# Ranking Fuzzy Numbers with a Distance Method using Circumcenter of Centroids and an Index of Modality

Authors: P. Phani Bushan Rao; N. Ravi Shankar
Journal: Advances in Fuzzy Systems
 (2011)

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## Abstract

Ranking fuzzy numbers are an important aspect of decision making in a fuzzy environment.  Since their inception in 1965, many authors have proposed different methods for ranking fuzzy numbers. However, there is no method which gives a satisfactory result to all situations.  Most of the methods proposed so far are nondiscriminating and counterintuitive. This paper proposes a new method for ranking fuzzy numbers based on the Circumcenter of Centroids and uses an index of optimism to reflect the decision maker's optimistic attitude and also an index of modality that represents the neutrality of the decision maker. This method ranks various types of fuzzy numbers which include normal, generalized trapezoidal, and triangular fuzzy numbers along with crisp numbers with the particularity that crisp numbers are to be considered particular cases of fuzzy numbers.


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## Body

## 1. Introduction

Ranking fuzzy numbers are an important tool in decision making. In fuzzy decision analysis, fuzzy quantities are used to describe the performance of alternatives in modeling a real-world problem. Most of the ranking procedures proposed so far in the literature cannot discriminate fuzzy quantities and some are counterintuitive. As fuzzy numbers are represented by possibility distributions, they may overlap with each other, and hence it is not possible to order them. It is true that fuzzy numbers are frequently partial order and cannot be compared like real numbers which can be linearly ordered. In order to rank fuzzy quantities, each fuzzy quantity is converted into a real number and compared by defining a ranking function from the set of fuzzy numbers to a set of real numbers which assign a real number to each fuzzy number where a natural order exists. Usually by reducing the whole of any analysis to a single number, much of the information is lost and hence an attempt is to be made to minimize this loss. Various ranking procedures have been developed since 1976 when the theory of fuzzy sets were first introduced by Zadeh [1]. Ranking fuzzy numbers were first proposed by Jain [2] for decision making in fuzzy situations by representing the ill-defined quantity as a fuzzy set. Since then, various procedures to rank fuzzy quantities are proposed by various researchers. Bortolan and Degani [3] reviewed some of these ranking methods [2, 4–14] for ranking fuzzy subsets. Chen [15] presented ranking fuzzy numbers with maximizing set and minimizing set. Dubois and Prade [16] presented the mean value of a fuzzy number. Lee and Li [17] presented a comparison of fuzzy numbers based on the probability measure of fuzzy events. Delgado et al. [18] presented a procedure for ranking fuzzy numbers. Campos and Muñoz [19] presented a subjective approach for ranking fuzzy numbers. Kim and Park [20] presented a method of ranking fuzzy numbers with index of optimism. Yuan [21] presented a criterion for evaluating fuzzy ranking methods. Heilpern [22] presented the expected value of a fuzzy number. Saade and Schwarzlander [23] presented ordering fuzzy sets over the real line. Liou and Wang [24] presented ranking fuzzy numbers with integral value. Choobineh and Li [25] presented an index for ordering fuzzy numbers. Chang and Lee [26] presented ranking of