research article a new method for defuzzification...

Research ArticleA New Method for Defuzzification and Ranking of FuzzyNumbers Based on the Statistical Beta Distribution

A Rahmani F Hosseinzadeh Lotfi M Rostamy-Malkhalifeh and T Allahviranloo

Department of Mathematics Science and Research Branch Islamic Azad University Tehran Iran

Correspondence should be addressed to M Rostamy-Malkhalifeh mohsen rostamyyahoocom

Received 25 March 2016 Revised 22 June 2016 Accepted 18 October 2016

Academic Editor RustomM Mamlook

Copyright copy 2016 A Rahmani et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Granular computing is an emerging computing theory and paradigm that deals with the processing of information granules whichare defined as a number of information entities grouped together due to their similarity physical adjacency or indistinguishabilityInmost aspects of human reasoning these granules have an uncertain formation so the concept of granularity of fuzzy informationcould be of special interest for the applications where fuzzy sets must be converted to crisp sets to avoid uncertainty This paperproposes a novel method of defuzzification based on the mean value of statistical Beta distribution and an algorithm for rankingfuzzy numbers based on the crisp number ranking system on RThe proposed method is quite easy to use but the main reason forfollowing this approach is the equality of left spread right spread andmode of Beta distribution with their corresponding values infuzzy numbers within (0 1) interval in addition to the fact that the resulting method can satisfy all reasonable properties of fuzzyquantity ordering defined by Wang et al The algorithm is illustrated through several numerical examples and it is then comparedwith some of the other methods provided by literature

1 Introduction

Granular computing is an emerging computing paradigmthat is concerned with the processing of information entitiescreated from the process of data abstraction and is intrinsi-cally linked with the adjustable nature of human perception[1 2] Information granularity and granular computing havebeen widely used in development of verbal and linguisticconcepts and particularly those concerned with fuzzy andrough sets [3] and are valuable assets for creating realisticmodels for real-world decision-making processes as theyprovide themeans to understand and solve abstract problemsof the real world with simplicity clarity good approximationand tolerance of uncertainty [1 4] Granular computing isthe science of building heterogeneous and multilevel modelsfor the processing of granular information by incorporatingdistinct concepts such as probabilistic sets rough sets andespecially fuzzy sets and their membership functions into asingle framework thereby allowing the verbal linguistic andhuman-centered concepts to be processed

Since the introduction of the term ldquogranular computingrdquoits related concepts have appeared in many different fields

such as artificial intelligence decision-making and clusteranalysis [5ndash7] Although there have been some works inregard with granular models granular computing is yet to befully and exclusively explored and its current structures andespecially those related to fuzzy sets seem to be underdevel-oped

In 1997 Zadeh [8] introduced a strong relationshipbetween granular data and fuzzy sets and developed thetheory of fuzzy information granulation to provide a newangle of approach for tackling the problem of ambiguityand uncertaintyThe theory of fuzzy information granulation(TFIG) is an informal method of using linguistic variablesand fuzzy IF-THEN rules to make rational decisions in anenvironment full of uncertainty Thus this article is focusedon this aspect of fuzzy theory and provides a method fordefuzzification of fuzzy sets to achieve certainty in solutionof real-world problems

The concept of fuzzy sets (referring to the sets withimprecise and ambiguous nature) was first introduced in1965 by Zadeh [9] He expanded the notion of membershipbeyond the ldquozero-onerdquo logic and utilized the dynamic infinite

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 6945184 8 pageshttpdxdoiorg10115520166945184

2 Advances in Fuzzy Systems

space between these values In themid-1980s Japanese indus-trialists used this worthwhile concept to develop a schemeof fully automated subway controls which demonstratedits real-world application encouraged a whole new wave ofresearchers to study its theoretical and practical potentials

The foundation of conventional mathematics is based onreal numbers and the process of defuzzifying and rankingfuzzy quantitiesmdashsuch as color or quality of goodsmdashplaysa significant role in data analysis economics and industrialsystems so an extensive amount of research has been dedi-cated to this specific subject

Following the previousworks [10ndash25] this study proposesa novel method for defuzzifying and ranking fuzzy numbersby using the mean value of Beta distribution Some of theprevious works in the literature are as follows

In 1980 Yager [19] proposed a method for ranking fuzzynumbers based on their corresponding centroid-index In1981 the same author published another article proposing amethod for ordering fuzzy subsets based on the integral ofthe mean of the level sets [20] His method was capable ofcomparing crisp numbers discrete fuzzy subsets and con-tinuous fuzzy subsets of the unit interval In 1993 Choobinehand Li [13] introduced a fuzzy number ordering method inwhich the membership function of fuzzy numbers did notneed to be normal and convex In addition their index hadsome additional properties that made it more suitable fordecision-making purposesThe concepts of expected intervaland expected value were introduced in 1992 by Heilpern[21] who then used them to order fuzzy numbers In 1998Cheng [11] proposed a centroid-index ranking method forcalculating the centroid-point of a fuzzy number In 2000Yao and Wu [17] used the decomposition principle and thecrisp ranking system on R to construct a ranking systemfor fuzzy numbers They used the decomposition principleto rewrite each fuzzy number as the union of all of its 120572-cuts (where 120572 isin (0 1)) and then used the centroid-pointof these 120572-cuts to calculate the signed distance between thetwo fuzzy numbers In 2001 Chen and Lu [18] developed anapproximate approach for ranking fuzzy numbers based onthe left and right dominance To do so they first defined theright and left limits of all 120572-cuts for all fuzzy numbers andthen defined the left and right dominance of a fuzzy numberover another as the average difference of the left and rightspreads They then used an optimality index 119861 isin [0 1] tointroduce the dominance of a fuzzy number over another asthe convex combination of its left and right dominance In2003 S-J Chen and S-M Chen [16] introduced a procedurefor ordering trapezoidal fuzzy numbers based on the centerof gravity The centroid-index ranking method presented in2005 by Yong and Qi [12] employed TOPSIS to order thetrapezoidal fuzzy numbers In 2006 Asady and Zendehnam[10] presented a method for ranking fuzzy numbers bydistance minimization The fuzzy number ranking methodproposed in 2008 by Chen andWang [22] used 120572-cuts for thispurpose

