1 Introduction

In deterministic design optimization, it is generally assumed that all design variables and system variables are precisely known; the influence of natural variability and data uncertainty on the optimality and feasibility of the design is not explicitly considered. However, real-life engineering problems are non-deterministic, and a deterministic assumption about inputs may lead to infeasibility or poor performance (Sim 2004). In recent years, many methods have been developed for design under uncertainty. Reliability-based design (e.g., Chiralaksanakul and Mahadevan 2005; Ramu et al. 2006; Agarwal et al. 2007; Du and Beiqing Huang 2007) and robust design (e.g., Parkinson et al. 1993; Du and Chen 2000; Doltsinis and Kang 2004; Huang and Du 2007) are two directions pursued by these methods. While reliability-based design aims to maintain design feasibility at desired reliability levels, robust design optimization attempts to minimize variability in the system performance due to variations in the inputs (Lee et al. 2008). In recent years, several methods have also been proposed to integrate these two paradigms of design under uncertainty (e.g., Du et al. 2004; Lee et al. 2008).

Taguchi proposed robust design methods for selecting design variables in a manner that makes the product performance insensitive to variations in the manufacturing process (Taguchi 1993). Taguchi’s methods have widespread applications in engineering; however, these methods are implemented through statistical design of experiments and cannot solve problems with multiple measures of performances and design constraints (Wei et al. 2009). With the introduction of nonlinear programming to robust design, it has become possible to achieve robustness in both performance and design constraints (Du and Chen 2000).

The essential elements of robust design optimization are: (1) maintaining robustness in the objective function (objective robustness); (2) maintaining robustness in the constraints (feasibility robustness); (3) estimating mean and measure of variation (variance) of the performance function; and (4) multi-objective optimization. The rest of this section briefly reviews the literature with respect to these four elements and establishes the motivation for the current study.

1.1 Objective robustness

In robust optimization, the robustness of the objective function is usually achieved by simultaneously optimizing its mean and minimizing its variance. Two major robustness measures are available in the literature: one is the variance, which is extensively discussed in the literature (Du and Chen 2000; Lee and Park 2001; Doltsinis and Kang 2004) and the other is based on the percentile difference (Du et al. 2004).

1.2 Feasibility robustness

Feasibility robustness i.e., robustness in the constraints can be defined as satisfying the constraints of the design in the presence of uncertainty. Du and Chen (2000) classified the methods of maintaining feasibility robustness into two categories: (i) methods that use probabilistic and statistical analysis, for example, a probabilistic feasibility formulation (Du and Chen 2000; Lee et al. 2008), and a moment matching formulation (Parkinson et al. 1993) and (ii) methods that do not require them, for example, worst case analysis (Parkinson et al. 1993), corner space evaluation (Sundaresan et al. 1995), and manufacturing variation patterns (MVP) (Yu and Ishii 1998).

A commonly used approach, the feasible region reduction method has been described in Park et al. (2006), which is general and does not require any normality assumption. This is a tolerance design method, where width of the feasible space in each direction is reduced by the amount kσ, where k is a user-defined constant and σ is the standard deviation of the performance function. A comparison study of the different constraint feasibility methods can be found in Du and Chen (2000).

1.3 Estimating mean and variance of the performance function

Various methods have been reported in the literature to estimate the mean and standard deviation of the performance function. These methods can be divided into three major classes: (i) Taylor series expansion methods, (ii) sampling-based methods and (iii) point estimate methods (Huang and Du 2007).

The Taylor series expansion method (Haldar and Mahadevan 2000; Du and Chen 2000; Lee and Park 2001) is a simple approach. However, for a nonlinear performance function, if the variances of the random variables are large, this approximation may result in large errors (Du et al. 2004). Sampling-based methods require information on distributions of the random variables, and are expensive. Efficient sampling techniques such as importance sampling, Latin hypercube sampling, etc. (Robert and Casella 2004) and surrogate models (Ghanem and Spanos 1991; Bichon et al. 2008; Cheng and Sandu 2009) may be used to reduce the computational effort. Point estimate method (Rosenblueth 1975) overcomes the difficulties associated with the computation of derivatives required in Taylor series expansion. Different variations of this point estimate method (Hong 1998; Zhao and Ono 2000; Zhao and Ang 2003) have been studied. A more recent approach to estimating the mean and variance of the performance function is the dimension reduction method (DRM) (Rahman and Xu 2004; Xu and Rahman 2004; Lee et al. 2008). DRM was developed to overcome the shortcomings associated with the Taylor series expansion and the sampling methods.

1.4 Multi-objective optimization

Robustness-based optimization considers two objectives: optimize the mean of the objective function and minimize its variation. An extensive survey of the multi-objective optimization methods can be found in Marler and Arora (2004). Among the available methods, the weighted sum approach is the most common approach to multi-objective optimization and has been extensively used in robust design optimization (Lee and Park 2001; Doltsinis and Kang 2004; Zou and Mahadevan 2006). Other methods include the ε-constraint method (Mavrotas 2009), goal programming (Zou and Mahadevan 2006), compromise decision support problem (Bras and Mistree 1993, 1995; Chen et al. 1996), compromise programming (CP) (Zeleny 1973; Zhang 2003; Chen et al. 1999) and physical programming (Messac 1996; Messac et al. 2001; Messac and Ismail-Yahaya 2002; Chen et al. 2000). Each of these methods has its own advantages and limitations.

Although there is now an extensive volume of literature for robust optimization methods and applications, all these methods have only been studied with respect to physical or natural variability represented by probability distributions. Uncertainty in system design also arises from other contributing factors. Sources of uncertainty may be divided into two types: aleatory and epistemic (Oberkampf et al. 2004). Aleatory uncertainty is irreducible. Examples include phenomena that exhibit natural variation like operating conditions, material properties, geometric tolerances, etc. In contrast, epistemic uncertainty results from a lack of knowledge about the system, or due to approximations in the system behavior models, or due to limited or subjective (e.g., expert opinion) data; it can be reduced as more information about the system is obtained.

One type of data uncertainty involves having limited data to properly define the distribution parameters of the random variables. This type of uncertainty may be reduced by collecting more data. In some cases of data uncertainty, distribution information of a random variable may only be available as intervals given by experts. The objective of this paper is to develop an efficient robust optimization methodology that includes both aleatory and epistemic uncertainty described through sparse point data and interval data.

A few studies on robust design optimization are reported in the literature to deal with epistemic uncertainty arising from lack of information. Youn et al. (2007) used a possibility-based method, and redefined the performance measure of robust design using the most likely values of fuzzy random variables. Dai and Mourelatos (2003) proposed two two-step methods for robust design optimization that can treat aleatory and epistemic uncertainty separately using a range method and a fuzzy sets approach. Most of the current methods of robust optimization for epistemic uncertainty need additional non-probabilistic formulations to incorporate epistemic uncertainty into the robust optimization framework, which may be computationally expensive. However, if the epistemic uncertainty can be converted to a probabilistic format, the need for these additional formulations is avoidable, and well-established probabilistic methods of robust design optimization can be used. Therefore, there is a need for an efficient robust design optimization methodology that deals with both aleatory and epistemic uncertainty.