fuzzy sets based on the concept of existence. Since then several methods have been proposed by various researchers which includes ranking fuzzy numbers using area compensation, distance method, maximizing and minimizing set, decomposition principle, and signed distance [27–30]. Wang and Kerre [31, 32] classified all the above ranking procedures into three classes. The first class consists of ranking procedures based on fuzzy mean and spread [6, 8–11, 19, 24, 25, 27], and second class consists ranking procedures based on fuzzy scoring [2, 5, 12, 15, 20, 33], whereas the third class consists of methods based on preference relations [4, 7, 13, 18, 21, 23, 34, 35] and concluded that the ordering procedures associated with first class are relatively reasonable for the ordering of fuzzy numbers specially the ranking procedure presented by Adamo [9] which satisfies all the reasonable properties for the ordering of fuzzy quantities. The methods presented in the second class are not doing well and the methods [21, 23, 34, 35] which belong to class three are reasonable. Later on, ranking fuzzy numbers by preference ratio [36], left and right dominance [37], fuzzy distance measure [38], area between the centroid point and original point [39], preference weighting function expectations [40], sign distance [41], fuzzy simulation analysis method [42], an area method using radius of gyration [43], distance minimization [44], and fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers [45]. Garcia and Lamata [46] modified the index of Liou and Wang [24] for ranking fuzzy numbers. The development in ranking fuzzy numbers can also be found in [47–58]. Most of the methods presented above cannot discriminate fuzzy numbers, and some methods do not agree with human intuition, whereas some methods cannot rank crisp numbers which are a special case of fuzzy numbers.In this paper, a new method is proposed which is based on Circumcenterr to rank fuzzy quantities. In a trapezoidal fuzzy number, first the trapezoid is split into three parts where the first, second, and third parts are a triangle, a rectangle, and a triangle respectively. Then the Centroids of these three parts are calculated followed by the calculation of the Circumcenterr of these Centroids. Finally, a ranking function is defined which is the Euclidean distance between the Circumcenterr point and the original point to rank fuzzy numbers. Most of the ranking procedures proposed in the literature use Centroid of trapezoid as reference point, as the Centroid is a balancing point of the trapezoid. But the Circumcenterr of Centroids can be considered a much more balancing point as this point is equidistant from all the vertices which are Centroids. Further, this method uses an index of optimism to reflect the decision maker’s optimistic attitude and also uses an index of modality that represent the neutrality of the decision maker.The work is organized as follows. Section2 briefly introduces the basic concepts and definitions of fuzzy numbers. Section 3 presents the proposed new method. In Section 4, the proposed method has been explained with examples which describe the advantages and the efficiency of the method which ranks generalized fuzzy numbers, images of fuzzy numbers, and even crisp numbers. In Section 5, the method demonstrates its robustness by comparing with other methods like Liou and Wang, Yager, and others where the methods cannot discriminate fuzzy quantities and do not agree with human intuition. Finally, the conclusions of the work are presented in Section 6.


