1 Introduction

Today, fuzzy numbers offer a good possibility to represent uncertainties. However, if such fuzzy numbers are used as uncertain parameters, e.g., in FEM analysis, the calculations become expensive and surrogate models [1] are used to minimize the computation time. To estimate their accuracy, the resulting outcome must be evaluated and the deviations from the original function must be defined. The use of fuzzy numbers inevitably produces results in the form of such fuzzy numbers again. A comparison between two fuzzy numbers is therefore required.

The problem of comparability of fuzzy numbers is not new and has already been analyzed in many ways in the literature. Two different characteristics are distinguished: similarity and ranking/ordering [2]. Within ranking methods the arithmetic operations < or > are used to represent the size relations. Different possibilities for defining relational operations can be found via probability theory [3] or other approaches like Yager’s methods [4,5,6], Chang’s method [7], Kerre’s method [8], Murakami–Maeda–Imamura methods [9], Nakamura’s method [10], or the Li and Lee method [11].

The similarity approaches calculate a value between 1 (full agreement) and 0 (no agreement) and thus evaluate the agreement between two fuzzy variables. There are also different methods for similarity, see e.g., Chen and Lin’s method [12], Hsieh and Lin’s method [13], Lee’s method [14], Chen and Chen’s method [15], Peng’s method [16], Sen’s method [17], or Gogoi’s method [18].

For an estimation of the deviation, a ranking (absolute size comparison) is not sufficient and a similarity value is difficult to use, because the value of similarity depends on the universe of the discourse. The universe of the discourse is in a normalized form and contains all possible fuzzy sets. Better suited and more intuitive is a consideration of the percentage deviation of the fuzzy variables. A consideration of deviation instead of a measure of similarity is new and thus offers a different way of comparing fuzzy numbers.

If the fuzzy number is given as a decomposed fuzzy number [19] in the so-called \(\alpha \)-cuts, this kind of representation must also be considered. Since none of the previously mentioned methods did this, in section 3 the O-index is introduced as a new method, which can also describe a percentage deviation for each \(\alpha \)-cut. The O-index can also be used to compare arbitrary fuzzy numbers, while many of the other methods can be applied only to special fuzzy sets, e.g., in [18] only interval-valued fuzzy numbers, in Sen et al. [17] only Gaussian fuzzy numbers, or in Chen et al. [15] only generally trapezoidal fuzzy sets are compared.

This paper presents well-known approaches taken from ranking and similarity studies, and discuss their advantages and disadvantages when used for percentage comparability. The concept of the new O-index is outlined in section 3, where the capability of the O-index is demonstrated by means of case studies. Finally, in section 4, the O-index is applied to a mechanical problem, showing its usefulness in ranking the accuracy of surrogate models in percentage terms.

2 Comparative Values from Literature

In order to better understand the newly developed comparative index, this section briefly presents some existing methods and points out their advantages and disadvantages. All methods presented in this section compare fuzzy numbers. However, in contrast to our method presented in the next section, these methods allow only a relative comparison and not a percentage deviation. Furthermore, a decomposition by means of \(\alpha \)-cuts is not, or is only insufficiently, considered.

From Chen et al. [15], the cases presented in Fig. 1 can be used with the similarity methods shown in Table 1.

Fig. 1
figure 1

Five case studies with the reference fuzzy number (solid lines) the deviating fuzzy number (dashed lines) and the indicated alpha-cut boundaries (crosses) following the examples Set6-Set10 from Chen et al. [15]

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Table 1 Similarity for the fuzzy number pairings Case A–E as shown in Fig. 1 from the examples in Chen et al. [15]
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In Lee’s method [14], the similarity U is computed using the left bounds \(a_{1}, a_{2}\), the right bounds \(b_{1}, b_{2}\) , and the core values \(c_{1}, c_{2}\). For fuzzy numbers this results in

$$\begin{aligned} U =1 - \frac{\vert a_{1}-a_{2}\vert + 2 \cdot \vert c_{1}-c_{2}\vert + \vert b_{1}-b_{2}\vert }{4 \cdot (\text {max}(b_{1}, b_{2}) - \text {min}(a_{1}, a_{2}))}. \end{aligned}$$
(1)

Therefore, case B is not defined (see Table 1), because the denominator in Eq. (1) is zero. Case C results in zero because the numerator and denominator are equal. Even with crisp numbers farther apart, the value for U would be zero. So crisp values cannot be compared using this method. In Hsieh-and-Chen’s method [13], the similarity U is calculated by using the graded mean integration, which for fuzzy numbers is

$$\begin{aligned} U = \frac{1}{1 + \vert \frac{a_{1} + 4c_{1} + b_{1}}{6} - \frac{a_{2} + 4c_{2} + b_{2}}{6} \vert }. \end{aligned}$$
(2)

This method treats fuzzy numbers with symmetric boundaries like a crisp value. This results in the same similarity values for the cases C–E.

