A multi-objective robust optimization approach for engineering design under interval uncertainty

2018 ◽  
Vol 35 (2) ◽  
pp. 580-603 ◽  
Author(s):  
Qi Zhou ◽  
Xinyu Shao ◽  
Ping Jiang ◽  
Tingli Xie ◽  
Jiexiang Hu ◽  
...  
Keyword(s):  
Robust Optimization ◽  
Engineering Design ◽  
Optimization Problems ◽  
Interval Uncertainty ◽  
Optimization Approach ◽  
Content Type ◽  
Computational Burden ◽  
Multi Objective ◽  
Design Alternatives ◽  
Kriging Metamodel

Purpose Engineering system design and optimization problems are usually multi-objective and constrained and have uncertainties in the inputs. These uncertainties might significantly degrade the overall performance of engineering systems and change the feasibility of the obtained solutions. This paper aims to propose a multi-objective robust optimization approach based on Kriging metamodel (K-MORO) to obtain the robust Pareto set under the interval uncertainty. Design/methodology/approach In K-MORO, the nested optimization structure is reduced into a single loop optimization structure to ease the computational burden. Considering the interpolation uncertainty from the Kriging metamodel may affect the robustness of the Pareto optima, an objective switching and sequential updating strategy is introduced in K-MORO to determine (1) whether the robust analysis or the Kriging metamodel should be used to evaluate the robustness of design alternatives, and (2) which design alternatives are selected to improve the prediction accuracy of the Kriging metamodel during the robust optimization process. Findings Five numerical and engineering cases are used to demonstrate the applicability of the proposed approach. The results illustrate that K-MORO is able to obtain robust Pareto frontier, while significantly reducing computational cost. Practical implications The proposed approach exhibits great capability for practical engineering design optimization problems that are multi-objective and constrained and have uncertainties. Originality/value A K-MORO approach is proposed, which can obtain the robust Pareto set under the interval uncertainty and ease the computational burden of the robust optimization process.

An on-line Kriging metamodel assisted robust optimization approach under interval uncertainty

2017 ◽  
Vol 34 (2) ◽  
pp. 420-446 ◽  
Author(s):  
Qi Zhou ◽  
Ping Jiang ◽  
Xinyu Shao ◽  
Hui Zhou ◽  
Jiexiang Hu
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Computational Cost ◽  
Interval Uncertainty ◽  
Optimization Approach ◽  
Content Type ◽  
Computational Burden ◽  
Design Alternative ◽  
Kriging Metamodel ◽  
On Line

Purpose Uncertainty is inevitable in real-world engineering optimization. With an outer-inner optimization structure, most previous robust optimization (RO) approaches under interval uncertainty can become computationally intractable because the inner level must perform robust evaluation for each design alternative delivered from the outer level. This paper aims to propose an on-line Kriging metamodel-assisted variable adjustment robust optimization (OLK-VARO) to ease the computational burden of previous VARO approach. Design/methodology/approach In OLK-VARO, Kriging metamodels are constructed for replacing robust evaluations of the design alternative delivered from the outer level, reducing the nested optimization structure of previous VARO approach into a single loop optimization structure. An on-line updating mechanism is introduced in OLK-VARO to exploit the obtained data from previous iterations. Findings One nonlinear numerical example and two engineering cases have been used to demonstrate the applicability and efficiency of the proposed OLK-VARO approach. Results illustrate that OLK-VARO is able to obtain comparable robust optimums as to that obtained by previous VARO, while at the same time significantly reducing computational cost. Practical implications The proposed approach exhibits great capability for practical engineering design optimization problems under interval uncertainty. Originality/value The main contribution of this paper lies in the following: an OLK-VARO approach under interval uncertainty is proposed, which can significantly ease the computational burden of previous VARO approach.

A Multi-Objective Robust Optimization Approach Under Interval Uncertainty Based on Kriging and Support Vector Machine

2018 ◽  
Author(s):  
Tingli Xie ◽  
Ping Jiang ◽  
Qi Zhou ◽  
Leshi Shu ◽  
Yang Yang
Keyword(s):  
Support Vector Machine ◽  
Robust Optimization ◽  
Kriging Model ◽  
Interval Uncertainty ◽  
Support Vector ◽  
Optimization Approach ◽  
Multi Objective ◽  
Large Numbers ◽  
Design Alternatives ◽  
Engineering Design Problems

Interval uncertainty can cause uncontrollable variations in the objective and constraint values, which could seriously deteriorate the performance or even change the feasibility of the optimal solutions. Robust optimization is to obtain solutions that are optimal and minimally sensitive to uncertainty. Because large numbers of complex engineering design problems depend on time-consuming simulations, the robust optimization approaches might become computationally intractable. To address this issue, a multi-objective robust optimization approach based on Kriging and support vector machine (MORO-KS) is proposed in this paper. Firstly, the feasible domain of main problem in MORO-KS is iteratively restricted by constraint cuts formed in the subproblem. Secondly, each objective function is approximated by a Kriging model to predict the response value. Thirdly, a Support Vector Machine (SVM) model is constructed to replace all constraint functions classifying design alternatives into two categories: feasible and infeasible. A numerical example and the design optimization of a microaerial vehicle fuselage are adopted to test the proposed MORO-KS approach. Compared with the results obtained from the MORO approach based on Constraint Cuts (MORO-CC), the effectiveness and efficiency of the proposed MORO-KS approach are illustrated.