In this paper, we propose robustness-based design optimization formulations that work under both aleatory and epistemic uncertainty using probabilistic representations of different types of uncertainty. Our proposed formulations deal with both sparse point and interval data without any assumption about probability distributions of the random variables.

The performance of robustness-based design can be defined by the mean and variation of the performance function. In our proposed formulations, we obtain the optimum mean value of the objective function (e.g., gross weight) while also minimizing its variation (e.g., standard deviation). Thus, the design will meet target values in terms of both design bounds and standard deviations of design objectives and design variables thereby ensure feasibility robustness.

A Taylor series expansion method is used in this paper to estimate the mean and standard deviation of the performance function, which requires means and standard deviations of the random variables. However, with sparse point data and interval data, it is impossible to know the true moments of the data, and there are many possible probability distributions that can represent these data (Zaman et al. 2010). In this paper, we propose methods for robustness-based design optimization that account for this uncertainty in the moments due to sparse point data and interval data and thereby include epistemic uncertainty into the robust design optimization framework. As collecting more data reduces uncertainty but increases cost, the effect of sample size on the optimality and the robustness of the solution is also studied. A method to determine the optimal sample size for sparse point data that will lead to the minimum scatter on solutions to the design problem is also presented in this paper.

In some existing methods for robust design under epistemic uncertainty, all the epistemic variables are considered as design variables (Youn et al. 2007). However, if the designer does not have any control on an epistemic variable (e.g., Young’s modulus in beam design), considering that variable as a design variable might lead to a solution that could underestimate the design objectives. Therefore, in this paper, we propose a general formulation for robust design that considers some of the epistemic variables as non-design variables, which leads to a conservative design under epistemic uncertainty. An example of epistemic uncertainty in a design variable is the geometric dimension of a component, whose manufactured value is different from the design value. This difference might be specified as an interval by an expert, or only a few instances of historic values of this difference might be available.

Note that the proposed robustness-based design optimization method is general and capable of handling a wide range of application problems under data uncertainty. The proposed methods are illustrated for the conceptual level design process of a two-stage-to-orbit (TSTO) vehicle, where the distributions of the random inputs are described by sparse point and/or interval data.

The rest of the paper is organized as follows. Section 2 proposes robustness-based design optimization framework for sparse point data and interval data. In Section 3, we illustrate the proposed methods for the conceptual level design process of a TSTO vehicle. Section 4 provides conclusions and suggestions for future work.

2 Proposed methodology

2.1 Deterministic design optimization

In a deterministic optimization formulation, all design variables and system variables are considered deterministic. No random variability or data uncertainty is taken into account. The deterministic optimization problem is formulated as follows:

$$ \begin{array}{rll} \label{eq1} && {\mathop {\min }\limits_{\textbf{\emph x}} } \quad {f({\textbf{\emph x}})} \nonumber\\ [-1pt] && {s.t.} \quad{\textbf{\emph{LB}}}\le g_i \left({\textbf{\emph x}}\right)\le {{\textbf{\emph{UB}}}\,\,\,\,\mbox{for all }i} \nonumber\\ [-1pt] &&{\phantom{\quad}}\textbf{\emph{lb}}\le {\textbf{\emph x}} \le {\textbf{\emph{ub}}} \end{array}$$
(1)

where \(f({\textbf{\emph x}})\) is the objective function, x is the vector of design variables, g i (x) is the ith constraint, LB and UB are the vectors of lower and upper bounds of constraints g i ’s and lb and ub are the vectors of lower and upper bounds of design variables.

In practice, the input variables might be uncertain and solutions of this deterministic formulation could be sensitive to the variations in the input variables. Robustness-based design optimization takes this uncertainty into account. The optimal design points obtained using the deterministic method could be used as initial guesses in robustness-based optimization.

2.2 Robustness-based design optimization

In the proposed methodology, we use variance as a measure of variation of the performance function in order to achieve objective robustness, the feasible region reduction method to achieve feasibility robustness, a first-order Taylor series expansion to estimate the mean and variance of the performance function, and a weighted sum method for the aggregation of multiple objectives. This combination of methods is only used for the sake of illustration. Other approaches can be easily substituted in the proposed methodology. The robustness-based design optimization problem can now be formulated as follows:

$$\begin{array}{*{20}c} {\mathop {\min }\limits_d \,\,\,f\left( {\mu ,\sigma } \right)} \hfill & = \hfill & {\left( {w^* \mu _f + v^* \sigma _f } \right)} \hfill \\ {s.t.} \hfill & {} \hfill & {\user2{LB} + k\sigma \left( {g_i \left( {\user2{d},z} \right)} \right) \leqslant E\left( {g_i \left( {\user2{d},z} \right)} \right)} \hfill \\ {} \hfill & \leqslant \hfill & {\user2{UB - }k\sigma \left( {g_i \left( {\user2{d},z} \right)} \right) \leqslant \,\,\,\,\,\text{for}\,\text{all}\,i} \hfill \\ {} \hfill & {} \hfill & {lb_i + k\sigma \left( {x_i } \right) \leqslant d_i \leqslant ub_i - k\sigma \left( {x_i } \right)} \hfill \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\,i} \hfill & = \hfill & {1,2,...nrdv,} \hfill \\ {} \hfill & {} \hfill & {lb_i \leqslant d_i \leqslant ub_i } \hfill \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\,i} \hfill & \text{ = } \hfill & {1,2...,nddv} \hfill \\ \end{array} $$
(2)

where μ f and σ f are the mean value and standard deviation of the objective function, respectively; \(\boldsymbol{d}\) is the vector of deterministic design variables as well as the mean values of the uncertain design variables \(\boldsymbol{x}\); nrdv and nddv are the numbers of the random design variables and deterministic design variables, respectively; and \(\boldsymbol{z}\) is the vector of non-design input random variables, whose values are kept fixed at their mean values as a part of the design. \(\emph{w} \ge 0\) and \(\emph{v} \ge 0\) are the weighting coefficients that represent the relative importance of the objectives μ f and σ f in (2); \(g_{i}(\boldsymbol{d}, \boldsymbol{z})\) is the ith constraint; \(E(g_{i}(\boldsymbol{d}, \boldsymbol{z}))\) is the mean and \(\sigma(g_{i}(\boldsymbol{d}, \boldsymbol{z}))\) is the standard deviation of the ith constraint. \(\boldsymbol{LB}\) and \(\boldsymbol{U\!B}\) are the vectors of lower and upper bounds of constraints g i ’s; \(\boldsymbol{lb}\) and \(\boldsymbol{ub}\) are the vectors of lower and upper bounds of the design variables; \(\sigma({\textbf{\emph x}})\) is the vector of standard deviations of the random variables and k is some constant. The role of the constant k is to adjust the robustness of the method against the level of conservatism of the solution. It reduces the feasible region by accounting for the variations in the design variables and is related to the probability of constraint satisfaction. For example, if a design variable or a constraint function is normally distributed, k = 1 corresponds to the probability 0.8413, k = 2 to the probability 0.9772, etc. The choice of k simply indicates that we would like to restrict the feasible region to k standard deviations away from the mean, and this choice is made by the designer.