In Chen-and-Lin’s method [12] the similarity

$$\begin{aligned} U =1 - \frac{\vert a_{1}-a_{2}\vert + \vert c_{1}-c_{2}\vert + \vert b_{1}-b_{2}\vert }{3} \end{aligned}$$
(3)

is calculated with the same variables as in Lee’s method [14]. However, even in this method, fuzzy numbers with symmetric boundaries are treated like a crisp value; therefore, the same similarity values are calculated for cases C–E.

In Chen and Chen’s method [15], the similarity is calculated with an extension to Chen-and-Lin’s method [12], where the center of gravity is additionally used. As a result, the cases C–E differ significantly. However, the Chen and Chen method [15] fails when using non-general trapezoidal fuzzy numbers. Moreover, with this method, the calculation of the center of gravity heavily weights the differences in the membership functions, leading to, for example, exaggerated deviations in case D, although the absolute values hardly differ from case C.

Many methods from ranking are only conditionally suitable for a comparison of two fuzzy numbers [20]. Yager’s methods can be found in Yager [4,5,6]. Yager’s F1 index computes the geometric centroid of normalized fuzzy numbers. A disadvantage of this method is that the fuzzy numbers are reduced to one value (geometric centroid). This means that fuzzy numbers, which are totally different in their appearance, can have the same center of gravity and thus be assumed to be identical. Yager’s F3 index calculates an area that is dependent on the core value and the support. Even in this variant, fuzzy numbers with different core values and different support can produce the same values and thus cannot be compared.

Chang’s method [7] is close to Yager’s F1 index and therefore exhibits the same problems. Other methods, such as Kerre’s method [8], the Murakami–Maeda–Imamura methods [9], Nakamura’s method [10], or the Lee and Li method [11] also fail in their non-unique or insufficient distinction of fuzzy numbers.

Newer methods, e.g., [21,22,23] are limited to a comparison of (special) fuzzy numbers. Very good and promising approaches can be found in [24,25,26]. Here, a L–R decomposition takes place, which implies a consideration of the fuzzy boundaries. Furthermore, areas are used for the estimation, which distinguish the compared fuzzy numbers. These ideas are also implemented and extended in the new index in section 3 such that a percentage comparison can be made.

The multitude of available comparison methods shows the importance of such methods in fuzzy logic, where a simple size comparison is often sufficient for decision making. The decisive innovation of the following index resides in its possibility to rank a decomposed fuzzy number, i.e., its \(\alpha \)-cuts, as a whole or individually in percentage comparison.

3 O-Index

The O-index is calculated in two variants; on the one hand as a deviation of a fuzzy number and on the other hand as a deviation of a decomposed fuzzy number. An explanation of these terms can be found at the beginning of the respective sections  and 3.2.

3.1 Index for Fuzzy Numbers

A fuzzy number is characterized by the attribute that exactly one value can be assigned to the membership function \(\mu \) at the position \(\mu = 1\). This value is defined as the core value. All other membership function values \(\mu \ne 1\) are assigned at least one value, but can contain any number of values, such that, for each membership function value \( \mu \ne 1\), an interval

$$\begin{aligned} X^{(j)}=[a^{(j)},b^{(j)}] \end{aligned}$$
(4)

is defined. Furthermore, the boundary values of the membership functions obey

$$a^{(j-\Delta _{\mu })} \le a^{(j)} \le a^{(j+\Delta _{\mu })} \text { and } b^{(j-\Delta _{\mu })} \ge b^{(j)} \ge b^{(j+\Delta _{\mu })}.$$
(5)

Thus, a fuzzy number is defined as a convex fuzzy set. The best known “special case” of a fuzzy number is the triangular fuzzy number or linear fuzzy number, whose membership function is linear. Another special form of the fuzzy number is the real number. Here, the boundary values of the membership function for all \(\mu \in [0,1]\) correspond to the core value at \(\mu = 1\).

In order to be able to compare two fuzzy numbers, two factors are decisive: On the one hand the core values, on the other hand the support, which describes the uncertainty. Here, it is already apparent that the real number without uncertainty is a special case, since the support of the fuzzy number is the core value. As already presented in section 2, area-based values also serve for the comparison of two fuzzy numbers. These approaches then fail for this special case with no areas under the membership functions. The O-index can cover these special cases. An explanation how the O-index compares two fuzzy numbers is given in the following sections.

3.1.1 Theoretical basics

Fig. 2
figure 2

Partial areas from a 1st summand and b 2nd summand to calculate the reference area from Eq. (6)

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One fuzzy number is used as a reference; here, the reference area is calculated as follows: The area between the origin (\(x=0\)) and the core value is assigned the height \(\omega ~ \text {with}~ \omega \in \mathbb {R}^{+}\), see Fig. 2a. The \(\omega \) serves as a factor for how much the core value should be weighted in relation to the uncertainty that is governed by the width of the support. If \(\omega \) is chosen to be very small, the influence of the uncertainties is larger and vice versa.

