# Revisiting the Interval and Fuzzy TOPSIS Methods: Is Euclidean Distance a Suitable Tool to Measure the Differences between Fuzzy Numbers?

Authors: Hosein Arman; Abdollah Hadi-Vencheh; Reza Kiani Mavi; Mehdi Khodadadipour; Ali Jamshidi
Journal: Complexity
 (2022)

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## Abstract

Euclidean distance (ED) calculates the distance betweenn-coordinate points that n equals the dimension of the space these points are located. Some studies extended its application to measure the difference between fuzzy numbers (FNs).This study shows that this extension is not logical because although an n-coordinate point and an FN are denoted the same, they are conceptually different. An FN is defined by n components; however, n is not equal to the dimension of the space where the FN is located. This study illustrates this misapplication and shows that the ED between FNs does not necessarily reflect their difference. We also revisit triangular and trapezoidal fuzzy TOPSIS methods to avoid this misapplication. For this purpose, we first defuzzify the FNs using the center of gravity (COG) method and then apply the ED to measure the difference between crisp values. We use an example to illustrate that the existing fuzzy TOPSIS methods assign inaccurate weights to alternatives and may even rank them incorrectly.


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## Body

## 1. Introduction

The Euclidean distance (ED) measures the distance between two points in ann-dimensional space. The ED is calculated based on the Pythagorean theorem expresses that the square of the straight line distance between two points in an n-dimensional space is equal to the sum of the squares of the differences between their components. Some studies extend ED application to measure the difference between fuzzy numbers (FNs). For example, Chen [1] extends the ED to measure the difference between triangular FNs (TFNs). Also, Chen et al. [2]; Wan and Li [3]; and Seiti and Hafezalkotob [4] use the ED to calculate the difference between trapezoidal FNs (TrFNs). Some researchers like Yue [5] use the ED formula to measure the difference between interval values. An interval value, also called a gray number (GN), can be considered an FN in which all values in the given interval have a membership degree of 1.One of the most well-known techniques extended for fuzzy environments based on ED is the technique for order preferences by similarity to the ideal solution (TOPSIS). TOPSIS is a multi-attribute decision-making (MADM) technique proposed by Hwang and Yoon [6]. This method ranks the alternatives based on their EDs from positive- and negative-ideal solutions (PIS and NIS). TOPSIS has been extended for fuzzy environments. Different fuzzy TOPSIS (FTOPSIS) methods have been developed for type-1 fuzzy sets. By type-1 fuzzy sets, we mean the fuzzy sets proposed by Zadeh [7] for the first time.Type-1 FTOPSIS methods usually define the fuzzy PIS (FPIS) and fuzzy NIS (FNIS), and rank the alternatives based on their distances from the FPIS and FNIS. The most common distance used in the literature for this purpose is the ED (See the FTOPSIS methods proposed by Chen [1]; Chen et al. [2]; Mokhtarian and Hadi-Vencheh [8]; Huang and Peng [9]; Buyukozkan and Cifci [10]; Gok [11]; Wang et al. [12]; Baykasoglu and Golcuk [13]; and Seiti and Hafezalkotob [4]. Most of these methods have been developed for TFNs, although some researchers like Chen et al. [2] and Seiti and Hafezalkotob [4] developed FTOPSIS methods for TrFNs.FTOPSIS methods have also been developed for other types of fuzzy sets; some methods extended the ED for measuring the differences between FNs. For example, Li et al. [14] and Chen and Hong [15] developed TOPSIS for intuitionistic and interval type-2 FNs, respectively. Ye and Li [16] proposed an extended FTOPSIS method by utilizing the possibility theory. Yu et al. [17] extended TOPSIS under the interval-valued Pythagorean fuzzy environment. Mathew et al. [18] calculated the ED between alternatives and the spherical fuzzy positive and negative ideal solutions.This study shows that extending the ED to measure the differences between FNs, including TFNs and TrFNs, suffer from ED misapplication. This misapplication causes the difference between two FNs to be incorrectly measured. In fact, the values obtained from the ED do not necessarily represent the real differences between FNs. In other words, extending the ED to measure the difference between FNs is not logical because although ann-coordinate point and an FN are denoted the same, they are conceptually different. As a result, the techniques extending the ED to measure the differences between FNs also suffer from the same misapplication. This misapplication, in turn, causes a computational error; therefore, the results of these techniques are unreliable.The existing FTOPSIS methods usually misapply the ED to measure the differences between FNs; therefore, they may assign the wrong weights to alternatives. To avoid this misapplication, we suggest that instead of using the ED formula, the difference between two FNs is considered equal to the difference between their centroids. To show the application of this suggestion, we present a gray TOPSIS (GTOPSIS) method that considers the difference between the centroids of GNs as their difference. We also revisit the triangular and trapezoidal FTOPSIS methods. In the revisited methods, the differences between FNs are considered equal to the differences between their centroids.The rest of this paper is organized as follows: Section2 illustrates the misapplication of using the ED to measure the difference between FNs. Section 3 presents a GTOPSIS method and revisits the FTOPSIS methods for TFNs and TrFNs. Section 4 provides a numerical example to compare the results of a classical FTOPSIS with its revised version proposed in this study. Section 5 gives the conclusion.


## 2. The ED Misapplication

This section is divided into four subsections. Subsection2.1 reviews the formulas obtained by extending the ED to measure the differences between FNs. Subsection 2.2 analyzes the ED. Subsection 2.3 discusses and illustrates the misapplication of ED to measure the difference between FNs. Subsection 2.4 obtains the differences between FNs using two different approaches: using the ED and calculating the difference between FNs based on their centroids. This subsection compares the results of these methods and uses some numerical examples to illustrate how the ED leads to a wrong difference between FNs.


### 2.1. Measuring the Differences between FNs Using the ED

The ED is defined below.Definition 1.
LetX1 × X2 × , … × Xn be a universal set of an n-coordinate system, and P1=x11,…,x1n and P2=x21,…,x2n be two points of ℜn. The ED between these points, dP1,P2, is obtained as follows:(1)dP1,P2=∑i=1nx1i−x2i21/2.
The ED has been used to measure the differences between GNs. To review this application, letA=a1,a2 and B=b1,b2 be two GNs. Yue [5] computes the difference between A and B as follows:(2)dA,B=a1−b12+a2−b22.
Some studies use the ED to measure the differences between FNs. For example, Chen [1] extends the ED to measure the difference between two TFNs, denoted as K=k1,k2,k3 and L=l1,l2,l3, as follows:(3)dK,L=13k1−l12+k2−l22+k3−l32.
Chen et al. [2] use the ED to calculate the difference between two TrFNs, M=m1,m2,m3,m4 and N=n1,n2,n3,n4, as follows:(4)dM,N=14m1−n12+m2−n22+m3−n32+m4−n42.
Note that other formulas in the literature have been developed based on the ED formula to measure the difference between FNs. For example, Wan and Li [3] and Seiti and Hafezalkotob [4] measure the difference between TrFNs M and N as follows:(5)dM,N=16m1−n12+2m2−n22+2m3−n32+m4−n42.