Note that the robust design formulation in (2) is a standard nonlinear multi-objective optimization formulation. The optimality conditions of such a formulation have been extensively described in the literature including Cagan and Williams (1993) and Marler and Arora (2004).

In the proposed formulation, the performance functions considered are in terms of the model outputs. The means and standard deviations of the objective and constraints are estimated by using a first-order Taylor series approximation as follows:

$$ \label{eq3} \mbox{Performance}\;\mbox{function:}\;Y=g\left( {X_1 ,X_2 ,....,X_n } \right) $$
(3)
$$ \label{eq4} \begin{array}[b]{ll} {\kern-4pt} \mbox{First-order approximate mean of y:}\\[10pt] \;\;E\left( {Y^\prime} \right) \approx g\left( {\mu _{X_1 } ,\mu _{X_2 } ,....,\mu _{X_n } } \right) \end{array} $$
(4)
$$ \label{eq5} \begin{array}[b]{ll} {\kern-4pt} \mbox{First-order variance of y:}\\[10pt] \;\;Var\left( {Y^\prime } \right)\approx \displaystyle\sum\limits_{i=1}^n {\left( {\frac{\partial g}{\partial X_i }} \right)} ^2Var\left( {X_i } \right)\\[16pt] {\kern49pt}\;\; +\mathop {\displaystyle\sum\limits_{i=1}^n {\displaystyle\sum\limits_{j=1}^n {\frac{\partial g}{\partial X_i }\frac{\partial g}{\partial X_j }} } }\limits_{i\ne j} Co\emph{v}\left( {X_i ,X_j } \right) \end{array} $$
(5)

Note that in (4) and (5), Y ′ has been used instead of Y to indicate that the mean and variance of Y are obtained by approximation method.

The implementation of (2) requires that variances of the random design variables X i and the means and variances of the random non-design variables Z i be precisely known, which is possible only when a large number of data points are available. In practical situations, only a small number of data points may be available for the input variables. In other cases, information about random input variables may only be specified as intervals, as by expert opinion. This is input data uncertainty, causing uncertainty regarding the distribution parameters (e.g., mean and variance) of the inputs X i and Z i . Robustness-based optimization has to take this into account. In the following subsections, we propose a new methodology for robustness-based design optimization that accounts for data uncertainty.

2.3 Robustness-based design optimization under data uncertainty

The inclusion of epistemic uncertainty in robust design adds another level of complexity in the design methodology. The design variables \(\boldsymbol{d}\) and/or the input random variables \(\boldsymbol{z}\) in (2) might have epistemic uncertainty. Since the designer does not have any control on the non-design epistemic variables \(\boldsymbol{z}\), the design methodology has to employ a search among the possible values of such epistemic variables in order to find an optimal solution. In such case, we get a conservative robust design. The robustness-based design optimization problem can now be formulated with the following generalized statement, where the objective is to minimize the worst case cost (i.e., upper bound of the cost, which is due to the epistemic uncertainty):

$$ \begin{array}{rll} \label{eq6} {\kern-5.4pt} &&{\mathop {\mbox{min}}\limits_\textbf{\emph{d}} \left( {\mathop {\mbox{max}}\limits_{\boldsymbol\mu _{\boldsymbol{z}} } f\left( {\mu ,\sigma } \right)=\left( {\emph{w}\ast \mu _f + \emph{v}\ast \sigma _f } \right)} \right)} \nonumber\\ [5pt] && s.t.\quad \boldsymbol{L\!B} +k\sigma \left( {g_i \left(\boldsymbol{d}, \boldsymbol{\mu}_z \right)} \right)\le E\left( {g_i \left(\boldsymbol{d}, \boldsymbol{z} \right)}\right) \nonumber\\ [5pt] && \le \boldsymbol{U\!B}-k\sigma \left({g_i \left( {\boldsymbol{d}, \boldsymbol{\mu}_z } \right)} \right)\quad \mbox{for}\;\mbox{all}\;i \nonumber\\ [5pt] &&{\boldsymbol{lb} + k\boldsymbol{\sigma} \left( \boldsymbol{x} \right)\le \boldsymbol{d} \le \boldsymbol{ub}-k \boldsymbol{\sigma} \left(\boldsymbol{x}\right)} \nonumber\\ [5pt] && {\boldsymbol{Z}_{\boldsymbol{l}} \le \boldsymbol{\mu} _{\boldsymbol{z}} \le \boldsymbol{Z}_{\boldsymbol{u}} } \end{array} $$
(6)

where \(\boldsymbol{Z}_{\boldsymbol{l}}\) and \(\boldsymbol{Z}_{\boldsymbol{u}}\) are the vectors of lower and upper bounds of the decision variables \(\boldsymbol{\mu}_{\boldsymbol{z}}\) of the inner loop optimization problem.

Note that in this formulation, the outer loop decision variables \(\boldsymbol{d}\) may consist of stochastic design variables as well as epistemic design variables. The outer loop optimization is a design optimization problem, where a robust design optimization is carried out for a fixed set of non-design epistemic variables. The inner loop optimization is the analysis for the non-design epistemic variables, where the optimizer searches among the possible values of the non-design epistemic variables to calculate the upper bound of the objective function value.

The nested formulation does not guarantee to converge and even if it converges, it is computationally very expensive. In nested approach, for every iteration of the epistemic analysis, the design optimization problem under aleatory uncertainty has to be repeated. Therefore, we un-nest the design optimization problem from the epistemic analysis and thereby achieve computational efficiency. This nested optimization problem can be decoupled and expressed as:

$$ \begin{array}{rll} &&{\boldsymbol{d}^{\ast}}= \mathop {\arg \min}\limits_{\boldsymbol{d}} \,\left(\emph{w}\ast \mu _f \left(\boldsymbol{d}, \boldsymbol{\mu}_{\boldsymbol{z}}^{\ast} \right) + \emph{v} \ast \sigma _f \left(\boldsymbol{d}, \boldsymbol{\mu}_{\boldsymbol{z}}^{\rm {\bf \ast }} \right) \right) \nonumber\\ [5pt] &&s.t.\quad \boldsymbol{L\!B} + k\sigma \left( {g_i \left(\boldsymbol{d}, \boldsymbol{\mu}_{\boldsymbol{z}}^{\rm {\bf \ast }} \right)} \right)\le E\left( {g_i \left(\boldsymbol{d}, \boldsymbol{z} \right)} \right) \nonumber\\ [5pt] && \quad\quad \le \boldsymbol{U\!B}-k\sigma \left( {g_i \left(\boldsymbol{d}, \boldsymbol{\mu} _{\boldsymbol{z}}^{\rm {\bf \ast }} \right)} \right)\,\,\mbox{ for all }i \nonumber\\ [5pt] &&{\qquad\quad \boldsymbol{l\!b} + k \boldsymbol{\sigma} \left(\boldsymbol{x}\right)\le \boldsymbol{d} \le \,\boldsymbol{u\!b}-k\boldsymbol{\sigma} \left(\boldsymbol{x}\right)} \end{array}$$
(7)
$$ \begin{array}{rll} \label{eq8}{\kern-4.6pt} &&{\boldsymbol{\mu}_{\boldsymbol{z}}^\ast =\mathop {\arg \max }\limits_{{\mu}_{\boldsymbol{z}} } \,\left({\emph{w}\ast \mu _f \left(\boldsymbol{d}^{{\bf \ast }}, \boldsymbol{\mu}_{\boldsymbol{z}} \right)+ \emph{v} \ast \sigma _f \left({\boldsymbol{d}^{\bf \ast}, \boldsymbol{\mu}_{\boldsymbol{z}} } \right)} \right)} \nonumber\\ [5pt] &&s.t. \quad \boldsymbol{L\!B} + k\sigma \left( {g_i \left(\boldsymbol{d}^{\bf\ast}, \boldsymbol{\mu}_{\boldsymbol{z}} \right)} \right) \le E\left( {g_i \left(\boldsymbol{d}, \boldsymbol{z} \right)} \right) \nonumber\\ [5pt] &&{\kern36pt} \le \boldsymbol{U\!B}-k\sigma \left( {g_i \left(\boldsymbol{d}^{\ast}, \boldsymbol{\mu}_{\boldsymbol{z}}\right)} \right)\,\,\mbox{ for all }i \nonumber\\ [5pt] &&{\kern25pt} {\boldsymbol{Z}_{\boldsymbol{l}} \le \boldsymbol{\mu}_{\boldsymbol{z}} \le \, \boldsymbol{Z}_{{u}} } \end{array}$$
(8)