The area under the membership function, which describes the uncertainty, is called the membership area in the following. The membership area is shown in Fig. 2b. This area is calculated by the integral from the left to the right boundary of the fuzzy number. The partial area from Fig. 2a and this membership area are summed up and thus form the reference area

$$\begin{aligned} \text {S}_{\text {Ref}} = \text {core}(p_{\text {Ref}})*\omega + \int _{a_{\text {Ref}}}^{b_{\text {Ref}}} \mu _{\text {Ref}}(x) ~\text {d}x. \end{aligned}$$
(6)

In the next step, the deviating area of the second fuzzy number (comparison fuzzy number) to the reference fuzzy number must be calculated. It is not sufficient to calculate the included areas of the two uncertainties in the form

$$\begin{aligned} \text {S}_{\text {Dev}} = \int _{a_{\text {min}}}^{b_{\text {max}}} \vert \mu _{\text {Ref}}(x)- \mu _{\text {Cmp}}(x) \vert ~\text {d}x \end{aligned}$$
(7)

for the simple reason that this area is limited to a maximum of the addition of the areas of the two fuzzy numbers. Thus, the possible deviation of the fuzzy numbers is upper bounded. A percentage deviation should not be limited upward, so the deviating area cannot be limited. For this reason, the left and right limits of the fuzzy numbers are considered separately. To calculate the bounded area between the left and right boundaries of the fuzzy numbers, the inverse function of the membership function, defined as \(\mu _{}^{\langle -1 \rangle }(x)\), is integrated. These partial areas are then represented by

$$\begin{aligned} \begin{aligned} \text {S}_{\text {Dev,l}} =&\int _{0}^{1} \vert \mu _{\text {Ref}}^{\langle -1 \rangle }(x)- \mu _{\text {Cmp}}^{\langle -1 \rangle }(x) \vert ~\text {d}x, \\&\text {for } x \in [\text {min}(a_{\text {Ref}},a_{\text {Cmp}}),\text {max}(\text {core}(p_{\text {Ref}}), \\&\text {core}(p_{\text {Cmp}}))] \end{aligned} \end{aligned}$$
(8)

for the left boundary, and

$$\begin{aligned} \begin{aligned} \text {S}_{\text {Dev,r}} =&\int _{0}^{1} \vert \mu _{\text {Ref}}^{\langle -1 \rangle }(x)- \mu _{\text {Cmp}}^{\langle -1 \rangle }(x) \vert ~\text {d}x \\&\text {for } x \in [\text {min}(\text {core}(p_{\text {Ref}}),\text {core}(p_{\text {Cmp}})), \\&\text {max}(b_{\text {Ref}},b_{\text {Cmp}})] \end{aligned} \end{aligned}$$
(9)

for the right boundary, as represented in Fig. 3.

Fig. 3
figure 3

Included areas of fuzzy numbers for (a) the left and (b) the right boundaries from Eqs. (8) and (9). The red dashed lines indicate the second fuzzy number, which is used for the comparison

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As can be seen from the areas in Fig. 3, the upper part between the two fuzzy numbers is assigned to both the left and the right partial area. This should be prevented, or rather this area has to be included only once in the deviating area of the fuzzy numbers. Therefore the area

$$\begin{aligned} \begin{aligned} \text {S}_{\text {Dev,d}} =&\int _{x_{S}}^{1} \vert \mu _{\text {Ref}}^{\langle -1 \rangle }(x)- \mu _{\text {Cmp}}^{\langle -1 \rangle }(x) \vert ~\text {d}x ~\\&\text {for } {\left\{ \begin{array}{ll} \mathbb {D}:= x \in [\text {core}(p_{\text {Ref}}),\text {core}(p_{\text {Cmp}})], \\ \text {if}~\text {core}(p_{\text {Ref}}) \le \text {core}(p_{\text {Cmp}}) \\ \mathbb {D}:= x \in [\text {core}(p_{\text {Cmp}}), \text {core}(p_{\text {Ref}})], \\ \text {if}~\text {core}(p_{\text {Ref}}) > \text {core}(p_{\text {Cmp}}) \end{array}\right. } \end{aligned} \end{aligned}$$
(10)

is subtracted again. For the lower limit of the integral, the intersection point \(x_{S}\), two cases must be distinguished:

  1. (1)

    \(b_{\text {Cmp}} \le a_{\text {Ref}}~ \text {or } b_{\text {Ref}} \le a_{\text {Cmp}}\); this means that since the boundaries do not intersect, it follows that \(x_{S}=0\).