### 2.2. The ED Properties

The ED calculates the distance between twon-coordinate points located in an n-dimensional space. For this purpose, first, it calculates n distinct distances between the same dimensions of given points and then combines these distances using equation (1). For example, let X × Y be a universal set of a two-coordinate system, and A1=x1,y1 and A2=x2,y2 be two ordered points. The differences between x-values and y-values of these points, called the horizontal and vertical distances, are obtained as x1−x2 and y1−y2, respectively. The ED between points P1 and P2 is calculated by combining their horizontal and vertical distances as follows:(6)dA1,A2=x1−x22+y1−y221/2.The dashed lines in Figure 1(a) show the horizontal and vertical distances between pointsA1 and A2, and the solid line indicates the ED between them. Equation (6) uses the Pythagorean theorem to calculate the ED between two points in a two-coordinate system based on their horizontal and vertical distances. According to this theorem, the square of the hypotenuse in a right triangle is equal to the sum of the squares of the other two sides.Figure 1
Mutual interchange property of ED.
(a)(b)ED has been used for different purposes, including calculating the distance between two points in ann-dimensional space and measuring the difference between two FNs. However, using the ED for some purposes may come with some flaws. We provide a definition below to determine whether the ED is an appropriate tool for the given purpose.Definition 2.
(The mutual interchange property). LetX1 × X2 ×, …, × Xn be a universal set of an n-coordinate system and P1=x11,…,x1n and P2=x21,…,x2n be two points in this system. By mutually interchanging the values of the same dimension i for these points, i.e., the values of x1i and x2i, for one or more i indexes, the new points P3 and P4 are created. If the real distance between the new points P3 and P4 is equal to the real distance between P1 and P2, we can use the ED for the given purpose.
When the values of the same dimensions are mutually interchanged for one or more dimensions, equation (1) always calculates the same ED. However, the real distance between the new points may change. If the real distance between the new points is changed (not changed), the ED is inconsistent (consistent) with the real situation and cannot (can) be used for the given purpose. For example, by mutually interchanging the x-values of points A1=x1,y1 and A2=x2,y2 in Figure 1(a), two new order points are created as A3=x2,y1 and A4=x1,y2 shown in Figure 1(b). The ED between A3 and A4 is calculated as follows:(7)dA3,A4=x2−x12+y1−y221/2.
Equations (6) and (7) indicate that although A3 and A4 are two different points from A1 and A2, the ED between A3 and A4 equals the ED between A1 and A2. This conclusion is consistent with the real distances between the points. It implies that calculating the distance between points in a two-coordinate system comes with the mutual interchange property. Therefore, the ED is an appropriate tool for this purpose.


### 2.3. Misapplying the ED to Measure the Difference between FNs

This section argues that the ED is not an appropriate tool to measure the difference between FNs. In particular, we show that the misapplications of equations (2)–(5) extended based on the ED to measure the differences between GNs, TFNs, and TrFNs. For this purpose, we show that these equations do not come with the mutual interchange property; therefore, they should not be used to measure the differences between FNs. For example, by mutually interchanging the lower bounds of two GNs A=a1,a2 and B=b1,b2, two new GNs are created as C=b1,a2 and D=a1,b2, provided that a1,b1≤a2,b2. The real difference between new GNs C and D is not necessarily equal to the difference between GNs A and B(See Example 1). However, equation (2) calculates equal differences between the GNs for both cases. This implies the ED is inconsistent with measuring the difference between GNs. This conclusion can be generated for FNs. We present Examples 2 and 3 to illustrate this inconsistency for TFNs and TrFNs, respectively.A question arises: why is ED consistent with measuring the distance between twon-coordinate points located in an n-dimensional space but inconsistent with measuring the difference between two FNs? To answer this question, consider that n naturally different components characterize an n-coordinate point; each component is measured based on a different dimension. For example, let A = (10, 20, 30) be a point in a three-dimensional space. Three naturally different components characterize this point: its length, width, and height; the values of these components are 10, 20, and 30, respectively. The number of components characterizing a point equals the number of dimensions of the space in which the point is located. As a result, the ED between two points in an n-dimensional space is calculated based on their differences obtained for the same dimensions based on the Pythagorean theorem.However, the number of components characterizing an FN is not equal to the number of dimensions of the space in which it is located. For example, the GN, TFN, and TrFN characterized with 2, 3, and 4 components, respectively, are located in a one-dimensional space, not in two-, three-, and four-dimensional spaces. In other words, an FN is characterized byn components of the same nature; all of them are measured based on only the same dimension. For example, let B = (10, 20, and 30) be a TFN representing the fuzzy set of young people. Although B is characterized by three components, i.e., 10, 20, and 30, they do not correspond to three different dimensions; these components have the same nature, i.e., age, and can be measured using only one dimension.Both the three-coordinate pointA and the TFN B are identified as (10, 20, 30). However, they are entirely conceptually different and should not be treated the same. Point A is a three-coordinate point located in a three-dimensional space, while B is a TFN located in a one-dimensional space. This clearly shows that although A and B look similar, they are conceptually quite different. Therefore, we cannot simply generalize operations that are inherently appropriate for points located in three-dimensional space to TFNs.We can also criticize using the ED to measure the difference between FNs from the extension principle perspective. According to this principle, each fuzzy relation has been generally developed based on a crisp relation. However, the ED between FNs has been extended improperly based on the crisp ED. In other words, the ED in a crisp environment measures the distance between twon-coordinate points; each coordinate corresponds to a unique dimension. In contrast, the ED in a fuzzy environment measures the difference between two n-component FNs; all these components together correspond to only one dimension. It is clear that a point in an n-dimensional space is conceptually completely different from an FN denoted with n components.A point in ann-dimensional space contains n heterogeneous components. For example, the three components of a point in a three-dimensional space are length, width, and height, representing three different characteristics. In contrast, an FN contains n homogenous components. For example, the three components of a TFN are the lower, middle, and upper values of the same variable; these components together represent the same characteristic. A question arises: on what logic has the ED been extended to measure the distance between one-coordinate FNs? The only answer to this question is the similarity between denoting a point in an n-dimensional space and an FN. For example, both a point in a three-dimensional space and a TFN are denoted as (a ,b, c). Despite this similarity, they are entirely different; therefore, the operations proposed for one of them cannot be extended to the other simply.