This paper aims to use statistical Beta distribution fordefuzzifying and ranking fuzzy numbers since it is the onlydistribution function that is bounded to (0 1) and is zerooutside this interval [26] To do so we can consider the

surface inscribed within the projection of fuzzy number inthe interval (0 1) as a statistical population In additionparameters of this distribution could be set such that theresulting left spread right spread andmode fully match theircorresponding values in the fuzzy number transformed tothe interval (0 1) We initiate the defuzzification process ofa fuzzy number by obtaining the mean value of its corre-sponding Beta distribution We then use simple arithmeticoperations to determine the crisp number corresponding tothat fuzzy number We also provide a very simple algorithmfor ordering fuzzy numbers based on their correspondingcrisp real values and domainsThe first stage of the algorithmis centripetal and considers the fuzzy number with greatercorresponding real value as the greater fuzzy number butwhen two fuzzy numbers have equal corresponding realvalues it considers the onewith righter intuitionistic position(nomatter how slight) as the greater number Ourmotivationfor writing this paper is to provide an easy and tangibledefuzzification method and a well-ordered algorithm for theranking of fuzzy numbers The method presented in thispaper satisfies all reasonable properties defined by Wangand Kerre [23 24] for ordering of triangular fuzzy numbersbut the mathematical complexity obstructed the process ofproving the last property for trapezoidal fuzzy numbers

The rest of this paper is organized as follows Section 2presents a brief introduction to fuzzy and statistical conceptsand operations used in the paper It also presents twotheorems that constitute the basis of our method Section 3describes the proposed method of defuzzifying trapezoidaland triangular fuzzy numbers using the mean value of Betadistribution This section also provides a fuzzy numberranking algorithm as well as ordinal properties to which thisranking method is applicable In Section 4 several exam-ples of fuzzy number ordering performed by the proposedmethod are presented Section 5 uses 2 numerical examplesto compare the proposed method of fuzzy number rankingwith other methods developed for this purpose The paperends with Section 6 presenting conclusions

2 Preliminaries

This section reviews some basic fuzzy and statistical conceptsused in the rest of the paper Two theorems that provide thebasis of ourmethod for obtaining crisp numbers correspond-ing to fuzzy numbers are introduced later in this section

21 Fuzzy Preliminaries

Definition 1 (see [10]) A fuzzy number is a fuzzy set in theform of 119877 rarr [0 1] that satisfies the following conditions

(1) is upper semicontinuous(2) (119909) is zero outside the interval [119897 119906](3) there exist real numbers 1198981 1198982 such that 119897 le 1198981 le1198982 le 119906 and

(31) (119909) is increasing on [119897 1198981](32) (119909) is decreasing on [1198982 119906](33) (119909) = 11198981 le 119909 le 1198982

Advances in Fuzzy Systems 3

If 1198981 = 1198982 = 119898 then the fuzzy number = (119897 119898 119906) iscalled the triangular fuzzy number and is defined as follows

(119909) =

119909 minus 119897119898 minus 119897 119897 le 119909 le 119898119906 minus 119909119906 minus 119898 119898 le 119909 le 1199060 otherwise

(1)

Moreover the fuzzy number = (119897 1198981 1198982 119906) is called atrapezoidal fuzzy number and is defined as follows

(119909) =

119909 minus 1198971198981 minus 119897 119897 le 119909 le 11989811 1198981 le 119909 le 1198982119906 minus 119909119906 minus 1198982 1198982 le 119909 le 1199060 otherwise

(2)

The set of all fuzzy numbers is denoted by119864 Furthermorethe set of all numbers that belong to the universal set 119877 suchthat (119909) gt 0 is known as the supporter set of the fuzzynumber 22 Statistical Preliminaries Let 119878 be the sample space ofa random trial with a given probability value and let 119883be the random variable defined as a real-valued functionon 119878 If 119883 is a discrete random variable the function119891(119909) = 119875(119883 = 119909) for any specific value of 119909 withinrange of 119883 is called probability distribution When 119883 is acontinuous random variable the function 119891(119909) is known asprobability density function of 119883 Probability distributionsand probability densities come in different types includinguniform density and Beta distribution just to name two

The random variable119883 is said to have Beta distribution ifand only if its probability density is as follows

119891 (119909) =

D (120572 + 120573)D (120572) sdot D (120573)119909

120572minus1 (1 minus 119909)120573minus1 0 lt 119909 lt 10 otherwise

(3)

where 120572 gt 0 and 120573 gt 0 are Beta distribution parameters Themean of Beta distribution is obtained as follows

