The idea behind this special supplement came out of discussions with the Editor, Dr. Michael McCarthy, about the need to present some recent advances in the area of risk-based and robust design. We felt, and still feel, that there is an ever increasing need to design mechanical and structural systems that are risk tolerant and robust. It was our belief that it would be useful if, by way of a dedicated special issue, we could help advance the understanding of risk-based and robust design in our mechanical design community. In this regard, the idea of the special issue was suggested to Dr. McCarthy who was immediately and enthusiastically responsive and supportive of the idea. We would like to specially thank him for that.

The papers in this supplement were submitted to the ASME Journal of Mechanical Design in response to a Call for Papers (CFP) for the special issue. The CFP was issued in June 2005 with a submission deadline in mid September 2005. We received around 30 papers. All papers went through at least two rounds of review following the guidelines and standard review procedures of the journal. The 19 papers that are presented in this special supplement are for the most part in the area of robust and reliability-based design. We therefore decided to call the special supplement as such. These papers are organized into four groups: (1) robust design; (2) reliability-based design; (3) understanding and managing uncertainty; and (4) collaborative and multilevel design.

There are six papers in the Robust Design group. An invited paper by Allen, Seepersad, and Mistree gives a literature survey of models and methods in the area of robust design. They classify robust design into three types, based on the source of uncertainty. These include: Type I, which refers to uncertainty or noise in the environment; Type II, which refers to uncertainty in design variables or control factors; and Type III, which refers to uncertainty introduced by modeling methods. They review numerous papers including those related to the Taguchi approach and its limitations and extensions, various stochastic and deterministic robust design models and methods, models in early stage of design, and those for multidisciplinary systems. They conclude their paper with a discussion of research challenges in robust design of multiscale systems and materials.

In the next paper, Rolander, Rambo, Joshi, Allen, and Mistree present a combined compromise decision support robust design approach with proper orthogonal decomposition (POD). POD is a reduced order modeling approach which, in the paper, is used for compact modeling of turbulent convection. The combined approach is demonstrated with an example for thermally efficient computer server cabinet configurations, with the computer servers cooled by forced air convection. The paper shows that the combined approach can handle variability well while improving heat dissipation and decreasing temperature using the same cooling infrastructure.

Next, Mourelatos and Lian consider both robustness and reliability in their approach. They present a probabilistic multi-objective optimization model wherein mean performance is traded-off against robustness for some target reliability level. They use a preference aggregation method to convert their multi-objective problem to a single-objective form and then use a reliability-based design optimization (RBDO) approach to solve the single-objective problem. In their approach, they use indifference points to select the most preferred solution without calculating the entire Pareto frontier.

The paper by Kumar, Keane, Nair, and Shahpar presents a multi-objective formulation that trades-off mean against variance of the performance to obtain robust solutions based on a limited computational budget. They demonstrate their approach with an example in aerodynamic design of compressor blades with erosion. For their example, they use a multi-objective genetic algorithm that is combined with computational fluid dynamics and surrogate models to obtain robust design solutions.

The next two papers are on the subject of deterministic, multi-objective robust optimization. The paper by Li, Azarm, and Boyars presents a deterministic approach for multi-objective and feasibility robust optimization. In this context, a design point is optimally robust, if it is (i) feasible and as best as possible in a multi-objectively and feasibly robust. For multi-objective robustness, the known range of parameters for a trial design point is mapped forward to the objective space to form its objective sensitivity region (OSR). Using a worst case approach, a trial design point is said to be robust if it is determined that an acceptable objective variation range for a solution point encloses the OSR. A similar approach is presented for feasibility robustness. The paper by Besharati, Luo, Azarm, and Kannan presents a somewhat similar but less conservative deterministic robust optimization approach. In particular, they present an integrated design and marketing framework for generation of a robust set of product design alternatives. In their approach, they account for variability in both the engineering design domain as well as in customer preferences from the marketing domain. The overall goal of this paper is to obtain design alternatives that (1) are multi-objectively optimal for nominal values of parameters; (2) are within a known acceptable range for each objective function; and (3) maintain feasibility even when the alternatives are subject to applications and environments that are different from nominal conditions in both design and marketing.

