Abstract
Taking into account that interval-valued fuzzy numbers can provide more flexibility to represent the imprecise information and interval-valued trapezoidal fuzzy numbers are widely used in practice, this paper devotes to seek an approximation operator that produces an interval-valued trapezoidal fuzzy number which is the nearest one to the given interval-valued fuzzy number, and the approximation operator preserves the core of the original interval-valued fuzzy number with respect to the weighted distance. As an application, we use the interval-valued trapezoidal approximation to handle fuzzy risk analysis problems, which overcome the drawback of existing fuzzy risk analysis methods.
1. Introduction
The theory of fuzzy set, proposed by Zadeh [1], has received a great deal of attention due to its capability of handling uncertainty. Uncertainty exists almost everywhere, except in the most idealized situations; it is not only an inevitable and ubiquitous phenomenon, but also a fundamental scientific principle. As a generalization of an ordinary Zadeh’s fuzzy set, the notion of interval-valued fuzzy sets was suggested for the first time by Gorzalczany [2] and Turksen [3]. It was introduced to alleviate some drawbacks of fuzzy set theory and has been applied to the fields of approximate inference, signal transmission and control, and so forth.
In 1998, Wang and Li [4] defined interval-valued fuzzy numbers and gave their extended operations. In practice, interval-valued trapezoidal fuzzy numbers are widely used in decision making, risk analysis, sensitivity analysis, and other fields [5–7]. In this paper, we are interested in approximating interval-valued fuzzy numbers by means of interval-valued trapezoidal fuzzy numbers to simplify calculations. The interval-valued trapezoidal approximation must preserve some parameters of the given interval-valued fuzzy number, such as -level set invariance, translation invariance, scale invariance, identity, nearness criterion, ranking invariance, and continuity. Considering that the core (-level set, where ) of an interval-valued fuzzy number is an important parameter in practical problems, we use the Karush-Kuhn-Tucher Theorem to investigate the interval-valued trapezoidal approximation of an interval-valued fuzzy number, which preserves its core.
The plan of this paper goes as follows. Section 2 contains some basic notations of interval-valued fuzzy numbers and the -level set of interval-valued fuzzy numbers is presented, which differs from [8]. Some results related to interval-valued fuzzy numbers are investigated, these results will be frequently referred to in the subsequent sections. Section 3 is devoted to seek an approximation operator that produces an interval-valued trapezoidal fuzzy number which is the nearest one to the given interval-valued fuzzy number among all interval-valued trapezoidal fuzzy numbers, and it preserves the core of the original interval-valued fuzzy number with respect to the weighted distance . In Section 4, some properties of the approximation operator such as translation invariance, scale invariance, identity, nearness criterion, ranking invariance, and distance property are discussed. As an application we also use the approximation operator to handle fuzzy risk analysis problems, which provides us with a useful way to deal with fuzzy risk analysis problems in Section 5.
2. Preliminaries
2.1. Fuzzy Numbers
In 1972, Chang and Zadeh [9] introduced the conception of fuzzy numbers with the consideration of the properties of probability functions. Since then, the theory of fuzzy numbers and its applications have expansively been developed in data analysis, artificial intelligence, and decision making. This section will remind us of the basic notations of fuzzy numbers and give readers a better understanding of the paper.
Definition 1 (see [11–13]). A fuzzy number is a subset of the real line , with the membership function such that the following holds.(i)is normal; that is, there is an with .(ii) is fuzzy convex; that is, , for any and .(iii) is upper semicontinuous; that is, is closed for any .(iv)The support of is bounded; that is, the closure of is bounded.We denote by the set of all fuzzy numbers on .
Let , whose membership function can generally be defined as [14]
where , is a nondecreasing upper semicontinuous function such that , . is a nonincreasing upper semicontinuous function satisfying , . and are called the left and the right side of , respectively.
For any , the -level set of a fuzzy number is a crisp set defined as [15]
The support or -level set of a fuzzy number is defined as
It is well known that every -level set of a fuzzy number is a closed interval, denoted as
where
It is obvious that and are the inverse functions of and , respectively.