  2. (2)

    \(b_{\text {Cmp}} \ge a_{\text {Ref}}~ \text {or } b_{\text {Ref}} \ge a_{\text {Cmp}}\); this means that the right and left boundary intersect, making it valid:

    $$\begin{gathered} x_{S} = \mu _{{{\text{Ref}}}}^{{\langle - 1\rangle }} (x) = \mu _{{{\text{Cmp}}}}^{{\langle - 1\rangle }} (x) \hfill \\ {\text{for }}\left\{ {\begin{array}{lll} {\rm{\mathbb{D}}: = x \in [{\text{core}}(p_{{{\text{Ref}}}} ),{\text{core}}(p_{{{\text{Cmp}}}} )],} \\ {{\text{if}}~{\text{core}}(p_{{{\text{Ref}}}} ) \le {\text{core}}(p_{{{\text{Cmp}}}} )} \\ {\rm{\mathbb{D}}: = x \in [{\text{core}}(p_{{{\text{Cmp}}}} ),{\text{core}}(p_{{{\text{Ref}}}} )],} \\ {{\text{if}}~{\text{core}}(p_{{{\text{Ref}}}} ) > {\text{core}}(p_{{{\text{Cmp}}}} )} \\ \end{array} } \right. \hfill \\ \end{gathered}$$
    (11)

Case (2) corresponds to the intersection of the fuzzy numbers with the highest associated value for \(\mu \). If the two fuzzy numbers have the same core value, the integral reduces to zero, because the intersection point \(x_{S}\) lies at \(\mu =1\).

The final result for the deviating area is then

$$\begin{aligned} S_{\text {Dev,all}} = \text {S}_{\text {Dev,l}} + \text {S}_{\text {Dev,r}} - \text {S}_{\text {Dev,d}}. \end{aligned}$$
(12)

The deviation between the two fuzzy numbers is finally given by the O-index in the form

$$\begin{aligned} \Delta _{\text {O-index}} = \frac{\text {S}_{\text {Dev,all}}}{\text {S}_{\text {Ref}}}. \end{aligned}$$
(13)

The index is therefore defined in such a way that, for identical fuzzy numbers, and thus \(\text {S}_{\text {Dev,all}} = 0\), the value is also zero. Since the O-index aims at a percentage comparison and the partial areas from Eq. (12) are included in the index as amounts, no exact statement can be made about how the limits of the two comparison fuzzy numbers behave. Thus, the index can assume equal values if the limits of the comparison fuzzy number are either larger or smaller than the reference. Examples (2) and (8) in the case studies have an almost identical O-index, but differ significantly in their characteristics. With the O-index-\(\alpha \) extended in Sect. 3.2, however, more detailed conclusions can be drawn about the limit characteristics.

3.1.2 Special Case: Real Number 0

In the special case, when the reference is the real number 0, the area \(\text {S}_{\text {Ref}}\) vanishes, which renders Eq. (13) undefined. To obtain this case, both summands of Eq. (6) must be zero. This holds, if either of the following conditions is satisfied:

  1. (1)

    \(\text {core}(p_{\text {Ref}}) = 0\) or the weight \(\omega = 0\). If this occurs, the reference area is restricted to its uncertainty; in other words, to the membership area. This constellation can be used specifically when investigating only the deviating uncertainty.

  2. (2)

    The fuzzy number has no uncertainty, such that the membership area becomes \(\int _{a_{\text {Ref}}}^{b_{\text {Ref}}} \mu (x) = 0\).

If both the above conditions are fulfilled, we are dealing with the real number 0. As in classical division, this case is not defined.

3.2 Index for Fuzzy Numbers with \(\alpha \)-Cut Decomposition

In fuzzy arithmetic the computational method of \(\alpha \)-cut decomposition is the usual method of dealing with fuzzy sets. This type of decomposition reduces the problem to individual intervals, which are assigned a fixed membership function value. A more detailed explanation of \(\alpha \)-cuts can be found in Hanss [19]. Due to this decomposition, the consideration of the areas from section  is no longer possible for comparison. Thus, calculation methods of interval arithmetic must be resorted to. Also, in interval arithmetic, there are different approaches to deal with a comparison, as Sevastjanov et al. [27] show. Therefore, the O-index method is adapted and can offer suitable comparison values for single \(\alpha \)-cuts.

3.2.1 Theoretical basics

Similar to Sect. 3.1, the percentage deviation of individual \(\alpha \)-cuts is obtained here. The principle of comparison of a reference value with a deviation value remains the same. However, the areas are replaced by distances. For each \(\alpha \)-cut a separate reference section must be calculated. Similar to Eq. (6), the interval length of the respective \(\alpha \)-cut is added to the weighted core value. Using

$$\begin{aligned} \begin{aligned} d_{\text {Ref},j} =\;&\text {core}(p_{\text {Ref}})*\omega + b_{\text {Ref},j}-a_{\text {Ref},j}, \\&\text {for } b_{\text {Ref},j} \ge a_{\text {Ref},j},\\&~\text {with } j=\frac{0}{m-1},\frac{1}{m-1},...,\frac{m-1}{m-1}, \end{aligned} \end{aligned}$$
(14)

where j is the counting index over the \(\alpha \)-cuts, the fuzzy number is decomposed into m intervals whose gaps \(\Delta \mu \) are uniformly distributed over the membership function \(\mu (x)\). The limits \(b_{\text {.},j} \text { resp. } a_{\text {.},j}\) are thereby the largest and smallest values of the \(\alpha \)-cut j.

