c) and 8(d)) By contrast in the homogeneous plane belowthe curve of the principal function the 1199041 and 1199042 domains arereduced (cut) from the left side of their COG (Figure 8(d))This makes the new COG (ie 119877) move toward the right side

Response

Response

Response

Response

120583

Under controlling or under analyzing variable of system

(a)

(b)

(c)

(d)

Figure 7 (a) Both sides are certain (b) left side is slightly uncer-taintified (c) left side is more uncertaintified (d) addition of uncer-tainty in the right side of (c)

of 119872 as far as 119889119903 The addition of the 1199041 domain makes theCOGmove left and the 1199042 domain is removed because of themovement of the COG to the right (Figure 8(c))The effect ofthe 1199041 and 1199042 domains will not cause COG to exit toward theright side of119872The uncertainty grows equally at both sides ofthe principal functionThe 1199041 and 1199042 domains in both Figures8(c) and 8(d) have similar effects on the movement of COG(in the same and opposite directions)

It is concluded that 119889119897 lt 119889119903 considering that the effect ofboth domains on COG displacement is in the same directionin Figure 8(c) and in opposite direction in Figure 8(d)Therefore the center of gravity of IT2MF function movestoward the right side of 119872 as shown in Figure 8(b)


M

y

Principalfunction

120583

(a)

MCL R

y

U

L

120583

(b)

ML

LeftembeddedMF

120583

y

dl

s1

s2

(c)

y

M R

RightembeddedMF

120583

drs1

s2

(d)

MC1L1 R1L2 C2 R2

U

L

120583

y

(e)

M CL R

y

120583

(f)

Figure 8 IT2MF that indicates the upper bound 1198711 is the lower bound 119898 is the principal function 119872 is the COG of 119898 119862 is the COG ofIT2MF

Figure 8(e) shows an IT2MF with the same princi-pal function as the IT2MF of Figure 8(a) The COG andswitching points of this IT2MF are 1198621 1198711 and 1198771 A solidcolor domain is added thus increasing uncertainty Giventhat the upper domain is considered the domain added tothe left side of 1198711 and T1 fuzzy function in which 1198711 is theCOG COG departs from 119872 toward the left In additionthe lower solid domain acts as the domain reduced fromthe T1 fuzzy function in which 1198771 is the COG This settingcauses the COG to depart from 119872 toward the right side Themovements of the right and left COGs do not help displace119862 Two solid domains are not added to or reduced from anidentical function however one domain was added to119880 andthe other was reduced from 119871 Considering that 119871 is alwayssmaller than 119880 (119871 lt 119880) 119889119903 is bigger than 119889119897 (119889119903 gt 119889119897) thus1198622 gt 1198621

The amount of uncertainty added to the right and leftsides of 119877 which is proven by the previous procedure movesthe COG toward the left side

4 Proposed Uncertainty Measurement Method

Theorem of Uncertainty Avoidance points out that ldquoTheresponse of system avoids uncertaintyrdquo In case of IT2FMsit means that the center of gravity of uncertaintified mem-bership functions is displaced toward the less uncertaintifieddomain or in other words toward themore clarified domainThe more the amount of imbalance of the created (oravailable) uncertainty on both sides of the center of gravityof principle membership function the more the displace-ment will tend toward certainty In other words if someuncertainty is injected to one side of the center of gravity ofIT2MF the center of gravity will be shifted toward the otherside

In this paper wemeasure the rate of uncertainty based onthe power of total uncertainty to push the response of systemto the opposite side considering the aforementioned theorem

In this method we add a completely certain member-ship function (with membership degree is equal to one)to IT2MF in the right hand in such a way that the COG


c

a

1

120583 IT2MF

Principal function = f(y)

(a)

a

c

1

120583

d

Extended IT2MF

Extended principal function

COG

120572

(b)

Figure 9 (a) A typical IT2MF (b) Extended principal function and extended IT2MF

of new established principal function is positioning in theconjunction point of the added part with earlier IT2MF

Considering the aforementioned theorem the COG ofnew established IT2MF must be displaced to the right sidein (Figure 9(b)) which is completely certain

Method 1 In Method 1 we use KM algorithm for calculatingCOG of IT2MF The procedure of calculation has beendescribed below in detail

(1) The domain of discourse of shown IT2MF inFigure 9(a) is from zero to 119910 = 119888 The principalfunction also has been shown in Figure 9(a)

(2) The principal function must be extended to the rightside by adding a T1 membership function with fixedand certainmembership degree of ldquoonerdquoWe lengthenthe added part to the right in Figure 9(b) so that theCOG of total new established principal function ispositioning in 119910 = 119888

(3) For obtaining ldquo119889rdquo (3) is written to show that 119910 = 119888 isthe COG of new principal function From this equa-tion the second-degree equation (4) is concluded inwhich just ldquo119889rdquo is unknown

119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)

Area with certain membership degree added to IT2MF

Figure 10 Extended IT2MF based on Method 2

(4) Using KM algorithm we calculate the COG ofextended IT2MF

(5) The direct distance between 119910 = 119888 and 119910 = (newIT2MF COG) which is shown with 120572 in Figure 9(b)is a criterion of pushing power of the IT2MF uncer-tainty We can compare the uncertainty inside of a setof IT2MFs which are put in a domain by calculating120572

Method 2 In this method we use an approximation forcalculating the COG of IT2MF during extracting a criterionfor measuring uncertainty

In this method we extend the IT2MF as describedin part 2 of Method 1 but not with the same length ofprincipal function In this method ldquo119889rdquo is multiplied by ldquo119899rdquoas shown in Figure 10 Enlarging ldquo119889rdquo to 119899 times helps us touse KM algorithm in closed form formula with acceptableapproximation for calculating COG of extended IT2MF Forlarger 119899 the error caused by this approximation is negligible


It should be considered that higher 119899 decreases the sensitivityof 120573 uncertainty measure criteria