## 2. Preliminaries

Definition 1 (Fuzzy Numbers and Membership Functions).
A fuzzy numberÃ is a fuzzy subset in support ℝ (real number) which is both “normal” and “convex” with membership function
(1)fÃ(x)={fÃL(x),a≤x≤b,w,b≤x≤c,fÃR(x),c≤x≤d,0,otherwise,
where 0<w≤1 is a constant, a, b, c, d are real numbers, and fÃL:[a,b]→[0,w], fÃR:[c,d]→[0,w] are two strictly monotonic and continuous functions from ℝ to the closed interval [0,w]. It is customary to write a fuzzy number as Ã=(a,b,c,d;w). If w=1, then Ã=(a,b,c,d;1) is a normalized fuzzy number, otherwise Ã is said to be a generalized or non-normal fuzzy number.
If the membership functionfÃ(x) is piecewise linear, then Ã is said to be a trapezoidal fuzzy number. The membership function of a trapezoidal fuzzy number is given by
(2)fÃ(x)={w(x-a)b-a,a≤x≤b,w,b≤x≤c,w(x-d)c-d,c≤x≤d,0,otherwise.
If w=1, then Ã=(a,b,c,d;1) is a normalized trapezoidal fuzzy number and Ã is a generalized or nonnormal trapezoidal fuzzy number if 0<w<1. The image of Ã=(a,b,c,d;w) is given by -Ã=(-d,-c,-b,-a;w).
As a particular case ifb=c, the trapezoidal fuzzy number reduces to a triangular fuzzy number given by Ã=(a,b,d;w). The value of “b” corresponds with the mode or core and [a,d] with the support. If w=1, thenÃ=(a,b,d) is a normalized triangular fuzzy number Ã is a generalized or nonnormal triangular fuzzy number if 0<w<1.
AsfÃL(x):[a,b]→[0,w] and fÃR(x):[c,d]→[0,w] are strictly monotonic and continuous functions, their inverse functions gÃL(y):[0,w]→[a,b] and gÃR(y):[0,w]→[c,d] are also continuous and strictly monotonic. Hence gÃL(y) and gÃR(y) are integrable on [0,w].Definition 2 (Liou and Wang’s Ranking Method).
Liou and Wang [24] ranked fuzzy numbers with total integral value. For a fuzzy number defined by Definition 1, the total integral value is defined as ITα(Ã)=αIR(Ã)+(1-α)IL(Ã) where IR(Ã)=∫01gÃR(y)dy and IL(Ã)=∫01gÃL(y)dy are the right and left integral values of Ã, respectively, and α∈[0,1]is the index of optimism which represents the degree of optimism of a decision maker. If α=0, the total integral value represents a pessimistic decision maker’s view point which is equal to left integral value. If α=1, the total integral value represents an optimistic decision maker’s view point and is equal to the right integral value and when α=0.5, the total integral value represents an moderate decision maker’s view point and is equal to the mean of right and left integral values. For a decision maker, the larger the value of α is, the higher is the degree of optimism.Definition 3 (Garcia and Lamata’s Ranking Method).
Garcia and Lamata [46] modified the index of Liou and Wang [24] for ranking fuzzy numbers. This method use an index of optimism to reflect the decision maker’s optimistic attitude, which is not enough to discriminate fuzzy numbers, but rather it also uses an index of modality that represents the neutrality of the decision maker. For a fuzzy number defined by Definition 1, Garcia and Lamata [46] proposed an index associated with the ranking as the convex combination Iβ,α(Ã)=βSM(Ã)+(1-β)ITα(Ã), where SM(Ã) is the area of the core of the fuzzy number which is equal to “b” for a triangular fuzzy number defined by Ã=(a,b,d;w) and the average value of the plateau in case of a trapezoidal fuzzy number given byÃ=(a,b,c,d;w),β∈[0,1] is the index of modality that represents the importance of central value against the extreme values, α∈[0,1] is the degree of optimism of the decision maker, andITα(Ã) has its own meaning as defined in Definition 2.