In the next step of the calculation, the distances between the limits are determined.

$$\begin{aligned} d_{\text {l},j} = \vert a_{\text {Ref},j}-a_{\text {Cmp},j} \vert~ \text {resp. } d_{\text {r},j} = \vert b_{\text {Ref},j}-b_{\text {Cmp},j} \vert , \text {with } j=\frac{0}{m-1},\frac{1}{m-1},...,\frac{m-1}{m-1}. \end{aligned}$$
(15)

Similar to the O-index for the total fuzzy number, an overlap of the distances can also occur here, but this should not be counted twice. Therefore,

$$ \begin{aligned} d_{\text {d},j} =&{\left\{ \begin{array}{ll} b_{\text {Ref},j}-a_{\text {Cmp},j}~\text {for } b_{\text {Ref},j} \ge a_{\text {Cmp},j}\\ b_{\text {Cmp},j}-a_{\text {Ref},j}~\text {for } b_{\text {Cmp},j} \ge a_{\text {Ref},j} \end{array}\right. } \\ \text {with } j=&\frac{0}{m-1},\frac{1}{m-1},...,\frac{m-1}{m-1}. \end{aligned} $$
(16)

is subtracted again. This results in the following expression for the deviating length

$$\begin{aligned} \begin{aligned} d_{\text {Dev},j} =&d_{\text {l},j} + d_{\text {r},j} - d_{\text {d},j} \\ \text {with } j=&\frac{0}{m-1},\frac{1}{m-1},...,\frac{m-1}{m-1}. \end{aligned} \end{aligned}$$
(17)

The differences of the \(\alpha \)-cuts for two fuzzy numbers are thus given by the O-index-\(\alpha \)

$$\begin{aligned} \begin{aligned} \Delta _{\text {O-index-}\alpha } =&\frac{d_{\text {Dev,j}}}{d_{\text {Ref,j}}} \\ \text {with } j=&\frac{0}{m-1},\frac{1}{m-1},...,\frac{m-1}{m-1}. \end{aligned} \end{aligned}$$
(18)

A special case \(d_{\text {Ref,j}} =0\) occurs whenever the core value of the reference fuzzy number is zero, because here in the \(\alpha \)-cut \(\mu _{\text {Ref}}(0) = 1\) the interval is reduced to a number and thus left and right limits are equal. This results in a reference length of zero, which is subsequently undefined in Eq. (18). If there is no uncertainty, it is again the real number 0, which was already considered undefined in section. However, if the fuzzy number with a core value zero has an uncertainty, then for all \(\alpha \)-cuts, except \(\mu _{\text {Ref}}(0) = 1\), a reference length is available and can therefore be calculated. The case \(\mu _{\text {Ref}}(0) = 1\) remains undefined and gives no result; compare example (10) in section 3.3.

3.3 Case Studies

To understand the behavior of the O-index, ten case studies, as shown in Fig. 4, will be discussed. The corresponding values of the O-index are listed in Table 2. Intuitively, \(\omega = 1\) is set as the weight of the core value. The fuzzy number with the solid line shows the reference. For the fuzzy number with the dashed line, the deviation to the reference is examined.

Fig. 4
figure 4

Ten case studies demonstrating the O-index with the reference fuzzy number (solid lines) the deviating fuzzy number (dashed lines) and the indicated alpha-cut boundaries (crosses)

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Table 2 O-index with \(\alpha \)-cut consideration for the fuzzy number pairings (1)–(10) shown in Fig. 4
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Case (1) describes two fuzzy numbers whose boundaries do not intersect or contact each other. This results in a significant deviation of 140%. The core value for the reference is 3, the core value of the comparison is 8, which corresponds to a deviation of 166.7% in the classical sense, as the O-index for \(\alpha \)-1 also shows. The membership areas of the two fuzzy numbers are the same, but the percentage uncertainty for 3 is larger than for 8 for the same area, so the slightly smaller deviation of 140% is reasonable. In the individual \(\alpha \)-cuts it can be seen that with the larger reference interval (\(\alpha \)-0), the deviations tend to become slightly smaller.