(1) Given that switching point in KM algorithm is posi-tioned in the right side of 119910 = 119888119910right KM is easilyobtainable from

119910right KM =int119888

0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)

(2) Given that switching point in KM algorithm is posi-tioned in the right side of 119910 = 119888119910leftKM is easilyobtainable from

119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)

(3) The COG of extended principal function is shown in

119910COG =119910right KM + 119910leftKM

2 (7)

(4) 120573 is introduced as a criterion for uncertainty measureby

120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)

5 Uncertainty Effect on Output

According to Theorem of Uncertainty Avoidance if uncer-tainty increases in one side of COG of principal function thenew COGmoves to the other side In other words we can seethis effect similar to the situation in which the membershipgrades of principal function are decreased in that side thatuncertainty increased Equation (9) is introduced to describethe behavior of this imbalanced internal uncertainty basedon Theorem of Uncertainty Avoidance and first principal ofuncertainty of Klir (refer to Section 2) This formula showsthat new membership degrees decrease after uncertaintyremoval The COG of 120583119889 is compatible with the COG of anIT2MF

120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)

1199041 ge 0 and 1199042 ge 0 are parameters that depend on variousIT2MF factors The overall formula which can be deducted

from this paper for the uncertainty effect on the MF andmembership degree should satisfy the following conditionsTheorem of Uncertainty Avoidance should be considered inthe overall formula because the resulted property is the mainreason to differentiate T1 and T2 fuzzy systems Anothercondition is the decrease of membership degrees The COGis displaced by data that are more deterministic with respectto the difference of uncertainty at both sides of the COG ofthe principal function In the last part attempts are madeto introduce the special formula whose output shows theeffect of uncertainty To reduce the error of the formula underspecial conditions wherein we insist the output is COG theformula is improved as in the following method The fol-lowing formula was introduced according to two conditionsmentioned previously and produces a proper output Thismethod has free parameters that can be determined by adesigner or the nature of the problem to obtain favorableresults such as the COG

Considering the definition of ARS in (2) ARS is alwaysbetween zero and one With respect to the performed simu-lations the increase in the distance of ARS from zero and onecorresponds to higher errors Thus the sensitivity of COGin medium ARS severely devaluates unreliable data Basedon this fact the behavior of the COG of IT2MF (ie (9)) isspecialized into (10) Equation (10) approximately producesthe COG of IT2MF

120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)

In the general condition of the proposed method theresponse of IT2FS is not always completely in accord withCOG We propose a technique that considers this difference(refer to Table 2) in real problems for designers that want touse the proposed method

The following results are obtained from themany calcula-tions and simulations conducted by the authors of this paperto extract the closed formula of COG and propose simple andeffective formulas

According to the aforementioned equations 120583119889119905119903 dependson the power of IT2MF No precise criterion exists to calcu-late IT2MF under these conditions The 1199041 and 1199042 domainsin (9) are not exactly equal to power but are the criteria ofpower 119878 is a better describer for showing the power of IT2MFthe rate of absence of uncertainty and the presence of highermembership degrees The answer will also be nearer to theCOG of IT2MF

Calculations and simulations at various different condi-tions lead to more complex results The COG can be shiftedby the proposed formula and by calculating theCOGby usingthe KM algorithm However one approach is slower or fasterthan the other approach

According to the results obtained from the simulation andcalculation inwhich IT2MF ismore asymmetric but has equaluncertainty areas in both sides of the COG of the principalfunction (10) acts worseWhen ARS distances from zero andone the behavior of theCOGdetector is intensified toweakenambiguous data


Table 2 Basic IT2FS words defuzzified output obtained by using the proposed method and difference percentage compared to the COGcalculated by the KM algorithm

UMF LMF KM Proposed ARS E(1) [0 0 014 197 1] [0 0 005 066 1] 047 049 027 10(2) [0 0 014 197 1] [0 0 001 013 1] 056 061 096 25(3) [0 0 026 263 1] [0 0 005 063 1] 063 066 047 11(4) [0 0 036 263 1] [0 0 005 063 1] 064 067 049 11(5) [0 0 064 247 1] [0 0 010 116 1] 066 066 024 00(6) [0 0 064 263 1] [0 0 009 099 1] 067 068 029 03(7) [059 150 200 341 1] [079 168 168 221 074] 175 174 045 03(8) [038 150 250 462 1] [109 183 183 221 053] 213 211 075 05(9) [009 125 250 462 1] [167 192 192 221 030] 219 228 099 02(10) [009 150 300 462 1] [179 228 228 281 040] 232 233 091 02(11) [059 200 325 441 1] [229 270 270 321 042] 259 259 093 00(12) [038 250 500 783 1] [288 361 361 421 035] 390 394 093 05(13) [117 350 550 783 1] [409 465 465 541 040] 456 457 095 01(14) [259 400 550 762 1] [429 475 475 521 038] 495 498 089 06(15) [217 425 600 783 1] [479 529 529 602 041] 513 513 098 00(16) [359 475 550 691 1] [486 503 503 514 027] 519 521 090 06(17) [359 475 600 741 1] [479 530 530 571 042] 541 541 099 00(18) [338 550 750 962 1] [579 650 650 721 041] 650 650 082 00(19) [438 650 800 941 1] [679 738 738 821 049] 716 715 082 02(20) [438 650 800 941 1] [679 738 738 821 049] 716 715 090 02(21) [438 650 825 962 1] [719 758 758 821 037] 725 721 086 07(22) [538 750 875 981 1] [779 822 822 881 045] 790 787 086 06(23) [538 750 875 983 1] [769 819 819 881 047] 791 788 045 06(24) [538 750 875 981 1] [779 830 830 921 053] 801 801 065 00(25) [538 750 900 981 1] [829 856 856 921 038] 803 797 090 13(26) [598 775 860 952 1] [803 836 836 917 057] 812 812 065 00(27) [737 941 10 10 1] [872 991 10 10 1] 930 931 024 03(28) [737 982 10 10 1] [974 998 10 10 1] 931 923 028 30(29) [737 959 10 10 1] [895 993 10 10 1] 934 935 028 03(30) [737 973 10 10 1] [934 995 10 10 1] 937 934 047 11(31) [737 982 10 10 1] [937 995 10 10 1] 938 934 048 15(32) [868 991 10 10 1] [961 997 10 10 1] 969 967 036 15Difference (error) RMS compared to KM 095

