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Article Type: Original Research

Ranking the Generalized Fuzzy Numbers Based on the Center of the Area

Subject Areas : International Journal of Decision Intelligence

Vahid Mohammadi ¹ , Esmaeil Mehdizadeh ² , Seyed Mojtaba Hejazi ³

1 - Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran
2 - Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran
3 - Department of Informatics Engineering, The International University of Valencia (VIU), Valencia, Spain

Received: 2023-10-28 Accepted : 2024-04-28 Published : 2024-09-22

Keywords: Generalized fuzzy numbers, Ranking fuzzy numbers, Center of area, Decision making, Uncertainty,

Abstract :

In decision-making contexts marked by uncertainty, the application of fuzzy numbers has emerged as a crucial tool. These numbers offer a mathematical framework for representing imprecise information, enabling a more nuanced approach to decision-making. Fuzzy numbers find widespread application in quantifying the inherent uncertainty present in decision-making contexts. When incorporating fuzzy numbers into decision-making procedures, the necessity to compare these fuzzy numbers becomes an unavoidable occurrence. Ranking fuzzy numbers is a challenging topic. In this paper, we propose a new method for ranking generalized fuzzy numbers based on the center of the area concept. First, we present the concepts of the presented method. Additionally, the proposed method can rank symmetric fuzzy numbers relative to the y-axis easily. Then the advantages of the proposed method are illustrated through several numerical examples. The results demonstrate that this approach is effective for ranking generalized fuzzy numbers and overcomes the shortcomings in recent studies. Finally, we checked the result of the presented method with other existing methods. The results show that the presented method has consistent results with less computational complexity.

References:

Full-Text:

Ranking the Generalized Fuzzy Numbers Based on the

Center of the Area

Vahid Mohammadia, Esmaeil Mehdizadeha,* Seyed Mojtaba Hejazib

a Department of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran

b Department of Informatics Engineering, The International University of Valencia (VIU), Valencia, Spain

Received 28 October 2023; Accepted 28 April 2024

Abstract

Keywords: Generalized fuzzy numbers; Ranking fuzzy numbers; Center of area; Decision making; Uncertainty

1.Introduction

While the centroid-based ranking approach has contributed valuable insights, it faces critiques that have spurred further exploration. Alternative ranking indices, such as those centered on the area between the centroid point and the original value, have been proposed to provide complementary perspectives on the importance of fuzzy numbers. These indices aim to capture the spatial relationship between the centroid and the original point, offering potential refinements to the ranking process. However, the implications of these methodologies in complex, real-world decision-making scenarios necessitate careful scrutiny. Additionally, considerations of the relative weights of horizontal and vertical components in the ranking process underscore the dynamic nature of these methodologies, driving ongoing efforts to enhance the effectiveness of generalized fuzzy number ranking. The process of ranking generalized fuzzy numbers based on the center of the area, while a valuable approach in decision-making under uncertainty, is not without its limitations. One notable concern arises from the tendency of this method to yield identical rankings for fuzzy numbers and their corresponding inverse images. This inherent limitation potentially hinders its practical applicability in complex decision-making scenarios. Furthermore, there are debates regarding the appropriate weighting of horizontal and vertical components in the ranking process, which can significantly impact the outcomes. Recognizing these drawbacks, there is a pressing need to address and eliminate these limitations to refine the methodology and enhance its effectiveness in real-world applications. This pursuit of refinement is crucial for advancing the field of generalized fuzzy number ranking and ensuring its practical utility in a wide range of decision-making contexts.The subsequent sections of this paper are structured as follows: Section 2 includes a literature review and the suggested drawbacks. Section 3 provides a concise overview of fundamental concepts and crucial definitions pertinent to our discourse. Section 4 outlines the centroid point method and expounds on its limitations. In Section 5, a novel approach for ranking generalized fuzzy numbers, grounded in the center of area principle, is introduced. Additionally, we offer numerical examples to demonstrate the merits of this proposed methodology. Lastly, Section 6 encapsulates the conclusion drawn from our findings.

