Towards a Better Understanding of Modeling Feasibility Robustness in Engineering Design

1999 ◽  
Author(s):  
Xiaoping Du ◽  
Wei Chen
Keyword(s):  
Robust Optimization ◽  
Robust Design ◽  
Optimization Problems ◽  
Statistical Analyses ◽  
Practical Significance ◽  
Worst Case Analysis ◽  
Engineering Practice ◽  
Worst Case ◽  
Feasibility Robustness ◽  
Six Sigma Approach

Abstract In robust design, it is important not only to achieve robust design objectives but also to maintain the robustness of design feasibility under the effect of variations (or uncertainties). However, the evaluation of feasibility robustness is often a computationally intensive process. Simplified approaches in existing robust design applications may lead to either over-conservative or infeasible design solutions. In this paper, several feasibility-modeling techniques for robust optimization are examined. These methods are classified into two categories: methods that require probability and statistical analyses (i.e., the probabilistic feasibility formulation and the moment matching method) and methods do not require probability and statistical analyses (i.e., the worst case analysis, the corner space evaluation, and the variation pattern method). Using illustrative examples, the effectiveness of each method is compared in terms of its efficiency and accuracy. Constructive recommendations are made to employ different techniques for modeling feasibility robustness under different circumstances. Under the framework of probabilistic robust optimization, we propose to use a most probable point (MPP) based importance sampling method, a method rooted in the field of reliability analysis, for evaluating the feasibility robustness. The advantages of this approach are discussed. Though our discussions are centered on robust design, the principles presented are also applicable for general probabilistic optimization problems. The practical significance of this work also lies in the development of efficient feasibility evaluation methods that can support quality engineering practice, such as the Six Sigma approach that is being widely used in American industry.

Towards a Better Understanding of Modeling Feasibility Robustness in Engineering Design

1999 ◽  
Vol 122 (4) ◽  
pp. 385-394 ◽  
Author(s):  
Xiaoping Du ◽  
Wei Chen
Keyword(s):  
Robust Design ◽  
Optimization Problems ◽  
Practical Significance ◽  
Engineering Practice ◽  
Probabilistic Optimization ◽  
Importance Sampling Method ◽  
Feasibility Evaluation ◽  
Feasibility Robustness ◽  
Computationally Intensive ◽  
Six Sigma Approach

In robust design, it is important not only to achieve robust design objectives but also to maintain the robustness of design feasibility under the effect of variations (or uncertainties). However, the evaluation of feasibility robustness is often a computationally intensive process. Simplified approaches in existing robust design applications may lead to either over-conservative or infeasible design solutions. In this paper, several feasibility-modeling techniques for robust optimization are examined. These methods are classified into two categories: methods that require probability and statistical analyses and methods that do not. Using illustrative examples, the effectiveness of each method is compared in terms of its efficiency and accuracy. Constructive recommendations are made to employ different techniques under different circumstances. Under the framework of probabilistic optimization, we propose to use a most probable point (MPP) based importance sampling method, a method rooted in the field of reliability analysis, for evaluating the feasibility robustness. The advantages of this approach are discussed. Though our discussions are centered on robust design, the principles presented are also applicable for general probabilistic optimization problems. The practical significance of this work also lies in the development of efficient feasibility evaluation methods that can support quality engineering practice, such as the Six Sigma approach that is being widely used in American industry. [S1050-0472(00)00904-1]

Feasibility Robust Optimization via Scenario Generation and Reduction

2018 ◽  
Author(s):  
Eliot Rudnick-Cohen ◽  
Jeffrey W. Herrmann ◽  
Shapour Azarm
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Computational Cost ◽  
Optimization Techniques ◽  
Scenario Generation ◽  
Test Problems ◽  
Optimization Approach ◽  
Uncertain Parameters ◽  
Worst Case ◽  
Robust Solution

Feasibility robust optimization techniques solve optimization problems with uncertain parameters that appear only in their constraint functions. Solving such problems requires finding an optimal solution that is feasible for all realizations of the uncertain parameters. This paper presents a new feasibility robust optimization approach involving uncertain parameters defined on continuous domains without any known probability distributions. The proposed approach integrates a new sampling-based scenario generation scheme with a new scenario reduction approach in order to solve feasibility robust optimization problems. An analysis of the computational cost of the proposed approach was performed to provide worst case bounds on its computational cost. The new proposed approach was applied to three test problems and compared against other scenario-based robust optimization approaches. A test was conducted on one of the test problems to demonstrate that the computational cost of the proposed approach does not significantly increase as additional uncertain parameters are introduced. The results show that the proposed approach converges to a robust solution faster than conventional robust optimization approaches that discretize the uncertain parameters.

