The Interval-Valued Trapezoidal Approximation of Interval-Valued Fuzzy Numbers and Its Application in Fuzzy Risk Analysis
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.
The theory of fuzzy set, proposed by Zadeh , 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  and Turksen . 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  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 . 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.
2.1. Fuzzy Numbers
In 1972, Chang and Zadeh  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  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  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 ). 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 ). 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  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  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
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 ,
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)
Proof. (i) See  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). Let be convex and differentiable functions. Then solves the convex programming problem:
if and only if there exist , such that(i);
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