This method has a high degree of freedom (9) and theuncertainty removingmethod can be defined by a simple lin-ear method or more complex methods based on conservativeor courageous logic and on the conditions and requirementsof different issues

Our method eases the defuzzifying of IT2MF to obtainaccurate results on the main feature of IT2FSs In case ofslight output differences between our method and the KMalgorithm no mathematic proof exists that shows that theoutputs gained by KM algorithm are better than our methodOn the contrary the existence of concepts and reasonsbehind our proposed method provides a designer with moreopportunities to manage parameters related to uncertaintyin IT2 fuzzy controllers in engineering and industrial affairsconveniently

6 Simulations and Investigations

Investigation 1 examines the affectivity of the proposeduncertainty measures Investigation 2 shows the illustrativeconcept of the effect of uncertainty on membership degreesInvestigation 3 shows the comparative outputs as a resultof applying the proposed formula and KM algorithm of32 basic IT2MF words A single-input single-output (SISO)fuzzy system is introduced in ldquoInvestigation 4rdquo to compareclearly the outputs created in different uncertainty imbalancesituations and by different methods Investigation 5 discusseson a comparison between collapsing method [43] and pro-posedmethod In Investigation 6 the effects of different typesof uncertainty on system output are investigated Examplesare chosen by selecting various forms of MFs and changing


a

b c d e

Figure 11 Common form of trapezoidal and triangle MFs

the differentmodes of antecedent and subsequentMFs to cre-ate various modes by using different ARSs and membershipdegrees and cover the generalities of the issueThe root meansquare of the differences and maximum error shows differentstates

The common form of the trapezoidal and triangular MFcan be described by a five-number vector [119886 119887 119888 119889 and119890] (Figure 11) This presentation of the common form oftrapezoidal and triangularMF is used in the rest of this paperThe distance between ldquo119887rdquo and ldquo119890rdquo that is (119890ndash119887) is called thesupport of MF (Figure 11)

Investigation 1 Calculate 120572 and 120573 for comparing uncertaintyof two IT2MFs shown in Figure 12 For calculating 120573 consider119899 = 3

According to (4) the new domain 119889 which is necessaryfor calculating 120572 and 120573 is calculable as follows

1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)

Calculating 120572 COG1 and COG2 based on KM algorithm are101 and 11 respectively then 1205721 and 1205722 are 001 and 01respectively

Two uncertainty measures have been shown in Figure 13Considering the area of surrounded region by upper andlower bounds it is clear that the uncertainty of Figure 12(a)is lesser than the uncertainty of Figure 12(b) before calcula-tion This fact is confirmed using the proposed method foruncertainty measurement Calculations for 120573 using (8) show1205731 = 0003 and 1205732 = 002 A comparison between 120572 and 120573

shows that the sensitivity of 120572 is more than 120573

Investigation 2 For IT2MF (UMF = [038 250 375 481 1]and LMF = [279 330 330 421 053]) calculate (a) ARS and(b) output value by using (10) (c) Calculate the differencepercentage between obtained value by using (12) and theCOG of IT2MF

Answer The COG of the principal function shown inFigure 14(a) is 119872 = 288 In Figure 14(b) the function of119906

= 120583119906 minus 120583119897 is observed wherein 119860 119897 = 715 119860119903 = 516and ARS = 072 are obtained from (2)

By using (10) the embedded fuzzy T1 shown inFigure 14(a) is obtained The COG of this fuzzy T1 is easilycalculable as COG (D3) = 30088

The COG of IT2MF in Figure 14(a) (ie 301) is obtainedby using the KM algorithm To calculate the differencepercentage between these two COGs we use the followingformula

119864 =

1003816100381610038161003816Output (10) minus COG (IT2MF)1003816100381610038161003816Sup (UMF)

times 100 (12)

where Sup is the support of MF (Figure 11)Here 119864 = 004 and (10) produces accurate answers to

COG (IT2MF)

Investigation 3WuandMendel showed in [18] the defuzzifiedoutput of 32 basic IT2 fuzzy words based on the KMalgorithm Here we calculate the defuzzified output by usingour proposed formula (ie (10) for obtaining T1MF and usingCOG for type-1 defuzzification) Table 2 shows those wordsby using the five parameters for introducing trapezoidal MFsshown in Figure 11 Equation (12) is used for calculating(119864) which shows the error percentages of (10) The absolutemaximum error of (10) for those words is 3

Investigation 4 A SISO single-rule system exists The specifi-cations of this system are as follows

Rule If 119883 is small then 119884 is small Implication = Max

119884 (output) MF is T1 small = [0 0 0 295 1]119883 (input) MF is the (a) T1MF shown in Figure 4(b)(b) standard IT2MF shown in Figure 5(a) (c)IT2MF shown in Figure 4(c) (d) IT2MF shown inFigure 5(c)

Calculate the output by using (9) and investigate the effectof uncertainty

The output is affected by various uncertainties separately(Figure 15) For example we want to discuss the effect ofdata uncertainty only Considering the IT2Mf of 119909 a lowerldquo119909rdquo (119909 lt 25) is more uncertaintified (compared with theconditions in which uncertainties are absent) A higher ldquo119909rdquo(7 gt 119909 gt 25) is more uncertaintified and 119909 gt 7 indicatesbalance If we look to the property of the rule used in thissystem and consider the bandwidth of uncertainty in inputMF higher values of ldquo119910rdquo are more certain if 119909 lt 2 andlower values of ldquo119910rdquo are more certain if 7 gt 119909 gt 25 in otherconditions ldquo119910rdquo does not change This reasoning is applicablefor other types of uncertainty

