Pareto Frontier Based Concept Selection Under Uncertainty,with Visualization

In a recent publication, we presented a new multiobjective decision-making tool for use in conceptual engineering design. In the present paper, we provide important developments that support the next phase in the evolution of the tool. These developments, together with those of our previous work, provide a concept selection approach that capitalizes on the benefits of computational optimization. Specifically, the new approach uses the efficiency and effectiveness of optimization to rapidly compare numerous designs, and characterize the tradeoff properties within the multiobjective design space. As such, the new approach differs significantly from traditional (non-optimization based) concept selection approaches where, comparatively speaking, significant time is often spent evaluating only a few points in the design space. Under the new approach, design concepts are evaluated using a so-calleds-Pareto frontier; this frontier originates from the Pareto frontiers of various concepts, and is the Pareto frontier for thesetof design concepts. An important characteristic of the s-Pareto frontier is that it provides a foundation for analyzing tradeoffs between design objectives and the tradeoffs between design concepts. The new developments presented in this paper include; (i) the notion ofminimally representingthe s-Pareto frontier, (ii) the quantification of concept goodness using s-Pareto frontiers, (iii) the development of an interactive design space exploration approach that can be used to visualizen-dimensional s-Pareto frontiers, and (iv) s-Pareto frontier-based approaches for considering uncertainty in concept selection. Simple structural examples are presented that illustrate representative applications of the proposed method.     

Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.     

Aggregate Objective Functions and Pareto Frontiers: Required Relationships and Practical Implications

This paper addresses the problem of capturing Pareto optimal points on non-convex Pareto frontiers, which are encountered in nonlinear multiobjective optimization problems in computational engineering design optimization. The emphasis is on the choice of the aggregate objective function (AOF) of the objectives that is employed to capture Pareto optimal points. A fundamental property of the aggregate objective function, the admissibility property, is developed and its equivalence to the coordinatewise increasing property is established. Necessary and sufficient conditions for such an admissible aggregate objective function to capture Pareto optimal points are derived. Numerical examples illustrate these conditions in the biobjective case. This paper demonstrates in general terms the limitation of the popular weighted-sum AOF approach, which captures only convex Pareto frontiers, and helps us understand why some commonly used AOFs cannot capture desirable Pareto optimal points, and how to avoid this situation in practice. Since nearly all applications of optimization in engineering design involve the formation of AOFs, this paper is of direct theoretical and practical usefulness.     

Multifidelity Response Surface Approximations for the Optimum Design of Diffuser Flows

Response surface methods use least-squares regression analysis to fit low-order polynomials to a set of experimental data. It is becoming increasingly more popular to apply response surface approximations for the purpose of engineering design optimization based on computer simulations. However, the substantial expense involved in obtaining enough data to build quadratic response approximations seriously limits the practical size of problems. Multifidelity techniques, which combine cheap low-fidelity analyses with more accurate but expensive high-fidelity solutions, offer means by which the prohibitive computational cost can be reduced. Two optimum design problems are considered, both pertaining to the fluid flow in diffusers. In both cases, the high-fidelity analyses consist of solutions to the full Navier-Stokes equations, whereas the low-fidelity analyses are either simple empirical formulas or flow solutions to the Navier-Stokes equations achieved using coarse computational meshes. The multifidelity strategy includes the construction of two separate response surfaces: a quadratic approximation based on the low-fidelity data, and a linear correction response surface that approximates the ratio of high-and low-fidelity function evaluations. The paper demonstrates that this approach may yield major computational savings.     

Multi-objective optimization based on meta-modeling by using support vector regression

Practical engineering design problems have a black-box objective function whose forms are not explicitly known in terms of design variables. In those problems, it is very important to make the number of function evaluations as few as possible in finding an optimal solution. So, in this paper, we propose a multi-objective optimization method based on meta-modeling predicting a form of each objective function by using support vector regression. In addition, we discuss a way how to select additional experimental data for sequentially revising a form of objective function. Finally, we illustrate the effectiveness of the proposed method through some numerical examples.     