2.Literature Review

In many instances, the data used for decision-making is only known approximately [1]. In 1965, Zadeh [2] introduced the concept of fuzzy set theory to address this issue. The ranking of fuzzy numbers holds significant importance within fuzzy set theory, decision-making processes, data analysis, and practical applications [3]. Jain explored methods for comparing and ranking fuzzy numbers in 1976 [4], employing the notion of maximizing sets for ordering them. Since then, various techniques for ranking quantities have been proposed by different researchers. Yager was the pioneer in utilizing the concept of the centroid for ranking fuzzy numbers in [5], where he employed the horizontal coordinate of the centroid point, , as the ranking index. However, this method does not accurately rank fuzzy numbers when is the same for different fuzzy numbers but their values differ, where represents the vertical coordinate of the centroid point of the fuzzy number [6]. Cheng [7] introduced a centroid index ranking method, which involves computing the distance between the centroid point of each fuzzy number and the original point, aiming to enhance Yager's [5] approach.

Cheng's method was not without its flaws. In this approach, the ranking of fuzzy numbers and corresponds to the ranking of their respective images, i.e., and (refer to Example 1). To address this limitation, Chu and Tsao [8] introduced the use of the area between the centroid point and the original point as a ranking index for fuzzy numbers. In this scheme, a larger area signifies a higher rank for the corresponding fuzzy number.

Wang and Lee [9] argued that in the method proposed by Chu and Tsao [8], the multiplication of values on the horizontal and vertical axes often diminishes the significance of the horizontal axis in fuzzy number ranking. They initially advocated for as a ranking criterion. Specifically, a larger corresponds to a higher-ranked fuzzy number, and in cases where is equal for two fuzzy numbers, should be utilized.

Wang et al. [10] illustrated that Wang and Lee's [9] approach was unable to distinguish between two fuzzy numbers sharing the same centroid point. They introduced the L-R deviation degree of a fuzzy number and proposed a ranking rule: the greater the left deviation degree and the smaller the right deviation degree, the higher the rank of the fuzzy number.

Nejad and Mashinchi [6] contended that the approach presented by Wang et al. [10] was not capable of correctly ranking fuzzy numbers in instances where either the left deviation degree, right deviation degree, or transfer coefficient of the fuzzy number is zero, or the transfer coefficient is one. They opted to use the areas on the left and right sides of fuzzy numbers for ranking. However, it's worth noting that this method fails to yield accurate rankings for symmetric fuzzy numbers, resulting in identical ranking orders [11].

Abbasbandy and Hajjari [12] proposed a ranking method for fuzzy numbers based on the center of gravity. Their approach is an adaptation of Wang and Lee's method [9]. Allahviranloo and Saneifard [13] utilized the concept of the center of gravity for the defuzzification of fuzzy numbers. Unfortunately, both Abbasbandy and Hajjari's method [12] and Allahviranloo and Saneifard's method [13] face difficulties in ranking symmetric fuzzy numbers relative to the y-axis.

We provide a comprehensive overview of various approaches to ranking fuzzy numbers, a critical aspect in decision-making under uncertainty. Overall, this literature review provides a comprehensive survey of the historical evolution, critiques, and refinements in the field of ranking fuzzy numbers, offering valuable insights for researchers and practitioners alike. To address the aforementioned limitations, this research presents a novel centroid point method that incorporates distance and area considerations for ranking fuzzy numbers.

3.Preliminaries

In this section, we brieﬂy give some basic notions and important deﬁnitions that are related to our discussion.

Definition 1. A fuzzy subset of the real line with membership functions onis called a fuzzy number if [14]:

(a) is normal, i.e., there exists an element such that,

(b) is fuzzy convex, i.e.,,

(d) is bounded, where and is a closure operator.

Definition 2. A fuzzy set, defined on the universal set of real numbers , is said to be a generalized fuzzy number if its membership function has the following characteristics [15]:

1. is continuous,

2. for all ,

3. strictly increasing on and strictly decreasing on,

4., for all, where.

