Similarity Measures of Sequence of Fuzzy Numbers and Fuzzy Risk Analysis
We present the methods to evaluate the similarity measures between sequence of triangular fuzzy numbers for making 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 type fuzzy number. By defining the algebraic operations on the type fuzzy numbers we can solve the equations in the form , where and are fuzzy number. By this way, we can build an algebraic structure on fuzzy numbers. Additionally, the generalized difference sequence spaces of triangular fuzzy numbers , , and , consisting of all sequences such that is in the spaces , , and , have been constructed, respectively. Furthermore, some classes of matrix transformations from the space and to and are characterized, respectively, where is any sequence space.
1. Preliminaries, Background, and Notation
The concept of fuzzy sets and fuzzy set operations were first introduced by . 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  introduced the class of bounded and convergent sequences of fuzzy numbers with respect to the Hausdorff metric. In , 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 . In , a method for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers is given. In , 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 [4–6]. 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 , , and . Additionally, we redefine fuzzy identity elements according to addition and multiplication for constructing an algebraic structure on the 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 , , and are linearly isomorphic to the spaces , , and , respectively. Finally, in Section 3, it is proved that the spaces , , and are complete. Section 4 is devoted to the calculation of the - and -duals of the spaces and . In Section 5, some classes of matrix transformations from the space and to and are characterized, respectively, where is any sequence space. In Section 6, we review the COG points of a sequence of fuzzy numbers  and existing similarity measures of fuzzy numbers [4–7]. 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 , , and are the set of all positive integers, all real numbers, and all bounded and closed intervals on the real line , respectively. For define the following metric: It can easily be seen that defines a metric on and the pair is a complete metric space . Let be a nonempty set. According to Zadeh a fuzzy subset of is a nonempty subset of for some function . Consider a function as a subset of a nonempty base space and denote the family of all such functions or fuzzy sets by . Let us suppose that the function satisfies the following properties:(1) is normal, that is, there exists an such that .(2) is fuzzy convex, that is, for any and , .(3) is upper semicontinuous.(4)The closure of , denoted by , is compact.
Then, the function is called a fuzzy number .
The properties (1)–(4) imply that, for each , the -cut set of the fuzzy number defined by is in . That is, the equality holds for each . 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 , and 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 to , , , , through all the text.
Let us consider any triangular type fuzzy number , as follows. If the function is the membership function of the triangular fuzzy number , then can be represented with the notation where and . If , then the fuzzy number is called symmetric fuzzy number and if , then is a real number. In general, the fuzzy number is called -type fuzzy number. After here, we deal with the set of triangular -type fuzzy numbers represented as which includes the classic triangular fuzzy numbers, through the text. Any element of the set will be denoted by . Additionally, for brevity, we call triangular -type fuzzy numbers triangular fuzzy numbers.
It means that, set of all -type fuzzy number is defined as in the following: The notations are called first, middle, and end points of triangular fuzzy number , respectively. In addition to this, the notation means that the height of the fuzzy number is 1 at the point . For every , the sets are different from each other and every element in the form of these sets belongs to . Another mean of the (4) is that for every , 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 .
Sometimes, the representation of fuzzy numbers with -cut sets induces errors according to algebraic operations. For example, if 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.
2. Algebraic Structure of the Set
Let and . Let us define addition, substraction, multiplication, division, and scalar multiplication on , respectively, as follows: where is nonzero fuzzy number and Let ; then . It means that 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 is equal to , according to addition and the determines a fuzzy number. With this idea, we can solve equations in the form . 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 and .
The second important matter is the topology on the set . Şengönül  has constructed a topology on by using the metric defined as follows:We can easily show that the set is a complete metric space with the metric .
Clearly, the representation (3) for is unique, and on the contrary there is a unique for every . We know that, generally in the practical applications, the spread of fuzziness should not be very large. So, the value of should be as small as possible. Theoretically, the value “ should be small” not necessary but it has to be in practical applications. For example, the “approximately 1” can be taken as but in the applications, generally, “approximately 1” is taken as , . This choice is more accurate than .
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 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 , .
Let us denote the set of all sequences of triangular fuzzy numbers by ; that is, where and for all .
The , , and 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.
We define the classical sets , , , and consisting of the bounded, convergent, null, and absolutely -summable sequences of fuzzy numbers, respectively, similar to ; it means thatWe should emphasize here that the sequence spaces of fuzzy numbers , , , and can be reduced to the classical sequence spaces of real numbers , and , respectively in the special case , where and . So, the properties and results related to the sequence spaces of , , , and are more general and more useful than the corresponding implications of the spaces , and , respectively.
Definition 2. Let ; is identity element of according to addition and algebraic operations in the sense of (5) and (9). The function is called norm on the set if it has the following properties:(9).(10), .(11).
If the function satisfies (9)–(11) then is called normed sequence space of the triangular fuzzy numbers. If is complete with respect to the norm then is called complete normed sequence space of the triangular fuzzy numbers.
Lemma 3. The sets , and are complete normed sequence spaces with the norm defined as follows:where is in any sets of .