Surrogate modeling of multiscale models using kernel methods

This work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods. The key element is to learn the interface between the different scales using a fast surrogate for the microscale model, which is given by multivariate kernel expansions. The expansions are computed using statistically representative samples of input and output of the microscale model. We apply both support vector machines and a vectorial kernel greedy algorithm as learning methods. We demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines. We consider, first, a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and, second, a model for simulation of saturation overshoots in porous media involving nonclassical shock waves. Copyright © 2014 John Wiley & Sons, Ltd.     

Generating Well-Distributed Sets of Pareto Points for Engineering Design Using Physical Programming

Engineering design generally involves two, possibly integrated, phases: (i) generating design options, and (ii) choosing the most satisfactory option on the basis of some determined criteria. The depth, or lack, of integration between these two phases defines different design approaches, and differing philosophical views from the part of researchers in the field of computational design. Optimization-Based Design (OBD) covers the spectrum of this depth of integration. While most OBD approaches strongly integrate these two phases, some employ computational optimization only in the first or second phase. Regardless of where a method or researcher lies in this philosophical spectrum, some requisite characteristics are fundamental to the effectiveness of OBD methods. In particular, (i) the Aggregate Objective Function (AOF) used in the optimization must have the ability to generate all Pareto solutions, (ii) the generation of any existing Pareto solutions must be possible with reasonable ease, and (iii) even changes in the AOF parameters should yield a well distributed set of Pareto solutions. This paper examines the effectiveness of physical programming (PP) with respect to the latter, yielding favorable conclusions. Previous papers have led to similarly positive conclusions with respect to the former two. This paper also presents a comparative study featuring PP and other popular methods, where PP is shown to perform favorably. A PP-based method for generating the Pareto frontier is presented.     

Sensitivity analysis for decision-making using the MORE method—A Pareto approach

Integrated Assessment Modelling (IAM) incorporates knowledge from different disciplines to provide an overarching assessment of the impact of different management decisions. The complex nature of these models, which often include non-linearities and feedback loops, requires special attention for sensitivity analysis. This is especially true when the models are used to form the basis of management decisions, where it is important to assess how sensitive the decisions being made are to changes in model parameters. This research proposes an extension to the Management Option Rank Equivalence (MORE) method of sensitivity analysis; a new method of sensitivity analysis developed specifically for use in IAM and decision-making. The extension proposes using a multi-objective Pareto optimal search to locate minimum combined parameter changes that result in a change in the preferred management option. It is demonstrated through a case study of the Namoi River, where results show that the extension to MORE is able to provide sensitivity information for individual parameters that takes into account simultaneous variations in all parameters. Furthermore, the increased sensitivities to individual parameters that are discovered when joint parameter variation is taken into account shows the importance of ensuring that any sensitivity analysis accounts for these changes.     

Multiobjective optimization with an adaptive weight determination scheme using the concept of hyperplane

Finding an optimum design that satisfies all performances in a design problem is very challenging. To overcome this problem, multiobjective optimization methods have been researched to obtain Pareto optimum solutions. Among the different methods, the weighted sum method is widely used for its convenience. However, since the different weights do not always guarantee evenly distributed solutions on the Pareto front, the weights need to be determined systematically. Therefore, this paper presents a multiobjective optimization using a new adaptive weight determination scheme. Solutions on the Pareto front are gradually found with different weights, and the values of these weights are adaptively determined by using information from the previously obtained solutions' positions. For an n-objective problem, a hyperplane is constructed in n -dimensional space, and new weights are calculated to find the next solutions. To confirm the effectiveness of the proposed method, benchmarking problems that have different types of Pareto front are tested, and a topology optimization problem is performed as an engineering problem. A hypervolume indicator is used to quantitatively evaluate the proposed method, and it is confirmed that optimized solutions that are evenly distributed on the Pareto front can be obtained by using the proposed method.     

Surrogate modelling and optimization using shape-preserving response prediction: A review

Computer simulation models are ubiquitous in modern engineering design. In many cases, they are the only way to evaluate a given design with sufficient fidelity. Unfortunately, an added computational expense is associated with higher fidelity models. Moreover, the systems being considered are often highly nonlinear and may feature a large number of designable parameters. Therefore, it may be impractical to solve the design problem with conventional optimization algorithms. A promising approach to alleviate these difficulties is surrogate-based optimization (SBO). Among proven SBO techniques, the methods utilizing surrogates constructed from corrected physics-based low-fidelity models are, in many cases, the most efficient. This article reviews a particular technique of this type, namely, shape-preserving response prediction (SPRP), which works on the level of the model responses to correct the underlying low-fidelity models. The formulation and limitations of SPRP are discussed. Applications to several engineering design problems are provided.     