Investigation 5 Calculate outputs by using collapsing method[43] and (9) for

(a) the MF of the upper and lower bands that are 08 and02 in the [0 1] range of the primary domain

(b) symmetric triangle MF UMF = [01 04 04 07 1]

and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)

Figure 12 Two IT2MFs with equal domains and different uncertainty bands

1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572

Figure 13 COG of extended IT2MF and position of 120572 and 120573 119889120573

=

3119889120572

(c) asymmetric GaussianMF UMF = [03 02] and LMF= [018 03]

Answer In case of symmetric IT2MFs the use of (9) causesno errorThe error is zerowhen the upper and lower bands areconsidered continuous functions or appliedwith few samplesThe defuzzified outputs for the upper and lower bands are05 and 04 respectively The collapsing method with 11 ormore samples is slightly different with the exact answer Forthe asymmetric Gaussian IT2MF the result obtained by (10)

for 100 samples is 033 the result decreaseswith increasing thenumber of samples By using continuous formula of the upperand lower MFs the output converges to 03219 The results in[43] are slightly different with the convergent answer

Investigation 6 Provide visualization on the effect of differenttypes of uncertainty by investigating the output of an SISOFLC The SISO inputs are selected from Section 2 of thispaper


119884 (output) MF is T1 small = [0 0 0 495 1]

119883 (input) MF is the (a) T1MF shown in Figure 4(b)the standard IT2MF shown in Figure 5(a) (c)IT2MF shown in Figure 4(c) (d) IT2MF shown inFigure 5(c)

In all the above conditions calculate the output by using(9) and investigate the effect of uncertainty

The output is affected by various uncertainties separately(Figure 16) Figures 16(a) 16(b) and 16(c) show that applieduncertainty causes rules to weakenWe use the following rulefor this case ldquoif the input is small the output is smallrdquo

However we considered the effect of uncertainty inFigure 16 We observed in all cases that the response afterapplying the uncertainty increased as well as in points inwhich input data has greater amount of uncertainty Outputalso significantly increasedThe result shows that the conceptof ldquolowrdquo in the output shifts toward the concept of ldquolargerdquo

Discussion on InvestigationsTheproposedmethod has a highdegree of freedomThe uncertainty removing method can bedefined by a simple linear method or more complex methodsbased on conservative or courageous logic and the conditionsand requirements of different issues

Our method eased the defuzzifying of IT2MF to obtainaccurate results on the feature of IT2FSs In case of slight


UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4

Uncertainty bandwidth function

(a) (b)

29 292 294 296 298 3

(c)COG of principal function COG of type-1 MF obtained after uncertainty removal

IT2FM COGPrincipal function

0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1

Final T1FM afteruncertainty removal

ARS = ArAl

ArAl

The resulted T1MF after uncertainty removal

Figure 14 (a) IT2MF and its principal and output-based imbalanced uncertainty (b) uncertain bandwidth function (c) COGs of principaloutput-based uncertainty imbalances and IT2MF (KM) functions

Data uncertainty only Fuzzy type-1 (certain)

Data uncertainty

All types ofuncertainty

reduce

32343638

442444648

5

0 2 4 6 8 10

y

x

Intrinsic and datauncertainties

Word intrinsicuncertainty

Figure 15 Effect of uncertainty on outputs of SISO (Investigation4)

output differences between our method and the KM algo-rithm no mathematical proof exists to show that the outputsgained by KM algorithm are better than our method On thecontrary the existence of concepts and reasons behind ourproposed method provides designers with more opportuni-ties to manage parameters related to uncertainty in T2 fuzzysystems in engineering and industrial affairs conveniently

Designers who work with fuzzy systems must haveinformation on the behavior of the fuzzifier inference anddefuzzifier methods used in the system No defuzzificationmethod exists that is suitable in all systems and all conditions

IfmethodAproduces better results thanmethodB B is betterthan A if conditions or systems change The experience of adesigner plays a major role in the selection of an appropriatemethod In our case understanding and absorbing systembehavior is easier

7 Conclusion

This paper presented the Theorem of Uncertainty Avoid-ance and used it for uncertainty measuring in IT2FMsThe proposed methods provide simple closed formulas forcalculating total uncertainty of a membership function Thispaper brought up a new vision to the problem of uncer-tainty measure The measurement is based on the powerof uncertainty to push the COG of principal function to acompletely certain domain The results of this paper providea new perspective on the relationship between uncertaintyand fuzzy system output For each T1MF uncertaintifiedfunctions are presented to be more complete than commonIT2MFs In addition we show that uncertainty reduces thevalue ofmembership degrees and the absolute value of wordsHigher uncertainty causes a higher reduction of values Forexample if the input and output of a system contain ldquolowrdquoldquomediumrdquo and ldquolargerdquo words after injecting uncertainty toinputs ldquolowrdquo will shift to ldquomediumrdquo ldquomediumrdquo to ldquolargerdquoand ldquolargerdquo to ldquomediumrdquo Results show that the uncertaintyreduces the value of the membership degree proportionallyThe concept of words is also shifted toward the oppositeneighbor words in the system output by the uncertainty of


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10

Response for MF input whichcovers internal uncertainty ofword only

Response forprincipal functionof final word

(a)

Response for MF input whichcovers data uncertainty only

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)

Response for MF input whichcovers all types of uncertainties

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


Response fortype 1 MF input

(d)

Figure 16 Effect of uncertainty on outputs of a simple fuzzy controller (Investigation 6)

system inputs proportionally The proposed technique foruncertainty removing can be considered as a closed formulafor calculating the COG of IT2MF with acceptable accuracyOn the contrary the existence of concepts and reasons behindthe new interpretation of uncertainty provides designersmore opportunities to manage parameters related to uncer-tainty in interval T2 fuzzy controllers in engineering andindustrial affairs conveniently

