An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.     

Application of reliability-based optimization to earth-moving machine: hydraulic cylinder components design process

Reliability-based design optimization (RBDO) incorporates probabilistic analysis into optimization process so that an optimum design has a great chance of staying in the feasible design space when the inevitable variability in design variables/parameters is considered. One of the biggest drawbacks of applying RBDO to practical problem is its high computational cost that is often impractical to industries. In search of the most suitable RBDO method for industrial applications, we first evaluated several existing RBDO approaches in details such as the double-loop RBDO, the sequential optimization and reliability assessment, and the response surface method. Then, based on industry needs, a platform incorporating/integrating the existing algorithm of optimization and reliability analysis is built for a practical RBDO problem. Effectiveness of the proposed RBDO approach is demonstrated using a simple cantilever beam problem and a more complicated industry problem.     

Conservative reliability-based design optimization method with insufficient input data

Reliability analysis and reliability-based design optimization (RBDO) require an exact input probabilistic model to obtain accurate probability of failure (PoF) and RBDO optimum design. However, often only limited input data is available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the PoF to be uncertain. Therefore, it is necessary to consider the PoF to follow a probability distribution. In this paper, the probability of the PoF is obtained with consecutive conditional probabilities of input distribution types and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to calculate the probability of the PoF. The probability of the PoF at a user-specified target PoF is defined as the conservativeness level of the PoF. The conservativeness level, in addition to the target PoF, will be used as a probabilistic constraint in an RBDO process to obtain a conservative optimum design, for limited input data. Thus, the design sensitivity of the conservativeness level is derived to support an efficient optimization process. Using numerical examples, it is demonstrated that the conservativeness level should be involved in RBDO when input data is limited. The accuracy and efficiency of the proposed design sensitivity method is verified. Finally, conservative RBDO optimum designs are obtained using the developed methods for limited input data problems.     

An improved two-stage framework of evidence-based design optimization

In this paper, an improved two-stage framework is presented to handle the evidence-based design optimization (EBDO) problem under epistemic uncertainty. The improvements include two aspects: (1) in the first stage, the equal areas method is employed to transform evidence variables into random variables, which avoids the assumption that unknown evidence variables and parameters obey the normal distribution. Then, a reliability-based design optimization (RBDO) problem with random variables is defined and solved by the sequential optimization and reliability assessment (SORA) method; (2) in the second stage, an improved algorithm is presented, which can calculate the plausibility of constraint violation more efficiently by continuously recording the minimum and maximum values of limit-state functions. The computational accuracy and efficiency of the improved framework are tested by numerical and engineering examples.     

An efficient reliability-based optimization scheme for uncertain linear systems subject to general Gaussian excitation

A very efficient methodology to carry out reliability-based optimization of linear systems with random structural parameters and random excitation is presented. The reliability-based optimization problem is formulated as the minimization of an objective function for a specified reliability. The probability that design conditions are satisfied within a given time interval is used as a measure of the system reliability. Approximation concepts are used to construct high quality approximations of dynamic responses in terms of the design variables and uncertain structural parameters during the design process. The approximations are combined with an efficient simulation technique to generate explicit approximations of the reliability measures with respect to the design variables. In particular, an efficient importance sampling technique is used to estimate the failure probabilities. The number of dynamic analyses as well as reliability estimations required during the optimization process are reduced dramatically. Several example problems are presented to illustrate the effectiveness and feasibility of the suggested approach.     

Unified uncertainty representation and quantification based on insufficient input data

Uncertainty quantification accuracy of system performance has an important influence on the results of reliability-based design optimization (RBDO). A new uncertain identification and quantification methodology is proposed considering the strong statistical variables, sparse variables, and interval variables simultaneously. Maximum likelihood function and Akaike information criterion (AIC) methods are used to identify the best-fitted distribution types and distribution parameters of sparse variables. The interval variables are represented with evidence theory. Finally, a unified uncertainty quantification framework considering the three types of uncertain design variables is put forward, and then the failure probability of system performance is quantified with belief and plausibility measures. The Kriging metamodel and random sampling method are used to reduce the computational complexity. Three examples are illustrated to verify the effectiveness of the proposed methodology.     

Optimum values of structural safety factors for a predefined reliability level with extension to multiple limit states

In the field of deterministic structural optimization, the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. This way, the resulting optimum solution may represent a lower level of reliability and thus a higher risk of failure. It is the objective of reliability-based design optimization (RBDO) to design structures that should be both economic and reliable. The coupling between mechanical modeling, reliability analyses and optimization methods leads to very high computational costs and weak convergence stability. Since the traditional RBDO solution is achieved by alternating between reliability and optimization iterations, the structural designers performing deterministic optimization do not consider the RBDO model as a practical tool for the design of real structures. Fortunately, a hybrid method based on simultaneous solution of the reliability and the optimization problem, has successfully reduced the computational time problem. The hybrid method allows us to satisfy a required reliability level, but the vector of variables here contains both deterministic and random variables. The hybrid RBDO problem is thus more complex than that of deterministic design. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a special optimization procedure. In this paper a new methodology is presented with the aim of finding a global solution to RBDO problems without additional computing cost for the reliability evaluation. The safety factor formulation for a single limit state case has been used to efficiently reduce the computational time . This technique is fundamentally based on a study of the sensitivity of the limit state function with respect to the design variables. In order to demonstrate analytically the efficiency of this methodology, the optimality condition is then used. The efficiency of this technique is also extended to multiple limit state cases. Two numerical examples are presented at the end of the paper to demonstrate the applicability of the new methodology.     

