Designing structures with polymorphic uncertainty: Enhanced decision making using information reduction measures to quantify robustness

The application of information reduction measures (IRMs) can provide valuable insight and enhance the process of design optimization when dealing with data uncertainty. For the engineering task of designing structures or products, an adequate modeling of data uncertainty is required. Therefore, a consideration of both aleatoric and epistemic uncertainty in combined form as polymorphic uncertain input variables is utilized. The resulting uncertain output quantities are post‐processed to provide relevant insights into robustness and performance for the design optimization. To this end, IRMs are applied, categorized into representative and uncertainty quantifying measures. Various IRMs exist, but clear recommendations or explanations of why certain uncertainty quantifying measures are chosen are scarce, although different features of uncertain quantities are considered with different measures. The aim of this contribution is to give an insight to commonly applied IRMs and the specific information of the uncertain quantity they reflect. Additionally, handling results of nested uncertainty analyses of polymorphic uncertain quantities regarding robustness towards aleatoric and epistemic uncertainty is investigated.


INTRODUCTION
The numerical design procedure of structures is a complex challenge.To optimize structures, the numerical simulation needs to depict the reality as accurately as possible.Therefore, the consideration of the input parameters' uncertainties is crucial, since deviations from the actual structure to the initial design are inevitable.The influences of these data uncertainties on the uncertainty of the structural response is investigated by a robustness evaluation.Especially for nonlinear system behavior, a small variation of the input parameters can lead to large change in the structural response.An adequate modeling of the input parameters' uncertainties is fundamental, taking into consideration both aleatoric and epistemic uncertainty in combined form as polymorphic uncertain variables.Uncertainty analyses yield uncertain output quantities, i.e., uncertain structural responses, which represent the magnitude of the variation resulting from uncertain input.
To include this information in design optimization, these uncertain output quantities are post-processed to gain relevant insights regarding the robustness as well as the performance.For this purpose, information reduction measures (IRMs), classified into representative measures and uncertainty quantifying measures, are applied.Various IRMs exist, but clear recommendations or explanations why certain uncertainty quantifying measures are selected in examples are scarce despite the fact that different characteristics of the uncertain output quantities are accounted for with different measures.
The focus of this contribution lies on investigating and comparing commonly applied IRMs to help users navigate the choice of appropriate measures for their task.This includes insights into, e.g., the effect of asymmetry of random or fuzzy quantities and interpretability.The aim is to provide a guide as to which "information" is reflected by different IRMs.Furthermore, the results of nested uncertainty analyses for structural designs with polymorphic uncertain variables are discussed.This includes the investigation of how the different origins of input parameter uncertainty -aleatoric and epistemic -contribute to the resulting value of each of the IRMs.Improvements regarding the interpretability of these values illustrated by an example.

THEORETICAL BASIS FOR ROBUSTNESS CONSIDERATION IN STRUCTURAL DESIGN OPTIMIZATION
When performing a deterministic design optimization utilizing numerical simulations, only the performance of a structure, i.e., the structural response, can be considered in the decision making process.The criticism for such an approach is the complete disregard of the various sources of uncertainty, which are present in reality, and their influence on the predicted response.By considering these uncertainties in the input parameters, it is possible to also predict and consider a design's robustness.
The term robustness is used interdisciplinary with different definitions.In the context of structural design, the term is commonly interpreted with regard to two aspects [1].Firstly, robustness can refer to the ability of a structure to maintain basic performance even under extreme or unforeseen events.The second aspect is the stability of the structural performance under common, foreseeable fluctuations.In this contribution, the focus lies on the second definition of robustness, and how to incorporate this information in the decision making process for structural design when dealing with polymorphic uncertainty.Concepts for uncertainty modeling as well as nested uncertainty analyses within the structural design optimization are briefly introduced in the following, as they constitute the basis for the robustness evaluation.

