Similarity Measures of Sequence of Fuzzy Numbers and Fuzzy Risk Analysis

Wepresent themethods to evaluate the similaritymeasures between sequence of triangular fuzzy numbers formaking contributions to fuzzy risk analysis. Firstly, we calculate the COG (center of gravity) points of sequence of triangular fuzzy numbers. After, we present the methods to measure the degree of similarity between sequence of triangular fuzzy numbers. In addition, we give an example to compare the methods mentioned in the text. Furthermore, in this paper, we deal with the (t 1 , t 2 ) type fuzzy number. By defining the algebraic operations on the (t 1 , t 2 ) type fuzzy numbers we can solve the equations in the form x+u (t1 ,t2) = V (t1 ,t2) , where


Preliminaries, Background, and Notation
The concept of fuzzy sets and fuzzy set operations were first introduced by [1].After his innovation many authors have studied various aspects of the fuzzy set theory and its applications, such as fuzzy topological spaces, similarity relations, and fuzzy mathematical programming.Matloka [2] introduced the class of bounded and convergent sequences of fuzzy numbers with respect to the Hausdorff metric.In [3], Nanda has studied the spaces of bounded and convergent sequences of fuzzy numbers and shown that these spaces are complete metric spaces.
Measuring the similarity between sequences of fuzzy numbers is very important subject of fuzzy decision making [4,5] and fuzzy risk analysis [6].In [7], a method for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers is given.In [8], a method for measuring the rate of aggregative risk in software development is presented.Recently, some methods have been introduced to calculate the degree of similarity between fuzzy numbers [4][5][6].However, these measures cannot determine the degree of similarity between sequences of fuzzy numbers.By generalizing these methods from fuzzy numbers to sequence of triangular fuzzy numbers, we give the methods to evaluate the similarity measures of sequence of triangular fuzzy numbers.In this paper, we also calculate the COG (center of gravity) points of a sequence of triangular fuzzy numbers.
It will not be right to regard this paper as a copy of classic summability theory because both a big generalization and definitions of fuzzy zero are presented in this paper.Therefore, the readers are advised to take these into consideration while reading the paper.Some important problems on sequence spaces of fuzzy numbers can be ordered as follows: (1) construct a sequence space of fuzzy numbers and compute -dual, -dual, and dual, (2) find some isomorphic spaces of it, (3) give some theorems about matrix transformations on sequence space of fuzzy numbers, and (4) study some inclusion problems and other properties.
In the present paper, we will define matrix domain of sequence spaces of triangular fuzzy numbers and introduce the sequence spaces of fuzzy numbers [ℓ ∞ (  )] B(r,ŝ) , [(  )] B(r,ŝ) , and [ 0 (  )] B(r,ŝ) .Additionally, we redefine fuzzy identity elements according to addition and multiplication for constructing an algebraic structure on the ( 1 ,  2 ) type sets of fuzzy numbers.
The rest of our paper is organized as follows.
In Section 2, some basic definitions and theorems related to the fuzzy numbers are given.In Section 3, we have introduced generalized difference sequence space of triangular fuzzy numbers and proved some inclusion relations on these sequence spaces.It is also established in Section 3 that the sequence spaces of triangular fuzzy numbers showed by [ℓ ∞ (  )] B(r,ŝ) , [(  )] B(r,ŝ) , and [ 0 (  )] B(r,ŝ) are linearly isomorphic to the spaces ℓ ∞ (  ), (  ), and  0 (  ), respectively.Finally, in Section 3, it is proved that the spaces [ℓ ∞ (  )] B(r,ŝ) , [(  )] B(r,ŝ) , and [ 0 (  )] B(r,ŝ) are complete.Section 4 is devoted to the calculation of the ()and ()-duals of the spaces [(  )] B(r,ŝ) and [ 0 (  )] B(r,ŝ) .In Section 5, some classes of matrix transformations from the space [(  )] B(r,ŝ) and (  ) to (  ) and [(  )] B(r,ŝ) are characterized, respectively, where (  ) is any sequence space.In Section 6, we review the COG points of a sequence of fuzzy numbers [7] and existing similarity measures of fuzzy numbers [4][5][6][7].In Section 7, we generalize the use of similarity measure methods to sequence of triangular fuzzy numbers.Furthermore, we use an example to compare the similarity measure methods between three sets of sequences of triangular fuzzy numbers.In the final section, we consider the similarity measure methods of sequence of triangular fuzzy numbers to deal with the fuzzy risk analysis problems.
In this section, we recall some of the basic definitions and notions in the theory of fuzzy numbers.Let us suppose that N, R, and   = { = [ − ,  + ] :  − ≤  ≤  + ,  − and  + ∈ R} are the set of all positive integers, all real numbers, and all bounded and closed intervals on the real line R, respectively.For ,  ∈   define the following metric: It can easily be seen that  defines a metric on   and the pair (  , ) is a complete metric space [9].Let  be a nonempty set.According to Zadeh a fuzzy subset of  is a nonempty subset {(, u()) :  ∈ } of  × [0, 1] for some function u :  → [0, 1] [10].Consider a function u : R → [0, 1] as a subset of a nonempty base space R and denote the family of all such functions or fuzzy sets by .Let us suppose that the function u satisfies the following properties: (1) u is normal, that is, there exists an  0 ∈ R such that u( 0 ) = 1.
(2) u is fuzzy convex, that is, for any ,  ∈ R and  ∈ (3) u is upper semicontinuous.
Then, the function u is called a fuzzy number [11].
The properties (1)-( 4) imply that, for each  ∈ [0, 1], the -cut set of the fuzzy number u defined by u() = { ∈ R : u() ≥ } is in   .That is, the equality u() = [u − (), u + ()] holds for each  ∈ [0, 1].We denote the set of all fuzzy numbers by .Any vector subspace of , the space of all complex valued sequences, is called a sequence space.We write ℓ ∞ , ,  0 , and ℓ  for the classical sequence spaces of all bounded, convergent, null, and absolutely -summable sequences, respectively.For simplicity in notation, the summation without limits runs from 0 to ∞, Let us consider any triangular ( 1 ,  2 ) type fuzzy number  ( 1 , 2 ) , as follows.If the function is the membership function of the triangular fuzzy number  ( 1 , 2 ) , then  ( 1 , 2 ) can be represented with the notation where The notations  −   4) is that for every  1 ,  2 ∈ R, there is no unique set of fuzzy numbers.Furthermore, there are infinitely-many sets of fuzzy numbers and these sets are different from each other according to structure of their elements.So, we can use the most appropriate one of these sets for our problem.In this study, we will take  1 ,  2 ∈ [0, 1).Sometimes, the representation of fuzzy numbers with cut sets induces errors according to algebraic operations.For example, if k is any fuzzy number, then is not equal to zero as expected in the classic mean.For avoiding this kind of problems we define fuzzy zero represented by  as in the following section.

