A Function Principle Approach to Jaccard RankingFuzzy Numbers

Nazirah RamliDepartment of Mathematics and Statistics

Faculty of Computer and Mathematical SciencesUniversiti Teknologi MARA Pahang

26400, Bandar Jengka, Pahang, MalaysiaEmail: nazirahr@pahang.uitm.edu.my

Daud MohamadDepartment of Mathematics

Faculty of Computer and Mathematical SciencesUniversiti Teknologi MARA Malaysia40450, Shah Alam, Selangor, Malaysia

Email: daud201@salam.uitm.edu.my

Abstract—Ranking of fuzzy numbers plays an important rolein practical use and has become a prerequisite procedure formany decision-making problems in fuzzy environment. Varioustechniques of ranking fuzzy numbers have been developed andone of them is based on the similarity measure technique. Jaccardindex similarity measure has been introduced in ranking thefuzzy numbers where the fuzzy maximum and fuzzy minimumare obtained by using the extension principle. However, thisapproach is only applicable to normal fuzzy numbers and,therefore, fails to rank the non-normal fuzzy numbers. Besidesthat, the extension principle does not preserve the type ofmembership function of the fuzzy numbers and also involveslaborious mathematical operations. In this paper, a simple vertexfuzzy arithmetic operation, namely, function principle is appliedin the Jaccard ranking index. This method is capable to rankboth normal and non-normal fuzzy numbers in a simpler manner.It has also improved the ranking results by the original Jaccardranking method and some of the existing ranking methods.

Index Terms—extension principle; function principle; Jaccardindex similarity measure; ranking fuzzy numbers

I. INTRODUCTION

In fuzzy environment, the ranking of fuzzy numbers isan important procedure for decision-making and generallybecomes one of the main issues in fuzzy theory. Varioustechniques of ranking fuzzy numbers have been developedsuch as distance index [1], signed distance [2] and [3],area index [4], index based on standard deviation [5], scorevalue [6], distance minimization [7] and centroid index [8].These methods range from the trivial to the complex, includingfrom one fuzzy number attribute to many fuzzy numbersattributes. The similarity measure concept using Jaccard indexwas also proposed in ranking fuzzy numbers. This method wasfirst introduced by Setnes and Cross [9] where the agreementbetween each pair of fuzzy numbers in similarity manner isevaluated. The extension principle (EP) concept is applied inobtaining the fuzzy maximum and fuzzy minimum which thenare used in determining the ranking of the fuzzy numbers.

However, the conventional arithmetic operations using theEP are only applicable to normal fuzzy numbers which meansthat the Jaccard index fails to rank non-normal fuzzy numbers.Besides that, the fuzzy arithmetic operations by EP will changethe membership function type of the fuzzy number and alsorequire complex and laborious mathematical operations [10].

In this paper, a simple vertex fuzzy arithmetic operation whichwas proposed by Chen [11] and named as function principle(FP) is applied to the Jaccard ranking index. This paper alsoproposes the degree of optimism concept in calculating thetotal fuzzy evidence instead of using the mean aggregationsince the ranking of fuzzy numbers is commonly implementedin the decision-making problems. Besides that, the usage ofthe mean aggregation in the original Jaccard ranking indexrepresents the neutral decision maker which is part of thedegree of optimism concept. This paper has improved notonly the Jaccard ranking method but some of the previousranking methods in ranking to both normal and non-normalfuzzy numbers.

II. PRELIMINARIES

A. Fuzzy Number

A fuzzy number is a fuzzy subset in the universe discoursethat is both convex and normal. The membership function ofa fuzzy number A can be defined as

fA(x) =

⎧⎪⎪⎨⎪⎪⎩

fLA(x) , a ≤ x ≤ b1 , b ≤ x ≤ c

fRA (x) , c ≤ x ≤ d0 , otherwise

where fLA : [a, b] → [0, 1], fR

A : [c, d] → [0, 1], fLA and

fRA are the left and the right membership functions of the

fuzzy number A respectively. Trapezoidal fuzzy numbers aredenoted as (a,b,c,d) and triangular fuzzy numbers which arespecial cases of trapezoidal fuzzy numbers with b=c denotedas (a,b,d).

