Multi-objective Robust Optimization Method Research Articles

Interval Pareto front-based multi-objective robust optimization for sensor placement in structural modal identification

Considering the multi-performance development of complex systems and the requirement of structural modal identification with typical uncertainties, the nominal single-objective optimization method is not suitable for sensor placement. Therefore, by combining conventional optimal sensor placement with non-probabilistic theory, this study proposes an uncertainty-oriented multi-objective robust optimization method for optimal sensor placement. The Fisher information matrix and ill-posedness comprise one eigenvalue-based optimization objective, and the mean and minimum off-diagonal values in the modal assurance criterion comprise another. Considering the high-cost limitation of the statistical method for handling uncertainties, uncertainty propagations are realized by a dimension-wise analysis with better accuracy and efficiency, thus avoiding the overestimation incurred by the classical Taylor expansion method. The multi-objective robust optimization is established by uncertain eigenvalue- and eigenvector-based indices with interval numbers and solved using the multi-objective optimization algorithm. Considering the solution sets located at the Pareto front, an interval possibility was developed using interval Pareto fronts to determine the optimal number of sensors. A numerical example demonstrated the validity of the proposed method with an optimal number of sensors and corresponding configurations.

6σ robust and optimal design of micro-motion platform based on kriging model

The two performance indexes of output displacement and inherent frequency are closely related to the design variables of the micro-motion platform. To address the problem that the design variables have certain fluctuations and the high nonlinearity between the design variables and the performance of the micro-motion platform, a multi-objective robust optimization method with high computational efficiency and high fitting accuracy is proposed. Based on the kriging agent model, the high-precision approximate functions of the key design variables with the output displacement and the inherent frequency are established by the Latin hypercube experimental design, respectively. With the maximum stress as the constraint, Monte Carlo simulation, 6σ robust design concept, and nondominated sorting genetic algorithm II (NSGA-II) multi-objective genetic algorithm are integrated to optimize the structure of the micro-motorized platform. The results show that compared with the traditional deterministic optimization method, the 6σ robust design considering parameter uncertainty effectively reduces the fluctuation of the performance of the micromotion platform and improves the resistance of the platform performance to disturbance under uncertainty.

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

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.

Multi-Objective Robust Optimization for a Dual-Flux-Modulator Coaxial Magnetic Gear

The conventional multi-objective deterministic optimization (MODO) design may be less meaningful or even unacceptable when considering the perturbations of design parameters and reliability of the optimization results. In order to overcome the above-mentioned drawback and improve both the torque capability and permanent magnet (PM) utilization efficiency of dual-flux-modulator coaxial magnetic gear (DFM-CMG) simultaneously, a robust optimization design method is presented. In this method, the sigma criteria with the Monte Carlo simulation (MCS) are adopted to obtain the statistic samples addressing the effects of parametric uncertainties of DFM-CMG on the optimization results. Both 2-D and 3-D finite-element (FE) models of a DFM-CMG are first established, and the 3-D FE model is proven more accurate by the experiment and used for further optimization. Through the parametric study, five parameters are selected as the key design variables to establish the quadratic polynomial regression metamodels. Finally, the multi-objective particle swarm optimization algorithm with MCS is employed to conduct the multi-objective robust optimization (MORO) for DFM-CMG. Although the stall torque per PM consumption achieved by MORO with six sigma ( $6\sigma$ ) is little lower than that achieved by MODO, it still has a 21.4% growth than that of the initial design under the same constraint of the stall torque. Furthermore, the reliability and stability of the MORO results are much higher than those of the MODO results. The MORO method is significantly effective as the torque performance and robustness of DFM-CMG could be improved simultaneously.

Green and Resilient Design of Electricity Supply Chain Networks: A Multiobjective Robust Optimization Approach

Smart grids provide great opportunities for design and implementation of sustainable and efficient electricity supply chains that are more resilient to disruptions. The problem with electricity supply chain network design (ESCND) using smart grid has not been broadly explored by researchers. This paper aims to approach this problem using a multiobjective robust optimization method. The three objectives are economic (profit maximization), environmental (greenhouse gas emissions minimization), and resilience (network resilience maximization). Efficiency maximization—power loss minimization—has also been taken into account using its corresponding cost element in the economic objective function. The proposed approach accounts for different unique smart grid components such as demand side management programs, microgrid structure, two-way distribution lines, and supplier–consumer nodes, while incorporating different interrelated decisions including facility location, capacity expansion, load allocation, and pricing. To solve the resultant model, a hybrid multiobjective robust optimization technique is proposed based on cutting plane and AUGMECON2 algorithms. The proposed model and solution approach are then applied to an actual case study and the produced results are thoroughly analyzed. We find that while economic and environmental objectives can be strictly conflicting, the implementation of smart grids can result in concurrent boost in both environmental performance and network resilience.