120583 = 120572(120572 + 120573) (4)

If 120572 ge 1 and 120573 ge 1 then the curve of Beta function will beunimodal When 120572 gt 120573 the curve is said to have negativeskewness and if 120572 lt 120573 then the skewness is positive For120572 = 120573 the curve of Beta function is called symmetric Tobetter understand the described concept see Figure 1 where1 lt 120572 lt 120573 lt 2 Given that the curve of Beta distributionis a unimodal one 1198911015840(119909) = 0 should have a unique solutionSolving the equation 1198911015840(119909) = 0 gives the following relationbetween Beta distribution parameters

120573 = (120572 minus 1) (1 minus 119909mod119909mod

) + 1 (5)

0 1

120572 120573

xmod

Figure 1 120572 lt 120573 and so the curve has positive skewness

where 119909mod isin (0 1) is the point at which 119891(119909) has themaximumvalueHence given both values of120572 and119909mod the120573parameter can be obtained from (5) and then the mean valueof Beta distribution can be calculated by (4)

Uniform density is a special case of Beta distributionTherandom variable119883 is said to have uniform density if and onlyif its probability density is as follows

119891 (119909) =

1119902 minus 119901 119901 lt 119909 lt 1199020 otherwise

(6)

Themean value of uniformdensity can be obtained by thefollowing equation

120583 = 119901 + 1199022 (7)

Using the above-mentioned statistical preliminaries acrisp number belonging to the interval (0 1) and corre-sponding to the triangular fuzzy number can be obtainedby (4) and (5) and a crisp number in the interval (0 1)and corresponding to the trapezoidal fuzzy number can beobtained by (4) (5) and (7)The following theorems can thenbe used to calculate the real number corresponding to eachfuzzy number in its domain

Theorem 2 Let 119886 119887 isin 119877 and lt 119887 and let [119886 119887] be an intervalon 119877 Then for every 120583 isin (119886 119887) there exists a unique number1205831015840 isin (119886 119887) and vice versaProof Suppose that 120583 isin (119886 119887) and let 1205831015840 = (120583 minus 119886)(119887 minus 119886)then we have 1205831015840 isin (0 1)

The uniqueness of 1205831015840 is proven by contradiction Assum-ing that 1205831015840 isin (0 1) and 12058310158401015840 isin (0 1) are two distinct numbers(1205831015840 = 12058310158401015840) corresponding to 120583 isin (119886 119887) the followingrelationships between 120583 and 1205831015840 and between 120583 and 12058310158401015840 areestablished

120583 = 1205831015840 (119887 minus 119886) + 119886120583 = 12058310158401015840 (119887 minus 119886) + 119886

(8)

The above equations imply that 1205831015840 = 12058310158401015840 which contra-dicts the initial assumption that is 1205831015840 = 12058310158401015840 Thus initialassumption is false and uniqueness of 1205831015840 is proven


Theorem 3 For two distinct intervals (119886 119887) and (119888 119889) where119886 minus 119888 = 119887 minus 119889 assume that 120583119886 isin (119886 119887) and 120583119888 isin (119886 119887) If1205831015840 = (120583119886 minus 119886)(119887 minus 119886) = (120583119888 minus 119888)(119889 minus 119888) then 120583119886 = 120583119888Proof Let (119886 119887) and (119888 119889) be two distinct intervalsThen onlyone of the following is true

(1) 119886 = 119888 and 119887 = 119889(2) 119886 = 119888 and 119887 = 119889(3) 119886 = 119888 and 119887 = 119889According to theoremrsquos assumption we have

(120583119886 minus 119886)(119887 minus 119886) = (120583119888 minus 119888)

(119889 minus 119888) (9)

In the first case we have 119886 = 119888 and 119887 = 119889 Assuming that119886 = 119888 it is supposed for contradiction that 120583119886 = 120583119888 Relation(9) yields 119887 = 119889 which contradicts the initial statementTherefore the contradictory assumption is false proving theclaim Statements (2) and (3) can be easily proved throughsimilar arguments

3 Using Beta Distribution for Defuzzifyingand Ranking Fuzzy Numbers

In this section for every fuzzy number themean value of itscorresponding Beta distribution in its domain is consideredas the crisp real number corresponding to Then theproposedmethod which is based on crisp ranking system on119877 is used to rank and order the fuzzy numbers

31 Defuzzification of Triangular Fuzzy Numbers Considerthe triangular fuzzy number = (119897 119898 119906) To obtain the crispreal number corresponding to the triangular fuzzy number = (119897 119898 119906) we first project on the interval (0 1) whichwill be in the form of 1015840 = ((119897 minus 119897)(119906 minus 119897) (119898 minus 119897)(119906 minus 119897) (119906 minus119897)(119906minus119897)) = (0 (119898minus119897)(119906minus119897) 1)Thenwe define the parameter120572 corresponding to the Beta distribution as follows

120572 = 119898 minus 119897119906 minus 119897 + 1 (10)

With the above definition firstly it is clear that 120572 ge 1 andif 119898 = 119897 then the Beta distribution curve will be unimodalSecondly if minus119897 = 119906minus119898 which denotes a symmetric triangularfuzzy number then 120572 = 120573 = 32 and the Beta distributioncurve will be symmetric Here the left skewness of the Betadistribution curve is the left side spread of triangular fuzzynumber divided by its support set