B. The Extension Principle (EP)

Based on Dubois and Prade [12], the extension principleintroduced by Zadeh [13] is defined as follows: Let X be aCartesian product of universes, X = X1 × X2 × . . . × Xr, andA1, A2, . . . , Ar be r fuzzy sets in X = X1 × X2 × . . . × Xr

respectively. Let f be a mapping from X = X1 ×X2 × . . .×Xr

to a universe Y such that y = f(x1, x2, . . . , xn). A fuzzy setB on Y induced from r fuzzy sets Ai through f is,

μB(y) = supy=f(x1,x2,...,xr)min(μA1(x1), μA2(x2), . . . , μAr(xr)).

2009 International Conference of Soft Computing and Pattern Recognition

978-0-7695-3879-2/09 $26.00 © 2009 IEEE

DOI 10.1109/SoCPaR.2009.71

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2009 International Conference of Soft Computing and Pattern Recognition

978-0-7695-3879-2/09 $26.00 © 2009 IEEE

DOI 10.1109/SoCPaR.2009.71

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C. The Function Principle (FP)

The function principle proposed by Chen [11] is defined asfollows: Let g be an arithmetical mapping from n dimensionreal number �n into real line �, and fg is a correspondingmapping from n dimension fuzzy numbers into fuzzy number.Suppose that Ai = (ai, bi, ci, di;hi), i = 1, 2, . . . , n be ntrapezoidal fuzzy numbers. The fuzzy number B on � inducedfrom these fuzzy numbers Ai through function fg is,

fg(A1, A2, . . . , An) = B = (a, b, c, d;h)

whereh = min{h1, h2, . . . , hn},Ai,s = min{x|μAi

(x) ≥ h},Ai,t = max{x|μAi

(x) ≥ h},T = {g(x1, x2, . . . , xn)|xi = ai or di, i = 1, 2, . . . , n},T1 = {g(x1, x2, . . . , xn)|xi = Ai,s or Ai,t, i = 1, 2, . . . , n},a = minT , b = minT1, c = maxT1, d = maxT ,minT ≤ minT1 and maxT1 ≤ maxT .

III. A REVIEW ON FUZZY JACCARD RANKING METHOD

Based on the psychological ratio model of similarity fromTversky [14], which is defined as

Sα,β(X,Y ) =f(X ∩ Y )

f(X ∩ Y ) + αf(X ∩ Y ) + βf(Y ∩ X),

various index of similarity measures have been proposedwhich depend on the value of α and β. For α = β = 1, thepsychological ratio model of similarity becomes the Jaccardindex similarity measure which is defined as

S1,1(X,Y ) =f(X ∩ Y )f(X ∪ Y )

.

Typically, the function f is taken to be the cardinalityfunction. In extending the Jaccard index similarity measure ofpsychology to similarity measure for fuzzy sets, the objects Xand Y described by the features are replaced with fuzzy sets Aand B which are described by the membership functions. Thefuzzy Jaccard index similarity measure is defined as

SJ(A,B) =|A ∩ B||A ∪ B|

where |A| denotes the cardinality of fuzzy set A, ∩ and ∪ canbe any t-norm and s-norm respectively.

In this section, the method in ranking fuzzy numbers usingJaccard similarity measure introduced by Setnes and Cross [9]is briefly reviewed. The procedure is presented as follows:

Step 1: For each pair of triangular fuzzy numbers Ai andAj where i, j = 1, 2, . . . , n , find the fuzzy minimum andfuzzy maximum between Ai and Aj by using the extensionprinciple.

Step 2: Calculate the evidences of E(Ai ≥ Aj),E(Aj ≤ Ai), E(Aj ≥ Ai) and E(Ai ≤ Aj) which aredefined based on fuzzy Jaccard index as

E(Ai ≥ Aj) = SJ(MAX(Ai, Aj), Ai),E(Aj ≤ Ai) = SJ(MIN(Ai, Aj), Aj),E(Aj ≥ Ai) = SJ(MAX(Ai, Aj), Aj),E(Ai ≤ Aj) = SJ(MIN(Ai, Aj), Ai).

MAX(Ai, Aj) and MIN(Ai, Aj) are the fuzzy maximumand fuzzy minimum between Ai and Aj respectively. Here,Cij and cji are used to represent E(Ai ≥ Aj) and E(Aj ≤Ai), respectively. Likewise, Cji and cij are used to denoteE(Aj ≥ Ai) and E(Ai ≤ Aj) respectively.

Step 3: Calculate the total evidences Etotal(Ai ≥ Aj)and Etotal(Aj ≥ Ai) which are defined based on the meanaggregation concept as

Etotal(Ai ≥ Aj) =Cij + cji

2and

Etotal(Aj ≥ Ai) =Cji + cij

2.