There are six papers in the Reliability-Based Design group. The paper by Chan, Skerlos, and Papalambros addresses the computational cost issue in RBDO. It presents an extension of deterministic monotonicity analysis concepts to probabilistic ones, based on the fact that several probability metrics are monotonic transformations. These concepts are used to construct active set strategies that are claimed to reduce the computational cost associated with handling inequality constraints, similarly to the deterministic case. A monotonicity strategy is presented as part of a sequential linear programming algorithm along with numerical examples.

The paper by Mourelatos and Zhou addresses the issue of design under uncertainty with incomplete information using principles from evidence theory. 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 cannot, therefore, be used. Design decisions are usually based on fuzzy information that is vague, imprecise qualitative, linguistic, or incomplete. A computationally efficient design optimization method is presented based on evidence theory, which can handle a mixture of epistemic and random uncertainties. The approach quickly identifies the vicinity of the optimal point and the active constraints by moving a hyper-ellipse in the original design space, using a RBDO algorithm. Examples demonstrate the presented evidence-based design optimization method.

The issue of design under uncertainty with incomplete information is also addressed in the paper by Du, Choi, Youn and Gorsich using possibility theory. It presents a possibility-based design optimization (PBDO) method which can handle a mixture of random and uncertain variables. The uncertain variables are modeled using membership functions as in fuzzy sets. The performance measure approach (PMA), which has been widely used in RBDO, is also used in the proposed PBDO method. A method for selecting the membership function that yields the least conservative optimum design is proposed by using the possibility-probability consistency theory in a least conservative condition. The proposed approach is demonstrated for design problems with a mixture of probabilistic and non-probabilistic input uncertainties. It is shown that a conservative optimum design is obtained compared with RBDO.

In yet a third paper in the area of design under uncertainty with incomplete information, Gunawan and Papalambros present a reliability-based optimization method for the case when limited information for uncertain variables or parameters is in the form of both finite samples and probability distributions. Their method adopts a Bayesian binomial inference technique to estimate reliability, and uses this estimate to maximize the confidence that the design will meet or exceed a target reliability. The method produces a set of Pareto trade-off designs instead of a single design, reflecting the levels of confidence about a design's reliability, given certain incomplete information. The design of an optimal piston-ring/cylinder-liner assembly under surface roughness uncertainty is used to demonstrate their approach.

A paper by Mahadevan and Rebba proposes a methodology to estimate errors in computational models and to include them in RBDO. Various sources of uncertainties and errors in model selection, approximations, and numerical solutions are considered. The solution approximation error is quantified based on the model itself, using the Richardson extrapolation method. The model error is quantified based on the comparison of model prediction with physical observations using an interpolated resampling approach. The proposed method is illustrated with numerical examples.

An innovative safety envelope concept for load tolerance is proposed by Wang and Kim. It identifies the capacity of the current design as a future reference for design upgrade, maintenance, and control. The safety envelope concept is applied in estimating the load tolerance of a structural part with respect to the fatigue life reliability. Based on an implicit function evaluation of the reliability, the boundary of the safety envelope is calculated numerically. The effect of different distribution types of random variables is investigated for identifying a conservative envelope. Probabilistic sensitivity information improves the efficiency. A stochastic response surface of logarithmic fatigue life with respect to the load capacity coefficient is constructed, and a Monte Carlo simulation calculates the reliability and its sensitivities.

There are four papers in the Understanding and Managing Uncertainty group. An invited paper by Apley, Liu, and Chen studies the effects of model interpolation uncertainty in robust design with computer experiments. Most existing approaches for treating interpolation uncertainty in computer experiments have been developed for deterministic optimization and are not applicable to design under uncertainty in which randomness is present in noise and/or design variables. A methodology is presented within a Bayesian framework for quantifying the impact of interpolation uncertainty in robust design considering the uncertain noise variables. A closed-form analytical expression is derived for a Bayesian prediction interval of the robust design objective function. The true response surface is considered as a realization of a random process, similarly to kriging and other Bayesian analyses of computer experiments. The proposed methodology is illustrated with the robust design of a simple container and an automotive engine piston with a mixture of continuous and discrete design variables.