### 2.4. Illustrating the ED Misapplication to Measure the Difference between FNs

The previous section concluded that using the ED to measure the difference between two FNs in a one-dimensional space is meaningless. Therefore, other approaches should be used to measure the difference between FNs. One of the approaches used for this purpose is to calculate the difference between the centroids of FNs. In the following, we first review the center of gravity (COG) method used to obtain the centroid of an FN. Then, we present a theorem to show that calculating the difference between two intervals using the ED and COG methods leads to different results. It is to be noted that the proposed theorem can be extended to FNs, including TFNs and TrFNs. Next, some numerical examples are given to illustrate the ED misapplication. To show this misapplication, we compare the results of two methods between FNs: the ED and the COG methods.Remark 1.
Yager [19] proposed the COG method to obtain the centroid of the FN A˜ with the membership function of μA˜x as follows:(8)CA˜=∫−∞+∞x.μA˜x.dx∫−∞+∞μA˜x.dx,where CA˜ is the centroid of A˜. Equation (8) can be used to find the centroids of different types of FNs. For example, consider Figures 2(a)–2(c) representing the normal TFN A˜=l,m,u, GN B˜=l,u, and TrFN C˜=l,m,m′,u, respectively.
Arman et al. [20] obtained the centroids of A˜, B˜, and C˜ using equation (8) as follows:(9)CA˜=l+m+u3,(10)CB˜=l+u2,(11)CC˜=13l+m+m′+u−m′×u−l×mm′+u−l+m.Figure 2
Membership function of three common FNs.
(a)(b)(c)Theorem 1.
LetA=a1,a2 and B=b1,b2 be two distinct GNs. Then, the difference between A and B using ED and COG leads to different results.Proof.
The centroids of GNsA and B using equation (10) are CA=a1+a2/2 and CB=b1+b2/2, respectively. By contrast, assume that the difference between A and B using the ED and COG methods is the same. Therefore a1−b12+a2−b22=a1+a2/2−b1+b2/2⇒a1−a2=b1−b2.
We know that although two new GNs are created by mutually interchanging the values of the same dimension of GNsA and B, the ED between new points is not changed. By mutually interchanging the lower bounds of A and B, two new GNs are created as C=b1,a2 and D=a1,b2, provided that a1,b1≤a2,b2. The centroids of GNs C and D using equation (10) are CC=b1+a2/2 and CD=a1+b2/2, respectively. Assume that the difference between C and D using the ED and COG methods leads to the same. Thus,(12)b1−a12+a2−b22=b1+a22−a1+b22⇒a1+a2=b1+b2.
Sincea1−a2=b1−b2 and a1+a2=b1+b2. So(13)a1−a2+a1+a2=b1−b2+b1+b2⇒a1=b1a1−a2−a1+a2=b1−b2−b1+b2⇒a2=b2.
This meansA = B, and this is a contradiction. Now, the proof is completed.Example 1.
LetA1 = [10, 90] and B1 = [80, 120] be two GNs. By interchanging the first components of these GNs, the new GNs A2 = [80, 90] and B2 = [10, 120] are created. The ED between GNs A2 and B2 is 76.16 using equation (2), exactly equal to that of GNs A1 and B1. On the other hand, the centroids of intervals A1, B1, A2, and B2 are obtained at 50, 100, 85, and 65, respectively, using equation (10). Therefore, the absolute difference between the centroids of A1 and B1 is CA1−CB1=50, while the absolute difference between the centroids of A2 and B2 is CA2−CB2=20. It implies that equation (2) is inconsistent with measuring the difference between GNs.Example 2.
Assume thatK1 = (1, 3, 5) and L1 = (2, 4, 6) are two TFNs (Figure 3(a)). The ED between TFNs K1 and L1 is 1 using equation (3). L1 is greater than K1 because all components of L1 are greater than their corresponding components of K1. This is confirmed by calculating the difference between the centroids of these TFNs. The centroids of K1 and L1 are 3 and 4, respectively, using equation (9); therefore, the difference between them is CK1−CL1=−1.
By mutually interchanging the third components ofK1 and L1, two new TFNs, K2 = (1, 3, 6) and L2 = (2, 4, 5), are created (Figure 3(b)). Compared to K1 and L1, the values of the first and second components of K2 and L2 have not changed; but the values of their third components have increased and decreased, respectively. Therefore, the FNs K2 and L2 are expected to be closer to each other compared to the FNs between K1 and L1. However, the ED between K2 and L2 is 1 using equation (3), exactly equal to the ED between K1 and L1. On the other hand, the centroids of K2 and L2 are 3.33 and 3.66, respectively, using equation (9); thus, the absolute difference between their centroids is CK2−CL2=0.33 that is smaller than CK1−CL1=1. It indicates that measuring the difference between TFNs based on their centroids is consistent with our expectation. This example shows that when the same elements of two TFNs are mutually interchanged, the real difference between new TFNs may change. However, equation (3) cannot discover this change and computes the same ED. As a result, the ED is not an appropriate tool for measuring the difference between two TFNs.Figure 3
Comparing TFNs when interchanging their third components.
(a)(b)Example 3.
Using the numbers 1 to 8, eight different two-member sets can be made, provided that the members in each set are TrFNs without using a number twice in each set. In other words, there are only eight separate sets asAi=a,b,c,d,e,f,g,h, i = 1, …, 8, provided that the numbers 1 to 8 appear only once in each set, and a<b<c<d and e<f<g<h (See Table 1).
The ED between two TrFNs is calculated equal to 1 using equations (4) and/or (5) for all sets. It means that the ED measures the difference between two TrFNs in each set equal to 1. However, it does not reflect reality. To prove that, we obtain the centroids of TrFNs using equation (11) and then calculate the difference between the centroids of TrFNs in each set. The results are given in Table 1. This table clearly shows that the differences between the centroids better reflect the real differences between TrFNs.Table 1
Comparing TrFNs using the ED and the COG methods.
SetTrFNsEuclidean distanceCentroidC(M) − C(N)TrFNMTrFNNC(M)C(N)1(1, 3, 5, 7)(2, 4, 6, 8)145−12(1, 3, 5, 8)(2, 4, 6, 7)14.2964.714−0.4183(1, 3, 6, 7)(2, 4, 5, 8)14.2224.810−0.5884(1, 4, 5, 7)(2, 3, 6, 8)14.194.778−0.5885(2, 3, 5, 7)(1, 4, 6, 8)14.2864.704−0.4186(2, 3, 5, 8)(1, 4, 6, 7)14.5834.4170.1667(2, 3, 6, 7)(1, 4, 5, 8)14.54.508(2, 4, 5, 7)(1, 3, 6, 8)14.54.50


## 2.1. Measuring the Differences between FNs Using the ED

The ED is defined below.Definition 1.
LetX1 × X2 × , … × Xn be a universal set of an n-coordinate system, and P1=x11,…,x1n and P2=x21,…,x2n be two points of ℜn. The ED between these points, dP1,P2, is obtained as follows:(1)dP1,P2=∑i=1nx1i−x2i21/2.
The ED has been used to measure the differences between GNs. To review this application, letA=a1,a2 and B=b1,b2 be two GNs. Yue [5] computes the difference between A and B as follows:(2)dA,B=a1−b12+a2−b22.
Some studies use the ED to measure the differences between FNs. For example, Chen [1] extends the ED to measure the difference between two TFNs, denoted as K=k1,k2,k3 and L=l1,l2,l3, as follows:(3)dK,L=13k1−l12+k2−l22+k3−l32.
Chen et al. [2] use the ED to calculate the difference between two TrFNs, M=m1,m2,m3,m4 and N=n1,n2,n3,n4, as follows:(4)dM,N=14m1−n12+m2−n22+m3−n32+m4−n42.
Note that other formulas in the literature have been developed based on the ED formula to measure the difference between FNs. For example, Wan and Li [3] and Seiti and Hafezalkotob [4] measure the difference between TrFNs M and N as follows:(5)dM,N=16m1−n12+2m2−n22+2m3−n32+m4−n42.


## 2.2. The ED Properties

The ED calculates the distance between twon-coordinate points located in an n-dimensional space. For this purpose, first, it calculates n distinct distances between the same dimensions of given points and then combines these distances using equation (1). For example, let X × Y be a universal set of a two-coordinate system, and A1=x1,y1 and A2=x2,y2 be two ordered points. The differences between x-values and y-values of these points, called the horizontal and vertical distances, are obtained as x1−x2 and y1−y2, respectively. The ED between points P1 and P2 is calculated by combining their horizontal and vertical distances as follows:(6)dA1,A2=x1−x22+y1−y221/2.The dashed lines in Figure 1(a) show the horizontal and vertical distances between pointsA1 and A2, and the solid line indicates the ED between them. Equation (6) uses the Pythagorean theorem to calculate the ED between two points in a two-coordinate system based on their horizontal and vertical distances. According to this theorem, the square of the hypotenuse in a right triangle is equal to the sum of the squares of the other two sides.Figure 1
Mutual interchange property of ED.
(a)(b)ED has been used for different purposes, including calculating the distance between two points in ann-dimensional space and measuring the difference between two FNs. However, using the ED for some purposes may come with some flaws. We provide a definition below to determine whether the ED is an appropriate tool for the given purpose.Definition 2.
(The mutual interchange property). LetX1 × X2 ×, …, × Xn be a universal set of an n-coordinate system and P1=x11,…,x1n and P2=x21,…,x2n be two points in this system. By mutually interchanging the values of the same dimension i for these points, i.e., the values of x1i and x2i, for one or more i indexes, the new points P3 and P4 are created. If the real distance between the new points P3 and P4 is equal to the real distance between P1 and P2, we can use the ED for the given purpose.
When the values of the same dimensions are mutually interchanged for one or more dimensions, equation (1) always calculates the same ED. However, the real distance between the new points may change. If the real distance between the new points is changed (not changed), the ED is inconsistent (consistent) with the real situation and cannot (can) be used for the given purpose. For example, by mutually interchanging the x-values of points A1=x1,y1 and A2=x2,y2 in Figure 1(a), two new order points are created as A3=x2,y1 and A4=x1,y2 shown in Figure 1(b). The ED between A3 and A4 is calculated as follows:(7)dA3,A4=x2−x12+y1−y221/2.
Equations (6) and (7) indicate that although A3 and A4 are two different points from A1 and A2, the ED between A3 and A4 equals the ED between A1 and A2. This conclusion is consistent with the real distances between the points. It implies that calculating the distance between points in a two-coordinate system comes with the mutual interchange property. Therefore, the ED is an appropriate tool for this purpose.