Case (2) shows an overlap of the right boundaries compared with the left boundaries of the reference. The core value for the reference is 6, the core value of the comparison is 4, which would correspond to a deviation of \((4-6)/6\), or approximately -33%. The amount of this deviation roughly corresponds to the 37.6% of the O-index. Since the reference here has a larger area of uncertainty, a few percent are added on top of the 33% deviation. The values of the individual \(\alpha \)-cuts at first increases with decreasing \(\mu \), until at \(\alpha \) \(-\)0.4 the maximum deviation is reached and then becomes slightly smaller. Here, two effects are apparent: First, the influence of the distance of the core value to the origin decreases with respect to the distance \(d_{\text {Dev},j}\) from Eq. (17) as the uncertainty increases. Second, the left limits of the reference get closer to the left limits of the comparison. While the first effect leads to decreasing values (compare example (1)), the second effect shows an increasing trend. Initially, the effect of convergence dominates, while for the lower \(\alpha \)-cuts the effect of increasing reference distance has a greater influence.

Case (3) shows two fuzzy numbers that have the same core value but different uncertainties. The membership area of the reference is a subset of the comparison. The value of the O-index with 62.5% is obtained exclusively from the area of uncertainty. This effect is clearly visible in the \(\alpha \)-cuts, because here the limits of the comparison become larger significantly faster. Thus, the O-index \(\alpha \)-1 = 0 and the O-index \(\alpha \)-0 = 100.

Case (4) shows the opposite situation to case (3), because the comparison is a subset of the reference. In addition, the right limits of the two fuzzy numbers lie on top of each other. This leads to a much smaller deviation of 11.1% overall. As in case (3), the O-index \(\alpha \) increases with decreasing \(\alpha \). However, due to the one-sided deviating boundary, it is much slower.

The special feature of case (5) is that the left boundary of the comparison intersects both the right and left boundaries of the reference. The core value of the comparison is therefore larger than the core value of the reference. For the maximum deviation of the left boundary this is then reversed. For the left boundary, there is a positive deviating surface, but also a negative deviating surface. So, in order that these areas do not eliminate each other (and thus suggest a too small deviation), the absolute value in Eqs. (8), (9), and (10) for the bounded areas, as well as in Eq. (15) for the distances between the limits, is required. This results in all O-index-\(\alpha \) values being about 74%–90% (except for core values), which is reasonable, because the fuzzy numbers are clearly different from each other.

In case (6), one of the two fuzzy numbers is not a triangular fuzzy number. Due to the different shapes, it is immediately recognizable that it is not the same fuzzy number. The areas of uncertainty are nevertheless the same size and the core value also matches. Methods that rely exclusively on the size of the area or the core value fail when making this comparison. Using the separate L–R consideration, a deviation of 16.7% results. Also the O-index-\(\alpha \) shows an agreement of the core value and increasing deviations with decreasing \(\alpha \).

In the same way as case (6), case (7) shows a coincidence of the core value and the maximum limits at \(\alpha \)-0. Here, however, the areas are not equal. From the O-index-\(\alpha \), it can be seen immediately where the fuzzy numbers differ. The O-index alone could indicate a parallel to case (4) because of its height of 12.9%. However, this can be clearly delimited by the O-index-\(\alpha \).

The special features of both cases (8) and (9) lie in a fuzzy number without uncertainty, which is nothing else than a real number. However, due to the area calculation of the reference, as well as the L–R consideration, such cases can also be classified, as the results in Table 2 show. For the limits of the real number the left and right limits are equal.

A final important case study is shown in case (10). The core value of the reference is zero. In addition, negative values occur in the left boundary. However, since absolute values and integrals are always computed in the calculation, no distinction between positive and negative values is necessary. The area of the reference fuzzy number from Eq. (6) reduces in this case to the uncertainty. With 93% deviation, this appears very high in contrast to case (2). The fuzzy numbers seem to have significantly larger overlaps at first. The small area \(\text {S}_{\text {Ref}}\) is responsible for this, where Eq. (13) becomes large very fast. However, this effect is also known from classical division when the denominator becomes small. In the O-index-\(\alpha \) calculation, the special case \(d_{\text {Ref,j}} =0\) occurs and thus the O-index-\(\alpha \)-1 is not defined. All other values, however, can be classified again due to their distance between left and right limits in the reference. Here, particularly large deviations occur because the denominator becomes extremely small.

