#### Abstract

This paper presents a novel analytical approach to evaluating continuous, nonmonotonic functions of independent fuzzy numbers. The approach is based on a parametric -cut representation of fuzzy numbers and allows for the inclusion of parameter uncertainties into mathematical models.

#### 1. Introduction

This paper continues the research from our previous work [1]. In [1], we formulated a practical analytical approach to evaluating continuous, monotonic functions of independent fuzzy numbers, which is based on an alternative formulation of the extension principle [2]. In this paper, we extend this approach to general, nonmonotonic functions of independent fuzzy numbers. For the theoretical background, we assume the reader is familiar with Sections 2 and 4 from [1].

#### 2. Preliminaries

In the remainder of this paper, we will use two typical fuzzy numbers in engineering, the triangular and the Gaussian fuzzy number. The definitions are provided below.

*Definition 1. *The* triangular fuzzy number* (TFN) [3] is defined by the membership functionwhere denotes the* modal value*, denotes the* left-hand,* and denotes the* right-hand spread* of . If , the TFN is called* symmetric*. Its -cuts result from the inverse functions of (2) with respect to :

*Definition 2. *The* Gaussian fuzzy number* (GFN) [3] is defined by the membership function where denotes the* modal value, * denotes the* left-hand,* and denotes the* right-hand standard deviation* of . If , the GFN is called* symmetric*. Its -cuts result in

#### 3. Analytical Approach

For evaluating continuous, nonmonotonic functions of independent fuzzy numbers, the authors in [4, 5] suggest including the extreme points as constant profiles into the computation. However, this is not enough and can lead to erroneous results, as was pointed out in [6]. More specifically, all permutations of the interval boundaries of , , with the components of the extreme points have to be considered as well.

Basically, our analytical approach can be classified into two parts depending on the monotonicity of : a general and an extended part.

##### 3.1. General Part

If the function is nonmonotonic in all , , we can obtain the analytical solution as follows.(1)Evaluate the function for all the permutations of the interval boundaries of , . For example, if , then compute (2)Evaluate the function for all the combinations of the interval boundaries of , , with the components of the extreme points , . For example, if and , then compute (3)If for certain , compute the corresponding .(4)Plot all solution candidates in the same diagram.(5)Finally, starting from the modal point at , construct the maximum envelope formed by the possible solution candidates for under the condition of convexity.

This* general* part of our approach requires a total of maximum and minimum function evaluations. It can be viewed as an analytical version of the* level interval algorithm* [6].

*Example 1. *The function with shall be evaluated for the two fuzzy numbers and . Since the function is nonmonotonic in both and in the domain with one (global) extremum at . Hence, the general part of our approach should be applied. The solution candidates for are We can see from their plots in Figure 1 that, starting from the modal point at , the left branch of the maximum envelope, illustrated by the gray area, is formed by for , by for , and by for , where the value corresponds to the intersection point between and and the value to the intersection point between and . Its right branch, on the other hand, is entirely formed by . Hence, the -cuts of are With , , , and , the membership function of yields

##### 3.2. Extended Part

Let the continuous function be (strictly) monotonic increasing in , , (strictly) monotonic decreasing in , , monotonic in , , and nonmonotonic in , , in the domain of interest, with . Then, the analytical solution can be obtained as follows.(1)Evaluate the function for , , and , , including all the permutations of the interval boundaries of , , and , , to compute the monotonic solution candidates for .(2)Evaluate the function for , , and , , including all the permutations of the interval boundaries of , , and , , to compute the monotonic solution candidates for .(3)Evaluate the function for , , and , , including all the combinations of the interval boundaries of , , with the components of the extreme points , , to compute the nonmonotonic solution candidates for .(4)Evaluate the function for , , and , , including all the combinations of the interval boundaries of , with the components of the extreme points , , to compute the nonmonotonic solution candidates for .(5)Plot all solution candidates in the same diagram.(6)Finally, starting from the modal point at , construct the maximum envelope formed by the possible solution candidates for under the condition of convexity.

This* extended* part of our approach requires a total of function evaluations.

*Example 2. *Now, the function with shall be evaluated for the two fuzzy numbers from Example 1. Since the function is nonmonotonic in with one (global) extremum at and (strictly) monotonic increasing in in the domain . Hence, the extended part of our approach should be applied. The monotonic solution candidates for are and for , The nonmonotonic solution candidate for is and for , We can see from their plots in Figure 2 that, starting from the modal point at , the left branch of the maximum envelope is formed by for and by for , where the value corresponds to their intersection point. Its right branch, on the other hand, is entirely formed by . Hence, the -cuts of are With , , , and , the membership function of yields

#### 4. Engineering Application

In order to illustrate the analytical approach in a more practical context, we consider a linear system with one degree of freedom consisting of a block with mass moving on a smooth surface as shown in Figure 3. The block is connected to a wall via a linear spring with spring constant . This system is governed by the following linear, homogeneous ordinary differential equation of second order with constant coefficients [7]: Here, denotes the* natural frequency* of the system. The general solution of (22) is given by where and denote the initial conditions.

We assume and to be uncertain, both described by fuzzy numbers. More specifically, the uncertain initial position is modeled by the (symmetric) triangular fuzzy number and the uncertain natural frequency by the (symmetric) Gaussian fuzzy number Furthermore, we assume . We are interested in the uncertain position of the mass after one period ().

Since is (strictly) monotonic increasing in and nonmonotonic in in the domain with an infinite number of local extrema , . Hence, the extended part of our approach should be applied. The monotonic solution candidates for are and for , The nonmonotonic solution candidates for , on the other hand, are and for ,

We can see from their plots in Figure 4 that the left branch of the maximum envelope is formed by and the right branch by . Hence, the -cuts of are Since in (32) is not invertible with respect to , it is not possible to give an analytical expression for the membership function of . However, the -cuts and the membership function are both equivalent representations of a fuzzy number. The inverted plots of (32) are illustrated in Figure 5.

#### 5. Conclusions

We extended our analytical approach from [1] to general, nonmonotonic functions of independent fuzzy numbers. It is based on an -cut formulation of the extension principle and allows for the inclusion of parameter uncertainties into mathematical models.

In further research activities, the influence of interdependency may be a subject of investigation.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This paper was supported by the German Research Foundation (DFG) and Hamburg University of Technology (TUHH) in the funding programme “Open Access Publishing.”