## 2.3. Misapplying the ED to Measure the Difference between FNs

This section argues that the ED is not an appropriate tool to measure the difference between FNs. In particular, we show that the misapplications of equations (2)–(5) extended based on the ED to measure the differences between GNs, TFNs, and TrFNs. For this purpose, we show that these equations do not come with the mutual interchange property; therefore, they should not be used to measure the differences between FNs. For example, by mutually interchanging the lower bounds of two GNs A=a1,a2 and B=b1,b2, two new GNs are created as C=b1,a2 and D=a1,b2, provided that a1,b1≤a2,b2. The real difference between new GNs C and D is not necessarily equal to the difference between GNs A and B(See Example 1). However, equation (2) calculates equal differences between the GNs for both cases. This implies the ED is inconsistent with measuring the difference between GNs. This conclusion can be generated for FNs. We present Examples 2 and 3 to illustrate this inconsistency for TFNs and TrFNs, respectively.A question arises: why is ED consistent with measuring the distance between twon-coordinate points located in an n-dimensional space but inconsistent with measuring the difference between two FNs? To answer this question, consider that n naturally different components characterize an n-coordinate point; each component is measured based on a different dimension. For example, let A = (10, 20, 30) be a point in a three-dimensional space. Three naturally different components characterize this point: its length, width, and height; the values of these components are 10, 20, and 30, respectively. The number of components characterizing a point equals the number of dimensions of the space in which the point is located. As a result, the ED between two points in an n-dimensional space is calculated based on their differences obtained for the same dimensions based on the Pythagorean theorem.However, the number of components characterizing an FN is not equal to the number of dimensions of the space in which it is located. For example, the GN, TFN, and TrFN characterized with 2, 3, and 4 components, respectively, are located in a one-dimensional space, not in two-, three-, and four-dimensional spaces. In other words, an FN is characterized byn components of the same nature; all of them are measured based on only the same dimension. For example, let B = (10, 20, and 30) be a TFN representing the fuzzy set of young people. Although B is characterized by three components, i.e., 10, 20, and 30, they do not correspond to three different dimensions; these components have the same nature, i.e., age, and can be measured using only one dimension.Both the three-coordinate pointA and the TFN B are identified as (10, 20, 30). However, they are entirely conceptually different and should not be treated the same. Point A is a three-coordinate point located in a three-dimensional space, while B is a TFN located in a one-dimensional space. This clearly shows that although A and B look similar, they are conceptually quite different. Therefore, we cannot simply generalize operations that are inherently appropriate for points located in three-dimensional space to TFNs.We can also criticize using the ED to measure the difference between FNs from the extension principle perspective. According to this principle, each fuzzy relation has been generally developed based on a crisp relation. However, the ED between FNs has been extended improperly based on the crisp ED. In other words, the ED in a crisp environment measures the distance between twon-coordinate points; each coordinate corresponds to a unique dimension. In contrast, the ED in a fuzzy environment measures the difference between two n-component FNs; all these components together correspond to only one dimension. It is clear that a point in an n-dimensional space is conceptually completely different from an FN denoted with n components.A point in ann-dimensional space contains n heterogeneous components. For example, the three components of a point in a three-dimensional space are length, width, and height, representing three different characteristics. In contrast, an FN contains n homogenous components. For example, the three components of a TFN are the lower, middle, and upper values of the same variable; these components together represent the same characteristic. A question arises: on what logic has the ED been extended to measure the distance between one-coordinate FNs? The only answer to this question is the similarity between denoting a point in an n-dimensional space and an FN. For example, both a point in a three-dimensional space and a TFN are denoted as (a ,b, c). Despite this similarity, they are entirely different; therefore, the operations proposed for one of them cannot be extended to the other simply.


## 2.4. Illustrating the ED Misapplication to Measure the Difference between FNs

The previous section concluded that using the ED to measure the difference between two FNs in a one-dimensional space is meaningless. Therefore, other approaches should be used to measure the difference between FNs. One of the approaches used for this purpose is to calculate the difference between the centroids of FNs. In the following, we first review the center of gravity (COG) method used to obtain the centroid of an FN. Then, we present a theorem to show that calculating the difference between two intervals using the ED and COG methods leads to different results. It is to be noted that the proposed theorem can be extended to FNs, including TFNs and TrFNs. Next, some numerical examples are given to illustrate the ED misapplication. To show this misapplication, we compare the results of two methods between FNs: the ED and the COG methods.Remark 1.
Yager [19] proposed the COG method to obtain the centroid of the FN A˜ with the membership function of μA˜x as follows:(8)CA˜=∫−∞+∞x.μA˜x.dx∫−∞+∞μA˜x.dx,where CA˜ is the centroid of A˜. Equation (8) can be used to find the centroids of different types of FNs. For example, consider Figures 2(a)–2(c) representing the normal TFN A˜=l,m,u, GN B˜=l,u, and TrFN C˜=l,m,m′,u, respectively.
Arman et al. [20] obtained the centroids of A˜, B˜, and C˜ using equation (8) as follows:(9)CA˜=l+m+u3,(10)CB˜=l+u2,(11)CC˜=13l+m+m′+u−m′×u−l×mm′+u−l+m.Figure 2
Membership function of three common FNs.
(a)(b)(c)Theorem 1.
LetA=a1,a2 and B=b1,b2 be two distinct GNs. Then, the difference between A and B using ED and COG leads to different results.Proof.
The centroids of GNsA and B using equation (10) are CA=a1+a2/2 and CB=b1+b2/2, respectively. By contrast, assume that the difference between A and B using the ED and COG methods is the same. Therefore a1−b12+a2−b22=a1+a2/2−b1+b2/2⇒a1−a2=b1−b2.
We know that although two new GNs are created by mutually interchanging the values of the same dimension of GNsA and B, the ED between new points is not changed. By mutually interchanging the lower bounds of A and B, two new GNs are created as C=b1,a2 and D=a1,b2, provided that a1,b1≤a2,b2. The centroids of GNs C and D using equation (10) are CC=b1+a2/2 and CD=a1+b2/2, respectively. Assume that the difference between C and D using the ED and COG methods leads to the same. Thus,(12)b1−a12+a2−b22=b1+a22−a1+b22⇒a1+a2=b1+b2.
Sincea1−a2=b1−b2 and a1+a2=b1+b2. So(13)a1−a2+a1+a2=b1−b2+b1+b2⇒a1=b1a1−a2−a1+a2=b1−b2−b1+b2⇒a2=b2.
This meansA = B, and this is a contradiction. Now, the proof is completed.Example 1.
LetA1 = [10, 90] and B1 = [80, 120] be two GNs. By interchanging the first components of these GNs, the new GNs A2 = [80, 90] and B2 = [10, 120] are created. The ED between GNs A2 and B2 is 76.16 using equation (2), exactly equal to that of GNs A1 and B1. On the other hand, the centroids of intervals A1, B1, A2, and B2 are obtained at 50, 100, 85, and 65, respectively, using equation (10). Therefore, the absolute difference between the centroids of A1 and B1 is CA1−CB1=50, while the absolute difference between the centroids of A2 and B2 is CA2−CB2=20. It implies that equation (2) is inconsistent with measuring the difference between GNs.Example 2.
Assume thatK1 = (1, 3, 5) and L1 = (2, 4, 6) are two TFNs (Figure 3(a)). The ED between TFNs K1 and L1 is 1 using equation (3). L1 is greater than K1 because all components of L1 are greater than their corresponding components of K1. This is confirmed by calculating the difference between the centroids of these TFNs. The centroids of K1 and L1 are 3 and 4, respectively, using equation (9); therefore, the difference between them is CK1−CL1=−1.
By mutually interchanging the third components ofK1 and L1, two new TFNs, K2 = (1, 3, 6) and L2 = (2, 4, 5), are created (Figure 3(b)). Compared to K1 and L1, the values of the first and second components of K2 and L2 have not changed; but the values of their third components have increased and decreased, respectively. Therefore, the FNs K2 and L2 are expected to be closer to each other compared to the FNs between K1 and L1. However, the ED between K2 and L2 is 1 using equation (3), exactly equal to the ED between K1 and L1. On the other hand, the centroids of K2 and L2 are 3.33 and 3.66, respectively, using equation (9); thus, the absolute difference between their centroids is CK2−CL2=0.33 that is smaller than CK1−CL1=1. It indicates that measuring the difference between TFNs based on their centroids is consistent with our expectation. This example shows that when the same elements of two TFNs are mutually interchanged, the real difference between new TFNs may change. However, equation (3) cannot discover this change and computes the same ED. As a result, the ED is not an appropriate tool for measuring the difference between two TFNs.Figure 3
Comparing TFNs when interchanging their third components.
(a)(b)Example 3.
Using the numbers 1 to 8, eight different two-member sets can be made, provided that the members in each set are TrFNs without using a number twice in each set. In other words, there are only eight separate sets asAi=a,b,c,d,e,f,g,h, i = 1, …, 8, provided that the numbers 1 to 8 appear only once in each set, and a<b<c<d and e<f<g<h (See Table 1).
The ED between two TrFNs is calculated equal to 1 using equations (4) and/or (5) for all sets. It means that the ED measures the difference between two TrFNs in each set equal to 1. However, it does not reflect reality. To prove that, we obtain the centroids of TrFNs using equation (11) and then calculate the difference between the centroids of TrFNs in each set. The results are given in Table 1. This table clearly shows that the differences between the centroids better reflect the real differences between TrFNs.Table 1
Comparing TrFNs using the ED and the COG methods.
SetTrFNsEuclidean distanceCentroidC(M) − C(N)TrFNMTrFNNC(M)C(N)1(1, 3, 5, 7)(2, 4, 6, 8)145−12(1, 3, 5, 8)(2, 4, 6, 7)14.2964.714−0.4183(1, 3, 6, 7)(2, 4, 5, 8)14.2224.810−0.5884(1, 4, 5, 7)(2, 3, 6, 8)14.194.778−0.5885(2, 3, 5, 7)(1, 4, 6, 8)14.2864.704−0.4186(2, 3, 5, 8)(1, 4, 6, 7)14.5834.4170.1667(2, 3, 6, 7)(1, 4, 5, 8)14.54.508(2, 4, 5, 7)(1, 3, 6, 8)14.54.50