The optimization problems in (7) and (8) are solved iteratively until convergence. Note that the first constraint (i.e., the robustness constraint) in (8) is required to ensure that the optimization is driven by all non-design epistemic variables, because sometimes the objective function may not be a function of all non-design epistemic variables. In cases when the objective function is the function of all non-design epistemic variables, this constraint is not required. Figure 1 illustrates the decoupled approach for robustness-based design optimization under both aleatory and epistemic uncertainty.

Fig. 1
figure 1

Decoupled approach for robustness-based design optimization

Full size image

Note that \(\boldsymbol{d}^{\ast}\) are fixed quantities in the optimization in (8) and \(\boldsymbol{\mu}_{\boldsymbol{z}}^{\ast}\) are the fixed quantities in the optimization in (7).

2.3.1 Robustness-based design with sparse point data

This section develops a methodology for robustness-based design optimization with sparse point data, using the formulations in (7) and (8). It is assumed that only sparse point data are available for the uncertain design variables as well as non-design epistemic variables. In using central limit theorem, it is customary to consider that the sample is reasonably large when the sample size n is greater than or equal to 30. In this paper, we assume a variable is described by the sparse point data if the sample size is less than 30. In practical problems, sparse data may refer to much smaller sample sizes.

When a variable, either design or non-design, is described by sparse point data, there is uncertainty about the mean and variance calculated from the samples. In the design optimization (7), the mean values of the design variables (either aleatory or epistemic) are controlled by the given design bounds. As in design optimization under aleatory uncertainty alone, here also it is assumed that the variances of the epistemic design variables do not change as their mean values change. However, since the mean values of the non-design variables cannot be controlled in the design optimization, the proposed robustness-based design optimization methodology accounts for the uncertainty about mean values of such epistemic variables through the optimization in (8).

The constraints on the non-design epistemic variables in (8) are implemented through the construction of confidence intervals about mean values. As these variables are described by the sparse point data, it is possible that the underlying distributions of the variables might have major deviations from normality. Therefore, we have used the Johnson’s modified t statistic (Johnson 1978) to construct the confidence bounds on mean values of the non-design epistemic variables as follows:

$$ \label{eq9}{\kern-2.5pt} \begin{array}{rl} \boldsymbol{Z}_{\boldsymbol{l}} &=\overline{\boldsymbol{z}} -t_{\alpha /2,n-1} \dfrac{\boldsymbol{s}}{\sqrt n }+\dfrac{\boldsymbol{\mu}_{\bf 3} }{6s^2n}\, \\[10pt] \boldsymbol{Z}_{\boldsymbol{u}} &=\overline{z} +t_{\alpha /2,n-1} \dfrac{\boldsymbol{s}}{\sqrt n }+\dfrac{\boldsymbol{\mu}_{\bf 3} }{6s^2n} \end{array} $$
(9)

where \({\overline z} \) is the vector of means of the epistemic variables, \(\boldsymbol{s}\) is the vector of sample standard deviations, n is the sample size of the sparse point data, \(\boldsymbol{\mu}_{\bf 3}\) is the third central moment \(\left( \mu _3 =\frac{1}{n}\sum\limits_{i=1}^n \left(x_i -\frac{1}{n}\sum\limits_{j=1}^n {x_j}\right)^{3}\right)\) and t α/2,n − 1 is obtained from the Student t distribution at (n − 1) degrees of freedom and α significance level. This modified statistic takes into account the skewness of the distribution and thus provides a better estimate of the confidence bound in the presence of limited data.

The proposed robustness-based design optimization methodology accounts for the uncertainty about the variances for all epistemic variables by first estimating confidence bounds on variances and then solving the optimization formulations in (7) and (8) using the upper bound variances for the input random variables x i and z i . Solving the optimization formulations in (7) and (8) using the upper bound variances for all the epistemic variables ensures that the resulting solution is least sensitive to the variations in the input random variables.

The chi-square distribution is a good assumption for the distribution of the variance, especially if the underlying population is normal. The two-sided (1 − α) confidence interval for the population variance σ 2 can be expressed as (Haldar and Mahadevan 2000):

$$ \label{eq10} \left[ {\frac{\left( {n-1} \right)s^2}{c_{1-\alpha /2,n-1} };\frac{\left( {n-1} \right)s^2}{c_{\alpha /2,n-1} }} \right] $$
(10)

where n is the sample size, s is the sample standard deviation of sparse point data, and c α/2,n − 1 is obtained from the chi-square distribution at (n − 1) degrees of freedom and α significance level. Note that (10) can still be used to obtain approximate confidence bounds for variance if the underlying population is not normal. However, in such cases, other approximation methods (Bonett 2006; Cojbasic and Tomovic 2007) can be used to obtain more reliable estimates of confidence bounds.

Note that (10) is used to estimate the confidence interval for the input random variables described by sparse point data. Once the confidence intervals for the input random variables are obtained using (10), the upper bound variances are used to estimate the variances of the performance function using first-order Taylor series approximation as discussed in Section 2.

The optimization formulation shown in (7) and (8) involves aggregation of multiple objectives. In the proposed formulations, the aggregate objective function consists of two types of objectives, expectation and standard deviation of model outputs. Since different objectives have different magnitudes, a scaling factor has to be used in the formulation.