3.4 Evaluation of the Examples from Section 2

Having demonstrated the advantages of the O-index on the basis of case studies (1)–(10), this section will again deal separately with the cases A–E from [15] shown in Fig. 1. Table 3 therefore lists the evaluations of the O-index for the cases A–E. Of course, the O-index results do not agree with any of the four methods shown in [15], since a deviation and not a similarity is calculated. Therefore, the value for Case B becomes 0 instead of 1, but this indicates the same result in the end. Case C can be calculated like a deviation of classical numbers \(\frac{0.3-0.2}{0.2} = \frac{1}{2}\). Case D shows identical values for the O-indices as case C. The left edge of the fuzzy number moves away from the crisp number at the same rate as the right edge approaches the crisp number. The mean deviation across all \(\alpha \) cuts remains the same, as in classical division of the crisp numbers. In contrast, Chen and Chen’s [15] value gives a significant deviation, which originates due to the influences of different centers of gravity. Compared to case C, this is an overestimated deviation. Case E shows slightly larger deviations for the O-index than cases C and D. In all three cases, the fuzzy numbers have the same core values but different supports. In the O-index-\(\alpha \) values in the higher \(\alpha \)-cuts of case E it can be seen that the reference distance from Eq. (14) increases slower than the deviating distance from Eq. (17). Due to the overlap starting at the \(\alpha \)-cut 0.4, the opposite effect takes place. Overall, this results in an O-index of 58%. Case A again shows the advantages of considering the \(\alpha \)-cut. In the middle, at \(\alpha \) \(-\)0.4, the fuzzy numbers are almost on an overlay. With a deviation from \(\alpha \) \(-\)0.4 to higher or lower \(\alpha \)-cuts, the deviation becomes larger again, which is also shown in the values of the O-index-\(\alpha \).

Table 3 O-index with \(\alpha \)-cut consideration for the fuzzy number pairings Case A–Case E shown in Fig. 1
Full size table

The O-index described here for cases A–E from [15] are more detailed than the method used there. It specifically describes them by looking the percentage deviation differently, and by the L–R consideration with other emphasis. If it is considered that a percentage deviation is strongly established in classical mathematics and also in everyday items (e.g., price quotations in the supermarket), the O-index offers an intuitive estimation of the result. An evaluation on the \(\alpha \)-cuts allows further information of the fuzzy numbers to be extracted from the O-index-\(\alpha \).

4 Application of Surrogate Modeling

In the following section, the intuitive handling and the possibility of using general fuzzy numbers of the O-index are shown by means of a comparison where the fuzzy numbers originate from the calculation of a mechanical problem.

4.1 Uncertainty analysis in a bending beam

An uncertainty analysis is performed on a clamped-free bending beam. The uncertainties are introduced using triangular fuzzy numbers for the Young’s modulus, the tip force, and the length of the beam. This results in a three-dimensional parameter space, which is described by triangular fuzzy numbers. The maximum possible deviations, and thus the complete parameter space, are in the membership function \(\mu = 0\). For higher values of \(\mu \) only a subspace is considered, which represents only a part of the complete parameter space. The maximum displacement at the end of the beam is evaluated as the result variable. The simulation, as shown in Fig. 5, is performed using the commercial program ABAQUS CAE.

Fig. 5
figure 5

Design and implementation of the bending beam in the commercial program Abaqus CAE

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This problem is taken from Oberleiter and Willner [28] and Reber and Oberleiter [29]. An important aspect in the uncertainty consideration is the multiple evaluation of the system. To enable this for a more complex system, the computation has to be substituted with a surrogate model. Approaches, to such a surrogate model, are for example presented in [1]. In the example presented here, the DACE presented in Lophaven et al. [30, 31] is used. The selection of a specific surrogate model is independent of the application of the O-index, but the index allows the estimation of the accuracy of the chosen surrogate. An index value that is as low as possible indicates a good surrogate.

Fig. 6
figure 6

Outcome fuzzy numbers (maximum displacement in y-direction) for the evaluation of the reference simulation compared with (1) surrogate model with Full factorial Design, (2) surrogate model with Latin Hybercube Sampling, and (3) surrogate model with Fuzzy Oriented Sampling Shift

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Figure 6 shows the reference evaluation of the simulation, as well as the evaluation of three surrogate models, each generated with different sampling plans. In the sampling plan, the possible parameter space is covered by sampling points, which are evaluated in the original system. The sampling points thus shape the approximation behavior of the surrogate model. A change in the sampling plan therefore also inevitably changes the surrogate model.

Table 4 Results of the O-index for the fuzzy number pairs shown in Fig. 6(1)–(3)
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The evaluation of the O-index is shown in Table 4. At this point it should be mentioned that the area calculation for \(\text {S}_{\text {Dev,all}} \text { and } \text {S}_{\text {Ref}} \) within the O-index can only be approximated. Due to the \(\alpha \)-cut decomposition, the membership is given by discrete points and is not available as a function. The changes, which are necessary in Eqs. (6)–(11), are trivial and nothing more than a simplified integral calculation. The O-index for the individual \(\alpha \)-cuts is unaffected. It can be assumed that the more \(\alpha \)-cuts there are and the more linear a membership function is, the better the approximation of the calculated areas will be.

If the plots and table values are now compared with each other, the difference between the surrogate models becomes clear. The indices over the total fuzzy numbers between 12.66–13.44% are very close to each other for all surrogate models. This shows that with the same number of sampling points of the surrogate models, the deviations are also of the same order of magnitude. However, the O-index over the \(\alpha \)-cuts clearly shows the influence of the sampling strategy.