## 3. Revisiting the Gray and Fuzzy TOPSIS Methods

This section is divided into three subsections. Subsection3.1 presents a new gray TOPSIS method to avoid misapplying the ED distance. Subsections 3.2 and 3.2 revise two triangular and trapezoidal FTOPSIS methods, respectively, proposed by Chen [1] and Chen et al. [2]. These methods misapply the ED to measure the differences between FNs. We revise these methods to avoid this misapplication.


### 3.1. The Interval value (Gray) TOPSIS

Consider the following gray comparison matrix.In this matrix,Aii=1,…,m and Cjj=1,…,n represent the alternative i and criterion j, respectively, and a¯ij=aijL,aijU is the interval (gray) value of alternative i for criterion j. Here, we propose a new gray TOPSIS method consisting of six steps as follows.Step 1. Normalizing the decision matrixAt this step, the decision matrixD¯=a¯ijm×n is converted into the normalized matrix N¯=n¯ijm×n using the linear scale transformation as follows:(14)n¯ij=aijLUj∗,aijUUj∗,Uj∗=maxiaijU,ifj∈B;n¯ij=Lj−aijL,Lj−aijU,Lj−=miniaijL,ifj∈C;whereB and C are the sets of benefit and cost criteria, respectively, and n¯ij=nijL,nijU is the normalized value of a¯ij=aijL,aijU.Step 2. Weighting the normalized matrixLetW¯=w¯j|j=1,…,n denote the vector of interval weights of criteria as w¯j=αj,βj is the triangular fuzzy weight of criterion j. Therefore, the weighted normalized matrix V¯=v¯ijm×n is obtained as V¯=W¯×N¯, in which each element v¯ij=w¯j.n¯ij is calculated as follows:(15)v¯ij=vijL,vijU=αj.nijL,βj.nijU,∀i,j.Step 3. Defining the ideal solutionsThe interval values of PIS and NIS for criterionj, shown as v¯j∗=vj∗L,vj∗U and v¯j−=vj−L,vj−U, respectively, can be defined as follows:(16)v¯j∗=1,1|j∈B,0,0|j∈C,j=1,…,n;v¯j−=0,0|j∈B,1,1|j∈C,j=1,…,n.Step 4. Transforming into crisp values based on the COG methodIn this step, the gray matrixV¯ is converted into the crisp matrix V. For this purpose, we use equation (10) to transform the gray value v¯ij∀i,j into the crisp value as follows:(17)vij=vijL+vijU2.This step also uses equation (10) to transform the interval values of PIS and NIS for criterion j into the crisp values as follows:(18)vj∗=vj∗L+vj∗U2,vj−=vj−L+vj−U2.Note that if a gray PIS (or a gray NIS) is defined as (1, 1) or (0, 0), its corresponding crisp value is 1 or 0, respectively.Step 5. Computing the EDsThe outcomes of Step 4 are a crisp weighted normalized matrixV=vijm×n and the crisp PIS and NIS vectors, shown as S∗=vj∗1×n and S−=vj∗1×n. In this step, we compute di∗ and di−, representing the EDs between the alternative i and the crisp PIS and NIS vectors. These distances are calculated as follows:(19)di∗=∑j=1nvij−vj∗2,i=1,..,m,di−=∑j=1nvij−vj−2,i=1,..,m.Step 6. Ranking the alternativesThis step calculates the relative closeness measures for alternatives as follows:(20)Ci+=di−di−+di∗,i=1,…,m,Ci+ is a utility measure. Therefore, the alternatives are ranked based on Ci+ ascendingly.