Acknowledgment

The authors would like to acknowledge the support of theUniversiti Sains Malaysia fellowship

References

[1] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-Irdquo Information Sciencesvol 8 no 3 pp 199ndash249 1975

[2] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning-IIrdquo Information Sciencesvol 8 no 4 pp 301ndash357 1975

[3] H Hagras ldquoType-2 FLCs a new generation of fuzzy con-trollersrdquo IEEE Computational Intelligence Magazine vol 2 no1 pp 30ndash43 2007

[4] R John and S Coupland ldquoType-2 fuzzy logic a historical viewrdquoIEEE Computational Intelligence Magazine vol 2 no 1 pp 57ndash62 2007


[5] F Liu and J M Mendel ldquoAggregation using the fuzzy weightedaverage as computed by the Karnik-Mendel algorithmsrdquo IEEETransactions on Fuzzy Systems vol 16 no 1 pp 1ndash12 2008

[6] F Liu and J M Mendel ldquoEncoding words into interval type-2 fuzzy sets using an interval approachrdquo IEEE Transactions onFuzzy Systems vol 16 no 6 pp 1503ndash1521 2008

[7] M Melgarejo and C A Pena-Reyes ldquoImplementing intervaltype-2 fuzzy processorsrdquo IEEE Computational Intelligence Mag-azine vol 2 no 1 pp 63ndash71 2007

[8] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice-Hall Upper-Saddle RiverNJ USA 2001

[9] J M Mendel ldquoType-2 fuzzy sets and systems an overviewrdquoIEEE Computational Intelligence Magazine vol 2 no 1 pp 20ndash29 2007

[10] J M Mendel ldquoAdvances in type-2 fuzzy sets and systemsrdquoInformation Sciences vol 177 no 1 pp 84ndash110 2007

[11] J M Mendel ldquoComputing with words zadeh turing popperand occamrdquo IEEE Computational Intelligence Magazine vol 2no 4 pp 10ndash17 2007

[12] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006

[13] J M Mendel H Hagras and R I John ldquoStandardbackground material about interval type-2 fuzzy logicsystems that can be used by all authorsrdquo httpieee-cisorgfilesstandardst2winpdf

[14] D Wu and J M Mendel ldquoUncertainty measures for intervaltype-2 fuzzy setsrdquo Information Sciences vol 177 no 23 pp5378ndash5393 2007

[15] D Wu and J M Mendel ldquoAggregation using the linguisticweighted average and interval type-2 fuzzy setsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1145ndash1161 2007

[16] J M Mendel and D Wu ldquoCardinality fuzziness variance andskewness of interval type-2 fuzzy setsrdquo in Proceedings of theIEEE Symposium on Foundations of Computational Intelligence(FOCI rsquo07) pp 375ndash382 April 2007


[18] D Wu and J M Mendel ldquoA comparative study of rankingmethods similarity measures and uncertainty measures forinterval type-2 fuzzy setsrdquo Information Sciences vol 179 no 8pp 1169ndash1192 2009

[19] R John ldquoType 2 fuzzy sets an appraisal of theory and appli-cationsrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 6 no 6 pp 563ndash576 1998

[20] G J Klir andB Parviz ldquoProbability-possibility transformationsa comparisonrdquo International Journal of General Systems vol 21no 3 pp 291ndash310 1992

[21] S Aminifar and A bin Marzuki ldquoHorizontal and vertical rulebases method in fuzzy controllersrdquo Mathematical Problems inEngineering vol 2013 Article ID 532046 9 pages 2013

[22] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)- An outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[23] G J Klir ldquoPrinciples of uncertainty what are they Why do weneed themrdquo Fuzzy Sets and Systems vol 74 no 1 pp 15ndash311995

[24] G J Klir and T A Folger Fuzzy Sets Uncertainty andInformation Prentice-Hall Englewood Cliffs NJ USA 1988

[25] D Harmanec ldquoMeasures of uncertainty and information Soci-ety for Imprecise Probability Theory and Applicationsrdquo 1999httpwwwsiptaorgdocumentationsummary measuresmainhtml

[26] R Christensen ldquoEntropyminimaxmultivariate statistical mod-eling I theoryrdquo International Journal of General Systems vol 11no 3 pp 231ndash277 1985

[27] R Christensen ldquoEntropyminimaxmultivariate statistical mod-eling II applicationsrdquo International Journal of General Systemsvol 12 no 3 pp 227ndash305 1985

[28] G J Klir ldquoA principle of uncertainty and information invari-ancerdquo International Journal of General Systems vol 17 no 2-3pp 249ndash275 1990


[30] L A Zadeh ldquoToward a theory of fuzzy information granulationand its centrality in human reasoning and fuzzy logicrdquo FuzzySets and Systems vol 90 no 2 pp 111ndash127 1997

[31] L A Zadeh ldquoFuzzy sets and information granularityrdquo inAdvances in Fuzzy Set Theory and Applications M Gupta RRagade and R Yager Eds pp 3ndash18 North-Holland PublishingAmsterdam The Netherlands 1979

[32] M Higashi and G Klir ldquoMeasures of uncertainty and infor-mation based on possibility distributionsrdquo in Fuzzy Sets ForIntelligent Systems pp 217ndash232 Morgan Kaufman PublishersSan Mateo Calif USA 1993

[33] J M Mendel ldquoFuzzy sets for words a new beginningrdquo inProceedings of the IEEE International conference on FuzzySystems pp 37ndash42 St Louis Mo USA May 2003

[34] N N Karnik and J M Mendel ldquoCentroid of a type-2 fuzzy setrdquoInformation Sciences vol 132 no 1ndash4 pp 195ndash220 2001

[35] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[36] O Linda and M Manic ldquoUncertainty modeling for intervaltype-2 fuzzy logic systems based on sensor characteristicsrdquo inProceedings of the IEEE Symposium onAdvances in Type-2 FuzzyLogic Systems (T2FUZZ rsquo11) pp 31ndash37 April 2011

