Ranking of fuzzy numbers based on centroid point and spread

Centroid and spread are commonly used approaches in ranking fuzzy numbers. Some experts rank fuzzy numbers using centroid or spread alone while others tend to integrate them together. Although a lot of methods for ranking fuzzy numbers that are related to both approaches have been presented, there are still limitations whereby the ranking obtained is inconsistent with human intuition. This paper proposes a novel method for ranking fuzzy numbers that integrates the centroid point and the spread approaches and overcomes the limitations and weaknesses of most existing methods. Proves and justifications with regard to the proposed ranking method are also presented. 5


Introduction
Ranking fuzzy numbers plays an important role in decision making in fuzzy environment.Many ranking methods have been presented in the literature since this idea was first introduced by [1].Among others were [2][3][4][5][6][7][8].Basically, ranking fuzzy numbers provides the appropriate technique to deal with fuzzy numbers for decision making problems [9].
The literature on ranking fuzzy numbers classifies ranking methods into four categories.One of them is fuzzy mean and spread.In ranking fuzzy numbers, the mean is generally specified as the centroid of the fuzzy numbers.The concept of centroid in ranking fuzzy numbers was first introduced in [10] and this was later followed by [11,12].However, the methods from [11,12] have limitations as they only consider the positive sign for both numerator and denominator.The method from [11] produces similar ranking order for positive and negative fuzzy numbers while the method from [12] produces same ranking order for a mirror image situa-combination of centroid and spread as a ranking fuzzy numbers method [18][19][20] were among the first that presented the ranking methods using both approaches.The method from [18] was unable to rank fuzzy numbers of different normality while the methods from [19,20] could only be applied to trapezoidal fuzzy numbers [9, [21][22][23] later proposed some adjustments to previous ranking methods but all of them were inconsistent with human intuition.
To overcome the drawbacks mentioned above, this paper introduces a new ranking method which integrates centroid point and spread approaches for ranking fuzzy numbers.This paper is organised as follows.Preliminaries are given in Section 2. These are followed by discussions on shortcomings of existing ranking methods in Section 3. Section 4 covers the validation and proves of the proposed ranking method.Section 5 discusses the applicability of the proposed ranking method to other cases of fuzzy numbers by comparing the results obtained with the ones from other existing methods.Finally, a conclusion is drawn in Section 6.

Theoretical preliminaries
Based on [9], some basic concepts used in this paper are illustrated as follows.

Trapezoidal fuzzy numbers
A trapezoidal fuzzy number can be represented by the following membership function given by For a trapezoidal fuzzy number, if a i2 = a i3 , then the fuzzy number is in the form of a triangular fuzzy number.However, if a i1 = a i2 = a i3 = a i4 for both triangular and trapezoidal fuzzy numbers, then both fuzzy numbers are said to be in the form of a singleton fuzzy number (crisp value).The length between a i1 and a i4 is known as the core of the fuzzy numbers.If the fuzzy number A has the property such that -1 < a i1 < a i2 < a i3 < a i4 < 1, then Ã is called a standardized generalized trapezoidal fuzzy number and is denoted as

Generalized trapezoidal fuzzy numbers
Furthermore if ãi2 = ãi3 then Ã is known as a standardized generalized triangular fuzzy number.Any generalized fuzzy number may be transformed into a standardized generalized fuzzy number by normalization as described in (2).
where k = max (a i1 , a i2 , a i3 , a i4 ) .Example 1: Consider the following sets of fuzzy numbers adopted from [9 ,22] and shown in Fig. 2. another case with fuzzy numbers obtained by [17]. 135 Example 2: Consider the following sets of fuzzy numbers adopted from [25] shown in Fig. 3.
A = (0.2, 0.5, 0.8; 1.0), B = (0.4,0.5, 0.6; 1.0) Although [17] method has solved the problem faced 136 by [9,22], this method has shortcomings when applied the ranking order obtained is equal ranking (A ≈ B) because the distance between the centroid for both fuzzy numbers is the same.Thus, [17] method produces unreasonable ranking order for this case with fuzzy numbers.
Example 3: Consider the following sets of fuzzy numbers adopted from [22] shown in Fig. 4.
A = (0.1, 0.3, 0.5; 1.0), B = (0.1, 0.3, 0.5; 0.8) Since w A > w B , the centroid point for fuzzy number A is greater than B. Therefore, it is obvious that the ranking order of fuzzy numbers which is consistent with human intuition for this example should be A B.
However, the application of the method from [25] to this example produced different ranking order for different degrees of optimism.Therefore, the method from [25] had pitfall in ranking fuzzy numbers for this example by giving ranking order that is unreasonable and inconsistent with human intuition.