## 3. Proposed Method

Definition 4.
The Centroid of a trapezoid is considered as the balancing point of the trapezoid (Figure1). Divide the trapezoid into three plane figures. These three plane figures are a triangle (APB), a rectangle (BPQC), and again a triangle (CQD), respectively. The Circumcenterr of the Centroids of these three plane figures is taken as the point of reference to define the ranking of generalized fuzzy numbers. The reason for selecting this point as a point of reference is that each Centroid point (G1 of triangle APB, G2of rectangle BPQC, and G3 of triangle CQD) are balancing points of each individual plane figure, and the Circumcenterr of these Centroid points is equidistant from each vertex (which are Centroids). Therefore, this point would be a better reference point than the Centroid point of the trapezoid.
Consider a generalized trapezoidal fuzzy numberÃ = (a,b,c,d;w) (Figure 1).
The Centroids of the three plane figures areG1=((a+2b)/3,w/3), G2=((b+c)/2,w/2), and G3=((2c+d)/3,w/3), respectively. Equation of the line G1G3⃡ is y=w/3 and G2 does not lie on the line G1G3⃡. Therefore, G1,G2 and G3 are non-collinear and they form a triangle.
We define the CircumcenterrSÃ(x0¯,y0¯) of the triangle with vertices G1, G2 and G3 of the generalized trapezoidal fuzzy number Ã=(a,b,c,d;w) as
(3)SÃ(x0¯,y0¯)=(a+2b+2c+d6,(2a+b-3c)(2d+c-3b)+5w212w).
As a special case, for triangular fuzzy number Ã=(a,b,d;w), that is, c=b the Circumcenterr of Centroids is given by
(4)SÃ(x0¯,y0¯)=(a+4b+d6,4(a-b)(d-b)+5w212w).Figure 1
Circumcenterr of Centroids.Definition 5.
For a generalized trapezoidal fuzzy numberÃ = (a,b,c,d;w), with Circumcenterr of Centroids SÃ(x0¯,y0¯)defined by (3), we define the index associated with the ranking as Iα(Ã)=αy0¯+(1-α)x0¯ where α∈[0,1] is the index of optimism which represents the degree of optimism of a decision maker. If α=0, we have a pessimistic decision maker’s view point which is equal to the distance of the Circumcenterr from y-axis. If α=1, we have an optimistic decision maker’s view point and is equal to the distance of the Circumcenterr from x-axis, and when α=0.5, we have the moderate decision maker’s view point and is equal to the mean of the distances of Circumcenterr from y and x axes. The larger the value of α is, the higher the degree of optimism of the decision maker. The index of optimism is  not alone sufficient to discriminate fuzzy numbers as this uses only the extreme values of the Circumcenterr of Centroids. Hence, we upgrade this by using an index of modality which represents the importance of central value along with  index of optimism.Definition 6.
For a generalized trapezoidal fuzzy numberÃ=(a,b,c,d;w), with Circumcenterr of Centroids SÃ(x0¯,y0¯) defined by (3), we define the index associated with the ranking as Iα,β(Ã)=β((x0¯+y0¯)/2)+(1-β)Iα(Ã) where β∈[0,1] is the index of modality which represents the importance of central value against the extreme values x0¯ and y0¯  and Iα(Ã) is the one which is defined in Definition 5. Here, β represents the weight of the central value and (1-β) is the weight associated with the extreme values x0¯ and y0¯.Definition 7.
For any decision maker whether pessimistic (α=0), optimistic (α=1), or neutral (α=0.5), the ranking function of the trapezoidal fuzzy number Ã=(a,b,c,d;w) which maps the set of all fuzzy numbers to a set of real numbers is defined as R(Ã)=x0¯2+y0¯2 which is the Euclidean distance from the Circumcenterr of the Centroids as defined in Definition 4 and the original point.
Using the above definitions we define  ranking between fuzzy numbers as follows
LetÃ and Ãj two fuzzy numbers, then(i)
ifR(Ãi)>R(Ãj), then Ãi>Ãj,(ii)
ifR(Ãi)<R(Ãj), then Ãi<Ãj,(iii)
ifR(Ãi)=R(Ãj) then in this case the discrimination of fuzzy numbers is not possible. In such cases we use Definition 6 to rank fuzzy numbers as Definition 5 alone is not sufficient to discriminate in all cases, that is, ifIα,β(Ãi)>Iα,β(Ãj), thenÃi>Ãj, and ifIα,β(Ãi)<Iα,β(Ãj) thenÃi<Ãj.