Advanced Multi-Objective Robust Optimization Under Interval Uncertainty Using Kriging Model and Support Vector Machine

2018 ◽  
Vol 18 (4) ◽  
Author(s):  
Tingli Xie ◽  
Ping Jiang ◽  
Qi Zhou ◽  
Leshi Shu ◽  
Yahui Zhang ◽  
...  
Keyword(s):  
Support Vector Machine ◽  
Robust Optimization ◽  
Engineering Design ◽  
Kriging Model ◽  
Support Vector ◽  
Svm Classifier ◽  
Optimization Approach ◽  
Design Problems ◽  
Multi Objective ◽  
Engineering Design Problems

There are a large number of real-world engineering design problems that are multi-objective and multiconstrained, having uncertainty in their inputs. Robust optimization is developed to obtain solutions that are optimal and less sensitive to uncertainty. Since most of complex engineering design problems rely on time-consuming simulations, the robust optimization approaches may become computationally intractable. To address this issue, an advanced multi-objective robust optimization approach based on Kriging model and support vector machine (MORO-KS) is proposed in this work. First, the main problem in MORO-KS is iteratively restricted by constraint cuts formed in the subproblem. Second, each objective function is approximated by a Kriging model to predict the response value. Third, a support vector machine (SVM) classifier is constructed to replace all constraint functions classifying design alternatives into two categories: feasible and infeasible. The proposed MORO-KS approach is tested on two numerical examples and the design optimization of a micro-aerial vehicle (MAV) fuselage. Compared with the results obtained from other MORO approaches, the effectiveness and efficiency of the proposed MORO-KS approach are illustrated.

A New Hybrid Algorithm for Multi-Objective Robust Optimization With Interval Uncertainty

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Shuo Cheng ◽  
Jianhua Zhou ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Hybrid Approach ◽  
Interval Uncertainty ◽  
Engineering Optimization ◽  
Objective Functions ◽  
Robust Solutions ◽  
Multi Objective ◽  
And Performance

Uncertainty is a very critical but inevitable issue in design optimization. Compared to single-objective optimization problems, the situation becomes more difficult for multi-objective engineering optimization problems under uncertainty. Multi-objective robust optimization (MORO) approaches have been developed to find Pareto robust solutions. While the literature reports on many techniques in MORO, few papers focus on using multi-objective differential evolution (MODE) for robust optimization (RO) and performance improvement of its solutions. In this article, MODE is first modified and developed for RO problems with interval uncertainty, formulating a new MODE-RO algorithm. To improve the solutions’ quality of MODE-RO, a new hybrid (MODE-sequential quadratic programming (SQP)-RO) algorithm is proposed further, where SQP is incorporated into the procedure to enhance the local search. The proposed hybrid approach takes the advantage of MODE for its capability of handling not-well behaved robust constraint functions and SQP for its fast local convergence. Two numerical and one engineering examples, with two or three objective functions, are tested to demonstrate the applicability and performance of the proposed algorithms. The results show that MODE-RO is effective in solving MORO problems while, on the average, MODE-SQP-RO improves the quality of robust solutions obtained by MODE-RO with comparable numbers of function evaluations.

Multiobjective Collaborative Robust Optimization (McRO) With Interval Uncertainty and Interdisciplinary Uncertainty Propagation

2007 ◽  
Author(s):  
Mian Li ◽  
Shapour Azarm
Keyword(s):  
Robust Optimization ◽  
Design Optimization ◽  
Multidisciplinary Design Optimization ◽  
Optimization Problems ◽  
Uncertainty Propagation ◽  
Multiple Objectives ◽  
Interval Uncertainty ◽  
Optimization Approach ◽  
Collaborative Robust Optimization ◽  
Fully Coupled

Real-world engineering design optimization problems often involve systems that have coupled disciplines with uncontrollable variations in their parameters. No approach has yet been reported for the solution of these problems when there are multiple objectives in each discipline, mixed continuous-discrete variables, and when there is a need to account for uncertainty and also uncertainty propagation across disciplines. We present a Multiobjective collaborative Robust Optimization (McRO) approach for this class of problems that have interval uncertainty in their parameters. McRO obtains Multidisciplinary Design Optimization (MDO) solutions which are as best as possible in a multiobjective and multidisciplinary sense. For McRO solutions, the sensitivity of objective and/or constraint functions is kept within an acceptable range. McRO involves a technique for interdisciplinary uncertainty propagation. The approach can be used for robust optimization of MDO problems with multiple objectives, or constraints, or both together at system and subsystem levels. Results from an application of the approach to a numerical and an engineering example are presented. It is concluded that the McRO approach can solve fully coupled MDO problems with interval uncertainty and can obtain solutions that are comparable to an all-at-once robust optimization approach.