Definition 3. A generalized fuzzy numberis said to be a generalized trapezoidal fuzzy number if its membership function is given by [16], as shown in Fig. 1:

Fig 1. Generalized trapezoidal fuzzy number

Definition 4. A generalized fuzzy number where is called a generalized triangular fuzzy number if its membership function is given by [16], as shown in Fig. 2:

Fig 2. Generalized triangular fuzzy number

Fig 3. Nonnegative fuzzy number

Definition 6.A Fuzzy number is called non-positive, if for all,[17],as shown in Fig. 4:

Fig 4.Non-positive fuzzy number

Definition 5. A Fuzzy number is called non-negative if, for all[17] as shown in Fig3:

Definition 7. The area is the size or the magnitude of a two-dimensional shape. The entire surface or the whole floor of any geometric shape of a fuzzy number such

as can be calculated as follows [18]:

(1)

4.A Review of the Centroid Point Method

Definition 8. Assume there are n fuzzy numbers where.The centroid point of a fuzzy number corresponded to a value on the horizontal axis and a value on the vertical axis. The centroid point of a fuzzy number was deﬁned as [7], [19]:

where and where the left and right membership functions of respectively, and and were inverse functionsof and respectively. For a non-normal trapezoidal fuzzy numberformulas lead to the following results, respectively [12]:

(2)

(3)

(4)

Definition 9. Suppose is the center of gravity for the desired fuzzy number. In this case, the distance of the center of gravity from the origin (original point) is obtained as follows [7]:

(5)

By using given in (6), Cheng [7] can rank fuzzy numbers. The considered the larger the value of , the better the ranking of .

Definition 10.The area between the centroid point) and original point of the fuzzy number was deﬁned as [8]:

(6)

(7)

Chu and Tsao [8] ranked fuzzy numbers according to the area covered. They considered the larger the value of , the better the ranking of . Wang and Lee [9] proposed a revised method based on the Cho and Tsao’s [8] method. They ranked the fuzzy numbers based on their values if they are different. In instances where they exhibit equality, a further comparison is conducted based on their respective ‘s values to establish their ranking.

Definition 11. Let E stand the set of non-normal fuzzy numbers, W be a constant provided that and be a function that is defined as [12]:

Abbasbandy and Hajjari[12] used to rank fuzzy numbers. They considered the larger the value of , the better the ranking of .

Definition 12. Assume there are fuzzy numbers. The maximum crisp valueis deﬁned as [13]:

i.e.,

(8)

Definition 13.The value of the fuzzy numbers , where, is defined as[13]:

(9)

Now, by using given in (9), for any two fuzzy numbers and , Allahviranloo and Saneifard [12] order are determined based on the following rules:

· if, and only if,

· if, and only if ,

· if, and only if .

Despite many efforts, most of the approaches with centroid points still have shortcomings. Now, the deficiency of the centroid point method is analyzed and discussed through the following examples.

Example 1. Consider fuzzy numbers, , as shown in Fig. 5.

According to Cheng’s method[7], we have:

(10)

This method cannot give a reasonable ranking for fuzzy numbers and their images.

Fig 5. Fuzzy numbers, , , and in Example 1

Example 2. Consider fuzzy numbers, adopted from[9], as shown in Fig. 6.

Capously, is smaller than. However, the ranking outcome by Chu and Tsao’s [8] method is contrary to one’s intuitions [9].

Fig 6.Fuzzy numbers and in Example 2

Example 3. Consider fuzzy numbers, adopted from[10], as shown in Fig. 7. According to Wang and Lee’s [9]method, we have:

This approach cannot differentiate two fuzzy numbers with the same centroid point [10].

Fig 7. Fuzzy numbers and in Example 3

Example 4. Consider fuzzy numbers, adopted from[9], as shown in Fig. 7. The ranking orders reported by Wang et al. [10] and NejadandMashinchi[6] are and , which is illogical[11].

