An accurate input statistical model has been assumed in most of reliability-based design optimization (RBDO) to concentrate on variability of random variables. However, only limited number of data are available to quantify the input statistical model in practical engineering applications. In other words, irreducible variability and reducible uncertainty due to lack of knowledge exist simultaneously in random design variables. Therefore, the uncertainty in reliability induced by insufficient data has to be accounted for RBDO to guarantee confidence of reliability. The uncertainty of input distributions is successfully propagated to a cumulative distribution function (CDF) of reliability under normality assumptions, but it requires a number of function evaluations in double-loop Monte Carlo simulation (MCS). To tackle this challenge, reliability measure approach (RMA) in confidence-based design optimization (CBDO) is proposed to handle the randomness of reliability following the idea of performance measure approach (PMA) in RBDO. Input distribution parameters are transformed to the standard normal space for most probable point (MPP) search with respect to reliability. Therefore, the reliability is approximated at MPP with respect to input distribution parameters. The proposed CBDO can treat confidence constraints employing the reliability value at the target confidence level that is approximated by MPP in P-space. In conclusion, the proposed method can significantly reduce the number of function evaluations by eliminating outer-loop MCS while maintaining acceptable accuracy.

There are two sorts of uncertainty inherent in engineering design, random uncertainty and epistemic uncertainty. Random, or stochastic, uncertainty deals with the randomness or predictability of an event. It is well understood, easily modelled using classical probability, and ideal for such uncertainties as variations in manufacturing processes or material properties. Epistemic uncertainty deals with our lack of knowledge, our lack of information, and our own and others’ subjectivity concerning design parameters. While there are many methods to incorporate random uncertainty in a design process, there are fewer that consider epistemic uncertainty. There are fewer still that attempt to incorporate both sorts of uncertainty, and those that do usually attempt to model both sorts using the same uncertainty model. Two methods, a range method and a fuzzy sets approach, are proposed to achieve designs that are robust to both epistemic uncertainty and random uncertainty. Both methods incorporate preference aggregation methods to achieve more appropriate trade-offs between performance and variability when considering both sorts of uncertainty. The proposed models for epistemic uncertainty are combined with existing models for stochastic uncertainty in a two-step process. An illustrative example incorporating subjectivity concerning design parameters is presented.

Engineering design decisions inherently are made under uncertainty. In this paper, we consider imprecise probabilities (i.e. intervals of probabilities) to express explicitly the precision with which something is known. Imprecision can arise from fundamental indeterminacy in the available evidence or from incomplete characterizations of the available evidence and designer’s beliefs. Our hypothesis is that, in engineering design decisions, it is valuable to explicitly represent this imprecision by using imprecise probabilities. We support this hypothesis with a computational experiment in which a pressure vessel is designed using two approaches, both variations of utility-based decision making. In the first approach, the designer uses a purely probabilistic, precise best-fit normal distribution to represent uncertainty. In the second approach, the designer explicitly expresses the imprecision in the available information using a probability box, or p-box. When the imprecision is large, this p-box approach on average results in designs with expected utilities that are greater than those for designs created with the purely probabilistic approach. In the context of decision theory, this suggests that there are design problems for which it is valuable to use imprecise probabilities.

Rapid Prototyping (RP) is the process of building three-dimensional objects, in layers, using additive manufacturing. Rapid Manufacturing (RM) is the use of RP technologies to manufacture end-use, or finished, products. At small lot sizes, such as with customized products, traditional manufacturing technologies become infeasible due to the high costs of tooling and setup. RM offers the opportunity to produce these customized products economically. Coupled with the customization opportunities afforded by RM is a certain degree of uncertainty. This uncertainty is mainly attributed to the lack of information known about what the customer’s specific requirements and preferences are at the time of production. In this paper, we present an overall method for selection of a RM technology under the geometric uncertainty inherent to mass customization. Specifically, we define the types of uncertainty inherent to RM (epistemic), propose a method to account for this uncertainty in a selection process (interval analysis), and propose a method to select a technology under uncertainty (Hurwicz selection criterion). We illustrate our method with an example on the selection of an RM technology to produce custom caster wheels.

Early in the engineering design cycle, it is difficult to quantify product reliability or compliance to performance targets due to insufficient data or information to model uncertainties. Probability theory can not be therefore, used. Design decisions are usually, based on fuzzy information that is vague, imprecise qualitative, linguistic or incomplete. Recently, evidence theory has been proposed to handle uncertainty with limited information as an alternative to probability theory. In this paper, a computationally efficient design optimization method is proposed based on evidence theory, which can handle a mixture of epistemic and random uncertainties. It quickly identifies the vicinity of the optimal point and the active constraints by moving a hyper-ellipse in the original design space, using a reliability-based design optimization (RBDO) algorithm. Subsequently, a derivative-free optimizer calculates the evidence-based optimum, starting from the close-by RBDO optimum, considering only the identified active constraints. The computational cost is kept low by first moving to the vicinity of the optimum quickly and subsequently using local surrogate models of the active constraints only. Two examples demonstrate the proposed evidence-based design optimization method.

Reliability-based design optimization is much more computationally expensive than deterministic design optimization. To alleviate the computational demand, the First Order Reliability Method (FORM) is usually used in reliability-based design. Since FORM requires a nonlinear transformation from non-normal random variables to normal random variables, the nonlinearity of a constraint function may increase. As a result, the transformation may lead to a large error in reliability calculation. In order to improve accuracy, a new reliability-based design method with Saddlepoint Approximation is proposed in this work. The strategy of sequential optimization and reliability assessment is employed where the reliability analysis is decoupled from deterministic optimization. The accurate First Order Saddlepoint method is used for reliability analysis in the original random space without any transformation, and the chance of increasing nonlinearity of a constraint function is therefore eliminated. The overall reliability-based design is conducted in a sequence of cycles of deterministic optimization and reliability analysis. In each cycle, the percentile value of the constraint function corresponding to the required reliability is calculated with the Saddlepoint Approximation at the optimal point of the deterministic optimization. Then the reliability analysis results are used to formulate a new deterministic optimization model for the next cycle. The solution process converges within a few cycles. The demonstrative examples show that the proposed method is more accurate and efficient than the reliability-based design with FORM.

Both aleatory and epistemic uncertainties exist in engineering applications. Aleatory uncertainty (objective or stochastic uncertainty) describes the inherent variation associated with a physical system or environment. Epistemic uncertainty, on the other hand, is derived from some level of ignorance or incomplete information about a physical system or environment. Aleatory uncertainty associated with parameters is usually modeled by probability theory and has been widely researched and applied by industry, academia, and government. The study of epistemic uncertainty in engineering has recently started. The feasibility of the unified uncertainty analysis that deals with both types of uncertainties is investigated in this paper. The input parameters with aleatory uncertainty are modeled with probability distributions by probability theory, and the input parameters with epistemic uncertainty are modeled with basic probability assignment by evidence theory. The effect of the mixture of both aleatory and epistemic uncertainties on the model output is modeled with belief and plausibility measures (or the lower and upper probability bounds). It is shown that the calculation of belief measure or plausibility measure can be converted to the calculation of the minimum or maximum probability of failure over each of the mutually exclusive subsets of the input parameters with epistemic uncertainty. A First Order Reliability Method (FORM) based algorithm is proposed to conduct the unified uncertainty analysis. Two examples are given for the demonstration. Future research directions are derived from the discussions in this paper.