### 3.2. Revisiting the Triangular FTOPSIS

Consider the following decision matrix filled with TFNs.In this matrix,Aii=1,…,m and Cjj=1,…,n represent the alternative i and criterion j, respectively, and a˜ij=aijL,aijM,aijU is the triangular fuzzy value of alternative i for criterion j. The revised triangular FTOPSIS method consists of six steps as follows.Step 1. Normalizing the decision matrixAt this step, the decision matrixD˜=a˜ijm×n is converted into the normalized matrix N˜=n˜ijm×n using the linear scale transformation as follows:(21)n˜ij=aijLUj∗,aijMUj∗,aijUUj∗,Uj∗=maxiaijU,ifj∈B;n˜ij=Lj−aijL,Lj−aijM,Lj−aijU,Lj−=miniaijL,ifj∈C;whereB and C are the sets of benefit and cost criteria, respectively, and n˜ij=nijL,nijM,nijU is the fuzzy normalized value of a˜ij=aijL,aijM,aijU.Step 2. Weighting the normalized matrixLetW˜=w˜j|j=1,…,n denote the vector of fuzzy weights of criteria as w˜j=αj,βj,χj is the triangular fuzzy weight of criterion j. Therefore, the weighted normalized matrix V˜=v˜ijm×n is obtained as V˜=W˜⊗N˜, in which each element v˜ij=w˜j.n˜ij is calculated as follows:(22)v˜ij=vijL,vijM,vijU=αj.nijL,βj.nijM,χj.nijU,∀i,j.Step 3. Defining the ideal solutionsThe FPIS and FNIS for criterionj, shown as v˜j∗=vj∗L,vj∗M,vj∗U and v˜j−=vj−L,vj−M,vj−U, respectively, can be defined as follows:(23)v˜j∗=1,1,1|j∈B,0,0,0|j∈C,j=1,…,n;v˜j−=0,0,0|j∈B,1,1,1|j∈C,j=1,…,n.Note that different approaches in the literature define the ideal solutions. We used the approach proposed by Chen [1]. However, the researchers can apply other approaches for future research.Step 4. Defuzzifying based on the COG methodIn this step, the weighted normalized matrixV˜ is converted into the crisp matrix V. For this purpose, we use equation (9) to defuzzify the triangular fuzzy value of v˜ij∀i,j as follows:(24)vij=vijL+vijM+vijU3.This step also uses equation (9) to defuzzify the FPIS and FNIS for criterion j as follows:(25)vj∗=vj∗L+vj∗M+vj∗U3,vj−=vj−L+vj−M+vj−U3.Note that if a FPIS (or a FNIS) is defined as (1, 1, 1) or (0, 0, 0), its corresponding crisp value is 1 or 0, respectively.Step 5. Computing the EDsThe outcomes of Step 4 are a crisp weighted normalized matrixV=vijm×n and the crisp PIS and NIS vectors, shown as S∗=vj∗1×n and S−=vj∗1×n. In this step, we compute di∗ and di−, representing the EDs between the alternative i and the crisp PIS and NIS vectors. These distances are calculated as follows:(26)di∗=∑j=1nvij−vj∗2,i=1,..,m,di−=∑j=1nvij−vj−2,i=1,..,m.Step 6. Ranking the alternativesThis step calculates the relative closeness measures for alternatives as follows:(27)Ci+=di−di−+di∗,i=1,…,m,Ci+ is a utility measure. Therefore, the alternatives are ranked based on Ci+ ascendingly.


### 3.3. Revisiting the Trapezoidal FTOPSIS

Consider the following decision matrix filled with TrFNs.In this matrix,a˜ij=aijL,aijM,aijM′,aijU is the trapezoidal fuzzy value of alternative i for criterion j. The revised trapezoidal FTOPSIS method consists of six steps as follows.Step 1. Normalizing the decision matrixThis step uses the linear scale transformation to convert the decision matrixD˜=a˜ijm×n into the normalized matrix N˜=n˜ijm×n as follows:(28)n˜ij=aijLUj∗,aijMUj∗,aijM′Uj∗,aijUUj∗,Uj∗=maxiaijU,ifj∈B;n˜ij=Lj−aijL,Lj−aijM,Lj−aijM′,Lj−aijU,Lj−=miniaijL,ifj∈C;whereB and C are the sets of benefit and cost criteria, respectively, and n˜ij=nijL,nijM,nijM′,nijU is the fuzzy normalized value of a˜ij=aijL,aijM,aijM′,aijU.Step 2. Weighting the normalized matrixAssume thatW˜=w˜j|j=1,…,n represents the vector of fuzzy weights of criteria in which w˜j=αj,βj,βj′,χj is the trapezoidal fuzzy weight of criterion j. Therefore, the weighted normalized matrix V˜=v˜ijm×n is obtained as V˜=W˜⊗N˜ so that each element v˜ij=w˜j.n˜ij is calculated as follows:(29)v˜ij=vijL,vijM,vijM′,vijU=αj.nijL,βj.nijM,βj′.nijM′,χj.nijU,∀i,j.Step 3. Defining the ideal solutionsThe FPIS and FNIS for criterionj, shown as v˜j∗=vj∗L,vj∗M,vj∗M′,vj∗U and v˜j−=vj−L,vj−M,vj−M′,vj−U, respectively, can be defined as follows:(30)v˜j∗=1,1,1,1|j∈B,0,0,0,0|j∈C,j=1,…,n;v˜j−=0,0,0,0|j∈B,1,1,1,1|j∈C,j=1,…,n..Note that we used the approach Chen et al. [2] proposed to define the ideal solutions. However, the other approaches in the literature can be used for this purpose in future research.Step 4. Defuzzifying based on the COG methodThis step converts the weighted normalized matrixV˜ into the crisp matrix V. For this purpose, we use equation (11) to defuzzify the trapezoidal fuzzy value of v˜ij∀i,j as follows:(31)vij=13vijL+vijM+vijM′+vijU−vijM′×vijU−vijL×vijMvijM′+vijU−vijL+vijM.This step also uses equation (11) to convert the FPIS v˜j∗ and FNIS v˜j− into the crisp PIS vj∗ and the crisp NIS vj−, respectively, as follows:(32)vj∗=13vj∗L+vj∗M+vj∗M′+vj∗U−vj∗M′×vj∗U−vj∗L×vj∗Mvj∗M′+vj∗U−vj∗L+vj∗M,vj−=13vj−L+vj−M+vj−M′+vj−U−vj−M′×vj−U−vj−L×vj−Mvj−M′+vj−U−vj−L+vj−M.Note that if a FPIS (or a FNIS) is defined as (1, 1, 1, 1) or (0, 0, 0, 0), its corresponding crisp value is 1 or 0, respectively.Step 5. Computing the EDsThe outcomes of Step 4 are the crisp matrixV=vijm×n and the crisp PIS and NIS vectors S∗=vj∗1×n and S−=vj∗1×n. This step computes di∗ and di−, representing the EDs between the alternative i and the crisp PIS and NIS vectors. These distances are calculated as follows:(33)di∗=∑j=1nvij−vj∗2,i=1,..,m,di−=∑j=1nvij−vj−2,i=1,..,m.Step 6. Ranking the alternativesThis step calculates the relative closeness measures for alternatives as follows:(34)Ci+=di−di−+di∗,i=1,…,m,Ci+ is a utility measure. Therefore, the alternatives are ranked based on Ci+ ascendingly.