Here, E≥(i, j) and E≥(j, i) are used to represent Etotal(Ai ≥Aj) and Etotal(Aj ≥ Ai), respectively.

Step 4: For two triangular fuzzy numbers, compare the totalevidences in Step 3 which will result the ranking of the twotriangular fuzzy numbers Ai and Aj as follows:

i. Ai � Aj if and only if E≥(i, j) > E≥(j, i).ii. Ai ≺ Aj if and only if E≥(i, j) < E≥(j, i).iii. Ai ≈ Aj if and only if E≥(i, j) = E≥(j, i).Step 5: For n triangular fuzzy numbers, develop n×n binary

ranking relation R>(i, j) , which is defined as

R>(i, j) ={

1 , E≥(i, j) > E≥(j, i)0 , otherwise

Step 6: Develop a column vector [Oi] where Oi is thetotal element of each row of R>(i, j) and is defined asOi =

∑nj=1 R>(i, j) for j = 1, 2, . . . , n .

Step 7: The total ordering of the triangular fuzzy numbersAi corresponds to the order of the elements [Oi] in the columnvector [Oi] .

IV. AN EXTENSION OF JACCARD RANKING METHOD

An extension of Jaccard procedure which applies the func-tion principle concept and at the same time improves Setnesand Cross’s [9] method can be expressed in a series of steps:

Step 1: For each pair of trapezoidal fuzzy numbers Ai andAj where i, j = 1, 2, . . . , n , find the fuzzy minimum andfuzzy maximum between Ai and Aj by using the functionprinciple. Suppose that A1 = (a1, b1, c1, d1;h1) and A2 =(a2, b2, c2, d2;h2). The fuzzy maximum of A1 and A2 is,

MAX(A1, A2) = (a, b, c, d;h)

whereh = min{h1, h2, . . . , hn},T = {max(a1, a2), max(a1, d2), max(d1, a2), max(d1, d2)},T1 = {max(A1,s, A2,s), max(A1,s, A2,t), max(A1,t, A2,s),

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max(A1,t, A2,t)},a = minT , b = minT1, c = maxT1 and d = maxT .

Similarly, the fuzzy minimum of A1 and A2 is,

MIN(A1, A2) = (a, b, c, d;h)

whereT = {min(a1, a2), min(a1, d2), min(d1, a2), min(d1, d2)},T1 = {min(A1,s, A2,s), min(A1,s, A2,t), min(A1,t, A2,s),min(A1,t, A2,t)},a = minT , b = minT1, c = maxT1 and d = maxT .

Step 2: Calculate the evidences of E(Ai ≥ Aj), E(Aj ≤Ai), E(Aj ≥ Ai) and E(Ai ≤ Aj) which are defined basedon fuzzy Jaccard index as

E(Ai ≥ Aj) = SJ(MAX(Ai, Aj), Ai),E(Aj ≤ Ai) = SJ(MIN(Ai, Aj), Aj),E(Aj ≥ Ai) = SJ(MAX(Ai, Aj), Aj),E(Ai ≤ Aj) = SJ(MIN(Ai, Aj), Ai).

MAX(Ai, Aj) and MIN(Ai, Aj) are the fuzzy maximumand fuzzy minimum between Ai and Aj respectively. Here,Cij and cji are used to represent E(Ai ≥ Aj) and E(Aj ≤Ai), respectively. Likewise, Cji and cij are used to denoteE(Aj ≥ Ai) and E(Ai ≤ Aj) respectively.

Step 3: Calculate the total evidences Etotal(Ai ≥ Aj) andEtotal(Aj ≥ Ai) which are defined based on the degree ofoptimism concept as

Etotal(Ai ≥ Aj) = βCij + (1 − β)cji

and

Etotal(Aj ≥ Ai) = βCji + (1 − β)cij

where β represents the degree of optimism. Here, E≥(i, j)and E≥(j, i) are used to represent Etotal(Ai ≥ Aj) andEtotal(Aj ≥ Ai) , respectively.

Step 4: For each pair of fuzzy numbers, compare the totalevidences in Step 3 which will result the ranking of the twofuzzy numbers Ai and Aj as follows:

i. Ai � Aj if and only if E≥(i, j) > E≥(j, i).ii. Ai ≺ Aj if and only if E≥(i, j) < E≥(j, i).iii. Ai ≈ Aj if and only if E≥(i, j) = E≥(j, i).Step 5: For n trapezoidal fuzzy numbers, develop n × n

binary ranking relation R>(i, j) , which is defined as

R>(i, j) ={

1 , E≥(i, j) > E≥(j, i)0 , otherwise

Step 6: Develop a column vector [Oi] where Oi is thetotal element of each row of R>(i, j) and is defined asOi =

∑nj=1 R>(i, j) for j = 1, 2, . . . , n .