Algebraic Structure of the Set 𝐹 𝑡
Let  ( 1 , 2 ) , V ( 1 , 2 ) ∈   and  ∈ R. Let us define addition, substraction, multiplication, division, and scalar multiplication on   , respectively, as follows: where V ( 1 , 2 ) is nonzero fuzzy number and Let is considered as the identity element of   according to operation which is given in (5).Therefore, we say that the inverse of the fuzzy number .From here fuzzy zeros of the sets   are as follows: This says to us that fuzzy zero is different for each element of the set   .
Theorem 1.All sets in the form   are linear spaces according to algebraic operations ( 5) and (9), where The second important matter is the topology on the set   .S ¸engönül [12] has constructed a topology on   by using the metric  :   ×   → R defined as follows: We can easily show that the set   is a complete metric space with the metric .
Clearly, the representation (3) for  ( 1 , 2 ) () is unique, and on the contrary there is a unique  ( 1 , 2 ) () for every − ( 1 , 2 ) = (− −  1 , −, − +  2 ).We know that, generally in the practical applications, the spread of fuzziness should not be very large.So, the value of max{|−V− 1 |, |−V|, |−V+ 2 |} should be as small as possible.Theoretically, the value "max{|−V− 1 |, |− V|, | − V +  2 |} should be small" not necessary but it has to be in practical applications.For example, the "approximately 1" can be taken as 1 ( 1 , 2 ) = (−4 −  1 , 1, 15 +  2 ) but in the applications, generally, "approximately 1" is taken as Furthermore, it must be emphasized that a fuzzy number is determined according to specific processes.For example, let  and  be two different specific systems.Then, if  is "approximately 1" for the system  then "approximately 1" may not be in the same sense for another system .So, algebraic properties of the systems  and  are different.This can be explained as follows.
Let us suppose that the spread of left and right fuzziness of every number  is equal to  in .Then fuzzy zero is equal to (−, 0, ), (0 ≤  < 1) for the system  and this fuzzy zero is unique for the system .
The function  : is called a sequence of triangular fuzzy numbers and is represented by ), [13].Let us denote the set of all sequences of triangular fuzzy numbers by (  ); that is, where    1 ≤   ≤    2 and   ( 1 , 2 ) ∈   for all  ∈ N [14].The    1 ,   , and    2 are called first, middle, and end points of general term of a sequence of fuzzy numbers, respectively.Each subspace of () is called a sequence space of fuzzy numbers.
Now, let (  ) and (  ) be two spaces of triangular fuzzy valued sequences and let A = (  ) be an infinite matrix of positive real numbers   , where ,  ∈ N.Then, we say that A defines a real-matrix mapping from (  ) to (  ) and we denote it by writing A : (  ) → (  ), if for every sequence and the series ∑       1 , ∑      , ∑       2 are convergent for all  ∈ N. By ((  ) : (  )), we denote the class of matrices A such that A : (  ) → (  ).Thus, A ∈ ((  ) : (  )) if and only if the series on the right side of ( 15) are convergent for each  ∈ N and every  ∈ (  ), and we have Let (  ) be a sequence space of triangular fuzzy numbers.Then, the set [(  )] A of sequences of triangular fuzzy numbers, defined as follows, is called the domain of an infinite matrix A in (  ): Let   ,   be nonzero real numbers for each  ∈ N and define the band matrix Here, r = (  ) and ŝ = (  ) are the convergent sequences.
Let us define the sequence of fuzzy numbers  = (  ( 1 , 2 ) ) which will be constantly used as the B(r, ŝ)-transform of a sequence of fuzzy numbers  * = (  ( 1 , 2 ) ); that is, where  −1 ( 1 , 2 ) = ,   ,   ∈ R − {0} for all  ∈ N. Now, we may begin with the following theorem which is essential in the text.
Proof.Since the others can be similarly proved, we consider only the case [ℓ ∞ (  )] B(r,ŝ) ≅ ℓ ∞ (  ).To prove this, we should show the existence of a linear bijection between the spaces [ℓ ∞ (  )] B(r,ŝ) and ℓ ∞ (  ).Consider the transformation defined , with the notation of ( 20 That is,  has the property homogeneity.Thus,  is linear. Let us take any  * ∈ ℓ ∞ (  ) and represent the sequence  * using B −1 (r, ŝ) as follows: where Then, we have That is,  is norm preserving.Consequently, the spaces It means that the matrix B(r, ŝ) is regular.
Proof.Let  * ∈ (  ) and consider the following equality: Which yields that D * ∈ [(  )] B(,) if and only if E * ∈ (  ).This step completes the proof.Now, right here, we give the following propositions which are obtained from Lemma 8 and Theorems 13 and 14.
Proposition 15.Let A = (  ) be an infinite matrix of real numbers.Then one has the following: B(r,ŝ) for each  ∈ N and holds for all  ∈ N.
( B(r,ŝ) also holds for all  ∈ N and Proposition 16.Let A = (  ) be an infinite matrix of real numbers.Then, with   ( 1 , 2 ) ∈   and  ∈ N holds.