Wang et al.’s [10] and Nejad and Mashinchi’s methods [6] cannot produce the correct ranking order under symmetric fuzzy number circumstances, and their methods lead to the same ranking orders.

Example 5. Let, be two triangular fuzzy numbers, as shown in Fig.8. The ranking orders reported by Abbasbandy and Hajjari [12] and Allahviranloo and Saneifard [13] are . Thus, these methods cannot rank symmetric fuzzy numbers relative to the y-axis.

Fig 8.Fuzzy numbers and in Example 5

5.Ranking Fuzzy Numbers Based on the Center of the Area

This section introduces a new method for ranking generalized fuzzy numbers. By this method, we will resolve the shortcomings discussed in the previous section by proposing the following definition.

Definition 14. If an area lies in theplane and is bounded by the curve, as shown in Fig.9, then its centroid will be in this plane, and we obtain formulas located in the center of an area, namely, [20]:

and

(11)

These integrals can be evaluated by performing a single integration if we use a rectangular strip for the differential area element. For example, if we consider a horizontal strip, Fig.10, then, and its centroid is located atand.

Fig 9.An area bounded by a curve in definition 8., [20]

Fig 10.A horizontal strip in definition 8, [20]

Definition 15. A composite structure comprises interconnected elementary bodies, typically of rectilinear or right triangular configuration. Such a structure can frequently be decomposed into its constituent parts, with known weights and respective centroids. This information obviates the necessity for integration in the computation of the overall center of gravity of the composite body. [20].

Therefore, the center of gravity for the entire composite shape can be calculated as follows:

(12)

Note: the above formulas can also be used to get the area center of the compound shapes [20].

Theorem 1. Suppose b is the length, and h is the height of a given rectangle (Fig. 11). The center of the rectangle area is equal to the intersection of two diameters, and its coordinates are as follows:

Fig11. Rectangular center area in Theorem 1

It is known that , , and . Then:

Proof:

(13)

Theorem 2. Suppose h is the height of the right triangle, and is its base (Fig. 12). In this case, the center coordinates are as follows:

(14)

Fig 12: Right triangle area center in Theorem 2

Proof: It is known that, , and . Then:

Note: is valid for any shape of the triangle [20].

Theorem 3: Now, assume that the right triangle (Fig. 13) is as follows:

Fig. 13: Right triangle area center in Theorem 3.

It is known that , , and . Then:

Now, we propose the algorithm for ranking the fuzzy numbers, in, based on the center of the area.

Algorithm:

Step 1: The first step is to calculate the area center of each generalized fuzzy number. Depending on whether the fuzzy number is nonnegative or non-positive, the following is applied:

(15)

(16)

Fig. 14: Nonnegative fuzzy number

Fig 15. Non-positive fuzzy number.

Where is the area center of the fuzzy number that can be calculated by using the formulas mentioned in equation (11), and is the least distance between the shapes of the fuzzy number from the origin of the coordinates. If a generalized fuzzy number is nonnegative then (Figure 14) and if a generalized fuzzy number is non-positive then (Figure 15).

Consider the scenario where the contour of the fuzzy number intersects the y-axis, implying that the fuzzy number is not strictly nonnegative or non-positive. Under such circumstances, the computation of the fuzzy number's area center proceeds as follows:

1) Divide the geometric contour of the fuzzy number into two distinct segments: the positive section and the negative section

2) Calculate the area of the positive section and the negative section.

3) Calculate the area center of the positiveand negative sections. For this purpose, divide the corresponding shape into simple shapes (right triangle and rectangles), calculate the area center of each shape, and finally obtain the area center of the positive and negative sections by using Equation 12.

4) The area center of the entire shapeis obtained by using the formulas in Definition 9 as follows:

(17)

* Corresponding Author. Email: emehdi@qiau.ac.ir

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Ranking the Generalized Fuzzy Numbers Based on the Center of the Area