2.3.2 Determination of optimal sample size for sparse point data

The optimal solutions depend on the sample size of the sparse data as will be discussed in Section 3.1. Therefore, it is of interest to determine the optimal sample size of the sparse data that leads to the solution of the design problem that is least sensitive to the variations of design variables. This will facilitate resource allocation decisions for data collection. The following two optimization formulations are solved iteratively until convergence for the optimal sample sizes of the epistemic design variables \(({\boldsymbol{n}_{\boldsymbol{d}}^{\rm {\bf \ast }} } )\) and epistemic non-design variables \(({\boldsymbol{n}_{\boldsymbol{e}}^\ast } )\). The formulations in (11) and (12) are the weighted sum formulations of a three-objective optimization problem, where the first and second objectives are the mean and standard deviation of the performance function respectively and the third objective is the total cost of obtaining samples for all the random variables.

$${\kern-4.6pt} {\begin{array}{*{20}l} \left[ {\boldsymbol{d}^\ast ,\boldsymbol{n}_{\boldsymbol{d}}^\ast } \right]=\arg \mathop {\min }\limits_{\boldsymbol{d},\boldsymbol{n}_{\boldsymbol{d}} } \left(\vphantom{\displaystyle\sum\limits_{j=1}^m} \emph{w}\ast E\left( {g_i \left( {\boldsymbol{d}, \boldsymbol{\mu}_{{z}}^\ast } \right)} \right) \right.\nonumber\\[16pt] {\kern88pt} + \, \emph{v} \ast \sigma \left( {g_i \left( {\boldsymbol{d}, \boldsymbol{\mu}_z^\ast } \right), \boldsymbol{n}_{\boldsymbol{d}}, \boldsymbol{n}_{\boldsymbol{e}}^\ast } \right)\nonumber\\[10pt] {\kern88pt} +\left( {1-\emph{w}-\emph{v}} \right)\\[10pt] \left. {\kern88pt} \ast \left( {\displaystyle\sum\limits_{j=1}^m {n_{d_j } c_{d_j } } + \displaystyle\sum\limits_{j=1}^q {n_{e_j }^\ast c_{e_j } } } \right) \right) \nonumber\\[18pt] s.t.\quad \quad \boldsymbol{L\!B}+k\sigma \left( {g_i \left( {\boldsymbol{d}, \boldsymbol{\mu}_z^\ast} \right), \boldsymbol{n}_{\boldsymbol{d}}, \boldsymbol{n}_{\boldsymbol{e}}^\ast } \right)\le E\left( {g_i \left( {\boldsymbol{d}, \boldsymbol{z}} \right)} \right) \nonumber\\[10pt] {\kern36pt} \le \boldsymbol{U\!B}-k\sigma \left( {g_i \left( {\boldsymbol{d}, \boldsymbol{\mu}_z^\ast} \right), \boldsymbol{n}_{\boldsymbol{d}}, \boldsymbol{n}_{\boldsymbol{e}}^\ast } \right)\;\mbox{for all}\;i \nonumber\\[10pt] {\qquad \quad \quad \boldsymbol{l\!b}+k\sigma \left( {\boldsymbol{x}, \boldsymbol{n}_{\boldsymbol{d}} } \right)\le \boldsymbol{d} \le \boldsymbol{u\!b} - k\sigma \left( {\boldsymbol{x}, \boldsymbol{n}_{\boldsymbol{d}} } \right)} \\[10pt] \end{array} } $$
$$ \begin{array}{rll} \label{eq11} &&{\displaystyle\sum\limits_{j=1}^m {n_{d_j } c_{d_j } } +\displaystyle\sum\limits_{j=1}^q {n_{e_j }^\ast c_{e_j } } \le C} \nonumber\\ && { \quad \quad n_{d_j } \le b_{d_j } \quad \mbox{for all}\;\mbox{j}} \end{array}$$
(11)
$$ {\kern-4.6pt} {\begin{array}{*{20}l} \left[ {\boldsymbol{\mu}_{\boldsymbol{z}}^\ast, \boldsymbol{n}_{\boldsymbol{e}}^\ast } \right]= \arg \mathop {\max }\limits_{\boldsymbol{\mu}_{\boldsymbol{z}}, \boldsymbol{n}_{\boldsymbol{e}} } \left(\vphantom{\displaystyle\sum\limits_{j=1}^m} \emph{w}\ast E\left( {g_i \left( {\boldsymbol{d}^\ast, \boldsymbol{\mu}_z } \right)} \right) \right.\\[16pt] {\kern88pt} +\, \emph{v} \ast \sigma \left( {g_i \left( {\boldsymbol{d}^\ast, \boldsymbol{\mu}_{\boldsymbol{z}} } \right), \boldsymbol{n}_{\boldsymbol{d}}^\ast, \boldsymbol{n}_{\boldsymbol{e}} } \right)\\[10pt] {\kern88pt} +\left( {1-\emph{w}-\emph{v}} \right) \\[10pt] {\kern88pt}\left. \ast \left( {\displaystyle\sum\limits_{j=1}^m {n_{d_j }^\ast c_{d_j } } +\displaystyle\sum\limits_{j=1}^q {n_{e_j } c_{e_j } } } \right) \right) \\[16pt] s.t.\quad \quad \boldsymbol{L\!B} +k\sigma \left( {g_i \left( {\boldsymbol{d}^\ast, \boldsymbol{\mu}_{\boldsymbol{z}} } \right), \boldsymbol{n}_{\boldsymbol{d}}^\ast, \boldsymbol{n}_{\boldsymbol{e}} } \right)\le E\left( {g_i \left( {\boldsymbol{d}, \boldsymbol{z}} \right)} \right) \\[10pt] {\kern27pt}\quad \le \boldsymbol{U\!B}-k\boldsymbol{\sigma} \left( {g_i \left( {\boldsymbol{d}^\ast, \boldsymbol{\mu}_{\boldsymbol{z}} } \right), \boldsymbol{n}_{\boldsymbol{d}}^\ast, \boldsymbol{n}_{\boldsymbol{e}} } \right)\;\mbox{for all}\;i \\[10pt] {\boldsymbol{Z}_{\boldsymbol{l}} \left( {\boldsymbol{n}_{\boldsymbol{e}} } \right)\le \boldsymbol{\mu}_{\boldsymbol{z}} \le \boldsymbol{Z}_{\boldsymbol{u}} \left( {\boldsymbol{n}_{\boldsymbol{e}} } \right)} \end{array} } $$
$$ \begin{array}{rll} \label{eq12} &&{\quad \quad \quad \quad \displaystyle\sum\limits_{j=1}^m {n_{d_j }^\ast c_{d_j } } +\displaystyle\sum\limits_{j=1}^q {n_{e_j } c_{e_j } } \le C} \nonumber\\ && {\quad \quad \quad \quad n_{e_j } \le b_{e_{\rm j} } \quad \mbox{for all}\;\mbox{j}} \end{array}$$
(12)

where \(\emph{w} \ge 0\) and \(\emph{v} \ge 0\) are the weighting coefficients that represent the relative importance of the objectives; \(n_{d_j }\) and \(n_{e_j }\) are the sample sizes and \(b_{d_j}\) and \(b_{e_j}\) are the maximum sample size possible for the jth design and non-design random variables, respectively. m and q are the number of design and non-design random variables, respectively. \(c_{d_j}\) and \(c_{e_j}\) are the cost of obtaining one sample for the jth random design and non-design variables, respectively and C is the total cost allocated for obtaining samples for all the random variables. Note that as in (8), the robustness constraint in (12) is only required if the objective function is not a function of all non-design epistemic variables. The optimization formulation presented above is a mixed-integer nonlinear problem. A relaxed problem is solved in Section 3.