Uncertainty modeling for input parameters
In the scientific community, there is consensus that uncertainty can be categorized into two types: aleatoric and epistemic [2][3][4].Inherent variability of the parameters, for example, variation of material properties or load variations, are considered as aleatoric uncertainty, which is irreducible.Probabilistic variables are the chosen method for modeling this type of uncertainty.On the other hand, epistemic uncertainty is the result of a lack of knowledge.This includes incomplete data, for example, caused by small sample sizes, as well as imprecise data, i.e., samples with a measurement error that is often not negligible.In [5,6], different modeling strategies are discussed, including evidence theory, possibility theory and interval analysis.In this contribution, possibility theory is applied.Epistemic uncertainty may be reduced by acquiring additional data, but not fully removed.Since most engineering tasks are confronted with both types of uncertainty, polymorphic uncertainty models can be used to combine them, as described in [4,7].Random variables  r , a basic uncertainty modeling method following probability theory, are based on a probability space (Ω, Σ, ) with Ω as the set of all elementary events  ∈ Ω and Σ as a -algebra as a set containing subsets of elements in Ω, assigning to each event a probability  ∶ Σ ↦ [0, 1].A random variable maps an elementary event to a real value  r ∶ Ω ↦ ℝ.It is described by a cumulative distribution function   = ( ≤ ) = ∫  −∞   (), with   denoting the probability density function (PDF), for which ∫ ∞ −∞   () = 1.The modeling of random variables requires a large amount of samples fulfilling the i.i.d.paradigm (independent and identically distributed).This is often not possible for engineering tasks, where only a smaller amount of imprecise data is available.
Fuzzy variables  f are another basic uncertainty modeling method, which, in contrast to random variables, follow possibility theory.A fuzzy variable is defined as a set  of pairs  = {(, ()) |  ∈ ℝ, 0 ≤ () ≤ 1}; ∃  | () = 1, assigning a possibility to each value, by assigning to each value  a possibility of occurrence using a membership function () ∶ ℝ → [0, 1].A special case of a fuzzy variable is an interval variable.The membership function can only assume the values {0, 1}, with () = 1 ∀  ∈ [ l ,  u ] for all values between the upper and lower bound.The term support of a fuzzy variable can be a useful descriptor, separating the possible domain ( > 0) from the impossible domain ( = 0), and is denoted as  supp = { | () > 0,  ∈ } with the support length | supp | =   0 ,u −   0 ,l .Compared to random variables, less information is contained in fuzzy variables, since the possibility of a realization is interpreted linguistically only and does not assign a value of probability.This is useful in order to avoid the false assumption of knowing the whole underlying probability distribution of a structural response when insufficient data is available to justify this.
Polymorphic uncertainty models aim to combine both aleatoric and epistemic uncertainty in form of imprecise probabilities, which can be considered as an extension of a probabilistic model in combination with possibility theory.Several models exist and have to be selected based on the individual task at hand.A guide for selecting an appropriate uncertainty model for different data situations can be found, e.g., in [7].
A popular choice is the so-called probability-box (p-box), which is introduced in [8,9].Unlike for the traditional random variable, the cumulative density function is not described by a function   , but by a lower and upper bound  ,l and  ,u , therefore, giving an interval for possible values  ,l ≤   ≤  ,u for a random variable.This reduces the restrictions based on selecting and fitting of the data to one predefined type of function.For  ,l and  ,u , predefined as well as empirical distributions can be used.In the context of a p-box, each realization of the random variable can be considered an interval A further extension of the random variable is the model of fuzzy randomness (fr).While for a p-box each realization of the random variable is defined as an interval, here, a realization is described by a fuzzy variable  ∶ Ω ↦  (ℝ), giving  = ([ ,l ,  ,u ]) ∈ ]0,1] .The definition of the fuzzy cumulative distribution function is  ,,l ≤  , ≤  ,,u for each -level, analogous to a p-box.
Another model consisting of both fuzzy and random variables is fuzzy-probability based randomness (fp-r), denoted in bunch parameter representation.Contrary to fuzzy randomness though is the mapping of the random variable to a real value  ∶ Ω ↦ ℝ.To combine epistemic uncertainty quantification with the stochastic model, the hyperparameters  of the distribution function are fuzzified . Therefore, each realization of  has, for each -level, an interval of probability   = ([ ,,l ,  ,,u ]) ∈ ]0,1] .Polymorphic uncertainty models not only have the capability of combining aleatoric and epistemic uncertainty when modeling the input parameters, they also allow for a separation of the contribution of each type of uncertainty in the structural response.In this contribution, the evaluation of robustness for design optimization is discussed in a general manner without limitation to a specific modeling method.