An often used fuzzy number is the trapezoidal fuzzy number, which is completely characterized by four real numbers , denoted by and with the membership function
We write as the family of all trapezoidal fuzzy numbers on .
2.2. Interval-Valued Fuzzy Numbers
This section is devoted to review basic concept of interval-valued fuzzy numbers, which will be used extensively throughout this paper.
Let be a closed unit interval; that is, and .
Definition 2 (see [16]). Let be an ordinary nonempty set. Then the mapping is called an interval-valued fuzzy set on . All interval-valued fuzzy sets on are denoted by .
An interval-valued fuzzy set defined on is given by
where . The interval-valued fuzzy set can be represented by an interval , and the ordinary fuzzy sets and are called a lower and an upper fuzzy set of , respectively.
Definition 3 (see [17]). If an interval-valued fuzzy set satisfies the following conditions:(i) is normal, that is, there is an with ,(ii) is convex, that is, and for any and ,(iii) and are upper semicontinuous,(iv)the support of and are bounded, that is, the closure of and are bounded,then is called an interval-valued fuzzy number on . All interval-valued fuzzy numbers on are denoted by .
For any , the lower fuzzy number and the upper fuzzy number can be represented as
respectively, where , and are nondecreasing upper semicontinuous functions, such that , , , and , and are nonincreasing upper semicontinuous functions fulfilling , , , and .
If , , , , , and , that is, , then the interval-valued fuzzy number is a fuzzy number.
For any , the -level set of an interval-valued fuzzy number is defined as
where , , and are the inverse functions of , , , and , respectively. If , then this definition coincides with (4). The core of is presented as
Theorem 4. Let . if and only if , for any .
Proof. If: If , then there exist , such that
Since for any , this implies that
where . By the monotonicity of , we have
Similarly, we can prove that for any . If , then . Therefore, for any ; that is, .
Only if: If , then there exist , , such that
Since for any , this implies that
where . By the monotonicity of , we have
Similarly, we can prove that for any .
This concludes the proof.
It is well known, interval-valued fuzzy numbers with simple membership functions are preferred in practice. However, as a particular of interval-valued fuzzy numbers, interval-valued trapezoidal fuzzy numbers could be wide applied in real mathematical modeling. Thus, the properties of the interval-valued trapezoidal fuzzy number are discussed as follows.
Definition 5 (see [6, 18–20]). Let . If , then is called an interval-valued trapezoidal fuzzy number. The lower trapezoidal fuzzy number is expressed as and the upper trapezoidal fuzzy number is expressed as An interval-valued trapezoidal fuzzy number can be represented as . The family of all interval-valued trapezoidal fuzzy numbers on is denoted as .
Theorem 6. Let . if and only if , , and .
2.3. The Weighted Distance of Interval-Valued Fuzzy Numbers
In 2007, Zeng and Li [21] introduced the weighted distance of fuzzy numbers and as follows: where the function is nonnegative and increasing on with and . The function is also called the weighting function. The property of monotone increasing of function means that the higher the cut level, the more important its weight in determining the distance of fuzzy numbers and . Both conditions and ensure that the distance defined by (20) is the extension of the ordinary distance in defined by its absolute value. That means, this distance becomes an absolute value in when a fuzzy number reduces to a real number. In applications, the function can be chosen according to the actual situation.
We will define the weighted distance of interval-valued fuzzy numbers as follows. It can be considered as a natural extension of the weighted distance of fuzzy numbers.
Definition 7. Let . The weighted distance of and is defined as If and , then .
Property 1. Let . Then if and only if and .
Theorem 8. is a metric space.
By the completeness of metric space , we can obtain the following conclusion.
Theorem 9. The metric space is complete.
2.4. The Ranking of Interval-Valued Fuzzy Numbers
The ranking of fuzzy numbers was studied by many researchers and it was extended to interval-valued fuzzy numbers because of its attraction and applicability. We will propose a ranking of interval-valued fuzzy numbers, which embodies the importance of the core of interval-valued fuzzy numbers.