Step 7: The total ordering of the trapezoidal fuzzy numbersAi corresponds to the order of the elements [Oi] in the columnvector [Oi] .

V. COMPARATIVE EXAMPLES

In this section, six sets of numerical examples are presentedto illustrate the validity and advantages of the Jaccard with FPranking method. Sets 4 and 6 are adopted from Cheng’s [1]while Set 5 is adopted from Wang et al.’s [15]. Sets 1, 2 and 3are self-designed numerical examples where Jaccard with FPcan discriminate the ranking while some other methods cannot.Sets 1 - 3 and Sets 4 - 6 involve normal and non-normal fuzzynumbers respectively.

• Set 1 : A = (0.2, 0.5, 0.9; 1), B = (0.1, 0.6, 0.8; 1).• Set 2 : A = (0.1, 0.2, 0.4, 0.5; 1), B = (0.2, 0.3, 0.4; 1).• Set 3 : A =(2, 6.5, 9, 12.5; 1), B = (5, 6, 13; 1), C = (1,

7, 10, 12; 1).• Set 4 : A = (3, 5, 7; 1), B = (3, 5, 7; 0.8).• Set 5 : A = (1, 2, 3; 1), B = (0.5, 2.5, 3; 27

28 ) .• Set 6 : A = (5, 7, 9, 10; 1), B = (6, 7, 9, 10; 0.6), C =

(7, 8, 9, 10; 0.4).

Tables 1 and 2 show the ranking results for normal and non-normal fuzzy numbers respectively.

TABLE ICOMPARATIVE RESULTS OF THE JACCARD FP RANKING INDEX WITH THE

EXISTING RANKING METHODS FOR SET 1 TO SET 3 (NORMAL FUZZY

NUMBERS)

Indices Fuzzy Set 1 Set 2 Set 3Numbers

A 0.726 0.583 7.466Cheng B 0.724 0.583 8.014

[1] C 7.328Result A � B A ≈ B B � A � C

A 0.525 0.3 7.5Yao & Wu B 0.525 0.3 7.5

[2] C 7.5Result A ≈ B A ≈ B A ≈ B ≈ C

A 1.05 0.6 15Abbasbandy & Asady B 1.05 0.6 15

[3] C 15Result A ≈ B A ≈ B A ≈ B ≈ C

A 0.262 0.15 3.766Chu & Tsao B 0.262 0.15 3.7633

[4] C 3.817Result A ≈ B A ≈ B C � A � B

A 1.423 1.206 7.564Chen & Chen B 1.386 1.267 8.223

[5] C 7.360Result A � B A ≺ B B � A � C

A 0.406 0.424 1.161Chen & Chen B 0.400 0.473 1.042

[6] C 1.317Result A � B A ≺ B C � A � B

A 0.525 0.3 7.5Asady & Zendehnam B 0.525 0.3 7.5

[7] C 7.5Result A ≈ B A ≈ B A ≈ B ≈ C

A 0.533 0.3, 0.5 7.449, 0.506Wang & Lee B 0.5 0.3, 0.5 8, 0.467

[8] C 7.310, 0.522Result A � B A ≈ B B � A � C

E≥(A, B) 0.895 0.583 0Original E≥(B, A) 0.867 0.583 2Jaccard 0

Result A � B A ≈ B B � A ≈ CE≥(A, B) 0.778 0.5 1

Jaccard FP E≥(B, A) 0.750 0.667 2β = 0 0

Result A � B A ≺ B B � A � CE≥(A, B) 0.778 0.583 1

Jaccard FP E≥(B, A) 0.750 0.583 2β = 0.5 0

Result A � B A ≈ B B � A � CE≥(A, B) 0.778 0.67 1

Jaccard FP E≥(B, A) 0.750 0.5 0β = 1 2

Result A � B A � B C � A � B

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TABLE IICOMPARATIVE RESULTS OF THE JACCARD FP RANKING INDEX WITH THE

EXISTING RANKING METHODS FOR SET 4 TO SET 6 (NON-NORMAL FUZZY

NUMBERS)

