A New Distance Measure for Trapezoidal Fuzzy Numbers

Abstract

We propose a new distance measure for the space of all trapezoidal fuzzy numbers using centroid point and left/right spread of trapezoidal fuzzy numbers. Moreover, the metric properties of suggested distance measure are investigated. Indeed, we show that for two arbitrary trapezoidal fuzzy numbers if the distance between centroid points and also the distance between left spreads and right spreads go to zero, then two given fuzzy numbers are equal. Consequently, we complete discussion about the relation between fuzzy number and its centroid which is the firstly discussed by Hadi-Vencheh and Allame (2010). To the best of our knowledge, this is first time in the literature that such metric is applied by centroid point.

1. Introduction

Fuzzy set theory [1] has been applied to many areas which need to manage uncertain and vague data. Such areas include approximate reasoning, decision making, optimization, and control, where ranking of fuzzy numbers is an important component of the decision process [2, 3].

There are several metrics to state the distance between fuzzy numbers. However, most of them have used the parametric form of fuzzy numbers or membership functions of given fuzzy numbers ([4–6] and references therein). But, in this paper we adopt a new approach to build a distance measure. To do so, we applied COG point to construct such distance measure. Before we proceed, we state some historical point of view about it.

Recently, Hadi-Vencheh and Allame [7] have suggested a relation between trapezoidal fuzzy numbers and their centroid formulas. To this end, they have solved the related linear system to characterize the components of each of the trapezoidal fuzzy numbers explicitly. However, they did not investigate the uniqueness of problem in this system. In other words, if the centroids and left/right spreads of two arbitrary trapezoidal fuzzy numbers are equal, do we conclude that these fuzzy numbers are identical?

To answer this essential question, we proposed a new distance measure for the space of trapezoidal fuzzy numbers, and metric properties of it are also stated.

The paper is organized as follows. In Section 2, we state some basic concepts. In Section 3, the centroid formula for trapezoidal fuzzy numbers is stated and also some required expressions are recalled using Hadi-Vencheh and Allame’s paper [7]. In Section 4, we propose our new distance measure using centroid formula and left/right spreads. Moreover, the metric properties of proposed distance are discussed. Finally, the paper ends with conclusion.

2. Basic Concepts

In this section, some basic concepts are given.

In the fuzzy theory, triangular and trapezoidal fuzzy numbers are used extensively [3], whose membership functions are, respectively, defined by 𝜇_𝐴(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂,𝑎₃−𝑥𝑎₃−𝑎₂,𝑎₂≤𝑥≤𝑎₃,𝜇0,otherwise,_𝐵(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂1,𝑎₂≤𝑥≤𝑎₃,𝑎₄−𝑥𝑎₄−𝑎₃,𝑎₃≤𝑥≤𝑎₄,0,otherwise. (1) For brevity, triangular and trapezoidal fuzzy numbers are often denoted as 𝐴=(𝑎₁,𝑎₂,𝑎₃) and 𝐵=(𝑎₁,𝑎₂,𝑎₃,𝑎₄). Clearly, triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers with 𝑎₂=𝑎₃.

An equivalent parametric definition is also given in [8] as follows.

Definition 1. A fuzzy number 𝑢 in parametric form is a pair (𝑢,𝑢) of functions 𝑢(𝑟), 𝑢(𝑟),0≤𝑟≤1, which satisfy the following requirements: (1)𝑢(𝑟) is a bounded nondecreasing left continuous function in (0,1] and right continuous at 0;(2)𝑢(𝑟) is a bounded nonincreasing left continuous function in (0,1] and right continuous at 0;(3)𝑢(𝑟)≤𝑢(𝑟),0≤𝑟≤1.

According to Zadeh’s extension principle, the operation of addition on 𝔼 (the space of all fuzzy numbers) is defined by (𝑢+𝑣)(𝑥)=sup_𝑦∈ℝmin𝑢𝑦,𝑣𝑥−𝑦,𝑥∈ℝ, (2) and scalar multiplication of a fuzzy number is given by (𝑥)=𝑢𝑥𝑘⊙𝑢)(𝑘,𝑘>0,_∼0,𝑘=0, (3) where 0^∼∈𝔼.