Determining the Center of Gravity of Sequence Space of Triangular Fuzzy Numbers and Similarity Measures between Fuzzy Numbers
In this section, we give the COG points of a sequence space of triangular fuzzy numbers by means of [7].Let us suppose that ) is a sequence of triangular fuzzy number.Then, the values  ⋆   and  ⋆   of the COG points of (  ( 1 , 2 ) ) are presented as follows: Based on (54) and (55), we can determine the COG points of a sequence of triangular fuzzy number (  ( 1 , 2 ) ) as ( ⋆   ,  ⋆   ).Let us suppose that u and k are two classical fuzzy numbers where u = (u 1 , u 2 , u 3 ) and k = (k 1 , k 2 , k 3 ); then the degree of similarity  between u and k is calculated by [6] as in the following: where (u, k) ∈ [0, 1].
In [5], Lee proposed a similarity measure for classical fuzzy numbers and used the similarity measure to deal with fuzzy opinions for group decision making, where the degree of similarity (u, k) between the triangular fuzzy numbers u and k can be calculated as follows: 1/ (57) and ‖‖ = max() − min() where  is the universe of discourse.The larger the value of , the higher the similarity between the components of .
In [4], Hsieh and Chen defined a similarity measure method where the degree of similarity (u, k) between classical fuzzy numbers u and k can be given as in the following: where (u 1 +4u 2 + u 3 )/6 and (k 1 +4k 2 + k 3 )/6 represent graded mean integration components of u and k, respectively.In addition to these similarity measure methods, S.-J.Chen and S.-M.Chen [7] introduced a simple center of gravity method represented as SCGM, to determine the center of gravity point of a generalized fuzzy number by using the concept of geometry.Besides, they gave a method to evaluate the degree of similarity between generalized fuzzy numbers.
In the present paper we generalize these formulas to sequence spaces of triangular fuzzy numbers.By this method, we will fill the big gap in the literature.

