Determination of sample size for input variables in RBDO through bi-objective confidence-based design optimization under input model uncertainty

Structural and Multidisciplinary Optimization - Reliability analysis accounting for only randomness of input variables often shows a significant error due to the lack of knowledge and insufficient...     

Robustness-based design optimization under data uncertainty

This paper proposes formulations and algorithms for design optimization under both aleatory (i.e., natural or physical variability) and epistemic uncertainty (i.e., imprecise probabilistic information), from the perspective of system robustness. The proposed formulations deal with epistemic uncertainty arising from both sparse and interval data without any assumption about the probability distributions of the random variables. A decoupled approach is proposed in this paper to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency. The proposed methods are illustrated for the upper stage design problem of a two-stage-to-orbit (TSTO) vehicle, where the information on the random design inputs are only available as sparse point data and/or interval data. As collecting more data reduces uncertainty but increases cost, the effect of sample size on the optimality and robustness of the solution is also studied. A method is developed to determine the optimal sample size for sparse point data that leads to the solutions of the design problem that are least sensitive to variations in the input random variables.     

Multi-objective ordinal optimization for simulation optimization problems

Ordinal optimization (OO) has been successfully applied to accelerate the simulation optimization process with single objective by quickly narrowing down the search space. In this paper, we extend the OO techniques to address multi-objective simulation optimization problems by using the concept of Pareto optimality. We call this technique the multi-objective OO (MOO). To define the good enough set and the selected set, we introduce two performance indices based on the non-dominance relationship among the designs. Then we derive several lower bounds for the alignment probability under various scenarios by using a Bayesian approach. Numerical experiments show that the lower bounds of the alignment probability are valid when they are used to estimate the size of the selected set as well as the expected alignment level. Though the lower bounds are conservative, they have great practical value in terms of narrowing down the search space.     

Multi-objective optimization for process control of the in-situ bioremediation system under uncertainty

In-situ bioremediation is a commonly used remediation technology to clean up the subsurface of a petroleum-contaminated site. The process control of such a system is complex and may involve more than one objective. This study discusses the development of a simulation model-based, dynamic, and multi-objective predictive control system for generating cost-effective control strategies for a bioremediation site, which involves substantial uncertain data. This control system was developed by enhancing the single objective control system presented in Hu et al. [2004. Dynamic process control for in-situ bioremediation system. In: Proceedings of the 2004 IEEE Canadian Conference on Electrical & Computer Engineering, May 4–7]. It shares the objective of minimizing overall cost as that presented in Hu et al. (2004), and it also has the added objective of maximizing system efficiency. An optimized linear interpolation method has been developed to handle the uncertainty involved in the changes of the hydraulic characteristics in groundwater transport simulation, and an interactive decision-making tool is built for multi-objective process control. The solution method includes generation of a set of optimal control strategies and costs to meet different efficiency requirements, normalization of the costs and efficiencies, and construction of the optimal control strategy to satisfy the decision maker's particular preferences on tradeoff between cost and efficiencies.The developed system has been applied on data, which is obtained from lab experiment and a hypothetical site. The results indicate that the optimized linear interpolation function could model inherent uncertainties that result from inadequacies in the chosen sampling points, and enhance overall accuracy of the simulation model. The results show that the control system could generate a set of control strategies, which assign different importance to each objective, thereby providing an optimal strategy to meet particular requirements of the decision maker.     

Identification of quasi-optimal regions in the design space using surrogate modeling

The use of Surrogate Based Optimization (SBO) is widely spread in engineering design to find optimal performance characteristics of expensive simulations (forward analysis: from input to optimal output). However, often the practitioner knows a priori the desired performance and is interested in finding the associated input parameters (reverse analysis: from desired output to input). A popular method to solve such reverse (inverse) problems is to minimize the error between the simulated performance and the desired goal. However, there might be multiple quasi-optimal solutions to the problem. In this paper, the authors propose a novel method to efficiently solve inverse problems and to sample Quasi-Optimal Regions (QORs) in the input (design) space more densely. The development of this technique, based on the probability of improvement criterion and kriging models, is driven by a real-life problem from bio-mechanics, i.e., determining the elasticity of the (rabbit) tympanic membrane, a membrane that converts acoustic sound wave into vibrations of the middle ear ossicular bones.     

Reliability-based design optimization with confidence level under input model uncertainty due to limited test data

For obtaining a correct reliability-based optimum design, the input statistical model, which includes marginal and joint distributions of input random variables, needs to be accurately estimated. However, in most engineering applications, only limited data on input variables are available due to expensive testing costs. The input statistical model estimated from the insufficient data will be inaccurate, which leads to an unreliable optimum design. In this paper, reliability-based design optimization (RBDO) with the confidence level for input normal random variables is proposed to offset the inaccurate estimation of the input statistical model by using adjusted standard deviation and correlation coefficient that include the effect of inaccurate estimation of mean, standard deviation, and correlation coefficient.     