In the Beta distribution corresponding to the projectionof fuzzy number = (119897 119898 119906) we have 119909mod = (119898 minus 119897)(119906 minus 119897)and by using (10) and substituting it into (5) we get

120573 = 119906 minus 119898119906 minus 119897 + 1 (11)

0 1 l m u

120583 = 120583998400 (u minus l) + l

Subtract by l and then divide by u minus l

120583998400

Multiply by u minus l and then sum with l

m minus l

u minus l

Figure 2 Transfer of the triangular fuzzy number to the interval(0 1) and vice versa

We use (4) (5) and (11) as shown below to calculate themean value of Beta distribution corresponding to the fuzzynumber

1205831015840 = 120572120572 + 120573

= (119898 minus 119897) (119906 minus 119897) + 1(119898 minus 119897) (119906 minus 119897) + 1 + (119906 minus 119898) (119906 minus 119897) + 1

= 119898 + 119906 minus 21198973 (119906 minus 119897)

(12)

The real number 120583 which is obtained (as shown below)by transferring 1205831015840 from the interval (0 1) to the interval (119897 119906)is considered as the real number corresponding to the fuzzynumber = (119897 119898 119906)

120583 = 1205831015840 (119906 minus 119897) + 119897 (13)

Figure 2 shows the manner of projecting the triangularfuzzy number on the interval (0 1) and Figure 3 shows themanner of defining the Beta function corresponding to theprojection of fuzzy number in the interval (01)

Remark 4 The crisp real number 120583 corresponding to thetriangular fuzzy number = (119897 119898 119906) is obtained from thefollowing relation

120583 = 119897 + 119898 + 1199063 (14)

32 Defuzzification of Trapezoidal Fuzzy Number Considerthe trapezoidal fuzzy number = (119897 1198981 1198982 119906) We use acombination of Beta distribution and uniform distribution todefuzzify this trapezoidal fuzzy number To start the interval(119897 119906) is partitioned as shown below

(119897 119906) = 1198601 cup 1198602 cup 1198603 (15)

where 1198601 = (119897 1198981) 1198602 = [1198981 1198982] and 1198603 = (1198982 119906) Wedefine the triangular fuzzy numbers corresponding to1198601 and1198603 as 1 = (119897 1198981 1198981) and 3 = (1198982 1198982 119906) and then obtain


0 1

asymp120572 asymp120573

mod =m minus l

u minus l

Figure 3 Graph of Beta function corresponding to the triangularfuzzy number

their corresponding real numbers denoted by 1198861 and 1198863 asfollows

1198861 = 119897 + 1198981 + 11989813

1198863 = 1198982 + 1198982 + 1199063

(16)

In order to obtain the real number 1198862 corresponding to theinterval 1198602 = [1198981 1198982] we use the uniform distribution andcalculate its mean value via relation (7) The mean of the realnumbers 1198861 1198862 and 1198863 is considered as the crisp real numbercorresponding to the trapezoidal fuzzy number

Remark 5 The crisp real number 120583 corresponding to thetrapezoidal fuzzy number = (119897 1198981 1198982 119906) is obtained bythe following relation

120583 = 2119897 + 71198981 + 71198982 + 211990618 (17)

33 The Fuzzy Number Ranking Algorithm Consider thefuzzy numbers = (119897 119898 119906) and 119887 = (119897 119898 119906) We takethe following steps to order these numbers

Step 1 Calculate the crisp real numbers 120583 and 120583 corre-sponding to and 119887Step 2

(A) If 120583 gt 120583 then ≻ 119887(B) If 120583 = 120583 and 119906 gt 119906 then ≻ (C) If 120583 = 120583 119906 = 119906 and119898 gt 119898 then ≻ (D) If 120583 = 120583 119906 = 119906 and119898 = 119898 then asymp Note that for trapezoidal fuzzy number we consider119898 =

1198982 and use the symbols ≻ ≺ and asymp to denote ldquogreater thanrdquoldquoless thanrdquo and ldquoequalsrdquo relations between fuzzy numbers

We consider the following reasonable properties for theordering approach see [23 24]

(1198601) For an arbitrary finite subset 119878 of 119864 and isin 119878 ≿

(1198602) For an arbitrary finite subset 119878 of 119864 and ( ) isin 1198782 ≿ 119887 and ≿ we should have asymp

(1198603) For an arbitrary finite subset 119878 of 119864 and ( ) isin 1198783 ≿ 119887 and ≿ we should have ≿

(1198604) For an arbitrary finite subset 119878 of 119864 and ( ) isin 1198782inf supp() gt sup supp() we should have ≿

(11986010158404) For an arbitrary finite subset 119878 of 119864 and ( ) isin 1198782inf supp() gt sup supp() we should have ≻

(1198605) Let 119878 and 1198781015840 be two arbitrary finite subsets of 119864 inwhich and are in 119878 cap 1198781015840 We obtain the rankingorder ≻ 119887 by Beta distribution method on 1198781015840 if andonly if ≻ by Beta distribution method on 119878

(1198606) Let 119887 + and + be elements of 119864 If ≿ 119887then + ≿ +

(11986010158406) Let 119887 + and + be elements of 119864 and = 0 If ≻ 119887 then + ≻ +

(1198607) Let sdot and sdot be elements of 119864 and ge 0 ≿ 119887 by Beta distribution method on implies sdot ≿ sdot by one on sdot sdot

While we could only prove the properties1198601 to11986010158406 for thetrapezoidal fuzzy numbers all of these properties are satisfiedfor the triangular fuzzy numbers