Similarity Measures between Sequence Spaces of Triangular Fuzzy Numbers
Assume that there are two sequences of triangular fuzzy numbers,  * = (  − 1 ,   ,   + 2 ) and then the degree of similarity ( * , V * ) between the sequences of triangular fuzzy numbers  * and V * can be calculated as in the following: where ( * , V * ) ∈ [0,1] and the notion of ℎ( * ) shows the highest membership degree of fuzzy number   .For all  ∈ N, ℎ( * ) is considered as 1 because of the validity of fuzzy number conditions, through all the text.The function  : () × () → R is called similarity degree between sequences of fuzzy sets  * and V * .If ( * , V * ) = 1, then we say that  * is completely similar to the sequence V * ; if 0 < ( * , V * ) =  < 1, then we say that the sequence  * is -similar to the sequence V * ; if  ≤ 0 we can say that  * is not similar to V * .Now, we introduce another similarity measure method for sequence of triangular fuzzy numbers as follows: We take  = 1,  1 = 0,  2 = 0.6 for examples, through all the text.One of the other similarity measure methods for sequence of triangular fuzzy numbers is given as follows: . (61) In [7], S.-J.Chen and S.-M.Chen introduced a simple center of gravity method to determine the center of gravity point of a generalized fuzzy number by using the concept of geometry.By means of this method, we will generalize this formula to the sequence of fuzzy numbers as follows: where  ⋆   and  ⋆ V  are calculated by (55) and ℎ( * ) = ℎ(V * ) = 1 since the fuzzy number property and (  * ,  V * ) are defined as follows: where   * and  V * are the lengths of the bases of the th place of the sequence of triangular fuzzy numbers  * and V * , respectively, defined as follows by means of [7]: Next, we deal with three sets of sequence spaces of triangular fuzzy numbers to compare similarity measures mentioned above.The calculation results are listed in Table 1.Now, we write the equations of three sets of sequences of triangular fuzzy numbers as in the following: SET I: ( 1 = 0, SET III: We can give a comparison of the computing conclusions of the similarity measures of the above equations with the methods mentioned above as in Table 1.[5] 0.333 0.166 0.5 Chen and Lin's Method [6] 0.6 0.5 0.9 Hsieh and Chen's Method [4]

Fuzzy Risk Analysis Based on the Similarity Measure of Sequence of Triangular Fuzzy Numbers
Fuzzy weighted mean method is used in [26,27] for introducing fuzzy risk analysis.According to [26] for every , subcomponent   is measured by two evaluating items represented as   and   that denote the probability of failure of the subcomponents   and severity of loss of the subcomponents   , respectively, and 1 ≤  ≤ 3. Kangari and Riggs [26] and Schmucker [27] use the linguistic terms ("absolutely high," "fairly-high," "high," "medium," "low," "fairly-low," and "absolutely-low") for determining the values of   and   .In [7], S.-J.Chen and S.-M.Chen use ninemember linguistic terms based on [28,29] to represent the linguistic terms in their paper.We show the linguistic terms and sequence of triangular fuzzy numbers in Table 3 and generalize   and   to sequences (   ) and (   ), respectively.In addition to these, we present the algorithm for fuzzy sequence risk analysis by means of [7] as follows.
Step 1. Use the fuzzy weighted mean method to integrate the evaluating items    and    of sequence of each subcomponent    , where 1 ≤  ≤ , to get the total risk  * of the sequence of component  * = (  ) shown as follows: Step 2. Use the similarity measure (62) to measure the degree of similarity between the sequences of triangular fuzzy number  * and Table 3 includes each linguistic term used for classifying the sequence space of triangular fuzzy numbers.
In the following, we use an example presented in [27].
Example 18.Let us consider that the component   consists of three subcomponents   1 ,   2 ,   3 , as shown in Figure 1.Now, we would like to measure the probability of failure  * of the component  * = (  ).Table 2 shows the linguistic values of the two evaluating items    and    , respectively, similar to [27], where the linguistic values are shown by sequence of triangular fuzzy numbers in Table 3.
In the following, we use the proposed fuzzy risk analysis method to consider the fuzzy risk analysis problem.
By using (62), the degree of similarity between the sequence of triangular fuzzy number  * = (  ) and the linguistic terms shown in Table 2 can be evaluated as follows: ( * , absolutely-low) = 0.1383, ( * , very-low) = 0.1541,

Table 1 :
A comparison between similarity measure methods.

Table 3 :
Linguistic terms and sequence of triangular fuzzy numbers.