[37] B Choi and F Chung-Hoon Rhee ldquoInterval type-2 fuzzy mem-bership function generation methods for pattern recognitionrdquoInformation Sciences vol 179 no 13 pp 2102ndash2122 2009

[38] J MMendel ldquoComputing with words and its relationships withfuzzisticsrdquo Information Sciences vol 177 no 4 pp 988ndash10062007

[39] DWu ldquoOn the fundamental differences between interval type-2 and type-1 fuzzy logic controllers in common systemsrdquo IEEETransactions on Fuzzy Systems vol 20 no 5 pp 832ndash848 2012

[40] W L Oberkampf S M DeLand B M Rutherford K VDiegert and K F Alvin ldquoError and uncertainty in modelingand simulationrdquo Reliability Engineering and System Safety vol75 no 3 pp 333ndash357 2002

[41] L J Lucas H Owhadi and M Ortiz ldquoRigorous verificationvalidation uncertainty quantification and certification throughconcentration-of-measure inequalitiesrdquo Computer Methods inApplied Mechanics and Engineering vol 197 no 51-52 pp 4591ndash4609 2008


[42] C J Roy and W L Oberkampf ldquoA comprehensive frameworkfor verification validation and uncertainty quantification inscientific computingrdquo Computer Methods in Applied Mechanicsand Engineering vol 200 no 25ndash28 pp 2131ndash2144 2011

[43] S Greenfield F Chiclana S Coupland and R John ldquoThecollapsing method of defuzzification for discretised intervaltype-2 fuzzy setsrdquo Information Sciences vol 179 no 13 pp2055ndash2069 2009

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


of uncertainty on displacing COG is presented as a noveluncertainty measure in this paper

The concept of uncertainty in fuzzy systems is interpretedin a new manner as illustrated in Figure 1 This researchconsiders the presence of uncertainty injectors and removersdifferent from conventional fuzzy systems (Figure 1) T1 fuzzysystems do not have these two blocks compared with theproposed fuzzy system The main difference between T1fuzzy systems and IT2FSs [13] is found in the fuzzifier anddefuzzifier blocks

Uncertainty removal is introduced to observe the effectof uncertainty on the output In Section 2 (Uncertainty) theconceptual descriptions regarding fuzzification block basedon the interpretations derived from this paper are discussedWe explain the application of these concepts in a systemwherein each variable corresponds to a number with oneinterval in each moment Our method attempts to find a T1fuzzy membership function for each IT2MF based on theeffect of uncertainty on certain data We also avoid usingiterative procedures

The direct role of uncertainty is introduced as the maindifference between T1FSs and T2FSs The justification ofapplication of IT2FLSs in fuzzy systems is highly depen-dent on uncertainty issue This paper aims to recognizedifferent types of uncertainty by introducing new definitionsand expressions thus creating informed and goal-orientedmaneuvers during the design process and increasing theapplication of interval T2 fuzzy controllers

Another objective of this paper is about new uncertaintymeasure for interval type-2 fuzzy membership functions(IT2MFs) Uncertainty measure is a necessity because to usefuzzy sets (FSs) as granules in general theory of uncertainty(GTU) which is introduced by Zadeh [22] it is necessaryto quantify the uncertainty associated with fuzzy sets (FS)because based on [23] ldquoonce uncertainty (and information)measures become well justified they can very effectively beutilized for managing uncertainty and associated informa-tion For example they can be utilized for extrapolatingevidence assessing the strength of relationship between givengroups of variables assessing the influences of given inputvariables on given output variables measuring the loss ofinformation when a system is simplified and the likerdquo

Klir and Folger [24] Klir and Parviz [20] and Harmanec[25] have developed three fundamental principles to guidethe use of uncertainty measures under different circum-stances

(1) ldquoThe principle of minimum uncertainty which statesthat solutionswith the least loss of information shouldbe selected can be used in simplification and conflictresolution problemsrdquo

(2) ldquoThe principle of maximum uncertainty which statesthat a conclusion should maximize the relevantuncertainty within constraints given by the verifiedpremises is widely used within classical probabilityframework [26 27]rdquo

(3) ldquoThe principle of uncertainty invariance which statesthat the amount of uncertainty should be preserved in

each transformation of uncertainty from one math-ematical framework to another is widely studied inthe context of probability-possibility transformation[28 29]rdquo

This paper tries to discuss the effect of uncertainty ofIT2FS based on the first principal of uncertainty

The remainder of this paper is organized as followsSection 2 discusses uncertainty and IT2MF efficacy andpresents improvements in this field Section 3 discusses theimbalance rate of internal uncertainty and its effect onsystem behavior Section 4 explains new uncertainty mea-sure Section 5 formulizes the uncertainty effect on systemoutput Sections 6 and 7 display the simulation analyses andconclusions respectively

2 Uncertainty

Zadeh points out in [22] that ldquouncertainty is an attribute ofinformationrdquo and introduced the general theory of uncer-tainty (GTU) ldquobecause existing approaches to representationof uncertain information are inadequate for dealing withproblems in which uncertain information is perception-based and is expressed in a natural languagerdquo He also statesthat ldquoIn GTU uncertainty is linked to information throughthe concept of granular structure a concept which plays a keyrole in human interaction with the real world [30ndash32]rdquo

Fuzzy systems are mainly applied for calculations thatuse lexical variables (ie CW) [33] Verbal interpretationsfor different operations are the key elements in CW Forfuzzification general T2 fuzzy systems are more effectivein modeling defined quantities by linguistic constraintsIT2FSs which have less complexity have discovered a fieldof application that is higher than T2 fuzzy sets This paperfocuses on IT2FLSs Figure 2 shows an example of an IT2MFThe membership degree of (119909 = 119886) takes the form of aninterval or a line segment [120583119897119886 120583119906119886] (Figure 2) The lengthof this line segment indicates the uncertainty bandwidth(119906