## 3.1. The Interval value (Gray) TOPSIS

Consider the following gray comparison matrix.In this matrix,Aii=1,…,m and Cjj=1,…,n represent the alternative i and criterion j, respectively, and a¯ij=aijL,aijU is the interval (gray) value of alternative i for criterion j. Here, we propose a new gray TOPSIS method consisting of six steps as follows.Step 1. Normalizing the decision matrixAt this step, the decision matrixD¯=a¯ijm×n is converted into the normalized matrix N¯=n¯ijm×n using the linear scale transformation as follows:(14)n¯ij=aijLUj∗,aijUUj∗,Uj∗=maxiaijU,ifj∈B;n¯ij=Lj−aijL,Lj−aijU,Lj−=miniaijL,ifj∈C;whereB and C are the sets of benefit and cost criteria, respectively, and n¯ij=nijL,nijU is the normalized value of a¯ij=aijL,aijU.Step 2. Weighting the normalized matrixLetW¯=w¯j|j=1,…,n denote the vector of interval weights of criteria as w¯j=αj,βj is the triangular fuzzy weight of criterion j. Therefore, the weighted normalized matrix V¯=v¯ijm×n is obtained as V¯=W¯×N¯, in which each element v¯ij=w¯j.n¯ij is calculated as follows:(15)v¯ij=vijL,vijU=αj.nijL,βj.nijU,∀i,j.Step 3. Defining the ideal solutionsThe interval values of PIS and NIS for criterionj, shown as v¯j∗=vj∗L,vj∗U and v¯j−=vj−L,vj−U, respectively, can be defined as follows:(16)v¯j∗=1,1|j∈B,0,0|j∈C,j=1,…,n;v¯j−=0,0|j∈B,1,1|j∈C,j=1,…,n.Step 4. Transforming into crisp values based on the COG methodIn this step, the gray matrixV¯ is converted into the crisp matrix V. For this purpose, we use equation (10) to transform the gray value v¯ij∀i,j into the crisp value as follows:(17)vij=vijL+vijU2.This step also uses equation (10) to transform the interval values of PIS and NIS for criterion j into the crisp values as follows:(18)vj∗=vj∗L+vj∗U2,vj−=vj−L+vj−U2.Note that if a gray PIS (or a gray NIS) is defined as (1, 1) or (0, 0), its corresponding crisp value is 1 or 0, respectively.Step 5. Computing the EDsThe outcomes of Step 4 are a crisp weighted normalized matrixV=vijm×n and the crisp PIS and NIS vectors, shown as S∗=vj∗1×n and S−=vj∗1×n. In this step, we compute di∗ and di−, representing the EDs between the alternative i and the crisp PIS and NIS vectors. These distances are calculated as follows:(19)di∗=∑j=1nvij−vj∗2,i=1,..,m,di−=∑j=1nvij−vj−2,i=1,..,m.Step 6. Ranking the alternativesThis step calculates the relative closeness measures for alternatives as follows:(20)Ci+=di−di−+di∗,i=1,…,m,Ci+ is a utility measure. Therefore, the alternatives are ranked based on Ci+ ascendingly.


## 3.2. Revisiting the Triangular FTOPSIS

Consider the following decision matrix filled with TFNs.In this matrix,Aii=1,…,m and Cjj=1,…,n represent the alternative i and criterion j, respectively, and a˜ij=aijL,aijM,aijU is the triangular fuzzy value of alternative i for criterion j. The revised triangular FTOPSIS method consists of six steps as follows.Step 1. Normalizing the decision matrixAt this step, the decision matrixD˜=a˜ijm×n is converted into the normalized matrix N˜=n˜ijm×n using the linear scale transformation as follows:(21)n˜ij=aijLUj∗,aijMUj∗,aijUUj∗,Uj∗=maxiaijU,ifj∈B;n˜ij=Lj−aijL,Lj−aijM,Lj−aijU,Lj−=miniaijL,ifj∈C;whereB and C are the sets of benefit and cost criteria, respectively, and n˜ij=nijL,nijM,nijU is the fuzzy normalized value of a˜ij=aijL,aijM,aijU.Step 2. Weighting the normalized matrixLetW˜=w˜j|j=1,…,n denote the vector of fuzzy weights of criteria as w˜j=αj,βj,χj is the triangular fuzzy weight of criterion j. Therefore, the weighted normalized matrix V˜=v˜ijm×n is obtained as V˜=W˜⊗N˜, in which each element v˜ij=w˜j.n˜ij is calculated as follows:(22)v˜ij=vijL,vijM,vijU=αj.nijL,βj.nijM,χj.nijU,∀i,j.Step 3. Defining the ideal solutionsThe FPIS and FNIS for criterionj, shown as v˜j∗=vj∗L,vj∗M,vj∗U and v˜j−=vj−L,vj−M,vj−U, respectively, can be defined as follows:(23)v˜j∗=1,1,1|j∈B,0,0,0|j∈C,j=1,…,n;v˜j−=0,0,0|j∈B,1,1,1|j∈C,j=1,…,n.Note that different approaches in the literature define the ideal solutions. We used the approach proposed by Chen [1]. However, the researchers can apply other approaches for future research.Step 4. Defuzzifying based on the COG methodIn this step, the weighted normalized matrixV˜ is converted into the crisp matrix V. For this purpose, we use equation (9) to defuzzify the triangular fuzzy value of v˜ij∀i,j as follows:(24)vij=vijL+vijM+vijU3.This step also uses equation (9) to defuzzify the FPIS and FNIS for criterion j as follows:(25)vj∗=vj∗L+vj∗M+vj∗U3,vj−=vj−L+vj−M+vj−U3.Note that if a FPIS (or a FNIS) is defined as (1, 1, 1) or (0, 0, 0), its corresponding crisp value is 1 or 0, respectively.Step 5. Computing the EDsThe outcomes of Step 4 are a crisp weighted normalized matrixV=vijm×n and the crisp PIS and NIS vectors, shown as S∗=vj∗1×n and S−=vj∗1×n. In this step, we compute di∗ and di−, representing the EDs between the alternative i and the crisp PIS and NIS vectors. These distances are calculated as follows:(26)di∗=∑j=1nvij−vj∗2,i=1,..,m,di−=∑j=1nvij−vj−2,i=1,..,m.Step 6. Ranking the alternativesThis step calculates the relative closeness measures for alternatives as follows:(27)Ci+=di−di−+di∗,i=1,…,m,Ci+ is a utility measure. Therefore, the alternatives are ranked based on Ci+ ascendingly.


## 3.3. Revisiting the Trapezoidal FTOPSIS

Consider the following decision matrix filled with TrFNs.In this matrix,a˜ij=aijL,aijM,aijM′,aijU is the trapezoidal fuzzy value of alternative i for criterion j. The revised trapezoidal FTOPSIS method consists of six steps as follows.Step 1. Normalizing the decision matrixThis step uses the linear scale transformation to convert the decision matrixD˜=a˜ijm×n into the normalized matrix N˜=n˜ijm×n as follows:(28)n˜ij=aijLUj∗,aijMUj∗,aijM′Uj∗,aijUUj∗,Uj∗=maxiaijU,ifj∈B;n˜ij=Lj−aijL,Lj−aijM,Lj−aijM′,Lj−aijU,Lj−=miniaijL,ifj∈C;whereB and C are the sets of benefit and cost criteria, respectively, and n˜ij=nijL,nijM,nijM′,nijU is the fuzzy normalized value of a˜ij=aijL,aijM,aijM′,aijU.Step 2. Weighting the normalized matrixAssume thatW˜=w˜j|j=1,…,n represents the vector of fuzzy weights of criteria in which w˜j=αj,βj,βj′,χj is the trapezoidal fuzzy weight of criterion j. Therefore, the weighted normalized matrix V˜=v˜ijm×n is obtained as V˜=W˜⊗N˜ so that each element v˜ij=w˜j.n˜ij is calculated as follows:(29)v˜ij=vijL,vijM,vijM′,vijU=αj.nijL,βj.nijM,βj′.nijM′,χj.nijU,∀i,j.Step 3. Defining the ideal solutionsThe FPIS and FNIS for criterionj, shown as v˜j∗=vj∗L,vj∗M,vj∗M′,vj∗U and v˜j−=vj−L,vj−M,vj−M′,vj−U, respectively, can be defined as follows:(30)v˜j∗=1,1,1,1|j∈B,0,0,0,0|j∈C,j=1,…,n;v˜j−=0,0,0,0|j∈B,1,1,1,1|j∈C,j=1,…,n..Note that we used the approach Chen et al. [2] proposed to define the ideal solutions. However, the other approaches in the literature can be used for this purpose in future research.Step 4. Defuzzifying based on the COG methodThis step converts the weighted normalized matrixV˜ into the crisp matrix V. For this purpose, we use equation (11) to defuzzify the trapezoidal fuzzy value of v˜ij∀i,j as follows:(31)vij=13vijL+vijM+vijM′+vijU−vijM′×vijU−vijL×vijMvijM′+vijU−vijL+vijM.This step also uses equation (11) to convert the FPIS v˜j∗ and FNIS v˜j− into the crisp PIS vj∗ and the crisp NIS vj−, respectively, as follows:(32)vj∗=13vj∗L+vj∗M+vj∗M′+vj∗U−vj∗M′×vj∗U−vj∗L×vj∗Mvj∗M′+vj∗U−vj∗L+vj∗M,vj−=13vj−L+vj−M+vj−M′+vj−U−vj−M′×vj−U−vj−L×vj−Mvj−M′+vj−U−vj−L+vj−M.Note that if a FPIS (or a FNIS) is defined as (1, 1, 1, 1) or (0, 0, 0, 0), its corresponding crisp value is 1 or 0, respectively.Step 5. Computing the EDsThe outcomes of Step 4 are the crisp matrixV=vijm×n and the crisp PIS and NIS vectors S∗=vj∗1×n and S−=vj∗1×n. This step computes di∗ and di−, representing the EDs between the alternative i and the crisp PIS and NIS vectors. These distances are calculated as follows:(33)di∗=∑j=1nvij−vj∗2,i=1,..,m,di−=∑j=1nvij−vj−2,i=1,..,m.Step 6. Ranking the alternativesThis step calculates the relative closeness measures for alternatives as follows:(34)Ci+=di−di−+di∗,i=1,…,m,Ci+ is a utility measure. Therefore, the alternatives are ranked based on Ci+ ascendingly.