Polynomial Response Surface Approximations for the Multidisciplinary Design Optimization of a High Speed Civil Transport

Surrogate functions have become an important tool in multidisciplinary design optimization to deal with noisy functions, high computational cost, and the practical difficulty of integrating legacy disciplinary computer codes. A combination of mathematical, statistical, and engineering techniques, well known in other contexts, have made polynomial surrogate functions viable for MDO. Despite the obvious limitations imposed by sparse high fidelity data in high dimensions and the locality of low order polynomial approximations, the success of the panoply of techniques based on polynomial response surface approximations for MDO shows that the implementation details are more important than the underlying approximation method (polynomial, spline, DACE, kernel regression, etc.). This paper selectively surveys some of the ancillary techniques—statistics, global search, parallel computing, variable complexity modeling—that augment the construction and use of polynomial surrogates.     

Use of Design Sensitivity Information in Response Surface and Kriging Metamodels

Metamodels based on responses from designed (numerical) experiments may form efficient approximations to functions in structural analysis. They can improve the efficiency of Engineering Optimization substantially by uncoupling computationally expensive analysis models and (iterative) optimization procedures. In this paper we focus on two strategies for building metamodels, namely Response Surface Methods (RSM) and kriging. We discuss key-concepts for both approaches, present strategies for model training and indicate ways to enhance these metamodeling approaches by including design sensitivity data. The latter may be advantageous in situations where information on design sensitivities is readily available, as is the case with e.g. Finite Element Models. Furthermore, we illustrate the use of RSM and kriging in a numerical model study and conclude with some remarks on their practical value.     

Response surface approximations for structural optimization

Response surface methodology can be used to construct global and midrange approximations to functions in structural optimization. Since structural optimization requires expensive function evaluations, it is important to construct accurate function approximations so that rapid convergence may be achieved. In this paper techniques to find the region of interest containing the optimal design, and techniques for finding more accurate approximations are reviewed and investigated. Aspects considered are experimental design techniques, the selection of the ‘best’ regression equation, intermediate response functions and the location and size of the region of interest. Standard examples in structural optimization are used to show that the accuracy is largely dependent on the choice of the approximating function with its associated subregion size, while the selection of a larger number of points is not necessarily cost-effective. In a further attempt to improve efficiency, different regression models were investigated. The results indicate that the use of the two methods investigated does not significantly improve the results. Finding an accurate global approximation is challenging, and sufficient accuracy could only be achieved in the example problems by considering a smaller region of the design space. © 1998 John Wiley & Sons, Ltd.     

Adapting the Hypervolume Quality Indicator to Quantify Trade‐offs and Search Efficiency for Multiple Criteria Decision Making Using Pareto Fronts

When choosing a best solution based on simultaneously balancing multiple objectives, the Pareto front approach allows promising solutions across the spectrum of user preferences for the weightings of the objectives to be identified and compared quantitatively. The shape of the complete Pareto front provides useful information about the amount of trade‐off between the different criteria and how much compromise is needed from some criterion to improve the others. Visualizing the Pareto front in higher (3 or more) dimensions becomes difficult, so a numerical measure of this relationship helps capture the degree of trade‐off. The traditional hypervolume quality indicator based on subjective scaling for multiple criteria optimization method comparison provides an arbitrary value that lacks direct interpretability. This paper proposes an interpretable summary for quantifying the nature of the relationship between criteria with a standardized hypervolume under the Pareto front (HVUPF) for a flexible number of optimization criteria, and demonstrates how this single number summary can be used to evaluate and compare the efficiency of different search methods as well as tracking the search progress in populating the complete Pareto front. A new HVUPF growth plot is developed for quantifying the performance of a search method on completeness, efficiency, as well as variability associated with the use of random starts, and offers an effective approach for method assessment and comparison. Two new enhancements for the algorithm to populate the Pareto front are described and compared with the HVUPF growth plot. The methodology is illustrated with an optimal screening design example, where new Pareto search methods are proposed to improve computational efficiency, but is broadly applicable to other multiple criteria optimization problems. Copyright © 2012 John Wiley & Sons, Ltd.     

