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.


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][6][7].In this paper, we are interested in approximating interval-valued fuzzy numbers by means of intervalvalued 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  = 1) 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 intervalvalued fuzzy numbers are investigated, these results will be frequently referred to in the subsequent sections.Section 3 is devoted to seek an approximation operator  : IF() → IF  () 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 intervalvalued 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

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.
(iv) The support of  is bounded; that is, the closure of { ∈  : () > 0} is bounded.
We denote by () the set of all fuzzy numbers on .
An often used fuzzy number is the trapezoidal fuzzy number, which is completely characterized by four real numbers  1 ≤  2 ≤  3 ≤  4 , denoted by  = { 1 ,  2 ,  3 ,  4 } and with the membership function We write   () as the family of all trapezoidal fuzzy numbers on .

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,  = [0, 1] 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 IF().
An interval-valued fuzzy set  defined on  is given by where 0 ≤   () ≤   () ≤ 1.The interval-valued fuzzy set  can be represented by an interval () = [  (x),   ()], and the ordinary fuzzy sets   :  →  and   :  →  are called a lower and an upper fuzzy set of , respectively.
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, intervalvalued 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][19][20]).Let  = [  ,   ] ∈ IF().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 Journal of Applied Mathematics

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 [0, 1] with (0) = 0 and ∫ 1 0 () = 1/2.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 (0) = 0 and ∫ 1 0 () = 1/2 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.

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 ,  ∈ IF().The ranking of ,  can be defined by the following formula: Example 11.Let We obtain core = {(, ) ∈  2 :  = 3,  ∈ [3,5]} and core = {(, ) ∈  2 :  = 3,  = 3}.By a direct calculation, we have  ⪰ .

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  : IF() → IF  () 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.
Remark 12.For any two different levels  1 and  2 ( 1 ̸ =  2 ), we obtain one and only one approximation operator which is invariant both in  1 -and  2 -level set.

Translation Invariance.
For  ∈ IF() and  ∈ , we define where 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.

Scale Invariance.
For  ∈ IF() and  ∈  \ {0}, we define When  > 0, ( When  < 0, ( We say that an approximation operator  is scale invariant if The continuity constraint means that if two interval-valued fuzzy numbers are close, then their interval-valued trapezoidal approximations also should be close.

Interval-Valued Trapezoidal Approximation Based on the
Weighted Distance.In this section, we are looking for an approximation operator  : IF() → IF  () 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).(42) According to the monotonicity of integration, we have That is Because ∫ 1 0 ( − 1) 2 () > 0, it follows that Theorem 15 (see [24]).Let , with respect to condition core = core(); 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 (54) We present the main result of the paper as follows. Theorem then we have ) , then we have ) , ) . ( then we have ) . ( 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  1 > 0 and  2 = 0, the solution of the system (66)-( 75) is ) , Firstly, we have from (55) that  1 > 0, and it follows from (56) that Then conditions (72), ( 73), (74), and (75) are verified.

Properties of the Interval-Valued Trapezoidal Approximation Operator
In this section we consider some properties of the approximation operator suggested in Section 3.  Example 27.Assume that the component  consists of three subcomponents  1 ,  2 , and  3 ; 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 1 ≤  ≤ 3.
Step 2. Calculate the probability of failure  of component .
By the interval-valued fuzzy number arithmetic operations defined as [8], we have Step 4. Calculate the similarity measure between the intervalvalued trapezoidal fuzzy number  * and the linguistic terms shown in Table It is obvious that   ( * , medium) ≈ 0.4748 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.

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.

Table 3 :
Interval-valued trapezoidal approximation of   and   .