3. Centroid Formula for Trapezoidal Fuzzy Number

Let us consider 𝐴 to be a given trapezoidal fuzzy number such that 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄); then the centroid point of 𝐴 is obtained by Wang et al. [9] by COG(𝐴)=𝛼_𝐴,𝛽_𝐴, (4) where 𝛼_𝐴=13𝑎₁+𝑎₂+𝑎₃+𝑎₄−𝑎₄𝑎₃−𝑎₁𝑎₂𝑎₄+𝑎₃−𝑎₁+𝑎₂,𝛽_𝐴=13𝑎1+³−𝑎₂𝑎₄+𝑎₃−𝑎₁+𝑎₂. (5)

Hence, for any triangular fuzzy number with a piecewise linear membership function, its centroid can be derived by 𝛼_𝐴=𝑎₁+𝑎₂+𝑎₄3,𝛽_𝐴=13. (6) A briefly review about the relation between trapezoidal fuzzy number and its centroid formula is given.

Let us consider the left and right spreads, (𝐿_𝐴,𝑅_𝐴), and COG point (𝛼_𝐴,𝛽_𝐴) of unknown fuzzy number 𝐴 is given. The goal is determining such unknown fuzzy number using mentioned information. To do this, we should solve the following linear system: 𝑎₂−𝑎₁=𝐿_𝐴,𝑎₄−𝑎₃=𝑅_𝐴,𝛼_𝐴=13𝑎₁+𝑎₂+𝑎₃+𝑎₄−𝑎₄𝑎₃−𝑎₁𝑎₂𝑎₄+𝑎₃−𝑎₁+𝑎₂,𝛽_𝐴=13𝑎1+³−𝑎₂𝑎₄+𝑎₃−𝑎₁+𝑎₂. (7) However, in [7], the authors proved that system (7) is linear. To this end, Hadi-Vencheh and Allame [7] have obtained the elements 𝑎₁,…,𝑎₄ using the following formula explicitly: 𝑎₁=𝛼_𝐴+𝑏²₀−𝑐²₀−𝑑²₀−𝑐₀𝑑₀3𝑐₀+𝑑₀−𝑏₀𝑎₂=𝑎₁+𝑏₀,𝑎₃=𝑎₁+𝑐₀,𝑎₄=𝑎₁+𝑑₀, (8) where 𝑏₀=𝐿_𝐴,𝑐₀=3𝛽_𝐴−1𝑅_𝐴+2−3𝛽_𝐴𝐿_𝐴31−2𝛽_𝐴,𝑑₀=𝑐₀+𝑅_𝐴. (9) Now, we provide some results which will be useful to prove the triangular inequality for proposed distance measure (12). The proofs are so easy, so we omitted them.

Lemma 2. Let one consider the real values 𝑎_𝑖, 𝑖=1,…,6 such that 𝑎₁≤𝑎₂, 𝑎₃≤𝑎₄, and 𝑎₅≤𝑎₆. Then, one has max𝑎₁,𝑎₃,𝑎₅≤max𝑎₂,𝑎₄,𝑎₆. (10)

Lemma 3. Let one consider the real values 𝑎_𝑖, 𝑖=1,…,6. Then, one has max𝑎₁+𝑎₂,𝑎₃+𝑎₄,𝑎₅+𝑎₆≤max𝑎₁,𝑎₃,𝑎₅+max𝑎₂,𝑎₄,𝑎₆. (11)

4. Proposed Distance Measure for Trapezoidal Fuzzy Numbers

In this section, we propose a new distance measure for trapezoidal fuzzy numbers using their centroids and left/right spread.