2.3.3 Robustness-based design with interval data

This section develops a methodology for robustness-based design optimization with interval data, using the formulations in (7) and (8). In this case, the only information available for one or more input random variables is in the form of single interval or multiple interval data. A discussion on the source and the nature of both single and multiple interval data can be found in Zaman et al. (2010).

The methodology for robustness-based design optimization with interval data is similar to sparse point data as described in Section 2.3.1. However, the estimation of mean values and variances for interval data is not straightforward. For interval data, the moments (e.g., mean and variance) are not single-valued, rather only bounds can be given (Zaman et al. 2010). The authors have proposed methods to compute the bounds of moments with both single and multiple interval data in Zaman et al. (2010). The methods for computing bounds of the first two moments for interval data are given later in this section. Once the bounds on the mean and variance of interval data are estimated, we use the upper bounds of sample variance to solve the formulations of robust design under uncertainty represented through single interval or multiple interval data. Therefore, the resulting solution becomes least sensitive to the variations in the uncertain variables.

For non-design epistemic variables described by interval data, the constraints on the decision variables in (8) are implemented through estimating the bounds of the means by the methods as described later in this section.

The following discussions briefly summarize the methods to estimate the bounds on the first two moments for single interval and multiple interval data, respectively.

Bounds on moments with single interval data

The methods for calculating bounds on the first two moments for single interval data are summarized in Table 1 below.

Table 1 Methods for calculating moment bounds for single interval data
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In Table 1, the formulas lead to the lower and upper endpoints of the interval as the lower and upper bounds for the first moment, respectively. The formulas also imply that the lower bound for the second moments is zero.

Bounds on moments with multiple interval data

The methods for calculating bounds on the first two moments for multiple interval data are summarized in Table 2 below.

Table 2 Methods for calculating moment bounds for multiple interval data
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Once the bounds on the mean and variance of interval data are estimated by the methods described above, we can now use these bounds to solve the formulations of robustness-based design optimization under uncertainty represented through single interval or multiple interval data. In the following section, we illustrate our proposed formulations for robustness-based design optimization with both sparse point and interval data.

3 Example problem

In this section, the proposed methods are illustrated for the conceptual level design process of a TSTO vehicle. The multidisciplinary system analysis consists of geometric modeling, aerodynamics, aerothermodynamics, engine performance analysis, trajectory analysis, mass property analysis and cost modeling (Stevenson et al. 2002). In this paper, a simplified version of the upper stage design process of a TSTO vehicle is used to illustrate the proposed methods. High fidelity codes of individual disciplinary analysis are replaced by inexpensive surrogate models. Figure 2 illustrates the analysis process of a TSTO vehicle.

Fig. 2
figure 2

TSTO vehicle concept

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The analysis outputs (performance functions) are Gross Weight (GW), Engine Weight (EW), Propellant Fraction Required (PFR), Vehicle Length (VL), Vehicle Volume (VV) and Body Wetted Area (BWA). Each of the analysis outputs is approximated by a second-order response surface and is a function of the random design variables Nozzle Expansion Ratio (ExpRatio), Payload Weight (Payload), Separation Mach (SepMach), Separation Dynamic Pressure (SepQ), Separation Flight Path Angle (SepAngle), and Body Fineness Ratio (Fineness). Each of the random variables is described by either sparse point data or interval data.

The objective is to optimize an individual analysis output (e.g., Gross Weight) while satisfying the constraints imposed by each of the design variables as well as all the analysis outputs. We note here that we have assumed independence among the uncertain input variables and thereby ignored the covariance terms in (5) to estimate the variance of the performance function in each of the following examples. The numerical values of the design bounds for the design variables and analysis outputs are given in Tables 3 and 4, respectively.

Table 3 Design bounds for the design variables
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Table 4 Design bounds for the analysis outputs
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3.1 Robustness-based design optimization with sparse point data

The methodology proposed in Section 2.3.1 is illustrated here for the TSTO problem. It is assumed that all the input variables x are described by sparse point data as given in Table 5. For this example, the input variable SepQ is assumed to be a non-design epistemic variable and all the remaining variables are assumed to be design variables. The design bounds for the respective design variables and the analysis outputs are given in Tables 3 and 4.

Table 5 Sparse point data for the random input variables
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The design problem becomes:

$$ \begin{array}{rll} \label{eq13} &&{\kern-53pt} d^\ast =\mathop {\arg \min }\limits_d \,( {\emph{w}\ast E( {GW} )+( {1-\emph{w}} )\ast \sigma ( {GW} )} ) \\s.t.\quad LB_1 &+&k\sigma ( {GW} )\le E( {GW} )\le UB_1 -k\sigma ( {GW} ) \\LB_2 &+&k\sigma ( {EW} )\le E( {EW} )\le UB_2 -k\sigma ( {EW} ) \\ LB_3 &+&k\sigma ( {PFR} )\le E( {PFR} )\le UB_3 -k\sigma ( {PFR} ) \\ LB_4 &+&k\sigma ( {VL} )\le E( {VL} )\le UB_4 -k\sigma ( {VL} )\\ LB_5 &+&k\sigma ( {VV} )\le E( {VV} )\le UB_5 -k\sigma ( {VV} ) \\ LB_6 &+&k\sigma ( {BWA} )\le E( {BWA} )\le UB_6 -k\sigma ( {BWA} ) \\ lb_i &+&k\sigma ( x )\le d_i \le \,ub_i -k\sigma (x) \\&&{\kern-25pt} \mbox{for } i=1,2,...,5 \end{array}$$
(13)
$$\begin{array}{rll} \label{eq14} && \mu _z^\ast =\mathop {\arg \max }\limits_{\mu _z } \,\left( {\emph{w}\ast E\left( {GW} \right)+\left( {1-\emph{w}} \right)\ast \sigma \left( {GW} \right)} \right) \\&& s.t. \;\;\quad Z_l \le \mu _{z_i } \le \,Z_u \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for}\,\,i=1 \end{array}$$
(14)

where the bounds Z l and Z u for the mean of the non-design epistemic variable SepQ are calculated by (9) as given in Section 2.3.1 using α = 0.05. Note that this significance level is chosen arbitrarily for the sake of illustration. Note that in (14), we do not use the robust design constraints, since the objective function in this case is a function of all non-design epistemic variables.

As mentioned earlier in Section 2, \(\emph{w} \ge 0\) is the weight parameter that represents the relative importance of the objectives and k is a constant that adjusts the robustness of the method against the level of conservatism of the solution. In this paper, k is assumed to be unity.

Variances of the random variables \(\boldsymbol{x}\) and \(\boldsymbol{z}\) are estimated as single point values. Confidence intervals for the variances are estimated for each random variable described by the sparse point data. The weight parameter w is varied (from 0 to 1) and the optimization problem in (13) and (14) are solved iteratively until convergence by the Matlab solver ‘fmincon’ for different sample sizes (n) of the sparse point data. The formulations are relaxed by assuming that standard deviations estimates of the variables do not change significantly as the sample size changes. Therefore, the same standard deviations as estimated from the data given in Table 5 are used in each case. As the sample size (n) changes, the confidence bounds on the variance also change (see (10)). Note that we have solved the proposed design optimization formulations for different sample sizes (5, 10, 15, 20, 25, and 30) of sparse point data. However, we have used only 10 data points for each input random variable in this paper in order to avoid listing a large amount of data. The use of confidence bounds on the variance demonstrates the presence of epistemic uncertainty in the formulations and reduces the effect resulting from the assumed constant standard deviations. However, the proposed formulations can be easily solved using the standard deviations calculated from the real data i.e., using 5, 10, 15, 20, 25, 30 data points.