= 120583119906 to 120583119897) on that point This paper emphasizes thehypothesis in [34] in which IT2MF serves as the expansion ofT1 fuzzy MF with an equal amount of uncertainty expansionat both ends A principal MF that is located in the middleof the upper and lower bands of uncertainty is definedfor each IT2MF where the membership degree of eachmember 120583119901 is the average of the upper and lowermembershipdegrees (120583119901 = (120583119906 + 120583119871)2) [34] For each T1MF (whichis referred to as ldquodeterministic membership degreesrdquo thatis one-to-one correspondence between each variable and itsmembership degree) an unlimited number of IT2MFs arecreated because of the presence of uncertainty on either sideof each point of the member function These IT2MFs arecalled uncertaintified MFs of the T1MF

Four sources with different nature were mentioned foruncertainty in [35]These sources introduce the equal IT2MFfor uncertainty modeling For instance the uncertainty thatis transmitted to the data of a local factor such as noise iscompletely different from the uncertainty that is an insep-arable part of a word in mind Therefore the uncertaintyshould be modeled separately or the description trend of


Inference engine

Type II

Inputs

Outputs

Uncertainty remover


Uncertaintifier


Fuzzifier

T1

DefuzzifierT1


120583




120583ux

120583lx

x










b

120583

a0 c d

Low

(a)

120583

a0 b c d

Medium

(b)

a0 b c

120583High

(c)















119860119877 (119860 119861) = ((119860

119861if 119861 lt 119860) (

119861

119860if 119861 lt 119860)) (1)





0

11

120583l

Low principal

1

(b)

120583 = 1 minus y11

120583

120583u


Dat

a

(a)

11

110 1 2 3 4 5 6 7 8 9 10

Low IT2MF

1 New principal

(c)

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10

a bc

120583

11


1

120583

110

120583u


9

120583l

(a)

0 1 2 3 4 5 6 7 8 9 10 11

120583Injection of

data uncertainty

1

07

(b)

0 1 2 3 4 5 6 7 8 9 10 11

120583Low IT2MF


07

1

(c)



and 119906

(119910) is a



ARS (119865) = 119860119877(int119872

119886

119880

119889119910 int119887

119872

119880

119889119910) (2)



M

ba y

120583

120583u

120583l

Principal function

(a)

ba y


function


function

M

usim


Al

Ar

(b)






Response

Response

Response

Response

120583


(a)

(b)

(c)

(d)





M

y

Principalfunction

120583

(a)

MCL R

y

U

L

120583

(b)

ML

LeftembeddedMF

120583

y

dl

s1

s2

(c)

y

M R

RightembeddedMF

120583

drs1

s2

(d)

MC1L1 R1L2 C2 R2

U

L

120583

y

(e)

M CL R

y

120583

(f)









c

a

1

120583 IT2MF


(a)

a

c

1

120583

d

Extended IT2MF


COG

120572

(b)








119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)











0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Inference engine

Type II

Inputs

Outputs

Uncertainty remover


Uncertaintifier


Fuzzifier

T1

DefuzzifierT1


120583




120583ux

120583lx

x










b

120583

a0 c d

Low

(a)

120583

a0 b c d

Medium

(b)

a0 b c

120583High

(c)















119860119877 (119860 119861) = ((119860

119861if 119861 lt 119860) (

119861

119860if 119861 lt 119860)) (1)





0

11

120583l

Low principal

1

(b)

120583 = 1 minus y11

120583

120583u


Dat

a

(a)

11

110 1 2 3 4 5 6 7 8 9 10

Low IT2MF

1 New principal

(c)

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10

a bc

120583

11


1

120583

110

120583u


9

120583l

(a)

0 1 2 3 4 5 6 7 8 9 10 11

120583Injection of

data uncertainty

1

07

(b)

0 1 2 3 4 5 6 7 8 9 10 11

120583Low IT2MF


07

1

(c)



and 119906

(119910) is a



ARS (119865) = 119860119877(int119872

119886

119880

119889119910 int119887

119872

119880

119889119910) (2)



M

ba y

120583

120583u

120583l

Principal function

(a)

ba y


function


function

M

usim


Al

Ar

(b)






Response

Response

Response

Response

120583


(a)

(b)

(c)

(d)





M

y

Principalfunction

120583

(a)

MCL R

y

U

L

120583

(b)

ML

LeftembeddedMF

120583

y

dl

s1

s2

(c)

y

M R

RightembeddedMF

120583

drs1

s2

(d)

MC1L1 R1L2 C2 R2

U

L

120583

y

(e)

M CL R

y

120583

(f)









c

a

1

120583 IT2MF


(a)

a

c

1

120583

d

Extended IT2MF


COG

120572

(b)








119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)











0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










b

120583

a0 c d

Low

(a)

120583

a0 b c d

Medium

(b)

a0 b c

120583High

(c)















119860119877 (119860 119861) = ((119860

119861if 119861 lt 119860) (

119861

119860if 119861 lt 119860)) (1)





0

11

120583l

Low principal

1

(b)

120583 = 1 minus y11

120583

120583u


Dat

a

(a)

11

110 1 2 3 4 5 6 7 8 9 10

Low IT2MF

1 New principal

(c)

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10

a bc

120583

11


1

120583

110

120583u


9

120583l

(a)

0 1 2 3 4 5 6 7 8 9 10 11

120583Injection of

data uncertainty

1

07

(b)

0 1 2 3 4 5 6 7 8 9 10 11

120583Low IT2MF


07

1

(c)



and 119906

(119910) is a



ARS (119865) = 119860119877(int119872

119886

119880

119889119910 int119887

119872

119880

119889119910) (2)



M

ba y

120583

120583u

120583l

Principal function

(a)

ba y


function


function

M

usim


Al

Ar

(b)






Response

Response

Response

Response

120583


(a)

(b)

(c)

(d)





M

y

Principalfunction

120583

(a)

MCL R

y

U

L

120583

(b)

ML

LeftembeddedMF

120583

y

dl

s1

s2

(c)

y

M R

RightembeddedMF

120583

drs1

s2

(d)

