An Extended Analytical Approach to Evaluating Monotonic Functions of Fuzzy Numbers

There is an increasing effort in the scientific community to provide suitable methods for the inclusion of uncertainties into mathematical models. One way to do so is to introduce parametric uncertainty by representing the uncertain model parameters as fuzzy numbers [1] and evaluating the model equations by means of Zadeh’s extension principle [2]. The evaluation of this classical formulation of the extension principle, however, turns out to be a highly complex task [3]. Fortunately, Buckley and Qu [4] provide an alternative formulation that operates on α-cuts and is applicable to continuous functions of independent fuzzy numbers. Powerful numerical techniques have been developed to implement this alternative formulation [5]. These techniques are particularly suitable for very complex simulation models [6]. In engineering design [7], however, the mathematical equations are usually less complex, and hence analytical methods might be more suitable for the inclusion of parameter uncertainties into the computations. For this purpose, a practical analytical approach to evaluating continuous, monotonic functions of independent fuzzy numbers was introduced by the authors [8], which is based on the alternative formulation of the extension principle. In this paper, we extend this approach in terms of computational efficiency depending on certain monotonicity conditions. An outline of this paper is as follows. In Section 2, we give a definition of fuzzy numbers and present two important types. In Section 3, we introduce the notion of a linguistic variable. In Section 4, we briefly recall Zadeh’s extension principle and introduce the alternative formulation based on α-cuts. In Section 5, we describe our extended analytical approach and give four illustrative examples. In Section 6, a practical engineering application is presented. Finally, in Section 7, some conclusions are drawn.


Introduction
There is an increasing effort in the scientific community to provide suitable methods for the inclusion of uncertainties into mathematical models.One way to do so is to introduce parametric uncertainty by representing the uncertain model parameters as fuzzy numbers [1] and evaluating the model equations by means of Zadeh's extension principle [2].The evaluation of this classical formulation of the extension principle, however, turns out to be a highly complex task [3].Fortunately, Buckley and Qu [4] provide an alternative formulation that operates on -cuts and is applicable to continuous functions of independent fuzzy numbers.Powerful numerical techniques have been developed to implement this alternative formulation [5].These techniques are particularly suitable for very complex simulation models [6].In engineering design [7], however, the mathematical equations are usually less complex, and hence analytical methods might be more suitable for the inclusion of parameter uncertainties into the computations.For this purpose, a practical analytical approach to evaluating continuous, monotonic functions of independent fuzzy numbers was introduced by the authors [8], which is based on the alternative formulation of the extension principle.In this paper, we extend this approach in terms of computational efficiency depending on certain monotonicity conditions.
An outline of this paper is as follows.In Section 2, we give a definition of fuzzy numbers and present two important types.In Section 3, we introduce the notion of a linguistic variable.In Section 4, we briefly recall Zadeh's extension principle and introduce the alternative formulation based on -cuts.In Section 5, we describe our extended analytical approach and give four illustrative examples.In Section 6, a practical engineering application is presented.Finally, in Section 7, some conclusions are drawn.

Fuzzy Numbers
Fuzzy numbers are a special class of fuzzy sets [9], which can be defined as follows [1].
A normal, convex fuzzy set x over the real line R is called fuzzy number if there is exactly one  ∈ R with  x() = 1 and the membership function is at least piecewise continuous.The value  is called the modal or peak value of x.
It is important to note that some authors consider normal, convex fuzzy sets with a core interval also as fuzzy numbers [10].In [3,6], these types of fuzzy numbers are denoted as fuzzy intervals.Furthermore, some authors define a fuzzy number having a compact support [11].Although all concepts presented in this paper can be extended to these definitions of fuzzy numbers, we stick to the definition from [1].
Theoretically, an infinite number of possible types of fuzzy numbers can be defined.However, only few of them are important for engineering applications [6].These typical fuzzy numbers shall be described in the following.2.1.Triangular Fuzzy Numbers.Due to its very simple, linear membership function, the triangular fuzzy number (TFN) is the most frequently used fuzzy number in engineering.In order to define a TFN with the membership function we use the parametric notation [6] x = tfn (,  L ,  R ) , where  denotes the modal value,  L denotes the left-hand, and  R denotes the right-hand spread of x (cf. Figure 1).If  L =  R , the TFN is called symmetric.Its -cuts () = [ L (),  R ()] result from the inverse functions of (1) with respect to : (3)