## 4. Examples

Example 8.
LetÃ=(0.1,0.3,0.3,0.5;1), B̃=(0.2,0.3,0.3,0.4;1), and C̃=(1,1,1,1;1).
Then,SÃ(x0¯,y0¯)=(0.3,0.4033), SB̃(x0¯,y0¯)=(0.3,0.4133), and SC̃(x0¯,y0¯)=(1,0.4166).
Therefore,R(Ã)=0.5026, R(B̃)=0.5107,andR(C̃)=1.0833⇒Ã<B̃<C̃.
It is observed that the above ranking order is unaltered even by using the index of modality proposed  in Definition6 whoever might be the decision maker.Example 9.
LetÃ=(0.1,0.3,0.3,0.5;1), B̃=(-0.5,-0.3,-0.3,-0.1;1).
Then,SÃ(x0¯,y0¯)=(0.3,0.4033)and SB̃(x0¯,y0¯)=(-0.3,0.4033).
Therefore,R(Ã)=0.5026, R(B̃)=0.5026, and discrimination of fuzzy numbers is not possible by using Definition 7.
Now, by  using Definition6, we have the following.  (i)
For a pessimistic decision make,(5)I0,β(Ã)=0.35165β+(1-β)(0.3),I0,β(B̃)=0.05165β+(1-β)(-0.3).
As 0.35165β+(1-β)(0.3)>0.05165β+(1-β)(-0.3)⇒Ã>B̃(ii)
For a optimistic decision maker,(6)I1,β(Ã)=0.35165β+(1-β)(0.4033),I1,β(B̃)=0.05165β+(1-β)(0.4033).
As 0.35165β+(1-β)(0.4033)>0.05165β+(1-β)(0.4033)⇒Ã>B̃(iii)
For a neutral decision maker,(7)I0.5,β(Ã)=0.35165β+(1-β)(0.35165),I0.5,β(B̃)=0.05165β+(1-β)(0.05165).
As 0.35165β+(1-β)(0.35165)>0.05165β+(1-β)(0.05165)⇒Ã>B̃.Thus, we see that the ranking order is same in all the three cases.Example 10.
LetÃ=(0.1,0.3,0.3,0.5;1)and B̃=(-0.5,-0.3,-0.3,-0.1;1).
Then,-Ã=(-0.5,-0.3,-0.3,-0.1;1) and-Ã=(0.1,0.3,0.3,0.5;1),S-Ã(x0¯,y0¯)=(-0.3,0.4033) and S-B̃(x0¯,y0¯)=(0.3,0.4033).
ThereforeR(-Ã)=0.5026, R(-B̃)=0.5026, and discrimination of fuzzy numbers is not possible by using Definition 7.
Now, by  using Definition6 we have the following.(i)
For an pessimistic decision maker,(8)I0,β(-Ã)=0.05165β+(1-β)(-0.3),I0,β(-B̃)=0.35165β+(1-β)(0.3).
As 0.35165β+(1-β)(0.3)>0.05165β+(1-β)(-0.3)⇒-Ã<-B̃(ii)
For an optimistic decision maker,(9)I1,β(-Ã)=0.05165β+(1-β)(0.4033),I1,β(-B̃)=0.35165β+(1-β)(0.4033).
As 0.35165β+(1-β)(0.4033)>0.05165β+(1-β)(0.4033)⇒-Ã<-B̃(iii)
For a neutral decision maker,(10)I0.5,β(-Ã)=0.05165β+(1-β)(0.05165),I0.5,β(-B̃)=0.35165β+(1-β)(0.35165).
As 0.35165β+(1-β)(0.35165)>0.05165β+(1-β)(0.05165)⇒-Ã<-B̃.Thus, we see that the ranking order is same in all the three cases.
From Examples9 and 10, we see that Ã>B̃⇒-Ã<-B̃.Example 11.
LetÃ=(0.1,0.3,0.3,0.5;0.8), B̃=(0.1,0.3,0.3,0.5;1).
Then,SÃ(x0¯,y0¯)=(0.3,0.2533) and SB̃(x0¯,y0¯)=(0.3,0.4033).
Therefore,R(Ã)=0.3926 and R(B̃)=0.5026⇒Ã<B̃.
It is observed that the above ranking order is unaltered even by using the index of modality proposed in Definition6 irrespective of  decision maker.