MC1L1 R1L2 C2 R2

U

L

120583

y

(e)

M CL R

y

120583

(f)









c

a

1

120583 IT2MF


(a)

a

c

1

120583

d

Extended IT2MF


COG

120572

(b)








119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)











0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

11

120583l

Low principal

1

(b)

120583 = 1 minus y11

120583

120583u


Dat

a

(a)

11

110 1 2 3 4 5 6 7 8 9 10

Low IT2MF

1 New principal

(c)

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10

a bc

120583

11


1

120583

110

120583u


9

120583l

(a)

0 1 2 3 4 5 6 7 8 9 10 11

120583Injection of

data uncertainty

1

07

(b)

0 1 2 3 4 5 6 7 8 9 10 11

120583Low IT2MF


07

1

(c)



and 119906

(119910) is a



ARS (119865) = 119860119877(int119872

119886

119880

119889119910 int119887

119872

119880

119889119910) (2)



M

ba y

120583

120583u

120583l

Principal function

(a)

ba y


function


function

M

usim


Al

Ar

(b)






Response

Response

Response

Response

120583


(a)

(b)

(c)

(d)





M

y

Principalfunction

120583

(a)

MCL R

y

U

L

120583

(b)

ML

LeftembeddedMF

120583

y

dl

s1

s2

(c)

y

M R

RightembeddedMF

120583

drs1

s2

(d)

MC1L1 R1L2 C2 R2

U

L

120583

y

(e)

M CL R

y

120583

(f)









c

a

1

120583 IT2MF


(a)

a

c

1

120583

d

Extended IT2MF


COG

120572

(b)








119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)











0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M

ba y

120583

120583u

120583l

Principal function

(a)

ba y


function


function

M

usim


Al

Ar

(b)






Response

Response

Response

Response

120583


(a)

(b)

(c)

(d)





M

y

Principalfunction

120583

(a)

MCL R

y

U

L

120583

(b)

ML

LeftembeddedMF

120583

y

dl

s1

s2

(c)

y

M R

RightembeddedMF

120583

drs1

s2

(d)

MC1L1 R1L2 C2 R2

U

L

120583

y

(e)

M CL R

y

120583

(f)









c

a

1

120583 IT2MF


(a)

a

c

1

120583

d

Extended IT2MF


COG

120572

(b)








119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)











0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M

y

Principalfunction

120583

(a)

MCL R

y

U

L

120583

(b)

ML

LeftembeddedMF

120583

y

dl

s1

s2

(c)

y

M R

RightembeddedMF

120583

drs1

s2

(d)

MC1L1 R1L2 C2 R2

U

L

120583

y

(e)

M CL R

y

120583

(f)









c

a

1

120583 IT2MF


(a)

a

c

1

120583

d

Extended IT2MF


COG

120572

(b)








119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)











0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










c

a

1

120583 IT2MF


(a)

a

c

1

120583

d

Extended IT2MF


COG

120572

(b)








119888 =int119888

0119891 (119910) 119910 119889119910 + int

119889

1198881119910 119889119910

int119888

0119891 (119910) 119889119910 + int

119889

1198881 119889119910

=int119888

0119891 (119910) 119910 119889119910 + (12) (119889

2 minus 1198882)

int119888

0119891 (119910) 119889119910 + (119889 minus 119888)

(3)

1

21198892minus 119888119889 + (int

119888

0

119891 (119910) 119910 119889119910 minus int119888

0

119891 (119910) 119889119910 minus1

21198882+ 119888) = 0

(4)

a c

1

d

Extended IT2MF

COG

120573

2d 3d

Upper bound = u(y)

Lower bound = l(y)











0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0119906 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119906 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119906 (119910) 119889119910 + (119899119889 minus 119888)

(5)


119910leftKM =int119888

0119897 (119910) 119910 119889119910 + int

119899119889

1198881119910 119889119910

int119888

0119897 (119910) 119889119910 + int

119899119889

1198881 119889119910

=int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

int119888

0119897 (119910) 119889119910 + (119899119889 minus 119888)

(6)



2 (7)


120573 = 119910COG minus 119888 =int119888

0119906 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119906 (119910) 119889119910 + 2 (119899119889 minus 119888)

+int119888

0119897 (119910) 119910 119889119910 + (12) ((119899119889)

2minus 1198882)

2 int119888

0119897 (119910) 119889119910 + 2 (119899119889 minus 119888)

minus 119888

(8)



120583119889119905119903 = 120583119901 minus1

2 + 1199041119906

1+1199042 (9)




120583119889119905119903 = 120583119901 minus1

2 + (1ARS) lowast 120583119901119906

2 (10)














a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















a

b c d e






1

21198892minus 119889 + (int

1

0

05119910 119889119910 minus int1

0

05 119889119910 minus1

2+ 1) = 0

997904rArr1

21198892minus 119889 + (

1

4minus

1

2minus

1

2+ 1) = 0

997904rArr1

21198892minus 119889 +

1

4= 0 997904rArr 119889 = 1 +

radic2

2

(11)









119864 =


times 100 (12)


COG (IT2MF)










and LMF = [02 04 04 06 1]


1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

05

07

02

1

(a)

1

1

05

(b)


1

1

05

07

02

1

05

1205721

1205722

1205731

1205732

COG1 COG2

d120573d120572


=

3119889120572














UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










UMF

LMF

0

01

02

03

04

05

06

07

08

09

1

0 1 2 3 4


(a) (b)

29 292 294 296 298 3



0 1 2 3 4 50

01

02

03

04

05

06

07

08

09

1


ARS = ArAl

ArAl




Data uncertainty


reduce

32343638

442444648

5

0 2 4 6 8 10

y

x







7 Conclusion



32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(a)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(b)


32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10


(c)

32

34

36

38

4

42

44

46

48

5

0 2 4 6 8 10



(d)



Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article uncertainty in interval type-2 fuzzy...

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