Definition 4. Let one consider two trapezoidal fuzzy numbers 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄) and 𝐵=(𝑏₁,𝑏₂,𝑏₃,𝑏₄) with centroid points (𝛼_𝐴,𝛽_𝐴) and (𝛼_𝐵,𝛽_𝐵), left/right spreads (𝐿_𝐴,𝑅_𝐴) and (𝐿_𝐵,𝑅_𝐵), respectively. Then, the distance measure between two mentioned fuzzy numbers is as follows: 𝑑(𝐴,𝐵)=max𝛼_𝐴−𝛼_𝐵,𝛽_𝐴−𝛽_𝐵,𝐿_𝐴−𝐿_𝐵,𝑅_𝐴−𝑅_𝐵. (12)

Now, we state the metric properties of 𝑑 proposed by (12).

Theorem 5. Let one consider distance measure (12) under the assumptions of Definition 4. Then, one has (1)𝑑()(2)𝑑()(),(3)𝑑()(4)𝑑()()(),𝐴,𝐵≥0,𝐴,𝐵=𝑑𝐵,𝐴𝐴,𝐵=0⟺𝐴=𝐵,𝐴,𝐶≤𝑑𝐴,𝐵+𝑑𝐵,𝐶 (13) for all arbitrary trapezoidal fuzzy numbers 𝐴, 𝐵, and 𝐶.

Proof. The proofs of the first and the second properties are an easy work, so we omitted them. Now, we prove the third property. Let us consider 𝐴=𝐵; then we deduce that 𝛼_𝐴=𝛼_𝐵, 𝛽_𝐴=𝛽_𝐵, and 𝐿_𝐴=𝐿_𝐵, 𝑅_𝐴=𝑅_𝐵, and then using (12) we obtain 𝑑(𝐴,𝐵)=0. Conversely, Let us consider 𝑑(𝐴,𝐵)=0; then using (12) we have max𝛼_𝐴−𝛼_𝐵,𝛽_𝐴−𝛽_𝐵,𝐿_𝐴−𝐿_𝐵,𝑅_𝐴−𝑅_𝐴=0, (14) which leads to obtain 𝛼_𝐴−𝛼_𝐵𝛽=0,_𝐴−𝛽_𝐵𝐿=0,_𝐴−𝐿_𝐵𝑅=0,_𝐴−𝑅_𝐵=0. (15) Hence, we have 𝛼_𝐴=𝛼_𝐵,𝛽_𝐴=𝛽_𝐵,𝐿_𝐴=𝐿_𝐵,𝑅_𝐴=𝑅_𝐵. (16) Using obtained result (16), we prove that 𝑎_𝑖=𝑏_𝑖, 𝑖=1,2,3,4 which completes the proof of this part.
Now, we use the following notations: 𝑏₂=𝑏₁+𝑓₀,𝑏₃=𝑏₁+𝑔₀,𝑏₄=𝑏₁+ℎ₀. (17) Similar to the results obtained for 𝑎₁,…,𝑎₄, we have 𝑓₀=𝐿_𝐵, ℎ₀=𝑔₀+𝑅_𝐵, and 𝑔₀=3𝛽_𝐵−1𝑅_𝐵+2−3𝛽_𝐵𝐿_𝐵31−2𝛽_𝐵. (18) Obviously, using (16), we deduce that 𝑔₀=𝑐₀. Moreover, since 𝑑₀=𝑐₀+𝑅_𝐴 and ℎ₀=𝑔₀+𝑅_𝐵; we derive 𝑑₀=ℎ₀. Also, 𝑏₀=𝐿_𝐴 and 𝑓₀=𝐿_𝐵, using (16) we obtain 𝑏₀=𝑓₀. So, using obtained results we have 𝑏²₀−𝑐²₀−𝑑²₀−𝑐₀𝑑₀3𝑐₀+𝑑₀−𝑏₀=𝑓²₀−𝑔²₀−ℎ²₀−𝑔₀ℎ₀3𝑔₀+ℎ₀−𝑓₀. (19) Using 𝑎₁=𝛼_𝐴+𝑏²₀−𝑐²₀−𝑑²₀−𝑐₀𝑑₀3𝑐₀+𝑑₀−𝑏₀,𝑏₁=𝛼_𝐵+𝑓²₀−𝑔²₀−ℎ²₀−𝑔₀ℎ₀3𝑔₀+ℎ₀−𝑓₀, (20)(21) and (16), we deduce that 𝑎₁=𝑏₁.
Now, we demonstrate that 𝑎_𝑖=𝑏_𝑖, 𝑖=2,3,4. Since, 𝑎₂=𝑎₁+𝐿_𝐴 and 𝑏₂=𝑏₁+𝐿_𝐵. Using the fact that 𝐿_𝐴=𝐿_𝐵 and 𝑎₁=𝑏₁, we deduce that 𝑎₂=𝑏₂. Moreover, 𝑎₃=𝑎₁+𝑐₀ and 𝑏₃=𝑏₁+𝑔₀. Then, using the fact that 𝑐₀=𝑔₀ and 𝑎₁=𝑏₁ we obtain 𝑎₃=𝑏₃. Finally, since 𝑑₀=ℎ₀, we derive that 𝑎₄=𝑎₁+𝑑₀ and 𝑏₄=𝑏₁+ℎ₀ are equal. The proof of this part is completed.
Now, we prove the triangular inequality for the proposed distance measure (12). Let us consider three trapezoidal fuzzy numbers 𝐴=(𝑎₁,𝑎₂,𝑎₃,𝑎₄), 𝐵=(𝑏₁,𝑏₂,𝑏₃,𝑏₄), and 𝐶=(𝑐₁,𝑐₂,𝑐₃,𝑐₄). Then, using Lemmas 2 and 3 and (12), we have 𝑑()𝛼𝐴,𝐶=max_𝐴−𝛼_𝐶,𝛽_𝐴−𝛽_𝐶,𝐿_𝐴−𝐿_𝐶,𝑅_𝐴−𝑅_𝐶≤max𝛼_𝐴−𝛼_𝐵+𝛼_𝐵−𝛼_𝐶,𝛽_𝐴−𝛽_𝐵+𝛽_𝐵−𝛽_𝐶,𝐿_𝐴−𝐿_𝐵+𝐿_𝐵−𝐿_𝐶,𝑅_𝐴−𝑅_𝐵+𝑅_𝐵−𝑅_𝐶≤max𝛼_𝐴−𝛼_𝐵,𝛽_𝐴−𝛽_𝐵,𝐿_𝐴−𝐿_𝐵,𝑅_𝐴−𝑅_𝐵+max𝛼_𝐵−𝛼_𝐶,𝛽_𝐵−𝛽_𝐶,𝐿_𝐵−𝐿_𝐶,𝑅_𝐵−𝑅_𝐶=𝑑(𝐴,𝐵)+𝑑(𝐵,𝐶), (22) which completes the proof.