Nested uncertainty analysis for polymorphic uncertain input
The propagation of uncertainty is achieved through an uncertainty analysis, which allows for the quantification of the resulting output parameters' uncertainty.The selected modeling methods for the input parameter uncertainty are present in the output quantities as well, i.e., random input will lead to random output, fuzzy input will result in fuzzy output, and polymorphic uncertain input will yield polymorphic uncertain output.The uncertainty analysis is performed for basic uncertainty models, as a stochastic or a fuzzy analysis.If polymorphic uncertain parameters are present, a nested uncertainty analysis is carried out, as shown in Figure 1.For a task including random and fuzzy parameters, a nested analysis is to be performed as well, except in special cases where an interaction between the two types of uncertain parameters can be ruled out.For a stochastic analysis, sampling-based approaches can be utilized in form of a Monte-Carlo-Simulation, as applied in, e.g., [10,11].For the case of risk assessment, the failure probability and, therefore, the tail of the distribution is of interest.Methods like Sequential Importance Sampling, see, for example, [12], Subset Sampling and Line Sampling [13] can be applied.A fuzzy analysis is often performed as an -level optimization based on -discretization, as considered in [14].Alternatively, the resulting fuzzy output can be computed by using -level-free sampling-based methods, see, e.g., [15].

Evaluation of polymorphic uncertain structural responses
In order to evaluate the robustness for the design optimization, it is beneficial to reduce the uncertain output quantities to a few characteristic values using IRMs ℜ ∶  u ↦ ℝ. □ u is used here as a general notation for an uncertain quantity, and could be either a random, fuzzy or polymorphic uncertain quantity.The measures can be divided into two categories, depending on the characteristic of the uncertain quantity they quantify [4,11].Representative measures  yield the physical value to optimize a structure's performance, while uncertainty quantifying measures  can be used in an optimization of the robustness of a design.The reduction measures are only applicable on basic uncertainty models.Therefore, a reduction step is performed at the end of each basic analysis (stochastic or fuzzy), see Figure 1.By evaluating multiple measures ℜ for each design, the optimization task is extended, as shown in Figure 2.For each structural response   , which is investigated, the single-criterion optimization is extended to a multi-criteria optimization when considering the input parameter uncertainty.If a polymorphic uncertainty analysis is performed, the optimization becomes a nested task.In Section 4, an example is provided on how to analyze the nested results and utilize the information for design optimization.

CHARACTERISTICS OF COMMON INFORMATION REDUCTION MEASURES
Generally speaking, the term "information reduction" describes a process which reduces the extend, size or dimensionality of given information or data, in order to eliminate irrelevant information and only keep what is essential to the task at hand.In the context of uncertainty analysis and, therefore, in this contribution, IRMs are applied in order to reduce an uncertain output quantity to one or several numerical values, that contain the essential information needed for decision making in the design optimization.In this section, various commonly applied IRMs for random and for fuzzy variables are introduced and explored regarding the specific features that are captured by the different measures.Additionally, their comparability to other measures within the same categories is discussed.Their interpretability in the scope of decision making is investigated as well.The focus here is on comparing a specific structural performance, not, e.g., functional output.In such cases, a reduction to a specific value or feature of the function (e.g., maximum value) is necessary to apply the presented scheme.

Overview of commonly applied IRMs
Various commonly applied IRMs are presented, categorized into the two groups of representative measures , which describe the performance, and uncertainty quantifying measures , which represent the extend of uncertainty of the response.For both categories, different measures exist for both random and fuzzy variables, see, e.g., [11,16].An overview is given in Table 1.Note that the notation  r ,  r ,  f and  f is only used to highlight for which type of uncertain quantity (random or fuzzy) the measure is applied.In a general case, the □ u is omitted for simplicity.
F I G U R E 2 (Nested) optimization task, based on uncertainty modeling of input parameters.