基于代理模型和改进遗传算法的注塑翘曲优化

提出一种最小化制品翘曲的注塑工艺参数优化集成方法.以空调柜机顶盖注塑制品开发为例,该方法使用Moldflow软件分析制品的翘曲变形,运用田口方法确定与制品翘曲量密切相关的工艺因素,然后采用响应曲面法(RSM)和改进的精英保留自适应遗传算法(EAGA)相结合的方法,建立主要影响工艺参数与制品翘曲量之间的关系模型,通过对模型寻优以实现对制品翘曲的优化.该方法的适用性在制品的实际生产中得到了验证.     

An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging

Variable-fidelity (VF) modelling methods have been widely used in complex engineering system design to mitigate the computational burden. Building a VF model generally includes two parts: design of experiments and metamodel construction. In this article, an adaptive sampling method based on improved hierarchical kriging (ASM-IHK) is proposed to refine the improved VF model. First, an improved hierarchical kriging model is developed as the metamodel, in which the low-fidelity model is varied through a polynomial response surface function to capture the characteristics of a high-fidelity model. Secondly, to reduce local approximation errors, an active learning strategy based on a sequential sampling method is introduced to make full use of the already required information on the current sampling points and to guide the sampling process of the high-fidelity model. Finally, two numerical examples and the modelling of the aerodynamic coefficient for an aircraft are provided to demonstrate the approximation capability of the proposed approach, as well as three other metamodelling methods and two sequential sampling methods. The results show that ASM-IHK provides a more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

Pitfalls of using a single criterion for selecting experimental designs

For surrogate construction, a good experimental design (ED) is essential to simultaneously reduce the effect of noise and bias errors. However, most EDs cater to a single criterion and may lead to small gains in that criterion at the expense of large deteriorations in other criteria. We use multiple criteria to assess the performance of different popular EDs. We demonstrate that these EDs offer different trade‐offs, and that use of a single criterion is indeed risky. In addition, we show that popular EDs, such as Latin hypercube sampling (LHS) and D‐optimal designs, often leave large regions of the design space unsampled even for moderate dimensions. We discuss a few possible strategies to combine multiple criteria and illustrate them with examples. We show that complementary criteria (e.g. bias handling criterion for variance‐based designs and vice versa) can be combined to improve the performance of EDs. We demonstrate improvements in the trade‐offs between noise and bias error by combining a model‐based criterion, like the D‐optimality criterion, and a geometry‐based criterion, like LHS. Next, we demonstrate that selecting an ED from three candidate EDs using a suitable error‐based criterion helped eliminate potentially poor designs. Finally, we show benefits from combining the multiple criteria‐based strategies, that is, generation of multiple EDs using the D‐optimality and LHS criteria, and selecting one design using a pointwise bias error criterion. The encouraging results from the examples indicate that it may be worthwhile studying these strategies more rigorously and in more detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Multidisciplinary Design Optimization with Quasiseparable Subsystems

Numerous hierarchical and nonhierarchical decomposition strategies for the optimization of large scale systems, comprised of interacting subsystems, have been proposed. With a few exceptions, all of these strategies lack a rigorous theoretical justification. This paper focuses on a class of quasiseparable optimization problems narrow enough for a rigorous decomposition theory, yet general enough to encompass many large scale engineering design problems. The subsystems for these problems involve local design variables and global system variables, but no variables from other subsystems. The objective function is a sum of a global system criterion and the subsystems' criteria. The essential idea is to give each subsystem a budget and global system variable values, and then ask the subsystems to independently maximize their constraint margins. Using these constraint margins, a system optimization then adjusts the values of the system variables and subsystem budgets. The subsystem margin problems are totally independent, always feasible, and could even be done asynchronously in a parallel computing context. An important detail is that the subsystem tasks, in practice, would be to construct response surface approximations to the constraint margin functions, and the system level optimization would use these margin surrogate functions. The purpose of the present paper is to present a decomposition strategy in a general context, provide rigorous theory justifying the decomposition, and give some simple illustrative examples.