In the next section we use the proposed method to orderseveral fuzzy numbers

4 Numerical Examples

In this section we first use themean value of Beta distributionto obtain the crisp real numbers corresponding to three setsof fuzzy numbers and then use the proposed algorithm toorder these numbers

Set 1 Consider three fuzzy numbers and as follows = (1 3 5) = (3 5 8) = (3 8 12)

(18)

The Beta distribution method yields their corresponding realnumbers as follows

120583 = 3120583 = 5333120583 = 7667

(19)

The proposed algorithm returns ≺ ≺ (see Figure 4)Set 2 Consider the fuzzy numbers = (2 3 4) and =(0 3 6) According to the proposed method the crisp realnumbers corresponding to and are equal (120583 = 120583 =3) but as Figure 5 shows based on the hatched area theproposed algorithm returns ≺


1 3 5 8 12

a b c

Figure 4 It is shown that ≺ ≺

0 2 3 4 6

a

b

Figure 5 It is shown that ≺ 119887

Set 3 Consider the following four fuzzy numbers and and their corresponding crisp real numbers obtained fromBeta distribution method

= (4 6 10 12) 997888rarr 120583 = 8 = (4 7 9 12) 997888rarr 120583 = 8 = (8 10 12 14) 997888rarr 120583 = 9778 = (6 8 10) 997888rarr 120583 = 8

(20)

It is clear that is greater than other fuzzy numbers because120583 is greater than other obtained crisp real numbers Also120583 =120583 = 120583 = 8 Meanwhile 119906 is less than 119906 and 119906 so is thesmallest fuzzy number in the set Since 119906 = 119906 to determinetheir ordering we need to compare their1198982 values Hence wehave 119887 ≺

The proposed method determines the ordering of theabove fuzzy numbers as ≺ ≺ ≺ (see Figures 6 and 7)As Figure 6 shows although 120583 = 120583 based on the hatchedarea the proposed algorithm announces that 119887 lt 5 The Comparison of Beta DistributionMethod with Other Existing Methods

In this section we compare our method of ranking fuzzynumbers with other methods developed for this purpose Wedo so by using two sets of fuzzy numbers

Set 1 Consider the set of fuzzy numbers containing =(01 03 05) = (02 03 04) and = (1 1 1) Table 1shows the ranking of these three numbers according to themethods of Chen and Wang [22] Yong and Qi [12] and S-JChen and S-M Chen [16] in addition to the results obtainedby the proposed method As can be seen the proposedmethod returns equal crisp real numbers for and 119887 but since119906 gt 119906 we conclude that ≻ The proposed algorithm

Table 1 Ranking results for the first set of fuzzy

Ranking method Set of fuzzy numbers Ranking results 119887 Chen and Wang 00992 01021 04737 ≺ 119887 ≺ Yong and Qi 06214 06244 1 ≺ 119887 ≺ S-J Chen andS-M Chen 12359 12674 2 ≺ 119887 ≺ Proposedmethod 03 03 1 ≺ ≺

104 6 8 127 9

a

b

d


104 6 8 12 14

a c

Figure 7 It is shown that ≺

which is based on the mean value of Beta distribution ranksthese three fuzzy numbers as ≺ ≺ contradicting theresult of other ranking methods used for comparison (seeFigure 8)

Set 2 Consider the following three fuzzy numbers and

= (0 04 07 08) 119887 = (02 05 09) = (01 06 08)

(21)

The ranking of these three fuzzy numbers obtained by themethods proposed in [10 13 15 17 20 23 25] and the resultof Beta distribution method are shown in Table 2

The method proposed in this paper calculates the realnumbers corresponding to fuzzy numbers and as0516 0533 and 05 and therefore determines the orderof these fuzzy numbers as ≺ ≺ which is onlyconsistent with the distance method proposed byWang thuscontradicting the results obtained via other methods For abetter understanding see Figures 9 and 10

6 Conclusion

In this paper we proposed a simple method for obtaining thecrisp real number corresponding to a fuzzy number by using


Table 2 Ranking results for the second set of fuzzy

Authors Set of fuzzy numbers Ranking results 119887 Choobineh and Li 05 05833 06111 ≺ ≺ Baldwin and Guild 04 042 042 ≺ asymp Yager 045 0525 055 ≺ ≺ Yao and Wu 0475 0525 0525 ≺ asymp Wang centroid method 01967 01778 01667 ≺ 119887 ≺ Wang distance 06284 06289 06009 ≺ ≺ Asady distance 0475 0525 0525 ≺ asymp Proposed method 0516 0533 05 ≺ ≺

a

b

c

01 02 03 04 05 1

Figure 8 It is shown that 119887 ≺ ≺

a

c

0 01 04 06 07 08 1


0 04 0907 08 1

a

b

02 05


the mean value of Beta distribution and showed that thiscrisp real number could be obtained via simple mathematicaloperationsWe also introduced a novel algorithm for rankingand ordering fuzzy numbers and reviewed the reasonableproperties established for this algorithm and ultimatelyshowed that the proposedmethod is a completemethodologyfor ranking fuzzy numbers In the end we used numericalexamples to demonstrate the performance of ourmethod andthen compared it with other methods developed with similarobjectives

Competing Interests

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

References

[1] W Pedrycz and S M Chen Granular Computing and Decision-Making Interactive and Iterative Approaches Springer Heidel-berg Germany 2015