Definition 10. Let . The ranking of , can be defined by the following formula:
Example 11. Let We obtain and . By a direct calculation, we have .
3. Weighted Interval-Valued Trapezoidal Approximation
3.1. Criteria for Interval-Valued Trapezoidal Approximation
If we want to approximate an interval-valued fuzzy number by an interval-valued trapezoidal fuzzy number, we must use an approximate operator which transforms a family of all interval-valued fuzzy numbers into a family of interval-valued trapezoidal fuzzy numbers ; that is, . Since interval-valued trapezoidal approximation could also be performed in many ways, we propose a number of criteria which the approximation operator should possess at least one. Reference [22] has given some criteria for the fuzzy number approximation, similarly we give some criteria for interval-valued trapezoidal approximation as follows.
3.1.1. -Level Set Invariance
An approximation operator is -level set invariant if
Remark 12. For any two different levels and , we obtain one and only one approximation operator which is invariant both in - and -level set.
Proof. Let , , . Then we can obtain one and only one interval-valued trapezoidal fuzzy number , where It is obvious that Hence and .
3.1.2. Translation Invariance
For and , we define where , ; that is, An approximation operator is invariant to translation if Translation invariance means that the relative position of the interval-valued trapezoidal approximation remains constant when the membership function is moved to the left or to the right.
3.1.3. Scale Invariance
For and , we define When , , ; that is, When , , ; that is, We say that an approximation operator is scale invariant if
3.1.4. Identity
This criterion states that the interval-valued trapezoidal approximation of an interval-valued trapezoidal fuzzy number is equivalent to that number; that is, if , then
3.1.5. Nearness Criterion
An approximation operator fulfills the nearness criterion if for any interval-valued fuzzy number its output value is the nearest interval-valued trapezoidal fuzzy number to with respect to the weighted distance defined by (21). In other words, for any , we have
Remark 13. We can verify that is closed and convex, so exists and is unique.
3.1.6. Ranking Invariance
A reasonable approximation operator should preserve the accepted ranking. We say that an approximation operator is ranking invariant if for any ,
3.1.7. Continuity
Let . An approximation operator is continuous if for any , there is ; when , we have The continuity constraint means that if two interval-valued fuzzy numbers are close, then their interval-valued trapezoidal approximations also should be close.
3.2. Interval-Valued Trapezoidal Approximation Based on the Weighted Distance
In this section, we are looking for an approximation operator which produces an interval-valued trapezoidal fuzzy number, that is, the nearest one to the given interval-valued fuzzy number and preserves its core with respect to the weighted distance defined by (21).
Lemma 14. Let , , . If function is nonnegative and increasing on with and , then we have(i)
(ii)
Proof. (i) See [23] the proof of Theorem 3.1.
(ii) Since is a nonincreasing function, we have for any . By , we can prove that
According to the monotonicity of integration, we have
That is
Because , it follows that
Theorem 15 (see[24]). Let be convex and differentiable functions. Then solves the convex programming problem:
if and only if there exist , such that(i);
(ii);
(iii);
(iv).
Suppose that , , . We will try to find an interval-valued trapezoidal fuzzy number , which is the nearest interval-valued trapezoidal fuzzy number of and preserves its core with respect to the weighted distance . Thus we have to find such real numbers , , , , , , and that minimize
with respect to condition ; that is,
It follows that
Making use of Theorem 4, we have
Using (47) and (50), together with Theorem 6, we only need to minimize the function
subject to
After simple calculations we obtain
subject to
We present the main result of the paper as follows.
Theorem 16. Let , , . is the nearest interval-valued trapezoidal fuzzy number to and preserves its core with respect to the weighted distance . Consider the following.(i)If then we have (ii)If then we have (iii)If then we have (iv)If then we have
Proof. Because the function in (53) and conditions (54) satisfy the hypothesis of convexity and differentiability in Theorem 15, after some simple calculations, conditions (i)–(iv) in Theorem 15 with respect to the minimization problem (53) in conditions (54) can be shown as follows:
(i) In the case and , the solution of the system (66)–(75) is
Firstly, we have from (55) that , and it follows from (56) that
Then conditions (72), (73), (74), and (75) are verified.
