A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibilty Robust Design Optimization

2005 ◽  
Author(s):  
Mian Li ◽  
Shapour Azarm ◽  
Art Boyars
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Continuous Functions ◽  
Robust Design Optimization ◽  
Deterministic Approach ◽  
Worst Case ◽  
New Approach ◽  
Multi Objective ◽  
Tolerance Region ◽  
Gradient Based

We present a deterministic, non-gradient based approach that uses robustness measures for robust optimization in multi-objective problems where uncontrollable parameters variations cause variation in the objective and constraint values. The approach is applicable for cases with discontinuous objective and constraint functions, and can be used for objective or feasibility robust optimization, or both together. In our approach, the parameter tolerance region maps into sensitivity regions in the objective and constraint spaces. The robustness measures are indices calculated, using an optimizer, from the sizes of the acceptable objective and constraint variation regions and from worst-case estimates of the sensitivity regions’ sizes, resulting in an outer-inner structure. Two examples provide comparisons of the new approach with a similar published approach that is applicable only with continuous functions. Both approaches work well with continuous functions. For discontinuous functions the new approach gives solutions near the nominal Pareto front; the earlier approach does not.

A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibility Robust Design Optimization

2005 ◽  
Vol 128 (4) ◽  
pp. 874-883 ◽  
Author(s):  
Mian Li ◽  
Shapour Azarm ◽  
Art Boyars
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Continuous Functions ◽  
Robust Design Optimization ◽  
Worst Case ◽  
New Approach ◽  
Multi Objective ◽  
Tolerance Region ◽  
Parameter Variations ◽  
Gradient Based

We present a deterministic non-gradient based approach that uses robustness measures in multi-objective optimization problems where uncontrollable parameter variations cause variation in the objective and constraint values. The approach is applicable for cases that have discontinuous objective and constraint functions with respect to uncontrollable parameters, and can be used for objective or feasibility robust optimization, or both together. In our approach, the known parameter tolerance region maps into sensitivity regions in the objective and constraint spaces. The robustness measures are indices calculated, using an optimizer, from the sizes of the acceptable objective and constraint variation regions and from worst-case estimates of the sensitivity regions’ sizes, resulting in an outer-inner structure. Two examples provide comparisons of the new approach with a similar published approach that is applicable only with continuous functions. Both approaches work well with continuous functions. For discontinuous functions the new approach gives solutions near the nominal Pareto front; the earlier approach does not.

An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

2021 ◽  
Vol 21 (4) ◽  
Author(s):  
Tingting Xia ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Computational Time ◽  
Pareto Solutions ◽  
Worst Case ◽  
Multi Objective ◽  
Pareto Points ◽  
Single Objective

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design. To find robust Pareto fronts, multi-objective robust optimization (MORO) methods with inner–outer optimization structures usually have high computational complexity, which is a critical issue. Generally, in design problems, robust Pareto solutions lie somewhere closer to nominal Pareto points compared with randomly initialized points. The searching process for robust solutions could be more efficient if starting from nominal Pareto points. We propose a new method sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points where MOOPs with uncertainties are solved in two stages. The deterministic optimization problem and robustness metric optimization are solved in the first stage, where nominal Pareto solutions and the robust-most solutions are identified, respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and robust-most points. The proposed SARPF method can reduce a significant amount of computational time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values, which can reduce computational efforts further. The global solvers, NSGA-II for multi-objective cases and genetic algorithm (GA) for single-objective cases, are used in corresponding optimization processes. Three examples with the comparison with results from the previous method are presented to demonstrate the applicability and efficiency of the proposed method.

A Bi-Level Multi-Objective Robust Design Optimization Method With Multiple Robustness Index Capabilities

2011 ◽  
Author(s):  
Todd Letcher ◽  
M.-H. Herman Shen
Keyword(s):  
Robust Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Optimization Method ◽  
Robust Design Optimization ◽  
Nsga Ii ◽  
Worst Case ◽  
Multi Objective ◽  
Robustness Index ◽  
Multiple Robustness

A multi-objective robust optimization framework that incorporates a robustness index for each objective has been developed in a bi-level approach. The top level of the framework consists of the standard optimization problem formulation with the addition of a robustness constraint. The bottom level uses the Worst Case Sensitivity Region (WCSR) concept previously developed to solve single objective robust optimization problems. In this framework, a separate robustness index for each objective allows the designer to choose the importance of each objective. The method is demonstrated on a commonly studied two-bar truss structural optimization problem. The results of the problem demonstrate the effectiveness and usefulness of the multiple robustness index capabilities added to this framework. A multi-objective genetic algorithm, NSGA-II, is used in both levels of the framework.