Research methodology
To overcome the limitations of existing methods, this study introduces a novel hybrid methodology for ranking fuzzy numbers method based on centroid point and spread (CPS).The centroid point is utilised in CPS due to the effectiveness of this approach in ranking various cases of fuzzy numbers which are suited to human intuition.The spread method, on the other hand, is integrated with the centroid point to cater for the pitfalls faced by the existing ranking methods, as already discussed in Section 3. The full illustration of the proposed new ranking method is presented below.
Since centroid is considered as the main factor in ranking fuzzy numbers by human intuition [17], the centroid method from [14] is used here as one of the components of CPS.This is due to the fact that this centroid method has the ability to deal with numerous types of fuzzy numbers as discussed in [17].Therefore, the centroid method from [14] is proposed here as one of the components in the CPS ranking method which is defined as follows.
Assume that a fuzzy number A is generally described as A = (a 1 , a 2 , a 3 , a 4 ; w A ), the horizontalx centroid equation of fuzzy number A, x A is calculated as and the verticaly centroid equation of fuzzy number A, y A is calculated as where A α i is the length of the α -cuts of fuzzy number A, As discussed in Section 3, there are some cases where the centroid method is unable to rank the fuzzy numbers appropriately, especially when fuzzy numbers of different spread are considered.Therefore, considering spread in the formulation is important.

Spread in ranking fuzzy numbers and decision making
The roles play by the spread can be in twofold.They   The distance along the x -axis from the centroid of 214 x -value is defined as Further on, the distance along the vertical y -axis from the centroid of y -value is defined as Therefore, spread of A, s(A) is defined as where i and ii are dist(a 4 -a 1 ) and y A respectively.
The following figure is the illustration of the pro-218 posed spread methodology.Then, by definition, we have dist(a 4 − a 3 ) + Therefore, the proposed ranking fuzzy numbers is defined as follows.
Definition 4. The CPS ranking index value is defined as

103[
It should be noted that in the normalization process 104 only the components of fuzzy numbers are changed 105 where a i1 , a i2 , a i3 , a i4 are changed to ãi1 , ãi2 , ãi3 , ãi4 106 but the height of the fuzzy number remains the same 107 proposed, there are still shortcom-111 ings demonstrated by the recently proposed methods in 112 ranking fuzzy numbers consistently with human intu-113 ition.In this section, limitations of the existing ranking 114 methods are discussed and analysed using three counter 115 examples shown below.It should be noted that all fuzzy 116 number examples used from this section onwards are 117 in the form of standardized generalized fuzzy numbers.118 Example 1 illustrates the limitations of [9, 22, 24] in producing a consistent ranking order for the following 120 cases with fuzzy numbers.

are 1 . 2 .
Capability in Ranking Fuzzy Numbers.Although, centroid point can rank almost all cases of fuzzy numbers, spread does gives great assistance when centroid point fails to rank the following fuzzy numbers cases a. Fuzzy numbers of different spreads.b.Embedded fuzzy numbers.Role in Decision Making 194 In decision making environment, the decision makers 195 can be categorised into three namely pessimistic, neu-196 tral and optimistic [5, 27].This implies that they have 197 different views in terms of the spread of fuzzy num-198 bers, although the fuzzy numbers they observe are of 199 the same situation.Therefore, the ranking order might 200 be differ from one to another which indicates that spread 201 is also important in the decision making process.202 Thus, it is crucial not only to consider centroid point 203 but also spread in ranking fuzzy numbers and decision 204 making applications.

2054. 2 .
Spread formula for fuzzy numbers 206 According to [9], the spread is not considered as 207 important as the centroid in ranking fuzzy numbers.208 However, the spread does provide great assistance to 209 the centroid when dealing with fuzzy numbers in certain 210 cases such as the ones presented in Section 3. There-211 fore, a new spread formula is proposed here based on 212 the distance from the centroid point. 213

2194. 3 .Property 1 :
Properties of spread method 220 The relevant properties considered for justifying the 221 spread in ranking fuzzy numbers depend on the useful-222 ness within the domain of research and the list of these 223 properties can be extended further.The applicability of 224 the proposed spread method in ranking fuzzy numbers 225 is illustrated using the following properties.Let A and B be trapezoidal and triangular generalized fuzzy numbers respectively.If A and B are embedded and having similar core, then s(A) > s(B).Proof: Since A and B are embedded and having similar core, hence we know that x A = x B and y A > y B .Then, from equation (1) we have i A = i B and ii A > ii B .Therefore, s(A) > s(B).

Figure 1 Property 2 :
Figure 1 is the best representation of this property.Property 2: If A is a vertical fuzzy numbers, then s(A) = 0. Proof: For any crisp (real) numbers, we know that a 1 = a 2 = a 3 = a 4 implies that i A = 0 and ii A = w/3.Therefore, s(A) = 0. Property 3: If A is an asymmetrical triangular fuzzy numbers then s(A) = i A × ii A .Proof: For any asymmetrical triangular fuzzy numbers, it is obvious that a 2 = a 3 / = x A .
H.L. Kwang and J.H. Lee, A method for ranking fuzzy numbers 399 and its application to decision-making, IEEE Transactions on If CPS (A) < CPS (B), then A ≺ B. (i.e.A is ranked lower than B).If CPS (A) = CPS (B), then A ≈ B. (i.e. the ranking for A and B is equal).obtained is in line with [9, 12, 17, 22, 26] where the 346 ranking order is consistent with human intuition.fuzzy sets, IEEE World Congress on Computational Intelli-397 gence 1 (1998), 773-778.398 [4]