Secondly, combing with (48), (55), and Lemma 14 (i), we can prove that
And on the basis of (50), we have
By making use of (48) and Lemma 14 (ii), we get
It follows from (49) that and are two trapezoidal fuzzy numbers.
Therefore by Theorem 6 and (50), is the nearest interval-valued trapezoidal approximation of in this case.
(ii) In the case and , the solution of the system (66)–(75) is
We have from (58) and (59) that and . Then conditions (72), (73), (74), and (75) are verified.
Furthermore, by making use of (48), (50), and (58) similar to (i), we can prove that
According to (48), (59), and Lemma 14 (ii), we obtain
This implies that
It follows from (49) that and are two trapezoidal fuzzy numbers.
Therefore by Theorem 6 and (50), is the nearest interval-valued trapezoidal approximation of in this case.
(iii) In the case and , the solution of the system (66)–(75) is
First, we have from (62) that . Also, it follows from (61), we can prove that
Then conditions (72), (73), (74), and (75) are verified.
Secondly, combing with (48) and Lemma 14 (i), we have
According to (48), (50), (62), and the second result of Lemma 14 (ii), similar to (ii), we can prove that . It follows from (49) that and are two trapezoidal fuzzy numbers.
Therefore by Theorem 6 and (50), is the nearest interval-valued trapezoidal approximation of in this case.
(iv) In the case and , the solution of the system (66)–(75) is
By (64), similar to (i) and (iii) we have and ; then conditions (72), (73), (74), and (75) are verified.
Furthermore, similar to (i) and (iii), we can prove that and are two trapezoidal fuzzy numbers.
Therefore by Theorem 6 and (50), is the nearest interval-valued trapezoidal approximation of in this case.
For any interval-valued fuzzy number, we can apply one and only one of the above situations (i)–(iv) to calculate the interval-valued trapezoidal approximation of it. We denote
It is obvious that the cases (i)–(iv) cover the set of all interval-valued fuzzy numbers and , , , and are disjoint sets. So the approximation operator always gives an interval-valued trapezoidal fuzzy number.
By the discussion of Theorem 16, we could find the nearest interval-valued trapezoidal fuzzy number for a given interval-valued fuzzy number. Furthermore, it preserves the core of the given interval-valued fuzzy number with respect to the weighted distance .
Remark 17. If , that is, , then our conclusion is consisten with [23].
Corollary 18. Let , where and
If and
then
If and
then
If and
then
If and
then
Proof. Let . We have ,,, and . It is obvious that ,, , and . Then by , we can prove that According to Theorem 16 the results can be easily obtained.
Example 19. Let and . Since that is, satisfies condition (i) of Corollary 18, we have Therefore, is the nearest interval-valued trapezoidal fuzzy number to , which preserves the core of .
Remark 20. Let . be not true in general.
Example 21. Let , , and . Then, according to condition (iv) of Corollary 18, we have Therefore, and . Based on Example 19, we known that . Further, we have from Theorem 6 that is not a trapezoidal interval-valued fuzzy number.
4. Properties of the Interval-Valued Trapezoidal Approximation Operator
In this section we consider some properties of the approximation operator suggested in Section 3.2. With respect to the criteria, translation invariance, scale invariance, identity, nearness criterion, and ranking invariance, we present the following results.
Theorem 22. The approximation operator which preserves the core of the initial interval-valued fuzzy number has the following properties. The operator is invariant to translations; that is, for any and , The operator is scale invariance; that is, for any and , The operator fulfills the identity criterion; that is, for any , The operator fulfills the nearness criterion with respect to the weighted distance ; that is, for every and such that . The operator is core invariance; that is, for any , The operator is ranking invariance; that is, for every .