An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

2020 ◽  
Author(s):  
Tingting Xia ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Pareto Solutions ◽  
Worst Case ◽  
Multi Objective ◽  
Pareto Points ◽  
Engineering Design Problems ◽  
Single Objective

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design problems. To find the robust Pareto fronts, multi-objective robust optimization methods with inner-outer optimization structures generally have high computational complexity, which is always an important issue to address. Based on the general experience, robust Pareto solutions usually lie somewhere near the nominal Pareto points. Starting from the obtained nominal Pareto points, the search process for robust solutions could be more efficient. In this paper, we propose a method that sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points. MOOPs are solved by the SARPF in two optimization stages. The deterministic optimization problem and the robustness metric optimization problem are solved in the first stage, and nominal Pareto solutions and the robust-most solutions can be found respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and the robust-most points. The proposed SARPF method can save a significant amount of computation time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values which can save computational efforts further. The global solvers, NSGA-II for the multi-objective case and genetic algorithm (GA) for the single-objective case, are used in corresponding optimization processes. Two examples with comparison to a previous method are presented for the applicability and efficiency demonstration.

Toward the Use of Pareto Performance Solutions and Pareto Robustness Solutions for Multi-Objective Robust Optimization Problems

2012 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Oscar Brito Augusto
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Robust Solutions ◽  
Multi Objective ◽  
Robust Optimization Problem ◽  
Design Solutions ◽  
Further Development

For Multi-Objective Robust Optimization Problem (MOROP), it is important to obtain design solutions that are both optimal and robust. To find these solutions, usually, the designer need to set a threshold of the variation of Performance Functions (PFs) before optimization, or add the effects of uncertainties on the original PFs to generate a new Pareto robust front. In this paper, we divide a MOROP into two Multi-Objective Optimization Problems (MOOPs). One is the original MOOP, another one is that we take the Robustness Functions (RFs), robust counterparts of the original PFs, as optimization objectives. After solving these two MOOPs separately, two sets of solutions come out, namely the Pareto Performance Solutions (PP) and the Pareto Robustness Solutions (PR). Make a further development on these two sets, we can get two types of solutions, namely the Pareto Robustness Solutions among the Pareto Performance Solutions (PR(PP)), and the Pareto Performance Solutions among the Pareto Robustness Solutions (PP(PR)). Further more, the intersection of PR(PP) and PP(PR) can represent the intersection of PR and PP well. Then the designer can choose good solutions by comparing the results of PR(PP) and PP(PR). Thanks to this method, we can find out the optimal and robust solutions without setting the threshold of the variation of PFs nor losing the initial Pareto front. Finally, an illustrative example highlights the contributions of the paper.

Robust Optimization of an Automotive Valvetrain Using a Multiobjective Genetic Algorithm

2003 ◽  
Author(s):  
Emre Kazancioglu ◽  
Guangquan Wu ◽  
Jeonghan Ko ◽  
Stanislav Bohac ◽  
Zoran Filipi ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Robust Optimization ◽  
Production Cost ◽  
Pareto Front ◽  
Total Production ◽  
Multi Objective ◽  
Engine Model ◽  
Multi Objective Genetic Algorithm ◽  
Current Design ◽  
Alternative Designs

A robust optimization of an automobile valvetrain is presented where the variation of engine performances due to the component dimensional variations is minimized subject to the constraints on mean engine performances. The dimensional variations of valvetrain components are statistically characterized based on the measurements of the actual components. Monte Carlo simulation is used on a neural network model built from an integrated high fidelity valvetrain-engine model, to obtain the mean and standard deviation of horsepower, torque and fuel consumption. Assuming the component production cost is inversely proportional to the coefficient of variation of its dimensions, a multi-objective optimization problem minimizing the variation in engine performances and the total production cost of components is solved by a multi-objective genetic algorithm (MOGA). The comparisons using the newly developed Pareto front quality index (PFQI) indicate that MOGA generates the Pareto fronts of substantially higher quality, than SQP with varying weights on the objectives. The current design of the valvetrain is compared with two alternative designs on the obtained Pareto front, which suggested potential improvements.

Multi-Objective Robust Design Optimization and Knowledge Mining of a Centrifugal Fan That Takes Dimensional Uncertainty Into Account

2008 ◽  
Author(s):  
Kazuyuki Sugimura ◽  
Shinkyu Jeong ◽  
Shigeru Obayashi ◽  
Takeshi Kimura
Keyword(s):  
Data Mining ◽  
Robust Optimization ◽  
Robust Design ◽  
Rough Set Theory ◽  
Optimization Techniques ◽  
Robust Design Optimization ◽  
Centrifugal Fan ◽  
Data Mining Techniques ◽  
Multi Objective ◽  
Design Variables

A new design approach named MORDE (multi-objective robust design exploration), in which multi-objective robust optimization techniques and data mining techniques are combined, is proposed in this paper. We first developed a widely applicable design framework for multi-objective robust optimization. In this framework, probabilistic representation of design variables are introduced and Kriging models are used to approximate relations between design variables with uncertainty and multiple design objectives. A multi-objective genetic algorithm optimizes the mean and standard deviation of the responses. We then applied the framework to the real-world design problem of a centrifugal fan used in a washer-dryer. Taking dimensional uncertainty into account, we optimized the means and standard deviations of the resulting distributions of fan efficiency and turbulent noise level. Steady Reynolds-averaged Navier Stokes simulations were used to build Kriging models that approximate these objective functions. With the obtained non-dominated solutions, we demonstrated how to analyze features of solutions and select design candidates. We also attempted to acquire design knowledge by applying several data mining techniques. Self-organizing map was used to visualize and reuse the high dimensional design data. Decision tree analysis and rough set theory were used to extract design rules to improve the product’s performance. We also discussed differences in types of rules, which were extracted by both methods.