## 4. Illustrative Example

This section uses a numerical example to illustrate that the FTOPSIS method proposed by Chen [1], and its revised version presented in this study assign different weights to alternatives and may even rank them differently. Assume that we aim to rank two alternatives, A and B, considering two attributes, C1 and C2. The values of these alternatives for each attribute are given in Table 2 as TFNs. In this table, the weights of attributes are also given as TFNs. Both C1 and C2 are benefit-type attributes.Table 2
Fuzzy triangular decision matrix.
AlternativeC1C2A(1, 3, 5)(3, 4, 8)B(2, 4, 6)(1, 3, 5)Weight(0.5, 0.6, 0.6)(0.3, 0.4, 0.4)Both the FTOPSIS methods proposed by Chen and its revised version have some common steps, including normalizing the decision matrix using equations (20) and (21), and obtaining the weighted normalized matrix using equation (22). The results of these steps are given in Table 3. Since both attributes C1 and C2 are of the benefit type, the FPIS and FNIS are considered to be (1, 1, 1) and (0, 0, 0), respectively, for both attributes.Table 3
The common steps of Chen’s method and its revised version.
AlternativeThe normalized matrixThe weighted normalized matrixC1C2C1C2A(0.167, 0.5, 0.833)(0.375, 0.5, 1)(0.083, 0.3, 0.583)(0.113, 0.2, 0.5)B(0.333, 0.667, 1)(0.125, 0.375, 0.625)(0.167, 0.4, 0.7)(0.038, 0.15, 0.313)After the common steps, Chen’s method and its revised version follow different steps described as follows.


### 4.1. Results Based on Chen’s FTOPSIS

Chen’s method obtains the EDs between the weighted normalized values and the FPIS and FNIS for each attribute, then calculates the relative closeness for alternatives,Ci+i=1,2, and accordingly ranks them. These calculations are given in Table 4. In this table, dij∗dij−, representing the ED between alternative i and FPIS (FNIS) for attribute j, is calculated using equation (3).Table 4
The required calculation based on Chen’s method.
AlternativeEDsC1C2SummationCi+RankingAdAj∗ using equation (3)0.7080.7481.4560.3251dAj− using equation (3)0.3820.3180.699BdBj∗ using equation (3)0.6180.8411.4590.3172dBj− using equation (3)0.4750.2010.677


### 4.2. Results Based on the Revised FTOPSIS

The revised FTOPSIS method presented in this study converts the fuzzy weighted normalized values into crisp values using equation (9), shown in Table 5 as vij. Also, C1 and C2 are both benefit-type attributes; therefore, both consider (1, 1, 1) and (0, 0, 0) the FPIS and FNIS, respectively. Therefore, the crisp PIS and NIS are obtained as A∗=1,1 and A−=0,0, respectively. Then, this method computes di∗ and di−, the EDs between the alternative i and the crisp PIS A∗ and NIS A−, respectively. Finally, it calculates the relative closeness of alternatives and accordingly ranks them. These calculations are given in Table 5.Table 5
The required calculation based on the revisited FTOPSIS.
AlternativeC1C2d∗ using equation (17)d− using equation (18)Ci+RankingAvA1 = 0.322vA2 = 0.2710.9960.4210.2972BvB1 = 0.422vB2 = 0.1671.0140.4540.3091In contrast to Chen’s method, our revised FTOPSIS method ranks alternative B as the best one.


## 4.1. Results Based on Chen’s FTOPSIS

Chen’s method obtains the EDs between the weighted normalized values and the FPIS and FNIS for each attribute, then calculates the relative closeness for alternatives,Ci+i=1,2, and accordingly ranks them. These calculations are given in Table 4. In this table, dij∗dij−, representing the ED between alternative i and FPIS (FNIS) for attribute j, is calculated using equation (3).Table 4
The required calculation based on Chen’s method.
AlternativeEDsC1C2SummationCi+RankingAdAj∗ using equation (3)0.7080.7481.4560.3251dAj− using equation (3)0.3820.3180.699BdBj∗ using equation (3)0.6180.8411.4590.3172dBj− using equation (3)0.4750.2010.677


## 4.2. Results Based on the Revised FTOPSIS

The revised FTOPSIS method presented in this study converts the fuzzy weighted normalized values into crisp values using equation (9), shown in Table 5 as vij. Also, C1 and C2 are both benefit-type attributes; therefore, both consider (1, 1, 1) and (0, 0, 0) the FPIS and FNIS, respectively. Therefore, the crisp PIS and NIS are obtained as A∗=1,1 and A−=0,0, respectively. Then, this method computes di∗ and di−, the EDs between the alternative i and the crisp PIS A∗ and NIS A−, respectively. Finally, it calculates the relative closeness of alternatives and accordingly ranks them. These calculations are given in Table 5.Table 5
The required calculation based on the revisited FTOPSIS.
AlternativeC1C2d∗ using equation (17)d− using equation (18)Ci+RankingAvA1 = 0.322vA2 = 0.2710.9960.4210.2972BvB1 = 0.422vB2 = 0.1671.0140.4540.3091In contrast to Chen’s method, our revised FTOPSIS method ranks alternative B as the best one.


## 5. Conclusion

The ED is the length of a straight line connecting two points in ann-dimensional space. It calculates n distinct differences between these points for n given dimensions and then combines them using the Pythagorean theorem. A misconception caused the ED to be applied to measure the difference between FNs. This misconception occurs because an n-coordinate point and an FN are denoted alike. Some studies considered only this similarity and generalized using the ED to measure the difference between FNs. However, they did not consider the fundamental differences between a point in an n-dimensional space and an FN denoted by n components. For example, although both a point in a three-dimensional space and a TFN are denoted exactly the same as (a, b, c), they have fundamental conceptual differences. Three components of a point in a three-dimensional space represent three completely different variables, while the three components of a TFN together represent the same variable. We showed that the ED used to measure the distance between two points in an n-dimensional space is inappropriately applied to measure the difference between two n-components FNs in a one-dimensional space.In the literature, different fields misapply the ED to measure the difference between FNs; one of the most widely used is fuzzy MADM. This study reviewed some FTOPSIS methods that misapplyied the ED to measure the difference between FNs and revisited two FTOPSIS methods for TFNs and TrFNs to avoid this misapplication. This study also presented a GTOPSIS method that uses the COG method to measure the difference between GNs instead of the ED method. It is suggested that future research revise the other FTOPSIS methods misapplyied the ED to measure the difference between FNs. This suggestion can be generalized to other fuzzy MADM techniques. Future research can also revisit the other fields in which the ED is misapplied to measure the difference between FNs.



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*Source: 7032662-2022-12-15.xml*