Proof. If , according to (28) and (48), we have
Similarly, we can prove that
Furthermore, we have from (28) and (29) that
Then, one can easily prove that the interval-valued fuzzy number satisfies one of conditions (i)–(iv) of Theorem 16 if and only if the interval-valued fuzzy number satisfies the same condition. In any case of Theorem 16, by making use of (111), we obtain
for every . Therefore, combine the above results with (109) and (110) and we have .
(ii) Let . If , combing with (32), similar to (i), we can prove that .
If , we have from (33) and (48) that
Similarly, we can prove that
Furthermore, it follows from (33) and (34) that
Thus, is in the case (i) of Theorem 16 if and only if is in the case (iii) of Theorem 16. Then making use of (115) and Theorem 16, we get
Therefore, combine the above results with (113) and (114) and according to (33) and (34) we have
Analogously, is in the case (ii) of Theorem 16 if and only if is in the case (ii) of Theorem 16. is in the case (iii) of Theorem 16 if and only if is in the case (i) of Theorem 16. is in the case (iv) of Theorem 16 if and only if is in the case (iv) of Theorem 16. In each case , , for every ; therefore, .
(iii) If , then is in the case (iv) of Theorem 16 and .
(iv) and (v) are the direct consequences of Theorem 16.
(vi) By (22) and Theorem 16, we can obtain the conclusion.
The continuity is considered the essential property for an approximation operator. However, the approximation operator given by Theorem 16 is not continuous, as the following example proves.
Example 23 (see [25]). Let us consider , , , such that and the sequence of fuzzy numbers is given by It is easy to check that the function is nondecreasing and ; therefore, is a fuzzy number, for any . Then, according to the weighted distance defined by (20), we have It is immediate that . Now, denote Because operator preserves the core of fuzzy number , by (48) we have By seeing Lemma 3 [25], we cannot have with respect to the weighted distance . It follows from Heine’s criterion that is discontinuous.
To overcome the handicap of discontinuity of the approximation operator we present the following distance property.
Lemma 24. Let , be a sequence of interval-valued trapezoidal fuzzy numbers. If , , , then with respect to the weighted distance , where .
Proof. It is similar to the proof of Lemma 2 in the paper [25].
Theorem 25. Let , , , and be a sequence of interval-valued fuzzy numbers, where , . If and are uniform convergent sequences to and , respectively, then with respect to the weighted distance .
Proof. We denote
Because , , and are uniform convergent sequences to , , and , respectively, we have
and by (48) we obtain
Considering the following cases.
(i) satisfies condition (i) of Theorem 16; the following situations are possible.
(ia) If
then there exists , when , satisfies condition (i) of Theorem 16. We have from (124) that
According to (125), and Lemma 24, we have with respect to the weighted distance .
(ib) If
then there exists , when , satisfies condition (i) or condition (ii) of Theorem 16 and
In both two cases, we can prove
Then according to (125), and Lemma 24, we have with respect to the weighted distance .
(ii) satisfies condition (ii) of Theorem 16. The proof is analogous with the proof of case (ia).
(iii) satisfies condition (iii) of Theorem 16. The proof is analogous with the proof of case (i).
(iv) satisfies condition (iv) of Theorem 16; the following situations are possible.
(iva) If
the proof is analogous with the proof of (ia).
(ivb) If
then there exists , when , satisfies condition (i), (ii), (iii), or (iv) of Theorem 16, and
In either cases among (i), (ii), (iii), and (iv), it follows from (133) that
Then according to (125) and Lemma 24, we have with respect to the weighted distance .
() If
the proof is analogous with the proof of (ib).
(ivd) If
the proof is analogous with the proof of (ib).
After we analyze all the cases, the theorem is proven.
Next we will give an example to illustrate Theorem 25.
Example 26. Let us consider interval-valued fuzzy number , , . We will determine with an error less than with respect to the weighted distance .
Let be a sequence of interval-valued fuzzy numbers and
From the Taylor formula we have
Therefore, for any , we can prove that
That is, and satisfy the hypothesis in Theorem 25.