4.2 Interpretation of results

The Full factorial Design (FFD) with 8 sampling points covers only the vertices of the parameter space. Thus, the \(\alpha \)-level \(\mu = 0\) is hit exactly (O-index-\(\alpha \)-0 = 0). This shows that the maximum uncertainty of the parameters of the bending beam also results in the maximum and minimum displacement. The shape of the reference result suggests either a strictly monotonically increasing or strictly monotonically decreasing effect on the displacement. Another effect of the FFD, which covers only the edge points with 8 sampling points, is the largest deviation in the \(\alpha \)-level \(\mu = 1\). This evaluation point has the maximum distance from the sampling points in the parameter space and can thus be approximated only poorly (see O-index-\(\alpha \)-1 = 5.97 for FFD instead of 2.36 and 0).

It appears that both of the Latin Hybercube Sampling (LHS) and the Fuzzy Oriented Sampling Shift (FOSS) [32] have hardly any influence on the O-indices. Here, several effects play a role. The LHS is a random arrangement of the sampling points, which is subject to conditions, such that the parameter space is covered as uniformly as possible. The FOSS method has an evaluation point on the \(\alpha \)-level \(\mu = 1\) and a subsequent point distribution over the LHS. However, the sampling points generated in this way are weighted so that the higher \(\alpha \)-cuts are better represented. Nevertheless, the evaluation of the O-indices in Table 4 shows the exact opposite. The values of the O-indices \(\alpha \) = 0.8–0.4 are better mapped with the LHS method, and the \(\alpha \) values 0.2 and 0 are better attained with the FOSS method. Since both methods have a random component in the distribution of their sampling points, the effect explained previously can deviate for individual surrogate models. This is especially true if only very few sampling points are used. In addition, the generated surrogate model has an influence, since it is not necessarily the case that the approximation becomes worse the farther the distance to a sampling point is. In [32] it is also shown, using the Root Mean Squared Error (RMSE), that averaged values show these effects.

At this point, it is useful to briefly discuss the RMSE. This is an error measure of the quality of the surrogate model compared to the reference simulation. The RMSE is calculated from the square root of the difference between the reference result y and the result of the substitute model \(\hat{y}\), as given in Eq. (19).

$$\begin{aligned} \text {RMSE} = \sqrt{\frac{1}{n} \sum _{z=1}^{z=n} (\hat{y}-y)^2} \end{aligned}$$
(19)

The absolute error deviation is thus already problem-specific, since an absolute error \(\text {RMSE} = 1\) for a problem with the value range \(\mathbb {W} \approx [-1;1]\) is much more critical than, for example, a problem with the value range \(\mathbb {W} \approx [-100;200]\). Furthermore, when evaluating the RMSE, the uncertainty consideration is completely lost, making the comparison of the reference with the surrogate model much more difficult to understand. Another advantage of the O-index is that, in contrast to the RMSE, the reference system and surrogate model do not have to be evaluated at the same points in order to be compared. This can save a lot of computing time.

Thus, the O-index offers a fast and beneficial solution for comparing two results in the form of fuzzy numbers. By using the values of the O-index-\(\alpha \), the deviation of the surrogate models can be made more precise and, if necessary, improved in an optimized way.

5 Conclusion

Both the case studies from section 3.3 as well as the mechanical problem of the bending beam show that, with the help of the O-index, a comparative quantity has been created that can represent the deviation of fuzzy numbers as a percentage, in an easily understandable way. The interpretation of the results of the bending beam shows the potential of the O-index-\(\alpha \). Due to the character of the fuzzy number and the associated deviations on the \(\alpha \)-cut level, statements about the behavior of the simulation can be made without knowledge of the underlying function. This is not possible with an evaluation of the RMSE or a comparison of fuzzy numbers using other methods, ignoring decomposed fuzzy numbers. The O-index thus offers more than just a similarity or ordering representation of two fuzzy numbers. It allows a statement about the degree of deviation and contains additional information compared to the conventional methods by an \(\alpha \)-cut consideration. It is therefore a powerful tool for the interpretation and decision making of uncertain results.

An improvement of the O-index can be done with respect to two points of view. Currently, only a comparison of two arbitrary fuzzy numbers is possible. An extension to general fuzzy sets is to be aimed for. Another kind of improvement would be in the field of application of the O-index. For this purpose, a possibility must be created to specify the effect of the percentage deviation. Thus, especially in risk management, it is necessary to define how severe the effect of a certain percentage deviation is. One way to implement this would be the addition of another summand to Eq. (6) resp. Eq (14). The reference area, particularly the reference distance, could be modified by this summand (then also for the case of the real number 0 and \(\text {core}(p_{\text {Ref}}) = 0\)) in such a way that the comparison takes into account how critical a deviation is. This assessment could play an important role in surrogate modeling, since, for example, changes in outcomes in mechanical problems can lead to changes in behavior, e.g., failure. In such a case, a purely percentage-based consideration may no longer be sufficient.