Multi-objective optimization for inventory control in two-level assembly systems under uncertainty of lead times

Supply planning for two-level assembly systems under lead time uncertainties is considered. It is supposed that the demand for the finished product and its due date are known. The assembly process at each level begins when all necessary components are in inventory. A holding cost at each level appears if some components needed to assemble the same semi-finished product arrive before beginning the assembly at this level. It is assumed also that the component lead time is a random discrete variable. The objective is to find the release dates for the components at level 2 in order to minimize the expected component holding costs and to maximize the customer service level for the finished product. For this new problem, we consider two multi-objective approaches, which are both based on genetic algorithms. They are evaluated with a variety of supply chain settings, and their respective performance is reported and commented. These two heuristics permitted to obtain interesting results within a reasonable computational time.     

A decoupled credibility-based design optimization method for fuzzy design variables by failure credibility surrogate modeling

Structural and Multidisciplinary Optimization - In order to make a good compromise of cost and safety with small data in the early structural design stage, a practical decoupled credibility-based...     

Multi-objective reliability-based design optimization for the VRB-VCS FLB under front-impact collision

Structural and Multidisciplinary Optimization - The front longitudinal beam (FLB) is the most important energy-absorbing and crashing-force-transmitting structure of a vehicle in a front crash. Its...     

The integration of design of experiments, surrogate modeling and optimization for thermoscience research

This paper presents an integrated approach for the solution of complex optimization problems in thermoscience research. The cited approach is based on the design of computational experiments (DOE), surrogate modeling, and optimization. The DOE/surrogate modeling techniques under consideration include: A-optimal/classical linear regression, Latin hypercube/artificial neural networks, and Latin hypercube/Sugeno-type fuzzy models. These techniques are coupled with both local (modified Newtons method) and global (genetic algorithms) optimization methods. The proposed approach proved to be an effective, efficient and robust modeling and optimization tool in the context of a case study, and holds promise for use in larger scale optimization problems in thermoscience research.     

Ensemble of surrogate based global optimization methods using hierarchical design space reduction

For computationally expensive black-box problems, surrogate models are widely employed to reduce the needed computation time and efforts during the search of the global optimum. However, the construction of an effective surrogate model over a large design space remains a challenge in many cases. In this work, a new global optimization method using an ensemble of surrogates and hierarchical design space reduction is proposed to deal with the optimization problems with computation-intensive, black-box objective functions. During the search, an ensemble of three representative surrogate techniques with optimized weight factors is used for selecting promising sample points, narrowing down space exploration and identifying the global optimum. The design space is classified into: Original Global Space (OGS), Promising Joint Space (PJS), Important Local Space (ILS), using the newly proposed hierarchical design space reduction (HSR). Tested using eighteen representative benchmark and two engineering design optimization problems, the newly proposed global optimization method shows improved capability in identifying promising search area and reducing design space, and superior search efficiency and robustness in identifying the global optimum.     

Aircraft wing box optimization considering uncertainty in surrogate models

Aerospace design often involves computationally expensive physics based analysis methods such as Computational Fluid Dynamics (CFD) or the Finite Element Method (FEM). Since conceptual design optimization can require many function evaluations, simplified analysis methods are typically used. Designs optimized with simplified analysis methods may be found to violate design goals when subjected to the high fidelity approaches later in the design process. This paper presents how the uncertainty introduced by an approximation model in the conceptual design of the wing box of a generic light jet can be assessed and managed by applying Reliability Based Design Optimization (RBDO) in order to ensure that a feasible solution is obtained. Additionally, the performance of several alternative RBDO approaches are benchmarked using the wing box conceptual design problem.     

Reliability measure approach for confidence-based design optimization under insufficient input data

Structural and Multidisciplinary Optimization - In most of the reliability-based design optimization (RBDO) researches, accurate input statistical model has been assumed to concentrate on the...     

A surrogate based multistage-multilevel optimization procedure for multidisciplinary design optimization

Optimization procedure is one of the key techniques to address the computational and organizational complexities of multidisciplinary design optimization (MDO). Motivated by the idea of synthetically exploiting the advantage of multiple existing optimization procedures and meanwhile complying with the general process of satellite system design optimization in conceptual design phase, a multistage-multilevel MDO procedure is proposed in this paper by integrating multiple-discipline-feasible (MDF) and concurrent subspace optimization (CSSO), termed as MDF-CSSO. In the first stage, the approximation surrogates of high-fidelity disciplinary models are built by disciplinary specialists independently, based on which the single level optimization procedure MDF is used to quickly identify the promising region and roughly locate the optimum of the MDO problem. In the second stage, the disciplinary specialists are employed to further investigate and improve the baseline design obtained in the first stage with high-fidelity disciplinary models. CSSO is used to organize the concurrent disciplinary optimization and system coordination so as to allow disciplinary autonomy. To enhance the reliability and robustness of the design under uncertainties, the probabilistic version of MDF-CSSO (PMDF-CSSO) is developed to solve uncertainty-based optimization problems. The effectiveness of the proposed methods is verified with one MDO benchmark test and one practical satellite conceptual design optimization problem, followed by conclusion remarks and future research prospects.     