## 5. Comparative Study

Example 12.
Consider two fuzzy numbersA=(1,4,5)  and  B=(2,3,6).
By Liou and Wang method [24], it is clear that the two fuzzy numbers are equal for all the decision maker’s  as ITα(A)=4.5α+(1-α)2.5 and ITα(B)=4.5α+(1-α)2.5 which is not even true by intuition. By using our method we have
(11)S(A)=(3.6666,-0.5833),S(B)=(3.3333,-0.5833),R(A)=3.7127,R(B)=3.3839⟹A>B.
It is observed that the above ranking order is unaltered even by using the index of    modality proposed in Definition 6 from the decision maker’s view point.Example 13.
Consider four fuzzy numbersA1=(0.1,0.2,0.3;1), A2=(0.2,0.5,0.8;1), A3=(0.3,0.4,0.9;1), A4=(0.6,0.7,0.8;1) which were ranked earlier by Yager [10], Fortemps and Roubens [27], Liou and Wang [24], and Chen and Lu [37] as shown in Table 1.
It can be seen from Table1 that none of the methods discriminates fuzzy numbers. Yager [10] and Fortemps and Roubens  [27] methods failed to discriminate the fuzzy numbers A2 and A3, whereas the methods of  Liou and Wang [24] and Chen and Lu [37] cannot discriminate the fuzzy numbers A2, A3 and A1, A4.
By using our method, we have(12)S(A1)=(0.2,0.4133),S(A2)=(0.5,0.3866),S(A3)=(0.4666,0.4),S(A4)=(0.7,0.4133);R(A1)=0.4591,R(A2)=0.6320,R(A3)=0.6146,R(A4)=0.8129⟹A4>A2>A3>A1.
It is observed that the above ranking order is unaltered even by using the index of modality proposed in Definition 6 from the decision maker’s view point.Table 1
Comparison of various ranking methods.
MethodA1A2A3A4Ranking orderYager0.200.500.500.70A4>A2=A3>A1Fortemps and Roubens0.200.500.500.70A4>A2=A3>A1Liou and Wangα=10.250.650.650.75A4>A2=A3>A1α=0.50.200.500.500.70A4>A2=A3>A1α=00.150.350.350.65A4>A2=A3>A1Chen and Lu [37]β=1−0.200.000.00−0.20A2=A3>A1=A4β=0.5−0.200.000.00−0.20A2=A3>A1=A4β=0−0.200.000.00−0.20A2=A3>A1=A4Example 14.
LetÃ=(0.1,0.2,0.4,0.5;1) and B̃=(1,1,1,1;1). Cheng [28] proposed a ranking function which is the distance from centroid point and the original point where as  Chu and Tsao [39] proposed a ranking function which is the area between the centroid point and original point. Their centroid formulae are given by
(13)(w(d2-2c2+2b2-a2+dc-ab)+3(c2-b2)3w(d-c+b-a)+6(c-b),w3(1+(b+c)-(a+d)(1-w)(b+c-a-d)+2(a+d)w)w(d2-2c2+2b2-a2+dc-ab)+3(c2-b2)3w(d-c+b-a)+6(c-b)),(w(d2-2c2+2b2-a2+dc-ab)+3(c2-b2)3w(d-c+b-a)+6(c-b),w3(1+b+ca+b+c+d)w(d2-2c2+2b2-a2+dc-ab)+3(c2-b2)3w(d-c+b-a)+6(c-b)).
Both these Centroid formulae cannot rank crisp numbers which are a special case of fuzzy numbers as it can be seen from the above formulae that the denominator in the first coordinate of their Centroid formulae is zero, and hence Centroid of crisp numbers are undefined for their formulae. By using our method, we have
(14)SÃ(x0¯,y0¯)=(0.3,0.3633),SB̃(x0¯,y0¯)=(1,0.4166).
Therefore, R(Ã)=0.4711 and R(B̃)=1.0833⇒Ã<B̃.
It is observed that the above ranking order is unaltered even by using the index of modality proposed in Definition6 irrespective of decision maker choice.


## 6. Conclusions

This paper proposes a method that ranks fuzzy numbers which is simple and concrete. This method ranks trapezoidal  as well as triangular fuzzy numbers and their images. This method also ranks crisp numbers which are special case of fuzzy numbers whereas methods proposed by Cheng and Chu cannot rank crisp numbers as their centroid formulae are undefined for crisp numbers. This method uses an index of modality which represents the importance of central value against the extreme values, beside the decision maker’s degree of optimism.  This method which is simple in calculation not only gives satisfactory results to well-defined problems, but  also gives a correct ranking order to problems, whereas Yager index, Fortemps and Roubens,  Liou and Wang, and Chen and Lu indexes failed to discriminate fuzzy numbers, and this method also agrees with human intuition.



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