The subject of model uncertainty is also considered in the paper by Martin and Simpson. In most cases, model uncertainty is ignored through the use of high fidelity models assuming that it is insignificant to the decision making process. The authors present a methodology for managing uncertainty during system-level conceptual design of complex multidisciplinary systems. Their approach is based upon quantifying the information available in a set of observations of computationally expensive subsystem models with more computationally efficient kriging models. In doing so, the computational expense of a Monte Carlo simulation in assessing the impact of the sources of uncertainty on system-level performance parameters becomes tractable. The use of a kriging surrogate model introduces model uncertainty, which is included in the proposed methodology. The methodology is demonstrated as a decision making tool for the design of a satellite system.

The next two papers use the concept of imprecise probabilities. The paper by Aughenbaugh and Paredis expresses the precision with which something is known using a probability p-box. A computational experiment is presented where a pressure vessel is designed using a purely probabilistic approach and a probability p-box approach. It is shown that when the imprecision is large, the p-box approach results in designs with expected utilities that are greater than those for designs created with the purely probabilistic approach, suggesting that there are design problems for which it is valuable to use imprecise probabilities. The paper by Ling, Aughenbaugh, and Paredis uses imprecise probabilities to effectively manage resources for supporting design decisions utilizing a cost-benefit trade-off. In their approach, they use principles of information economics in guiding decisions on information collection. They also explore ways to assist designers in bounding the value of information in the case of distributions with unknown parameters.

Finally, there are three papers in Collaborative and Multilevel Design group. The paper by Liu, Chen, Kokkolaras, Papalambros, and Kim extends previous work in analytical target cascading (ATC) to probabilistic ATC (PATC). PATC is essentially an approach for multilevel design optimization that accounts for random variables. Some of the implementation issues discussed include representation of probabilistic targets, matching responses with linking variables under uncertainty, and coordination strategies. The paper verifies the accuracy of the PATC against a probabilistic all-in-one (PAIO) approach and shows that within the limits of experimentations and when matching the first two moments of random responses and linking variables with assigned targets, PATC converges to the same optimal solution as PAIO does.

The paper by Gu, Renaud, and Penninger presents a mathematical construct for bilevel collaborative optimization under uncertainty. In the paper, they use an implicit approach for estimating the system performance under uncertainties in a collaborative optimization setting. This implicit approach is then used as the basis for a robust collaborative optimization (RCO) framework. The proposed approach maintains disciplinary autonomy and at the same time considers performance uncertainties to ensure feasibility robustness for the system. They demonstrate a preliminary application of the RCO on an aircraft concept sizing problem.

The last paper in the supplement is by Farhang-Mehr and Tumer, who present an approach for managing resources in a collaborative engineering design environment under uncertainty and risk. Their paper presents a probabilistic risk-based design decision-making method, referred to as RUBIC, which uses techniques from portfolio optimization and resource management for system risk mitigation. The RUBIC method is based on the concept that allocation of a unit of resource to mitigate a certain risk in the system contributes to the overall system risk reduction. They demonstrate the proposed approach with a satellite reaction wheel example.

We hope these papers stimulate further research ideas and advances in this important area of robust and reliability-based design. We would like to sincerely thank all authors who responded to our CFP, for their timely contributions and their hard work, whether or not their paper appeared in the special supplement. We also would like to thank all reviewers of this special issue for their very prompt response to our sometimes multiple requests. We apologize for our numerous short deadlines. Also, special thanks are due to Dr. Kemper Lewis who coordinated the review of two of our own papers. Finally, we would like to thank Ms. Tanya Eberhard who patiently helped us with numerous details of the special issue.

Enjoy reading the papers!