Multidisciplinary modeling and surrogate assisted optimization for satellite constellation systems

Satellite constellation system design is a challenging and complicated multidisciplinary design optimization (MDO) problem involving a number of computation-intensive multidisciplinary analysis models. In this paper, the MDO problem of a constellation system consisting of small observation satellites is investigated to simultaneously achieve the preliminary design of constellation configuration and the satellite subsystems. The constellation is established based on Walker-δ configuration considering the coverage performance. Coupled with the constellation configuration, several disciplines including payload, power, thermal control, and structure are taken into account for satellite subsystems design subject to various constraints (i.e., ground resolution, power usage, natural frequencies, etc.). Considering the mixed-integer and time-consuming behavior of satellite constellation system MDO problem, a novel sequential radial basis function (RBF) method using the support vector machine (SVM) for discrete-continuous mixed variables notated as SRBF-SVM-DC is proposed. In this method, a discrete-continuous variable sampling method is utilized to handle the discrete variables, i.e., the number of orbit planes and number of satellites, in the satellite constellation system MDO problem. RBF surrogates are constructed and gradually refined to represent the time-consuming simulations during optimization, which can efficiently lead the search to the optimum. Finally, the proposed SRBF-SVM-DC utilized to solve the satellite constellation system MDO problem is compared with a conventional integer coding based genetic algorithm (ICGA). The results show that SRBF-SVM-DC significantly decreases the system mass by about 28.63% subject to all the constraints, which greatly reduces the cost of the satellite constellation system. Moreover, the computational budget of SRBF-SVM-DC is saved by over 85% compared with ICGA, which demonstrates the effectiveness and practicality of the proposed surrogate assisted optimization approach for satellite constellation system design.     

Using augmented Lagrangian particle swarm optimization for constrained problems in engineering">Using augmented Lagrangian particle swarm optimization for constrained problems in engineering

The comparatively new stochastic method of particle swarm optimization (PSO) has been applied to engineering problems especially of nonlinear, non-differentiable, or non-convex type. Its robustness and its simple applicability without the need for cumbersome derivative calculations make PSO an attractive optimization method. However, engineering optimization tasks often consist of problem immanent equality and inequality constraints which are usually included by inadequate penalty functions when using stochastic algorithms. The simple structure of basic particle swarm optimization characterized by only a few lines of computer code allows an efficient implementation of a more sophisticated treatment of such constraints. In this paper, we present an approach which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrange multiplier method, for constraint handling where ill conditioning is a far less harmful problem and the correct solution can be obtained even for finite penalty factors. We describe the basic PSO algorithm and the resulting method for constrained problems as well as the results from benchmark tests. An example of a stiffness optimization of an industrial hexapod robot with parallel kinematics concludes this paper and shows the applicability of the proposed augmented Lagrange particle swarm optimization to engineering problems.     

Bayesian optimization for learning gaits under uncertainty

Designing gaits and corresponding control policies is a key challenge in robot locomotion. Even with a viable controller parametrization, finding near-optimal parameters can be daunting. Typically, this kind of parameter optimization requires specific expert knowledge and extensive robot experiments. Automatic black-box gait optimization methods greatly reduce the need for human expertise and time-consuming design processes. Many different approaches for automatic gait optimization have been suggested to date. However, no extensive comparison among them has yet been performed. In this article, we thoroughly discuss multiple automatic optimization methods in the context of gait optimization. We extensively evaluate Bayesian optimization, a model-based approach to black-box optimization under uncertainty, on both simulated problems and real robots. This evaluation demonstrates that Bayesian optimization is particularly suited for robotic applications, where it is crucial to find a good set of gait parameters in a small number of experiments.     

Semimyopic measurement selection for optimization under uncertainty

The following sequential decision problem is considered: given a set of items of unknown utility, an item with as high a utility as possible must be selected ("the selection problem"). Measurements (possibly noisy) of item features prior to selection are allowed at known costs. The goal is to optimize the overall sequential decision process of measurements and selection. Value of information (VOI) is a well-known scheme for selecting measurements, but the intractability of the problem typically leads to using myopic VOI estimates. In the selection problem, myopic VOI frequently badly underestimates the VOI, leading to inferior measurement policies. In this paper, the strict myopic assumption is relaxed into a scheme termed semimyopic, providing a spectrum of methods that can improve the performance of measurement policies. In particular, the efficiently computable method of "blinkered" VOI is proposed, and theoretical bounds for important special cases are examined. Empirical evaluation of "blinkered" VOI in the selection problem with normally distributed item values shows that it performs much better than pure myopic VOI.