[2] Y Y Yao ldquoGranular computingrdquo Computer Science vol 31 pp1ndash5 2004

[3] J MMendel ldquoA comparison of three approaches for estimating(synthesizing) an interval type-2 fuzzy set model of a linguisticterm for computingwithwordsrdquoGranular Computing vol 1 no1 pp 59ndash69 2016

[4] A Skowron A Jankowski and S Dutta ldquoInteractive granularcomputingrdquo Granular Computing vol 1 no 2 pp 95ndash113 2016

[5] G Wilke and E Portmann ldquoGranular computing as a basis ofhuman-data interaction a cognitive cities use caserdquo GranularComputing vol 1 no 3 pp 181ndash197 2016

[6] L Livi and A Sadeghian ldquoGranular computing computationalintelligence and the analysis of non-geometric input spacesrdquoGranular Computing vol 1 no 1 pp 13ndash20 2016

[7] M Antonelli P Ducange B Lazzerini and F MarcellonildquoMulti-objective evolutionary design of granular rule-basedclassifiersrdquo Granular Computing vol 1 pp 37ndash58 2016

[8] 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

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

[10] B Asady and A Zendehnam ldquoRanking fuzzy numbers bydistance minimizationrdquo Applied Mathematical Modelling vol31 no 11 pp 2589ndash2598 2007

[11] C-H Cheng ldquoA new approach for ranking fuzzy numbers bydistancemethodrdquo Fuzzy Sets and Systems vol 95 no 3 pp 307ndash317 1998

[12] D Yong and L Qi ldquoA TOPSIS-based centroid-index rankingmethod of fuzzy numbers and its application in decision-makingrdquo Cybernetics and Systems vol 36 no 6 pp 581ndash5952005

[13] F Choobineh andH Li ldquoAn index for ordering fuzzy numbersrdquoFuzzy Sets and Systems vol 54 no 3 pp 287ndash294 1993

[14] H J Zimmermann Fuzzy Set Theory and Its ApplicationsKluwer-Nijhoff Boston Mass USA 2nd edition 1991

[15] J F Baldwin and N C Guild ldquoComparison of fuzzy sets on thesame decision spacerdquo Fuzzy Sets and Systems vol 2 no 3 pp213ndash231 1979

[16] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onsimilarity measures of generalized fuzzy numbersrdquo IEEE Trans-actions on Fuzzy Systems vol 11 no 1 pp 45ndash56 2003

[17] J-S Yao and K Wu ldquoRanking fuzzy numbers based ondecomposition principle and signed distancerdquo Fuzzy Sets andSystems vol 116 no 2 pp 275ndash288 2000

[18] L-H Chen and H-W Lu ldquoAn approximate approach forranking fuzzy numbers based on left and right dominancerdquoComputers and Mathematics with Applications vol 41 no 12pp 1589ndash1602 2001

[19] R R Yager ldquoOn a general class of fuzzy connectivesrdquo Fuzzy Setsand Systems vol 4 no 3 pp 235ndash242 1980


[20] R R Yager ldquoA procedure for ordering fuzzy subsets of the unitintervalrdquo Information Sciences vol 24 no 2 pp 143ndash161 1981

[21] S Heilpern ldquoThe expected value of a fuzzy numberrdquo Fuzzy Setsand Systems vol 47 no 1 pp 81ndash86 1992

[22] S-M Chen and C-H Wang ldquoFuzzy risk analysis based onranking fuzzy numbers using 120572-cuts belief features and sig-nalnoise ratiosrdquo Expert Systems with Applications vol 36 no3 pp 5576ndash5581 2009

[23] X Wang and E E Kerre ldquoReasonable properties for theordering of fuzzy quantities (I)rdquo Fuzzy Sets and Systems vol118 no 3 pp 375ndash385 2001

[24] X Wang and E E Kerre ldquoReasonable properties for theordering of fuzzy quantities (II)rdquo Fuzzy Sets and Systems vol118 no 3 pp 387ndash405 2001

[25] Y-M Wang J-B Yang D-L Xu and K-S Chin ldquoOn thecentroids of fuzzy numbersrdquo Fuzzy Sets and Systems vol 157no 7 pp 919ndash926 2006

[26] J E FreundMathematical Statistics Prentice-Hall 5th edition1992

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Advances in

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Applied Computational Intelligence and Soft Computing

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space between these values In themid-1980s Japanese indus-trialists used this worthwhile concept to develop a schemeof fully automated subway controls which demonstratedits real-world application encouraged a whole new wave ofresearchers to study its theoretical and practical potentials

The foundation of conventional mathematics is based onreal numbers and the process of defuzzifying and rankingfuzzy quantitiesmdashsuch as color or quality of goodsmdashplaysa significant role in data analysis economics and industrialsystems so an extensive amount of research has been dedi-cated to this specific subject

Following the previousworks [10ndash25] this study proposesa novel method for defuzzifying and ranking fuzzy numbersby using the mean value of Beta distribution Some of theprevious works in the literature are as follows