Improving Multi-Objective Robust Optimization Under Interval Uncertainty Using Worst Possible Point Constraint Cuts

2009 ◽  
Author(s):  
W. Hu ◽  
M. Li ◽  
S. Azarm ◽  
S. Al Hashimi ◽  
A. Almansoori ◽  
...  
Keyword(s):  
Robust Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Computational Effort ◽  
Interval Uncertainty ◽  
Master Problem ◽  
Multi Objective ◽  
Acceptable Range ◽  
Function Calls ◽  
Order Of Magnitude

Many real-world engineering design optimization problems are multi-objective and have uncertainty in their parameters. For such problems it is useful to obtain design solutions that are both multi-objectively optimum and robust. A robust design is one whose objective and constraint function variations under uncertainty are within an acceptable range. While the literature reports on many techniques in robust optimization for single objective optimization problems, very few papers report on methods in robust optimization for multi-objective optimization problems. The Multi-Objective Robust Optimization (MORO) technique with interval uncertainty proposed in this paper is a significant improvement, with respect to computational effort, over a previously reported MORO technique. In the proposed technique, a master problem solves a relaxed optimization problem whose feasible domain is iteratively confined by constraint cuts determined by the solutions from a sub-problem. The proposed approach and the synergy between the master problem and sub-problem are demonstrated by three examples. The results obtained show a general agreement between the solutions from the proposed MORO and the previous MORO technique. Moreover, the number of function calls for obtaining solutions from the proposed technique is an order of magnitude less than that from the previous MORO technique.

scholarly journals A Distributed Multifactorial Particle Swarm Optimization Approach

2021 ◽  
Author(s):  
Ahlem Aboud ◽  
Nizar Rokbani ◽  
Seyedali Mirjalili ◽  
Abdulrahman M. Qahtani ◽  
Omar Almutiry ◽  
...  
Keyword(s):  
Particle Swarm Optimization ◽  
Optimization Problems ◽  
Beta Function ◽  
Particle Swarm ◽  
Search Space ◽  
Optimization Approach ◽  
Multi Objective Optimization ◽  
Swarm Optimization ◽  
Multi Objective ◽  
New System

<p>Multifactorial Optimization (MFO) and Evolutionary Transfer Optimization (ETO) are new optimization challenging paradigms for which the multi-Objective Particle Swarm Optimization system (MOPSO) may be interesting despite limitations. MOPSO has been widely used in static/dynamic multi-objective optimization problems, while its potentials for multi-task optimization are not completely unveiled. This paper proposes a new Distributed Multifactorial Particle Swarm Optimization algorithm (DMFPSO) for multi-task optimization. This new system has a distributed architecture on a set of sub-swarms that are dynamically constructed based on the number of optimization tasks affected by each particle skill factor. DMFPSO is designed to deal with the issues of handling convergence and diversity concepts separately. DMFPSO uses Beta function to provide two optimized profiles with a dynamic switching behaviour. The first profile, Beta-1, is used for the exploration which aims to explore the search space toward potential solutions, while the second Beta-2 function is used for convergence enhancement. This new system is tested on 36 benchmarks provided by the CEC’2021 Evolutionary Transfer Multi-Objective Optimization Competition. Comparatives with the state-of-the-art methods are done using the Inverted General Distance (IGD) and Mean Inverted General Distance (MIGD) metrics. Based on the MSS metric, this proposal has the best results on most tested problems.</p>

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.

scholarly journals Multi-Objective Robust Optimization Using a Postoptimality Sensitivity Analysis Technique: Application to a Wind Turbine Design

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Ricardo Soto ◽  
Broderick Crawford
Keyword(s):  
Sensitivity Analysis ◽  
Robust Optimization ◽  
Wind Turbine ◽  
Optimization Design ◽  
Optimization Problems ◽  
Design Environment ◽  
Multi Objective ◽  
Pareto Ranking ◽  
Analysis Technique ◽  
Robust Optimization Design

Toward a multi-objective optimization robust problem, the variations in design variables (DVs) and design environment parameters (DEPs) include the small variations and the large variations. The former have small effect on the performance functions and/or the constraints, and the latter refer to the ones that have large effect on the performance functions and/or the constraints. The robustness of performance functions is discussed in this paper. A postoptimality sensitivity analysis technique for multi-objective robust optimization problems (MOROPs) is discussed, and two robustness indices (RIs) are introduced. The first one considers the robustness of the performance functions to small variations in the DVs and the DEPs. The second RI characterizes the robustness of the performance functions to large variations in the DEPs. It is based on the ability of a solution to maintain a good Pareto ranking for different DEPs due to large variations. The robustness of the solutions is treated as vectors in the robustness function space (RF-Space), which is defined by the two proposed RIs. As a result, the designer can compare the robustness of all Pareto optimal solutions and make a decision. Finally, two illustrative examples are given to highlight the contributions of this paper. The first example is about a numerical problem, whereas the second problem deals with the multi-objective robust optimization design of a floating wind turbine.

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