Uncertainty quantifying measures as indicators for robustness
By using uncertainty quantifying measures the user is able to compare designs regarding their robustness.Larger values of ( u ) correspond to lower robustness, so for ( D1  ) < ( D2  ), Design 1 will be preferred.
A measure for the robustness of a structural response regarding the input parameters' uncertainty is often denoted as  = ( u )∕( u ), which corresponds to the ratio of the input to the output parameter uncertainty (see, e.g., [1]).A large value of  indicates a robust structure.A possible issue when it comes to decision making can occur in cases of design dependent input uncertainty.If, e.g., the production in a section of the domain can be done more accurately than in another, the resulting values for ( u ) are going to be different.Consider two exemplary Designs 1 and 2, with  D1 = 1∕1 = 1 and  D2 = 2∕1.5 = 1.33.Applying this measure  for the robustness evaluation, then Design 2 would be preferred, since  D1 <  D2 , even though the quantified uncertainty of the result quantity ( D2 u ) = 1.5 for Design 2 is actually larger than ( D1  ) = 1 for Design 1. Also, the comparability of the robustness measure to the performance value is lost.
Other methods for scaling the result of the uncertainty quantifying measure ( u ) can be considered as well.For example, ( u )∕( u ) yields the ratio of uncertainty to performance.This can be useful if the applied measure for  preserves some relation to the physical domain, like the interquantile range or the fuzzy area.Otherwise, this scaling can distort the perception of the actual extend of uncertainty.Considering, for example, variance versus standard deviation, the variance would yield a much smaller value, therefore, by using this measure the ratio of uncertainty to performance will be much smaller as well.This can lead to the false assumption that the extend of uncertainty is almost negligible compared to the performance.
Since the focus of this contribution lies on preserving interpretability of the results, the value of ( u ) is used as the robustness measure.For a relative comparison of the designs, a scaling to ÛD = ( u D )∕ max(( u D )) can be applied, where the maximum value will result in 1.A reciprocal value ǓD = min(( u D ))∕( u D ) can be used to make the result interpretation even more intuitive so that the highest value of the robustness measure ( Ǔmax = 1) also corresponds to the most robust structure.This scaling is applied in the example shown in Section 4.

Comparison of uncertainty quantifying measures
While selecting a fitting representative measure   , like mean and centroid value, is mostly intuitive, it can be a more challenging task to choose an appropriate uncertainty quantifying measure   .Therefore, different measures were investigated regarding their characteristics such as domain, computational effort and interpretability of the robustness value when comparing it to the performance value.Additionally, the effect that various attributes of  u can have on the measures' values are analyzed.The results are shown in Table 2.It should be noted that these results are subjective and may differ for other users.
The interpretability can be related to the information about the domain.For example, entropy of a random variable can assume negative and positive values, which makes the result difficult to interpret when comparing the robustness to TA B L E 2 Characteristics of   .−, o, + as not good, neutral, good.o, ↑, ↑↑ as neutral, increase, strong increase (↓ decrease, respectively).
Note: † compared to other measures for same type of variable (random or fuzzy) * dependent on chosen quantiles  1 ,  2 .
the performance.For fuzzy variables, a useful consideration is the domain limitation for a specific result  f  .The upper limit of the domain can be expressed using the width of the support of the fuzzy quantity | ,supp |.While area and entropy domain limits are (scaled) values of the support width, the quadratic value of the support width for bound of the fuzzy variance domain reduces the interpretability.The entropy value introduced by De Luca and Termini [18] only considers the extend of the fuzzy quantity for membership function values 0 < ( f ) < 1, and neglects the core, i.e., values of  for which ( f ) = 1.
Comparing the computational effort relatively for random and for fuzzy quantities, the interquantile range can be more expensive to compute, if the chosen quantiles are very small and very large, respectively, i.e., the tail is of interest.The entropy of a random quantity can be the very challenging to approximate, see, e.g., [20].
Specific features of the uncertain output quantities affect the values of the various measures   in different ways, as is shown in the right part of Table 2.For example, if the distribution of a random quantity is multimodal, the entropy is not affected by increasing distance between the modes, whereas the interquantile range is increased linearly and the measures of variance, standard deviation and coefficient of variation quadratically.Asymmetry of random and fuzzy quantities will raise the values of variance.Uncertain quantities with a "long tail", i.e., a large extend of realizations  with low values of () and (), respectively, yield an increase in the variances as well, and in some cases the interquantile range, dependent on the chosen quantiles.Fuzzy quantities with a membership function with nonlinear slope for 0 < () < 1 will not influence the support length, but will affect the area, the entropy and especially the variance.
A main goal of this contribution is to gain insight into the polymorphic uncertain result quantities regarding the effect of aleatoric and of epistemic uncertainty.Therefore, measures with values which are to a certain extend comparable to each other and have a good interpretability are preferred.Some of the given measures in Table 1 are specific only to one type of uncertain quantity (random or fuzzy), for example, an area can only quantify the uncertainty of a fuzzy quantity, since the value of a random variable would always result in 1.Other measures are adaptions of measures for random to become measures for fuzzy quantities, for example, fuzzy variance.While the concepts of these measures are the same, the values those measures can result in may differ greatly based on the fact that the fuzzy membership ( f ) ∈ [0, 1], while probability range is ( r ) ∈ [0, ∞].