Multi-objective robust optimization of foam-filled bionic thin-walled structures

Bio-inspired design has drawn increased attention in recent years for the excellent structural properties of biological system. In our recent work, a bionic thin-walled structure (BTS), which was inspired from the structural characteristic of horsetail, was found to have excellent crashworthiness (Yin et al., 2015) [1]. In order to further improve the crashworthiness of the BTS, a foam-filled bionic thin-walled structure (FBTS) was investigated using the software LS-DYNA in this study. And, the FBTS was optimized by a multi-objective deterministic optimization (MDO) method. The MDO result indicates that the FBTS performed better than the corresponding traditional structure. However, the deterministic optimal design is likely to become unacceptable when considering the uncertainties of design parameters. To solve this problem, a multi-objective robust optimization (MRO) method which employs ensemble metamodel, NSGA-II, “3-sigma” robust design and Monte Carlo simulation (MCS) was developed. Then, the FBTS was optimized by this MRO method. The comparison of the Pareto fronts of the MDO and MRO shows that the robust optimal FBTS is more reliable than the deterministic optimal FBTS. The robust optimal FBTS not only has excellent crashworthiness but also has high reliability. Therefore, the robust optimal FBTS is a kind of excellent and reliable energy absorber in impact engineering.

Multi-objective robust optimization method for the modified epoxy resin sheet molding compounds of the impeller

Abstract A kind of modified epoxy resin sheet molding compounds of the impeller has been designed. Through the test, the non-metal impeller has a better environmental aging performance, but must do the waterproof processing design. In order to improve the stability of the impeller vibration design, the influence of uncertainty factors is considered, and a multi-objective robust optimization method is proposed to reduce the weight of the impeller. Firstly, based on the fluid-structure interaction, the analysis model of the impeller vibration is constructed. Secondly, the optimal approximate model of the impeller is constructed by using the Latin hypercube and radial basis function, and the fitting and optimization accuracy of the approximate model is improved by increasing the sample points. Finally, the micro multi-objective genetic algorithm is applied to the robust optimization of approximate model, and the Monte Carlo simulation and Sobol sampling techniques are used for reliability analysis. By comparing the results of the deterministic, different sigma levels and different materials, the multi-objective optimization of the SMC molding impeller can meet the requirements of engineering stability and lightweight. And the effectiveness of the proposed multi-objective robust optimization method is verified by the error analysis. After the SMC molding and the robust optimization of the impeller, the optimized rate reached 42.5%, which greatly improved the economic benefit, and greatly reduce the vibration of the ventilation system.

The multi-objective robust optimization of the loading path in the T-shape tube hydroforming based on dual response surface model

In this study, a dual response surface model-based multi-objective robust optimization method is introduced to deal with the uncertainties in the tube hydroforming process. The objective of this study is to maximize the protrusion height and minimize the thinning ratio; meanwhile, the variations of the objectives should be minimized. A valid finite element model obtained from experimental result and LS-DYNA is employed to simulate the T-shape tube hydroforming process. To improve computation efficiency, radial basis function combined with Latin hypercube and orthogonal design sampling strategies is employed to construct dual response surface model, which are the mean and standard deviation response of the hydroforming process, respectively. The robust Pareto solutions can be obtained using NSGA-II; meanwhile, the ideal point method is used to obtain the most satisfactory solution from the Pareto solutions for the design engineers. As a conclusion, a significant improvement of the robustness can be achieved; however, the mean performance of the protrusion height has to be sacrificed.

Comparison of Robust Optimization Methods Applied to Hypersonic Vehicle Design

The use of various robust optimization methods to solve engineering problems with environmental parameter uncertainty was investigated. Comparisons were made between a novel multi-objective-based robust optimization formulation and conventional robust regularization-based and aggregation-based methods. The results, performance, and philosophies of each method are discussed. A hypersonic vehicle design problem was used as a test bed for the methods presented here. A focus was put on studying the effect of uncertain roughness-induced boundary-layer transition locations on vehicle controllability. The robust regularization results show that a flat wedgelike vehicle design is best for a worst-case scenario, whereas a pyramidal-shaped vehicle design minimizes the expected detrimental effects on vehicle controllability. The analysis proved that the multi-objective robust optimization method was able to identify these two types of results within an overall set of results while also capturing tradeoffs within the design space.