Sequential Cooperative Robust Optimization (SCRO) for Multi-Objective Design Under Uncertainty

2015 ◽  
Author(s):  
J. M. Hamel
Keyword(s):  
Robust Optimization ◽  
Uncertainty Analysis ◽  
Computational Efficiency ◽  
System Optimization ◽  
Surrogate Models ◽  
Optimization Techniques ◽  
Robust Design Optimization ◽  
Optimization Approach ◽  
Deterministic System ◽  
Multi Objective

The optimal design of systems under uncertainty is a critical challenge faced by design engineers. Robust optimization is a well-studied and widely used technique for the design of engineering systems that possess uncertainty, and numerous robust optimization techniques have been presented in recent years. The majority of the robust optimization techniques presented in the literature suffer from a computational efficiency challenge, either due to the expense of obtaining objective or constraint function uncertainty information, or due to the fact that many robust optimization approaches (with a few notable exceptions) require that a potentially expensive uncertainty analysis calculation (e.g. Monte-Carlo simulation) be nested within an already potentially expensive optimization solver (e.g. a genetic algorithm). Additionally, many robust optimization approaches focus solely on design problems that possess a single design objective, and the robust techniques that do consider problems with multiple design objectives often require various simplifying assumptions or are even more computationally expensive to implement. Clearly there are opportunities for improvement in the area of robust optimization, and this paper presents a new robust design Optimization approach called Sequential Cooperative Robust Optimization (SCRO), which uses both a sequential approach and multi-objective optimization techniques in an effort to decouple the deterministic system optimization problem from the associated uncertainty analysis problem. The SCRO approach first fits surrogate models to the system objective and constraint functions, in addition to system sensitivity functions, using as few function calls as possible in order to improve computational efficiency. The approach then performs a series of sequential multi-objective optimizations using the developed surrogate models. These optimizations work to find points in the design space that are optimal with respect to deterministic performance and both objective and feasibility robustness metrics based on predicted system sensitivities. The SCRO approach has the potential to find solutions not available to other robust optimization approaches, and can be more efficient than other more traditional robust optimization techniques due to its use of surrogate approximation and a sequential framework.

Non-Gradient Based Parameter Sensitivity Estimation for Robust Design Optimization

2003 ◽  
Author(s):  
S. Gunawan ◽  
S. Azarm
Keyword(s):  
Design Optimization ◽  
Robust Design ◽  
Parameter Sensitivity ◽  
Robust Design Optimization ◽  
Optimization Approach ◽  
Case Scenario ◽  
Worst Case ◽  
Design Alternative ◽  
Sensitivity Estimation ◽  
Gradient Based

We present a method for estimating the parameter sensitivity of a design alternative for use in robust design optimization. The method is non-gradient based: it is applicable even when the objective function of an optimization problem is non-differentiable and/or discontinuous with respect to the parameters. Also, the method does not require a presumed probability distribution for parameters, and is still valid when parameter variations are large. The sensitivity estimate is developed based on the concept that associated with each design alternative there is a region in the parameter variation space whose properties can be used to predict that design’s sensitivity. Our method estimates such a region using a worst-case scenario analysis and uses that estimate in a bi-level robust optimization approach. We present a numerical and an engineering example to demonstrate the applications of our method.

Analysis and design of electrical machines with material uncertainties in iron and permanent magnet

2017 ◽  
Vol 36 (5) ◽  
pp. 1326-1337 ◽  
Author(s):  
Min Li ◽  
Mohammad Hossain Mohammadi ◽  
Tanvir Rahman ◽  
David Lowther
Keyword(s):  
Permanent Magnet ◽  
Robust Design ◽  
Electrical Machines ◽  
Robust Design Optimization ◽  
Worst Case ◽  
Content Type ◽  
Multi Objective ◽  
Material Uncertainties ◽  
Ipm Machine ◽  
The Impact

Purpose Manufacturing processes, such as laminations, may introduce uncertainties in the magnetic properties of materials used in electrical machines. This issue, together with magnetization errors, can cause serious deterioration in the performance of the machines. Hence, stochastic material models are required for the study of the influences of the material uncertainties. The purpose of this paper is to present a methodology to study the impact of magnetization pattern uncertainties in permanent magnet electric machines. Design/methodology/approach The impacts of material uncertainties on the performances of an interior permanent magnet (IPM) machine were analyzed using two different robustness metrics (worst-case analysis and statistical study). In addition, two different robust design formulations were applied to robust multi-objective machine design problems. Findings The computational analyses show that material uncertainties may result in deviations of the machine performances and cause nominal solutions to become non-robust. Originality/value In this paper, the authors present stochastic models for the quantification of uncertainties in both ferromagnetic and permanent magnet materials. A robust multi-objective evolutionary algorithm is demonstrated and successfully applied to the robust design optimization of an IPM machine considering manufacturing errors and operational condition changes.

Sign in / Sign up

Export Citation Format

Share Document