Indices Fuzzy Numbers Set 4 Set 5 Set 6

A 5.025 2.062 7.731Cheng B 5.016 2.064 8.006

[1] C 8.502Result A � B A ≺ B A ≺ B ≺ C

A 5 2 7.25Yao & Wu B ∗ ∗ ∗

[2] C ∗Result − − −

A 10 4 15.5Abbasbandy & Asady B ∗ ∗ ∗

[3] C ∗Result − − −

A 2.5 1.000 3.899Chu & Tsao B 2 1.021 2.400

[4] C 1.700Result A � B A ≺ B A � B � C

A 5.516 2.718 8.022Chen & Chen B 5.603 2.650 8.591

[5] C 9.304Result A ≺ B A � B A ≺ B ≺ C

A 0.044 0.228 0.387Chen & Chen B 0.035 0.126 0.338

[6] C 0.8Result A � B A � B B ≺ A ≺ C

A 5 2 7.25Asady & Zendehnam B ∗ ∗ ∗

[7] C ∗Result − − −

A 0.5 0.5 7.714Wang & Lee B 0.4 0.51 8

[8] C 8.5Result A � B A ≺ B A ≺ B ≺ C

E≥(A, B) ∗ ∗ ∗Original E≥(B, A) ∗ ∗ ∗Jaccard ∗

Result − − −E≥(A, B) 0.909 0.712 2

Jaccard FP E≥(B, A) 0.880 0.805 0β = 0 1

Result A � B A ≺ B A � C � BE≥(A, B) 0.894 0.687 0

Jaccard FP E≥(B, A) 0.894 0.802 1β = 0.5 2

Result A ≈ B A ≺ B A ≺ B ≺ CE≥(A, B) 0.880 0.662 0

Jaccard FP E≥(B, A) 0.909 0.800 1β = 1 2

Result A ≺ B A ≺ B A ≺ B ≺ C′∗′: the ranking method cannot calculate the ranking value

′−′: no conclusion for the ranking result

VI. DISCUSSION

Based on Table 1, the Jaccard ranking index with FP pro-duces consistent consequences for all types of decision makersfor Set 1, which give result as A � B. The ranking resultsare also consistent with the original Jaccard ranking method,Cheng’s [1], Chen and Chen’s [5], Chen and Chen’s [6] andWang and Lee’s [8] ranking methods. However, Yao andWu’s [2], Chu and Tsao’s [4], Abbasbandy and Asady’s [3] andAsady and Zendehnam’s [7] methods cannot discriminate theranking between the two fuzzy numbers. Although, the fuzzymaximum and fuzzy minimum for the Jaccard with FP andthe original Jaccard ranking methods are not equal, they havethe same three vertices. The three vertices for fuzzy maximumare 0.2, 0.6 and 0.9 while the vertices for fuzzy minimum are0.1, 0.5 and 0.8. The fuzzy maximum and fuzzy minimum forthe Jaccard ranking index with FP are still in the triangulartype, but for the original Jaccard ranking index, the shapes ofmembership functions have changed.

For Set 2, almost all previous methods in Table 1 cannotdiscriminate the ranking between A and B except for Chenand Chen’s [5] and Chen and Chen’s [6]. The original Jaccardranking method also cannot discriminate the ranking betweenthe two fuzzy numbers which is consistent with the statementmade by Cross and Sudkamp [16] where for two fuzzynumbers with the same core and one is symmetrical includedin the other, the sets are regarded as equal even though theyare not identical. On the other hand, as for Jaccard rankingindex with FP, the ranking result depends on the index ofoptimism β. The use of different values of β , 0 or 1, producesa reverse consequence, while neutral decision maker treats thefuzzy number equivalent. Optimistic decision maker (β = 1)produces A � B, while pessimistic decision maker (β = 0)yields A ≺ B. Both the Jaccard with FP and the originalJaccard ranking methods have the same fuzzy maximum andfuzzy minimum where MAX(A,B) = (0.2, 0.3, 0.4, 0.5; 1)and MIN(A,B) = (0.1, 0.2, 0.3, 0.4; 1).

For Set 3, Yao and Wu’s [2] and Abbasbandy andAsady’s [3] and Asady and Zendehnam’s [7] methods treatthe three fuzzy numbers equally. Cheng’s [1], Chen andChen’s [5] and Wang and Lee’s [8] methods produce theordering as B � A � C , while Chu and Tsao’s [4]and Chen and Chen’s [6] methods produce the ranking asC � A � B . However, the original Jaccard ranking methodcannot discriminate the ranking between A and C. For theJaccard ranking index with FP, the ranking result dependson the index of optimism β . For β = 0 and β = 0.5, itproduces the ranking as B � A � C which is consistent withCheng’s [1], Chen and Chen’s [5] and Wang and Lee’s [8]methods, while for β = 1 , the ranking result is consistentwith Chu and Tsao’s [4] and Chen and Chen’s [6] methods.