Gaussian Fuzzy Numbers.
Another widely used fuzzy number in engineering is the Gaussian fuzzy number (GFN), which is based on the normal distribution from probability theory.In order to define such a GFN with the membership function we use the parametric notation [6] x = gfn (,  L ,  R ) , where  denotes the modal value,  L denotes the lefthand, and  R denotes the right-hand standard deviation of (cf. Figure 2).If

Linguistic Variables
In decision analysis, linguistic variables are of particular importance [12].A linguistic variable  L is a collection of subsets containing the following elements: (i)  : set of syntactic rules (e.g., in terms of a grammar) for the linguistic quantification of  L ; (ii) : set of terms   ,  ∈ N, resulting from ; (iii) : set of semantic rules that assign every term   to its (physical) meaning in terms of a fuzzy number t ; (iv) : (physically relevant) universal set with the (crisp) elements .For an easier handling with linguistic variables, they can be transformed into the unit interval [0, 1].These types of linguistic variables are referred to as normalized linguistic variables [12].

Extension Principle
Zadeh's extension principle [2] allows for extending any real-valued function to a function of fuzzy numbers.More specifically, let x1 , . . ., x be  independent or noninteractive fuzzy numbers, and let  : R  → R be a function with In case of interdependency between x1 , . . ., x , the minimum operator should be replaced by a suitable triangular norm [13].In this paper, however, we restrict ourselves to independent fuzzy numbers.
The evaluation of this classical formulation of the extension principle turns out to be a highly complex task [3].Fortunately, Buckley and Qu [4] provide an alternative formulation that operates on -cuts.
The extended analytical approach, which is presented in the next section, is based on this alternative formulation of the extension principle.

Extended Analytical Approach
Basically, our extended analytical approach can be classified into three parts depending on the monotonicity of : a reduced [8], a general [8], and an extended part.

Reduced Part.
Let the continuous function  be (strictly) monotonic increasing in   ,  = 1, . . ., , and (strictly) monotonic decreasing in   ,  = 1, . . ., ℓ, in the domain of interest, and let  + ℓ = .Then, the minimum values of  inside of every subdomain Ω() are always found at the left boundaries of   () and the right boundaries of   () and its maximum values at the right boundaries of   () and the left boundaries of   (), respectively.In such case, the -cuts with   () = [ L  (),  R  ()],  = 1, . . ., .If ( 9) is invertible with respect to , then the membership function of ỹ yields This reduced part of our approach can be viewed as an analytical version of the short transformation method [14].Basically, it is equivalent to Lemma 3 from [15] or Corollary 2 from [16].

General Part.
Unfortunately, the reduced part of our approach is only valid if the function  does not change its monotonicity within the domain of interest.However, we know from [17,18] that the global extrema of any monotonic function  are always found at the corner points of Ω().
Hence, in order to obtain the analytical solution, we can always proceed as follows.
(3) The analytical solution then corresponds to the maximum envelope formed by the possible solution candidates.
This general part of our approach can be viewed as an analytical version of the reduced transformation method [19].Basically, it is equivalent to Lemma 2 from [15] or Corollary 1 from [16].
Example 2. Next, the function  2 : R 2 + → R with shall be evaluated for the two fuzzy numbers from Example 1.Since the function  2 changes its monotonicity within the domain supp( x1 ) × supp( x2 ) = (0, 5) × (0, 4).Hence, the general part of our approach should be applied.The solution candidates for  2 () are We can see from their plots in Figure 4 that the left branch of the maximum envelope, illustrated by the gray area, is formed by  RL 2 for 0 <  ≤ 0.5 and by  LL 2 for 0.5 <  ≤ 1, where the value 0. With where (21)