In 1980 Yager [19] proposed a method for ranking fuzzynumbers based on their corresponding centroid-index In1981 the same author published another article proposing amethod for ordering fuzzy subsets based on the integral ofthe mean of the level sets [20] His method was capable ofcomparing crisp numbers discrete fuzzy subsets and con-tinuous fuzzy subsets of the unit interval In 1993 Choobinehand Li [13] introduced a fuzzy number ordering method inwhich the membership function of fuzzy numbers did notneed to be normal and convex In addition their index hadsome additional properties that made it more suitable fordecision-making purposesThe concepts of expected intervaland expected value were introduced in 1992 by Heilpern[21] who then used them to order fuzzy numbers In 1998Cheng [11] proposed a centroid-index ranking method forcalculating the centroid-point of a fuzzy number In 2000Yao and Wu [17] used the decomposition principle and thecrisp ranking system on R to construct a ranking systemfor fuzzy numbers They used the decomposition principleto rewrite each fuzzy number as the union of all of its 120572-cuts (where 120572 isin (0 1)) and then used the centroid-pointof these 120572-cuts to calculate the signed distance between thetwo fuzzy numbers In 2001 Chen and Lu [18] developed anapproximate approach for ranking fuzzy numbers based onthe left and right dominance To do so they first defined theright and left limits of all 120572-cuts for all fuzzy numbers andthen defined the left and right dominance of a fuzzy numberover another as the average difference of the left and rightspreads They then used an optimality index 119861 isin [0 1] tointroduce the dominance of a fuzzy number over another asthe convex combination of its left and right dominance In2003 S-J Chen and S-M Chen [16] introduced a procedurefor ordering trapezoidal fuzzy numbers based on the centerof gravity The centroid-index ranking method presented in2005 by Yong and Qi [12] employed TOPSIS to order thetrapezoidal fuzzy numbers In 2006 Asady and Zendehnam[10] presented a method for ranking fuzzy numbers bydistance minimization The fuzzy number ranking methodproposed in 2008 by Chen andWang [22] used 120572-cuts for thispurpose

This paper aims to use statistical Beta distribution fordefuzzifying and ranking fuzzy numbers since it is the onlydistribution function that is bounded to (0 1) and is zerooutside this interval [26] To do so we can consider the

surface inscribed within the projection of fuzzy number inthe interval (0 1) as a statistical population In additionparameters of this distribution could be set such that theresulting left spread right spread andmode fully match theircorresponding values in the fuzzy number transformed tothe interval (0 1) We initiate the defuzzification process ofa fuzzy number by obtaining the mean value of its corre-sponding Beta distribution We then use simple arithmeticoperations to determine the crisp number corresponding tothat fuzzy number We also provide a very simple algorithmfor ordering fuzzy numbers based on their correspondingcrisp real values and domainsThe first stage of the algorithmis centripetal and considers the fuzzy number with greatercorresponding real value as the greater fuzzy number butwhen two fuzzy numbers have equal corresponding realvalues it considers the onewith righter intuitionistic position(nomatter how slight) as the greater number Ourmotivationfor writing this paper is to provide an easy and tangibledefuzzification method and a well-ordered algorithm for theranking of fuzzy numbers The method presented in thispaper satisfies all reasonable properties defined by Wangand Kerre [23 24] for ordering of triangular fuzzy numbersbut the mathematical complexity obstructed the process ofproving the last property for trapezoidal fuzzy numbers

The rest of this paper is organized as follows Section 2presents a brief introduction to fuzzy and statistical conceptsand operations used in the paper It also presents twotheorems that constitute the basis of our method Section 3describes the proposed method of defuzzifying trapezoidaland triangular fuzzy numbers using the mean value of Betadistribution This section also provides a fuzzy numberranking algorithm as well as ordinal properties to which thisranking method is applicable In Section 4 several exam-ples of fuzzy number ordering performed by the proposedmethod are presented Section 5 uses 2 numerical examplesto compare the proposed method of fuzzy number rankingwith other methods developed for this purpose The paperends with Section 6 presenting conclusions

2 Preliminaries

This section reviews some basic fuzzy and statistical conceptsused in the rest of the paper Two theorems that provide thebasis of ourmethod for obtaining crisp numbers correspond-ing to fuzzy numbers are introduced later in this section

21 Fuzzy Preliminaries

Definition 1 (see [10]) A fuzzy number is a fuzzy set in theform of 119877 rarr [0 1] that satisfies the following conditions

(1) is upper semicontinuous(2) (119909) is zero outside the interval [119897 119906](3) there exist real numbers 1198981 1198982 such that 119897 le 1198981 le1198982 le 119906 and

(31) (119909) is increasing on [119897 1198981](32) (119909) is decreasing on [1198982 119906](33) (119909) = 11198981 le 119909 le 1198982



(119909) =


(1)


(119909) =


(2)



119891 (119909) =

D (120572 + 120573)D (120572) sdot D (120573)119909


(3)


120583 = 120572(120572 + 120573) (4)



) + 1 (5)

0 1

120572 120573

xmod




119891 (119909) =


(6)


120583 = 119901 + 1199022 (7)




120583 = 1205831015840 (119887 minus 119886) + 119886120583 = 12058310158401015840 (119887 minus 119886) + 119886

(8)






(119889 minus 119888) (9)





120572 = 119898 minus 119897119906 minus 119897 + 1 (10)



120573 = 119906 minus 119898119906 minus 119897 + 1 (11)

0 1 l m u

120583 = 120583998400 (u minus l) + l


120583998400


m minus l

u minus l



1205831015840 = 120572120572 + 120573


= 119898 + 119906 minus 21198973 (119906 minus 119897)

(12)


120583 = 1205831015840 (119906 minus 119897) + 119897 (13)