Remark 6. Indeed, we show that if the difference between centroids and also the difference between left spreads and right spreads of two given trapezoidal fuzzy numbers go to zero, we can conclude that the mentioned fuzzy numbers are identical. To the best of our knowledge, this is the first time in the literature that each fuzzy number is characterized using its centroid formula and its left/right spread uniquely. Also, our discussion completes Hadi-Vencheh and Allame’s paper [7] about the relation between fuzzy number and its centroid formula.

5. Conclusion

Recently, Hadi-Vencheh and Allame [7] have proposed the relation between trapezoidal fuzzy number and its centroid formula. To do so, they proved that one can easily obtain such relation by solving linear system (7). However, they did not consider the uniqueness of solution. In other words, if the centroids and left/right spreads of two trapezoidal fuzzy numbers are equal, how can we ensure that these fuzzy numbers are identical? This paper answered this question.

So, we proposed a new distance measure based on the differences between centroids and left/right spreads. Consequently, we proved the metric properties of it. To the best of our knowledge, this is the first time in the literature that such construction is applied. Indeed, the uniqueness of solution in this area is so important which has not been investigated yet.

For future research, we will apply (12) to obtain some new similarity measures and ranking fuzzy numbers [10–16] and risk analysis [17] and also will use it to determine the nearest approximation of fuzzy numbers [18].