Extended Part.
A drawback of the general part of our approach is the fact that a total of 2  function evaluations have to be carried out to compute the possible solution candidates.However, if some of the variables do not change their monotonicity within the domain of interest, that is, if  + ℓ =  < , we can adapt our approach as follows.
(3) Plot these solution candidates in the same diagram.
(4) The analytical solution then corresponds to the maximum envelope formed by the possible solution candidates.
This extended part of our approach requires a total of 2 −+1 function evaluations.Note that, for  = 1, the general and the extended part both lead to 2  function evaluations.
Example 3. Now, the function  3 : R 2 + → R with shall be evaluated for the two fuzzy numbers from Example 1.Since the function  3 is (strictly) monotonic increasing in  1 but changes its monotonicity in  2 within the domain supp( x1 ) × supp( x2 ) = (0, 5) × (0, 4).Hence, the extended part of our approach should be applied.Note that here,  = 1.The solution candidates for  L 3 () are and for  R 3 (), We can see from their plots in Figure 5 that the left branch of the maximum envelope is formed by  LL 3 for 0 <  ≤ 0.1 and by  LR 3 for 0.1 <  ≤ 1, where the value 0.1 corresponds to their intersection point.Its right branch, on the other hand, is entirely formed by  RL 3 .Hence, the -cuts  the case study from [20], where the material for an automotive bumper beam has to be selected.Here, two alternative materials (polymer composite and aluminum alloy) have to be evaluated against the criteria low weight and low cost using the normalized linguistic variable value scale from Figure 7.The corresponding linguistic weights and ratings are summarized in Table 1.
For computing the fuzzy overall rating r of each alternative , we use the fuzzy weighted average where r denotes the fuzzy rating of the alternative  according the criterion  and w denotes the fuzzy weight of the criterion  (see Table 1).Since is (strictly) monotonic increasing in   but may change its monotonicity in   within the domain (0, 1) 2 .Hence, the extended part of our approach should be applied.The solution candidates for We can see from their plots in Figure 8 that the left branch of the maximum envelope is formed by  LRLL

Conclusions
The proposed extended analytical approach is a very practical tool for the inclusion of parameter uncertainties into mathematical models.It is valid for continuous, monotonic functions of independent fuzzy numbers but can also be applied to fuzzy intervals as defined in [3,6].
An analytical solution has the advantage that the degrees of membership of the fuzzy output can be computed for any value within the support, whereas a numerical solution only provides a finite number of values.Furthermore, our approach also allows a symbolic processing of uncertainties.
In further research activities, this approach shall be generalized to nonmonotonic functions of independent fuzzy numbers, where the influence of interdependency shall be investigated as well.

Figure 3
Figure 3 illustrates a possible description of the linguistic variable color.It is based on the continuous spectrum of the wave length  of visible light:  = { ∈ R | 380 ≤  ≤ 780} nm.By subjective color perception, the colors   are chosen from the set  = {violet, blue, cyan, green, yellow, red} of possible colors.Each term   ∈  is represented as a fuzzy number t over the universal set .For an easier handling with linguistic variables, they can be transformed into the unit interval [0, 1].These types of linguistic variables are referred to as normalized linguistic variables[12].

Figure 3 :
Figure 3: Possible description of the linguistic variable color according to [6].
5 corresponds to the intersection point between  RL 2 and  LL 2 .Its right branch, on the other hand, is formed by  LL 2 for 0 <  ≤ 0.02 and by  RR 2 for 0.02 <  ≤ 1, where the value 0.02 corresponds to the intersectionFigure 4: Solution candidates from Example 2.point between  LL 2 and  RR 2 .Note that the value 0.02 is only approximate.Hence, the -cuts  2

Table 1 :
Linguistic weights of the criteria and ratings of the alternative materials.