120583 = 119897 + 119898 + 1199063 (14)


(119897 119906) = 1198601 cup 1198602 cup 1198603 (15)



0 1


mod =m minus l

u minus l



1198861 = 119897 + 1198981 + 11989813

1198863 = 1198982 + 1198982 + 1199063

(16)



120583 = 2119897 + 71198981 + 71198982 + 211990618 (17)




















(18)


120583 = 3120583 = 5333120583 = 7667

(19)



1 3 5 8 12

a b c


0 2 3 4 6

a

b




(20)







104 6 8 127 9

a

b

d


104 6 8 12 14

a c




= (0 04 07 08) 119887 = (02 05 09) = (01 06 08)

(21)



6 Conclusion





a

b

c

01 02 03 04 05 1


a

c

0 01 04 06 07 08 1


0 04 0907 08 1

a

b

02 05



Competing Interests


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





(119909) =


(1)


(119909) =


(2)



119891 (119909) =

D (120572 + 120573)D (120572) sdot D (120573)119909


(3)


120583 = 120572(120572 + 120573) (4)



) + 1 (5)

0 1

120572 120573

xmod




119891 (119909) =


(6)


120583 = 119901 + 1199022 (7)




120583 = 1205831015840 (119887 minus 119886) + 119886120583 = 12058310158401015840 (119887 minus 119886) + 119886

(8)






(119889 minus 119888) (9)





120572 = 119898 minus 119897119906 minus 119897 + 1 (10)



120573 = 119906 minus 119898119906 minus 119897 + 1 (11)

0 1 l m u

120583 = 120583998400 (u minus l) + l


120583998400


m minus l

u minus l



1205831015840 = 120572120572 + 120573


= 119898 + 119906 minus 21198973 (119906 minus 119897)

(12)


120583 = 1205831015840 (119906 minus 119897) + 119897 (13)



120583 = 119897 + 119898 + 1199063 (14)


(119897 119906) = 1198601 cup 1198602 cup 1198603 (15)



0 1


mod =m minus l

u minus l



1198861 = 119897 + 1198981 + 11989813

1198863 = 1198982 + 1198982 + 1199063

(16)



120583 = 2119897 + 71198981 + 71198982 + 211990618 (17)




















(18)


120583 = 3120583 = 5333120583 = 7667

(19)



1 3 5 8 12

a b c


0 2 3 4 6

a

b




(20)







104 6 8 127 9

a

b

d


104 6 8 12 14

a c




= (0 04 07 08) 119887 = (02 05 09) = (01 06 08)

(21)



6 Conclusion





a

b

c

01 02 03 04 05 1


a

c

0 01 04 06 07 08 1


0 04 0907 08 1

a

b

02 05



Competing Interests


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







(119889 minus 119888) (9)





120572 = 119898 minus 119897119906 minus 119897 + 1 (10)



120573 = 119906 minus 119898119906 minus 119897 + 1 (11)

0 1 l m u

120583 = 120583998400 (u minus l) + l


120583998400


m minus l

u minus l



1205831015840 = 120572120572 + 120573


= 119898 + 119906 minus 21198973 (119906 minus 119897)

(12)


120583 = 1205831015840 (119906 minus 119897) + 119897 (13)



120583 = 119897 + 119898 + 1199063 (14)


(119897 119906) = 1198601 cup 1198602 cup 1198603 (15)



0 1


mod =m minus l

u minus l



1198861 = 119897 + 1198981 + 11989813

1198863 = 1198982 + 1198982 + 1199063

(16)



120583 = 2119897 + 71198981 + 71198982 + 211990618 (17)




















(18)


120583 = 3120583 = 5333120583 = 7667

(19)



1 3 5 8 12

a b c


0 2 3 4 6

a

b




(20)







104 6 8 127 9

a

b

d


104 6 8 12 14

a c




= (0 04 07 08) 119887 = (02 05 09) = (01 06 08)

(21)



6 Conclusion





a

b

c

01 02 03 04 05 1


a

c

0 01 04 06 07 08 1


0 04 0907 08 1

a

b

02 05



Competing Interests


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 1


mod =m minus l

u minus l



1198861 = 119897 + 1198981 + 11989813

1198863 = 1198982 + 1198982 + 1199063

(16)



120583 = 2119897 + 71198981 + 71198982 + 211990618 (17)




















(18)


120583 = 3120583 = 5333120583 = 7667

(19)



1 3 5 8 12

a b c


0 2 3 4 6

a

b




(20)







104 6 8 127 9

a

b

d


104 6 8 12 14

a c




= (0 04 07 08) 119887 = (02 05 09) = (01 06 08)

(21)



6 Conclusion





a

b

c

01 02 03 04 05 1


a

c

0 01 04 06 07 08 1


0 04 0907 08 1

a

b

02 05



Competing Interests


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1 3 5 8 12

a b c


0 2 3 4 6

a

b




(20)







104 6 8 127 9

a

b

d


104 6 8 12 14

a c




= (0 04 07 08) 119887 = (02 05 09) = (01 06 08)

(21)



6 Conclusion





a

b

c

01 02 03 04 05 1


a

c

0 01 04 06 07 08 1


0 04 0907 08 1

a

b

02 05



Competing Interests


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






a

b

c

01 02 03 04 05 1


a

c

0 01 04 06 07 08 1


0 04 0907 08 1

a

b

02 05



Competing Interests


References



































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in


















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article a new method for defuzzification...

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