Robust Optimization for Time-Cost Tradeoff Problem in Construction Projects

Construction projects are generally subject to uncertainty, which influences the realization of time-cost tradeoff in project management. This paper addresses a time-cost tradeoff problem under uncertainty, in which activities in projects can be executed in different construction modes corresponding to specified time and cost with interval uncertainty. Based on multiobjective robust optimization method, a robust optimization model for time-cost tradeoff problem is developed. In order to illustrate the robust model, nondominated sorting genetic algorithm-II (NSGA-II) is modified to solve the project example. The results show that, by means of adjusting the time and cost robust coefficients, the robust Pareto sets for time-cost tradeoff can be obtained according to different acceptable risk level, from which the decision maker could choose the preferred construction alternative.

Multi-objective Robust Optimization of Injection Molding Process Parameters Based on TOPSIS

Present approaches for the robust optimization of injection molding process parameters cannot thoroughly analyze the influence of uncontrollable factors due to the lack of outer experimental runs.And they are often single-objective optimization with no comprehensive evaluation on the characteristics of molded parts.To overcome these shortcomings,a novel multi-objective robust optimization method is proposed on the basis of parameter design using inner-outer array,analysis of signal to noise ratio(SNR) and the technique for order preference by similarity to ideal solution(TOPSIS).The principal process parameters that have the most significant influence on the part quality are firstly screened out by executing two-level orthogonal experiments and analysis of variance(ANOVA),and then both inner and outer orthogonal arrays can be designed by taking the errors of principal process parameters as noise factors.The quadratic polynomial models for predicting the part quality of given process parameter scheme are formulated on the basis of uniform experimental design and stepwise regression analyses,which are then utilized to efficiently compute the values of quality indices corresponding to all parameter schemes arranged according to the inner-outer array.Thus the SNR of every chosen quality index corresponding to all the parameter schemes of the inner array can be efficiently obtained,based on which the preference of all inner schemes are sorted and the most robust one can be determined.The feasibility of the proposed method is demonstrated by the robust optimization of the process parameters for the injection of a mobile phone panel.

Multiobjective robust optimization method for drawbead design in sheet metal forming

It is recognized that fracture and wrinkling in sheet metal forming can be eliminated via an appropriate drawbead design. Although deterministic multiobjective optimization algorithms and finite element analysis (FEA) have been applied in this respect to improve formability and shorten design cycle, the design could become less meaningful or even unacceptable when considering practical variation in design variables and noises of system parameters. To tackle this problem, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on drawbead design, where the six sigma principle is adopted to measure the variations, a dual response surface method is used to construct surrogate model and a multiobjective particle swarm optimization is developed to generate robust Pareto solutions. In this paper, the procedure of drawbead design is divided into two stages: firstly, equivalent drawbead restraining forces (DBRF) are obtained by developing a multiobjective robust particle swarm optimization, and secondly the DBRF model is integrated into a single-objective particle swarm optimization (PSO) to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust. Thus, the presented method provides an effective solution to geometric design of drawbead for improving product quality.

Short-term scheduling of chemical process including uncertainty

This paper addresses the scheduling of chemical process with uncertainty. A multiobjective robust optimization method is proposed to identify Pareto optimal solutions, where normal boundary intersection technique is utilized to trace the Pareto optimal surface in the objective space. The problem is also addressed using parametric mixed integer linear programming where uncertain parameters appear on the right hand side of the constraints. For the case of multiple uncertain parameters, a new algorithm is proposed that does not require the construction of the LP tableaus but relies on the comparison between solutions at leaf nodes resulting in the set of optimal solutions and their critical regions.

SHORT-TERM SCHEDULING OF CHEMICAL PROCESS INCLUDING UNCERTAINTY

This paper addresses the short-term scheduling of chemical process with uncertainty considerations. A multiobjective robust optimization method is proposed to identify Pareto optimal solutions, where Normal boundary intersection (NBI) technique is utilized in order to trace the Pareto optimal surface in the objective space, on which each point represents a trade-off between the various objectives. The issue is also addressed using parametric mixed integer linear programming (pMILP) analysis where uncertain parameters are present on the right hand side (RHS) of the constraints. For the case of multiple uncertain parameters, a new algorithm of multiparametric linear programming (mpLP) is proposed that does not require the construction of the LP tableaus but relies on the comparison between solutions at leaf nodes. Given the range of uncertain parameters, the output of this proposed framework is a set of optimal integer solutions and their corresponding critical regions and optimal functions.