In each case, the optimization problems converged in less than 5 iterations. Here, ‘fmincon’ uses a sequential quadratic programming (SQP) algorithm. The estimate of the Hessian of the Lagrangian is updated using the BFGS formula at each iteration. The convergence properties of SQP have been discussed by many authors including Fletcher (1987) and Panier and Tits (1993).

The solutions are obtained by solving the problem using the upper confidence bound for the variances of the random variables \(\boldsymbol{x}\) and \(\boldsymbol{z}\). The solutions are presented in Fig. 3.

Fig. 3
figure 3

Robustness-based design optimization with sparse data for different sample sizes (n)

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It is seen in Fig. 3 that the solutions become more conservative (i.e., the mean and standard deviation of GW assume higher values) as we add uncertainty to the design problem. It is also seen from Fig. 3 that as the sample size (n) increases, both the standard deviation and mean of GW decrease. As gathering more data reduces data uncertainty, the solutions become less sensitive (i.e., the standard deviation of GW assumes lower value) to the variations of the input random variables as the sample size (n) increases. Also, looking at the mean of GW, it is seen that as the uncertainty decreases with sample size, the optimum mean weight required is less.

3.2 Determination of optimal sample size for sparse point data

The optimal sample size formulations are illustrated here for the TSTO design problem. The formulations are relaxed by assuming that standard deviations of the data do not change significantly as sample size changes. As in Section 3.1, the use of confidence bounds on the variance demonstrates the presence of epistemic uncertainty in the formulations and reduces the effect resulting from the assumed constant standard deviations. However, a more general problem can be solved using the proposed method by considering the standard deviations for each input random variable as optimization decision variables. This will eliminate the need for any further assumption.

To make the problem simpler, we first relax the integer requirement on the optimal sample size n and then round off the solution for n to the nearest integer value. The input variable SepQ is assumed to be a non-design epistemic variable and all the remaining variables are assumed to be design variables. The design bounds for the respective design variables and the analysis outputs remain the same as in Tables 3 and 4 respectively.

Therefore, the design problem becomes as follows:

$$ \begin{array}{rll} \label{eq15} &&\!\left[{\boldsymbol{d}^\ast, \boldsymbol{n}_{\boldsymbol d}^\ast } \right]=\arg \mathop {\min }\limits_{{\boldsymbol d},{\boldsymbol n_d} }\,\big( \emph{w}\ast E\left( {GW} \right) +\ \emph{v}\ast \sigma \left( {GW} \right) \nonumber\\ [2pt] &&{\kern12pt} +\left( {1-\emph{w}-\emph{v}} \right) \ast \big( 5n_{d1} +10n_{d2} +5n_{d3} +5n_{d4}\nonumber\\ [3pt] &&{\kern81pt} + 4n_{d5} +6n_e^\ast \big) \big) \nonumber\\ [3pt] &&s.t.\quad\;\; LB_1 +\sigma ( {GW} )\le E( {GW} )\le UB_1 -\sigma ( {GW} ) \nonumber\\ [4pt] &&{\kern32pt} {{\it LB}_2 +\sigma ( {\it EW} )\le E( {\it EW} )\le {\it UB}_2 -\sigma ( {\it EW} )} \nonumber\\ [4pt] &&{\kern32pt} { {\it LB}_3 +\sigma ( {\it PFR} )\le E( {\it PFR} )\le {\it UB}_3 -\sigma ( {\it PFR} )} \nonumber\\ [4pt] &&{\kern32pt}{ {\it LB}_4 +\sigma ( {\it VL} )\le E( {\it VL} )\le {\it UB}_4 -\sigma ( {\it VL} )} \nonumber\\ [4pt] &&{\kern32pt}{ {\it LB}_5 +\sigma ( {\it VV} )\le E( {\it VV} )\le {\it UB}_5 -\sigma ( {VV} )} \nonumber\\ [4pt] &&{\kern32pt}{ {\it LB}_6 +\sigma ( {\it BWA} )\le E( {\it BWA} )\le {\it UB}_6 -\sigma ( {\it BWA} )} \nonumber\\ [4pt] &&{\kern36pt}{ {\it lb}_i +k\sigma ( {x_i } )\le d_i \le \,{\it ub}_i -k\sigma ( {x_i } )}\nonumber\\ [4pt] &&{\kern32pt}{\text{for\ \ all}\;\;\; i=1,2,\ldots,5} \nonumber\\ [4pt] &&{\kern32pt}{ 5n_{d1} +10n_{d2} +5n_{d3} +5n_{d4} + 4n_{d5} +6n_e^\ast \!\le\! 1050} \nonumber\\ [4pt] &&{\kern37pt} {n_{d_j } \le 30\,\,\,\,\,\,\,\,\,\,\mbox{for\;\; j}=1,2,\ldots ,5} \end{array}$$
(15)
$$ \begin{array}{rll} \label{eq16} &&\! \left[ {\mu _z^\ast ,n_e^\ast } \right]=\arg \mathop {\max }\limits_{\mu _z ,ne} \,\big( \emph{w}\ast E\left( {GW} \right) + \emph{v} \ast \sigma \left( {GW} \right)\nonumber\\ [2pt] &&{\kern12pt} +\left( {1-\emph{w}-\emph{v}} \right) \ast \big( {5n_{d1}^\ast +10n_{d2}^\ast +5n_{d3}^\ast}{ + 5n_{d4}^\ast }\nonumber\\ [2pt] &&{\kern81pt}+\,4n_{d5}^\ast +6n_e^\ast \big) \big) \nonumber\\ [2pt] &&{s.t.\quad \qquad Z_l \left( {n_e } \right)\le \mu _z \le Z_u \left( {n_e } \right)} \nonumber\\ [2pt] &&{\phantom{s.t.\quad \qquad} 5n_{d1}^\ast +10n_{d2}^\ast +5n_{d3}^\ast +5n_{d4}^\ast +4n_{d5}^\ast} \nonumber\\ [2pt] &&\phantom{s.t.\quad \qquad}{+\,6n_e \le 1050} \nonumber\\ [2pt] &&{\phantom{s.t.\quad \qquad} n_{d_j} \le 30\,\,\,\,\,\,\,\,\,\,\mbox{for j}=1} \end{array}$$
(16)

where n d1, n d2, n d3, n d 4, and n d5 correspond to random design variables ExpRatio, Payload, SepMach, SepAngle, and Fineness, respectively and n e corresponds to non-design epistemic variable SepQ. Note that the cost of obtaining a sample to each random variable and the total cost allocated for sampling are chosen arbitrarily for the sake of illustration.

We have solved this problem for different combinations of weights w and v and the optimal solutions are presented in Table 6. In each case, the optimization problems converged in less than 4 iterations. Note that in Table 6, the Final Objective Value is the weighted summation of each objective function value.