As Yao and Wu’s [2], Abbasbandy and Asady’s [3] andAsady and Zendehnam’s [7] methods cannot rank the non-normal fuzzy numbers, thus, the ranking result for Set 4 bythe aforementioned methods cannot be obtained. The origi-nal Jaccard ranking index also cannot rank the non-normalfuzzy numbers. Cheng’s [1], Chu and Tsao’s [4], Chen andChen’s [6] and Wang and Lee’s [8] methods produce orderingas A � B, while Chen and Chen’s [5] produces the rankingas A ≺ B. For the Jaccard index with FP, the ranking resultdepends on the index of optimism, β . For β = 0 , the rankingresult is consistent with Cheng’s [1], Chu and Tsao’s [4], Chenand Chen’s [6] and Wang and Lee’s [8] methods, while forβ = 1 , the ranking is consistent with Chen and Chen’s [5].The neutral decision maker ranks A and B equally.

Analogous with Set 4, Yao and Wu’s [2], Abbasbandy andAsady’s [3], Asady and Zendehnam’s [7] and the originalJaccard methods cannot rank the non-normal fuzzy numbersfor Set 5. The Jaccard with FP ranks the two fuzzy numbersconsistently for all types of decision makers which producesA ≺ B and this is also consistent with the ranking results byCheng’s [1], Chu and Tsao’s [4] and Wang and Lee’s [8].However, Chen and Chen’s [5] and Chen and Chen’s [6]indices produce a reverse consequence, A � B.

Similar to Sets 4 and 5, Yao and Wu’s [2], Abbasbandy

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and Asady’s [3], Asady and Zendehnam’s [7] and the originalJaccard methods cannot rank B and C in Set 6. Cheng’s [1],Chen and Chen’s [5] and Wang and Lee’s [8] methods produceordering as A ≺ B ≺ C while Chen and Chen’s [6] indexproduces result as B ≺ A ≺ C. In contrast, Chu and Tsao’s [4]gives ordering as A � B � C. The ranking results foroptimistic and neutral decision makers of Jaccard with FP areconsistent with Cheng’s [1], Chen and Chen’s [5] and Wangand Lee’s [8] methods while the pessimistic decision maker’sranking result is A � C � B.

VII. CONCLUSION

This paper improves the Jaccard index similarity measureproposed by Setnes and Cross [9] in ranking both normaland non-normal fuzzy numbers. Instead of using the EP indetermining the fuzzy maximum and fuzzy minimum, thispaper applies the FP operations which are capable to solvethe arithmetic operations of both normal and non-normal fuzzynumbers. It is found that, the vertices of the fuzzy maximumand fuzzy minimum for normal fuzzy numbers between theEP and FP are similar. In fact, in some cases such as forcomparable fuzzy numbers and one fuzzy number that isincluded in the other (Set 2), the fuzzy maximum and fuzzyminimum between the EP and FP are equal. Besides that,the type of membership function for fuzzy maximum andminimum is preserved and the calculation of operation is easierand simpler under the operation of FP than EP. Due to Chenet al. [17], the EP is observed as a form of convolution, whilethe FP is akin to a point wise operation.

The Jaccard ranking index with FP produces consistentresults with some of the previous ranking indices and in facthas improved some of the results. For instance, the Jaccardranking index with FP can rank normal fuzzy numbers effec-tively which cannot be distinguished by some of the previousranking methods such as Cheng’s [1], Yao and Wu’s [2],Chu and Tsao’s [4], Abbasbandy and Asady’s [3], Asady andZendehnam’s [7], Wang and Lee’s [8] and the original Jaccardranking index. Similarly, the Jaccard index with FP can alsorank successfully the non-normal fuzzy numbers which fail tobe ranked by Yao and Wu’s [2], Abbasbandy and Asady’s [3],Asady and Zendehnam’s [7] and the original Jaccard rankingindex. The usage of degree of optimism concept in aggregatingthe fuzzy total evidence instead of the mean aggregationrepresents the perspective of all types of decision makers.Thus, it can be concluded that the Jaccard index with FPcan rank both normal and non-normal fuzzy numbers in asimpler manner which has improved not only the originalJaccard ranking method but also some of the previous rankingmethods.

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