Sequential Quadratic Programming for Robust Optimization With Interval Uncertainty

2012 ◽  
Vol 134 (10) ◽  
Author(s):  
Jianhua Zhou ◽  
Shuo Cheng ◽  
Mian Li
Keyword(s):  
Quadratic Programming ◽  
Robust Optimization ◽  
Sequential Quadratic Programming ◽  
Optimization Problems ◽  
Critical Role ◽  
Interval Uncertainty ◽  
Design Solution ◽  
Novel Approach ◽  
Design Variables ◽  
Feasibility Robustness

Uncertainty plays a critical role in engineering design as even a small amount of uncertainty could make an optimal design solution infeasible. The goal of robust optimization is to find a solution that is both optimal and insensitive to uncertainty that may exist in parameters and design variables. In this paper, a novel approach, sequential quadratic programming for robust optimization (SQP-RO), is proposed to solve single-objective continuous nonlinear optimization problems with interval uncertainty in parameters and design variables. This new SQP-RO is developed based on a classic SQP procedure with additional calculations for constraints on objective robustness, feasibility robustness, or both. The obtained solution is locally optimal and robust. Eight numerical and engineering examples with different levels of complexity are utilized to demonstrate the applicability and efficiency of the proposed SQP-RO with the comparison to its deterministic SQP counterpart and RO approaches using genetic algorithms. The objective and/or feasibility robustness are verified via Monte Carlo simulations.

scholarly journals Robust Optimization Model with Shared Uncertain Parameters in Multi-Stage Logistics Production and Inventory Process

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 211
Author(s):  
Lijun Xu ◽  
Yijia Zhou ◽  
Bo Yu
Keyword(s):  
Robust Optimization ◽  
Optimization Model ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Uncertain Parameters ◽  
Worst Case ◽  
Robust Model ◽  
Multi Stage ◽  
Proposed Model ◽  
Dual Forms

In this paper, we focus on a class of robust optimization problems whose objectives and constraints share the same uncertain parameters. The existing approaches separately address the worst cases of each objective and each constraint, and then reformulate the model by their respective dual forms in their worst cases. These approaches may result in that the value of uncertain parameters in the optimal solution may not be the same one as in the worst case of each constraint, since it is highly improbable to reach their worst cases simultaneously. In terms of being too conservative for this kind of robust model, we propose a new robust optimization model with shared uncertain parameters involving only the worst case of objectives. The proposed model is evaluated for the multi-stage logistics production and inventory process problem. The numerical experiment shows that the proposed robust optimization model can give a valid and reasonable decision in practice.

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.

scholarly journals Non-Convex Feasibility Robust Optimization Via Scenario Generation and Local Refinement

2019 ◽  
Vol 142 (5) ◽  
Author(s):  
Eliot Rudnick-Cohen ◽  
Jeffrey W. Herrmann ◽  
Shapour Azarm
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Computational Cost ◽  
Optimal Solution ◽  
Optimization Techniques ◽  
Scenario Generation ◽  
Test Problems ◽  
Optimization Approach ◽  
Uncertain Parameters ◽  
Worst Case

Abstract Feasibility robust optimization techniques solve optimization problems with uncertain parameters that appear only in their constraint functions. Solving such problems requires finding an optimal solution that is feasible for all realizations of the uncertain parameters. This paper presents a new feasibility robust optimization approach involving uncertain parameters defined on continuous domains. The proposed approach is based on an integration of two techniques: (i) a sampling-based scenario generation scheme and (ii) a local robust optimization approach. An analysis of the computational cost of this integrated approach is performed to provide worst-case bounds on its computational cost. The proposed approach is applied to several non-convex engineering test problems and compared against two existing robust optimization approaches. The results show that the proposed approach can efficiently find a robust optimal solution across the test problems, even when existing methods for non-convex robust optimization are unable to find a robust optimal solution. A scalable test problem is solved by the approach, demonstrating that its computational cost scales with problem size as predicted by an analysis of the worst-case computational cost bounds.

Robust design of NMR magnet through worst case analysis

2011 ◽  
Vol 37 (2-3) ◽  
pp. 101-107 ◽  
Author(s):  
Angelo Ambrisi ◽  
Alessandro Formisano ◽  
Raffaele Martone ◽  
Ivan Yatchev
Keyword(s):  
Robust Design ◽  
Case Analysis ◽  
Worst Case Analysis ◽  
Worst Case

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.

A Genetic Model: Analysis and Application to MAXSAT

2000 ◽  
Vol 8 (3) ◽  
pp. 291-309 ◽  
Author(s):  
Alberto Bertoni ◽  
Marco Carpentieri ◽  
Paola Campadelli ◽  
Giuliano Grossi
Keyword(s):  
Combinatorial Optimization ◽  
Genetic Model ◽  
Optimization Problems ◽  
Case Analysis ◽  
Optimization Techniques ◽  
Worst Case Analysis ◽  
Worst Case ◽  
Average Case ◽  
Combinatorial Optimization Problems

In this paper, a genetic model based on the operations of recombination and mutation is studied and applied to combinatorial optimization problems. Results are: The equations of the deterministic dynamics in the thermodynamic limit (infinite populations) are derived and, for a sufficiently small mutation rate, the attractors are characterized; A general approximation algorithm for combinatorial optimization problems is designed. The algorithm is applied to the Max Ek-Sat problem, and the quality of the solution is analyzed. It is proved to be optimal for k≥3 with respect to the worst case analysis; for Max E3-Sat the average case performances are experimentally compared with other optimization techniques.

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