ROBUSTNESS EVALUATION OF EIGENVALUE OF A MULTI-STORY FRAME STRUCTURE
A three-story frame structure, shown in Figure 3, is investigated regarding the robustness of its eigenvalues  E .To illustrate the procedure, only  E2 is shown.The objective is to predict these eigenvalues as accurately as possible in order to avoid resonance, i.e., to minimize the uncertainty of the results.To simplify the example, it is assumed that axial  3. The spiderplot depicted in Figure 5 is a way to display the results of all four nested measures to compare designs relative to one another for each of the measures, but not of the measures with one another.The values are scaled to allow all measures to be displayed in the same plot.For all results including an uncertainty quantifying measure   , the reciprocal is plotted, as was described in Section 3.2.Therefore, the optimal design would be the one which reaches the highest values.
To compare the actual values of the measures also with one another, a barplot is chosen, see Figure 6.Here, the large bar shows the performance and the smaller bars the respective measures quantifying epistemic and aleatoric uncertainty.It can be seen, highlighted by the arrow in the Figure , that Design 2 has the best robustness regarding epistemic uncertainty, while Design 3 is more robust regarding aleatoric uncertainty.Utilizing this information, more data can be collected to reduce the epistemic uncertainty.Now, instead of intervals, symmetrical triangular fuzzy variables with the same support as the interval are used.As expected and highlighted in the barplot, the measure corresponding to the epistemic uncertainty is reduced.Meanwhile, there are no changes in the spiderplot, since it only allows for a relative comparison of the designs for each measure.While for the interval modeling Design 2 could be preferred, after reducing the epistemic uncertainty by gathering additional data Design 3 may be chosen as the more robust one.
F I G U R E 6 Barplot IRMs for  u E2 , values and colors see Table 3.
Only by using measures with good interpretability, like interquantile range  r iqr and area  f  , it is possible to evaluate if the aleatoric and epistemic uncertainty are relevant compared to the performance and if a further reduction of the epistemic uncertainty by gathering additional data, which can be expensive, may yield a significant improvement of the robustness.

CONCLUSION AND OUTLOOK
The robustness regarding input parameters' uncertainty within design optimization can be an important factor for structural design optimization.By utilizing polymorphic uncertainty modeling methods, both types of uncertainty (aleatoric and epistemic) are represented.The benefit of using probability and possibility theory for the different types is that the trace back towards the source of the uncertainty in the input is still possible after the uncertainty analysis.Therefore, the user can investigate if gathering additional data to reduce the extend of the epistemic uncertainty could lead to a significant improvement towards the design's robustness.This is only possible by utilizing IRMs, specifically uncertainty quantifying measures   , which have a good interpretability and comparability to the performance as well as to the other chosen measure   .Future work will include the investigation of the applicability of the presented procedure on more complex structural design optimization tasks with different objectives.Also, the consideration of correlation between output quantities will be explored.
stiffness and bending stiffness of the bars  =  B = ∞.The uncertain design parameters are  D = [ 1 ,  2 ,  3 ,  C ].The masses  1 ,  2 ,  3 [Mg] are modeled as intervals, and the bending stiffness of the columns  C [kNm 2 ] is modeled as a p-box.Units are omitted from hereon to improve readability.The affine transformation for   is performed by adding F I G U R E 3 Three-story frame.the interval [−0.1, 0.1] to the design values.A normal distribution with  f = [−25, 25],  = 250 is added to the design value of  C .An uncertainty analysis is conducted for three different designs,  D1 = [2.4,3.4, 1.4, 4 500],  D2 = [2.25,3.4, 3.4, 3 300] and  D3 = [1.4,3.4, 2.3, 5 700].As IRMs for the fuzzy quantities in the inner loop (compare Figures 1 and 2), expected value  f  and area  f  are selected, and for the random quantities in the outer loop the mean value  r  and the interquantile range  r iqr with  1 = 0.05,  2 = 0.95.The resulting p-boxes and IRMs are shown in Figure 4 and Table

FTA B L E 3
I G U R E 4 P-box for resulting  u E2 .Information reduction measures (IRMs) for resulting  u E2 .Colors highlight values displayed in Fig. 6.
Commonly used information reduction measures (IRMs).
TA B L E 1 Spiderplot with scaled values for  u E2 of IRMs given in Table