Table 6 Objective function values at optimal solutions and optimal sample sizes
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It is seen in Table 6 that the total cost incurred in obtaining samples is the minimum when we solve the problem giving the maximum importance on the total cost. In this case, we get the most conservative robust design i.e., the mean and the standard deviation of GW assume the maximum of all possible values. Note that the optimal sample size required is also the minimum in this case. As we give more importance on the mean and standard deviation of GW, the total cost and also the optimal sample size increase with a decrease in both the mean and standard deviation of GW. Note that setting of weights is an open part of multi-objective optimization. Like any multi-objective optimization, the proposed formulation should be used with care in selecting the weights.

We mention here that the issue of sampling scheme is not relevant here, because our calculations only use the standard deviation estimates. We do not use sampling schemes to simulate the real world data for each sample size. Instead, we have made the assumption that the estimate of standard deviation of the variable does not change with sample size. That is, we randomly generated samples for n = 10 (using a normal distribution for illustration), and simply used the same standard deviation estimate for other values of n. Thus the optimization result is affected by the standard deviation, and not the actual samples. However, the calculation of confidence bounds on variances takes the sample size n into account. The assumption of constant standard deviation for different sample sizes can be avoided if desired, but this does not change the proposed formulation. In that case, we will need to generate multiple sets of samples for each sample size n, and then use the averaged estimates of standard deviations in the formulation.

3.3 Robustness-based design optimization with sparse point and interval data

The methodology proposed in Section 2.1 is illustrated here for the same TSTO problem. Here, it is assumed that the design variable ExpRatio is described by sparse point data as given in Table 5, the design variable Payload is described by multiple interval data as given in Table 7 and the design variables SepMach and SepQ are described by single interval data as given in Table 8. The non-design epistemic variables SepAngle and Fineness are described by the sparse point data (as given in Table 5) and the single interval data (as given in Table 8), respectively. The design bounds for the respective design variables and the analysis outputs remain the same as in Tables 3 and 4.

Table 7 Multiple interval data for the random input variables
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Table 8 Single interval data for the random input variables
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The design problem is now formulated as follows:

$$ \begin{array}{rll} \label{eq17} &&{\boldsymbol{d}^{{\ast }}=\mathop {\arg \min }\limits_{\boldsymbol{d}} \,\left( {\emph{w}\ast E\left( {GW} \right)+\left( {1-w} \right)\ast \sigma \left( {GW} \right)} \right)} \nonumber \\ && {s.t.\quad LB_1 +k\sigma ( {\it GW} )\le E( {\it GW} )\le UB_1 -k\sigma ( {\it GW} )} \nonumber\\ && {\phantom{s.t.\quad} LB_2 +k\sigma ( {\it EW} )\le E( {\it EW} )\le UB_2 -k\sigma ( {\it EW} )} \nonumber\\ && {\phantom{s.t.\quad} LB_3 +k\sigma ( {\it PFR} )\le E( {\it PFR} )\le UB_3 -k\sigma ( {\it PFR} )} \nonumber\\ && {\phantom{s.t.\quad} LB_4 +k\sigma ( {\it VL} )\le E( {\it VL} )\le UB_4 -k\sigma ( {\it VL} )} \nonumber\\ && {\phantom{s.t.\quad} LB_5 +k\sigma ( {\it VV} )\le E( {\it VV} )\le UB_5 -k\sigma ( {\it VV} )} \nonumber\\ && {\phantom{s.t.\quad} LB_6 +k\sigma ( {\it BWA} )\le E( {\it BWA} )\le UB_6 -k\sigma ( {\it BWA} )} \nonumber\\ && {\phantom{s.t.\quad} lb_i +k\sigma \left( {x_i } \right)\le d_i \le \,ub_i -k\sigma \left( {x_i } \right)}\nonumber\\ &&{\phantom{s.t.\quad}\mbox{for }i=1,2,3,4} \end{array}$$
(17)
$$ \label{eq18}{\begin{array}[b]{@{}l} {\boldsymbol{\mu}_z^{{\ast }} =\mathop {\arg \max }\limits_{\boldsymbol{\mu} _{\boldsymbol{z}} } \,\left( {\emph{w}\ast E\left( {GW} \right)+\left( {1-\emph{w}} \right)\ast \sigma \left( {GW} \right)} \right)} \\ {s.t.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Z_l \le \mu _{z_i } \le \,Z_u \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for}\,\,i=\,1,2} \\ \end{array} } $$
(18)

where the bounds Z l and Z u for the mean value of the non-design epistemic variable SepAngle are calculated by (9) as given in Section 2.3.1 using α = 0.05 and those for the epistemic variable Fineness are calculated by the method described in Section 2.3.3. Note that in (18), we do not use the robust design constraints, since the objective function in this case is a function of all non-design epistemic variables.

Variances of the random variables ExpRatio and SepAngle are estimated as single point values. Confidence intervals for the variances are estimated for each random variable described by sparse point data. Bounds on the variances of the random variables SepMach, SepQ, Fineness, and Payload are estimated by the methods described in Sections 2.3.3. The free parameter w is varied (from 0 to 1) and the optimization problems in (17) and (18) are solved iteratively until convergence. In each case, the optimization problems converged in less than 5 iterations. The solutions are obtained by solving the problems using the upper confidence bound on sample variance for the random variables ExpRatio and SepAngle, and the upper bound on sample variances for the random variables Payload, SepMach, SepQ and Fineness. The solutions are presented in Fig. 4.

Fig. 4
figure 4

Robustness-based design optimization with sparse point and interval data

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Figure 4 shows the solutions of the conservative robust design in presence of uncontrollable epistemic uncertainty described through mixed data i.e., both sparse point data and interval data, which is seen frequently in many engineering applications.

4 Summary and conclusion

This paper proposed several formulations for robustness-based design optimization under data uncertainty. Two types of data uncertainty—sparse point data and interval data—are considered. The proposed formulations are illustrated for the upper stage design problem of a TSTO space vehicle. A decoupled approach is proposed in this paper to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency. As gathering more data reduces uncertainty but increases cost, the effect of sample size on the optimality and the robustness of the solution is also studied. This is demonstrated by numerical examples, which suggest that as the uncertainty decreases with sample size, the resulting solutions become more robust. We have also proposed a formulation to determine the optimal sample size for sparse point data that leads to the solution of the design problem that is least sensitive (i.e., robust) to the variations of design variables. In this paper, we have used the weighted sum approach for the aggregation of multiple objectives and to examine the trade-offs among multiple objectives. Other multi-objective optimization techniques can also be explored within the proposed formulations.

The major advantage of the proposed methodology is that unlike existing methods, it does not use separate representations for aleatory and epistemic uncertainties and does not require nested analysis. Both types of uncertainty are treated in a unified manner using a probabilistic format, thus reducing the computational effort and simplifying the optimization problem. The results regarding robustness of the design versus data size are valuable to the decision maker. The design optimization procedure also optimizes the sample size, thus facilitating resource allocation for data collection efforts. Due to the use of a probabilistic format to represent all the uncertain variables, the proposed robustness-based design optimization methodology facilitates the implementation of multidisciplinary robustness-based design optimization, which is a challenging problem in presence of epistemic uncertainty.