If , then
such that satisfies condition (iv) of Theorem 16. Furthermore, let
It is obvious that is decreasing, and by (141) we have
Therefore, we can conclude that
It follows that satisfies condition (iv) of Theorem 16. We denote
Using Theorem 16 (iv) together with (139), we have
Similarly, we can prove that
Combing (48), (139), and (140) we obtain
Therefore, by making use of (147), (148), and (149), we get
It is obvious that for , we have . For , case (iv) in Theorem 16 is applicable to compute the nearest interval-valued trapezoidal fuzzy number of interval-valued fuzzy number , and we obtain
Then we obtain an interval-valued trapezoidal approximation with an error less than .
5. Fuzzy Risk Analysis Based on Interval-Valued Fuzzy Numbers
Recently, a lot of methods have been presented for handling fuzzy risk analysis problems. However, these researches did not consider the risk analysis problems based on interval-valued fuzzy numbers. Following, we will use the approximation operator presented in Section 3.2 to deal with fuzzy risk analysis problems.
Assume that there is a component consisting of subcomponents and each subcomponent is evaluated by two evaluating items “probability of failure” and “severity of loss.” We want to evaluate the probability of failure and severity of loss of component . Assume that denotes the probability of failure and denotes the severity of loss of the subcomponent , respectively, where and are interval-valued fuzzy numbers and . The algorithm for dealing with fuzzy risk analysis is now presented as follows.
Step 1. Use the fuzzy weighted mean method to integrate the evaluating values and of each subcomponent , where .
Step 2. Transform interval-valued fuzzy numbers and into interval-valued trapezoidal fuzzy numbers and by means of the approximation operator .
Step 3. Use the interval-valued fuzzy number arithmetic operations defined as [8] to calculate the probability of failure of component : Without a doubt is an interval-valued trapezoidal fuzzy number.
Step 4. Transform the interval-valued trapezoidal fuzzy number into a standardized interval-valued trapezoidal fuzzy number : where , denotes the absolute value and denotes the upper bound and .
Step 5. Use the similarity measure of interval-valued fuzzy numbers introduced in [26] to calculate the similarity measure of and each linguistic term shown in Table 1. The larger the similarity measure, the higher the probability of failure of component .
Example 27. Assume that the component consists of three subcomponents , , and ; we evaluate the probability of failure of the component . There are some evaluating values represented by interval-valued fuzzy numbers shown in Table 2, where denotes the probability of failure and denotes the severity of loss of subcomponent , and .
Step 1. Let . According to Corollary 18, we obtain interval-valued trapezoidal fuzzy numbers and as shown in Table 3.
Step 2. Calculate the probability of failure of component . By the interval-valued fuzzy number arithmetic operations defined as [8], we have
Step 3. Transform the interval-valued trapezoidal fuzzy number into a standardized interval-valued trapezoidal fuzzy number ;
Step 4. Calculate the similarity measure between the interval-valued trapezoidal fuzzy number and the linguistic terms shown in Table 1, we have It is obvious that is the largest value; therefore, the interval-valued trapezoidal fuzzy number is translated into the linguistic term “medium.” That is, the probability of failure of the component is medium.
6. Conclusion
In this paper, we use the -level set of interval-valued fuzzy numbers to investigate interval-valued trapezoidal approximation of interval-valued fuzzy numbers and discuss some properties of the approximation operator including translation invariance, scale invariance, identity, nearness criterion, and ranking invariance. However, Example 23 proves that the approximation operator suggested in Section 3.2 is not continuous. Nevertheless, Theorem 25 shows that the interval-valued trapezoidal approximation has a relative good behavior. As an application, we use interval-valued trapezoidal approximation to handle fuzzy risk analysis problems, which provides us with a useful way to deal with fuzzy risk analysis problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Fund of China (61262022), the Natural Scientific Fund of Gansu Province of China (1208RJZA251), and the Scientific Research Project of Northwest Normal University (no. NWNU-KJCXGC-03-61).