Now, let and be two spaces of triangular fuzzy valued sequences and let be an infinite matrix of positive real numbers , where Then, we say that defines a real-matrix mapping from to and we denote it by writing , if for every sequence the sequence is in where and the series , , are convergent for all By , we denote the class of matrices such that . Thus, if and only if the series on the right side of (15) are convergent for each and every , and we have for all .
Let be a sequence space of triangular fuzzy numbers. Then, the set of sequences of triangular fuzzy numbers, defined as follows, is called the domain of an infinite matrix in : Let be nonzero real numbers for each and define the band matrix by Here, and are the convergent sequences.
It is easy to calculate that the inverse of the generalized difference matrix is given by
3. The Generalized Difference Sequence Spaces , , and
In this section, we wish to introduce the , , and spaces as the set of all sequences such that -transforms of them are in the spaces , , and , respectively; that is,
We should emphasize here that the sequence spaces , , and of triangular fuzzy numbers can be reduced to the sets , and , respectively, in the case for all , in the structure of generalized difference matrix. So, the properties and results related to the sequence spaces of , , and are more general and more extensive than the corresponding consequences of the spaces , and , respectively.
Let us define the sequence of fuzzy numbers which will be constantly used as the -transform of a sequence of fuzzy numbers ; that is, where , for all .
Now, we may begin with the following theorem which is essential in the text.
Theorem 4. The sequence spaces , , and are linearly isomorphic to the spaces , , and , respectively; that is, , , and .
Proof. Since the others can be similarly proved, we consider only the case . To prove this, we should show the existence of a linear bijection between the spaces and . Consider the transformation defined , with the notation of (20), from to by The equality , where , is clear. Let us suppose that ; then, That is, has the property homogeneity. Thus, is linear.
Let us take any and represent the sequence using as follows: where Then, we have That is, is norm preserving. Consequently, the spaces and are linearly isomorphic. It is clear here that if the spaces and are, respectively, replaced by the spaces and , and , then we obtain the fact that and This completes the proof.
Theorem 5. The sets , , and are complete normed sequence space of the triangular fuzzy numbers with the norm defined by
Proof. It was seen that, in Theorem 4, the sequence spaces of triangular fuzzy numbers , , and are linearly isomorphic to the spaces , , and , respectively. Additionally, since the matrix is normal (see, ) and , , and are complete module spaces, it is clear that the sequence spaces , , and are complete normed spaces with the norm defined in (25).
Theorem 6. Let be a sequence of triangular fuzzy numbers. If as , then as It means that the matrix is regular.
Proof. Let be a sequence of fuzzy numbers and as . Then for a given there exists a positive integer such that Therefore, we can writeIf we take , then we see that . That is, holds. This completes the proof.
Theorem 7. The inclusions and strictly hold.
Proof. To prove the validity of the inclusion , let us take any . Then, bearing in mind the regularity of the method (see, Theorem 6), we immediately observe that which means that . Hence, the inclusion holds. One can see by analogy that the strict inclusion also holds. This completes the proof.
Let and be an infinite matrix of fuzzy numbers and consider the following expressions: In , some matrix classes are characterized by Talo and Başar which are given in the following lemma.
Lemma 8 (see ). The following statements hold:(12), , if and only if (29) holds.(13) if and only if (32) holds.(14) if and only if (29) and (31) hold.(15) if and only if (29) and (31) hold with for all .(16) if and only if (30) and (31) hold.(17) if and only if (30) and (31) hold with for all .(18) if and only if (29), (33), and (31) hold with for all .
Analogously to Talo and Başar, we can prove the following propositions.
Proposition 9. if and only if (29) and (31) also hold with for all .
4. Real Duals of the Spaces , , and
In this section, we state and prove the theorems determining the - and -duals of the spaces , , and . For the sequence spaces and , define the set by where denotes all real valued sequences space. With the notation of (35), - and -duals of a sequence space , which are respectively denoted by and , are defined by , and . We will use a technique, in the proof of Theorems 10 and 12, which is used in [17–19].
Theorem 10. The -dual of the spaces , , and is the set
Proof. Since the proof is similar for the rest of the spaces, we determine only -dual of the set . Let and define the matrix via the sequence byBearing in mind relation (20) we immediately derive that From (38) we realize that whenever if and only if whenever . Then, by considering Part (12) of Lemma 8, we have which yields the consequence that .
Lemma 11 (see , Theorem ). Let be defined via a sequence and the inverse matrix of the triangle matrix by for all Then
Theorem 12. Define the sets and by Then, And .
Proof. It is clear from Lemmas 8 and 11.
5. Matrix Transformations
For the first time, Lorentz introduced the concept of dual summability methods for the limitation which depends on a Stieltjes integral and passed to the discontinuous matrix methods by means of a suitable step function . Later, many authors, such as Başar , Başar and Çolak , Kuttner , Lorentz and Zeller , and Şengönül and Başar  worked on the dual summability methods.
Let us suppose that the set is any of the sets , , and . In this section, we characterize the matrix mappings from into any given sequence space of triangular fuzzy numbers via the concept of the dual summability methods and vice versa, so we call it the sequential generalized difference dual summability methods. Let us suppose that the sequences and are connected with (20) and let -transform of the sequence be and let the -transform of the sequence be ; that is,