Extension of optimum safety factor method to nonlinear reliability-based design optimization

In the reliability-based design optimization (RBDO) model, the mean values of uncertain system variables are usually applied as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, a RBDO solution that reduces the structural weight in uncritical regions does not only provide an improved design but also a higher level of confidence in the design. In this paper, we present recent developments for the RBDO model relative to two points of view: reliability and optimization. Next, we develop several distributions for the hybrid method and the optimum safety factor methods (linear and nonlinear RBDO). Finally, we demonstrate the efficiency of our safety factor approach extended to nonlinear RBDO with application to a tri-material structure.     

A decoupling approach for evidence-theory-based reliability design optimization

For the problem of evidence-theory-based reliability design optimization (EBDO), this paper presents a decoupling approach which provides an effective tool for the reliability design of some complex structures with epistemic uncertainty. The approach converts the original nested optimization into a sequential iterative process including design optimization and reliability analysis. In each iteration step, through the uniformity algorithm, the original EBDO is firstly transformed to a conventional reliability-based design optimization (RBDO) and an optimal solution is obtained by solving it. At the solution, the first-order approximate reliability analysis method (FARM) is then used to perform the evidence-theory-based reliability analysis for each constraint. In addition, the RBDO solving and the evidence-theory-based reliability analysis are carried out alternately until reaching the convergence. Finally, two numerical examples and a practical engineering application show the effectiveness of the proposed method.     

Equivalent target probability of failure to convert high-reliability model to low-reliability model for efficiency of sampling-based RBDO

This study presents a methodology to convert an RBDO problem requiring very high reliability to an RBDO problem requiring relatively low reliability by appropriately increasing the input standard deviations for efficient computation in sampling-based RBDO. First, for linear performance functions with independent normal random inputs, an exact probability of failure is derived in terms of the ratio of the input standard deviation, which is denoted by $\boldsymbol {\delta } $ . Then, the probability of failure estimation is generalized for other types of random inputs and performance functions. For the generalization of the probability of failure estimation, two types of coefficients need to be determined by equating the probability of failure and its sensitivities with respect to the input standard deviation at the given design point. The sensitivities of the probability of failure with respect to the standard deviation are obtained using the first-order score function for the standard deviation. To apply the proposed method to an RBDO problem, a concept of an equivalent target probability of failure, which is an increased target probability of failure corresponding to the increased input standard deviations, is also introduced. Numerical results indicate that the proposed method can estimate the probability of failure accurately as a function of the input standard deviation compared to the Monte Carlo simulation results. As anticipated, the sampling-based RBDO using equivalent target probability of failure helps find the optimum design very efficiently while yielding reasonably accurate optimum design, which is close to the one obtained using the original target probability of failure.     

An efficient reliability analysis approach for structure based on probability and probability box models

In this paper, a new reliability analysis method for engineering structures is developed based on probability and probability box (p-box) models. Random variable distributions are used to deal with the uncertain parameters with sufficient information, while the p-box models are employed to deal with the uncertain-but-bounded variables. Due to the existence of the p-box parameters, a limit-state band will result and a complex nesting optimization problem will be involved in this reliability analysis. To reduce the computational burden, an efficient decoupling strategy is developed to solve the nesting optimization problem. Through interval analysis for the probability transformation process, the complex nesting optimization problem can be transformed to a single-layer optimization model. Then, the optimum solution and corresponding reliability index can be obtained by introducing a sequential quadratic programming (SQP) method. Four engineering numerical examples are investigated to demonstrate the effectiveness of the present method.     

Topology optimization of structure for dynamic properties considering hybrid uncertain parameters

In the design and manufacturing of mechanical components, the dynamic properties of continuum structure are one of the most significant performances. At the same time, the uncertainty is widespread in these dynamic problems. This paper presents a robust topology optimization methodology of structure for dynamic properties with consideration of hybrid uncertain parameters. The imprecise probability uncertainties including materials, geometry and boundary condition are treated as an interval random model, in which the probability distribution parameters of random variables are modeled as the interval variables instead of given precise values. Two dynamic properties, including dynamic-compliance and eigenvalue, are chosen as the objective function. In addition, different excitation frequency or eigenvalue is discussed. In this work, the bi-directional evolutionary structural optimization (BESO) method is adopted to find the optimal robust layout of the structure. A series of numerical examples is presented to illustrate the optimization procedure, and the effectiveness of the proposed method is demonstrated clearly.     

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function. Received July 25, 2000     

Component and system reliability-based topology optimization using a single-loop method

We perform reliability-based topology optimization by combining reliability analysis and material distribution topology design methods to design linear elastic structures subject to random inputs, such as random loadings. Both component reliability and system reliability are considered. In component reliability, we satisfy numerous probabilistic constraints which quantify the failure of different events. In system reliability, we satisfy a single probabilistic constraint which encompasses the component events. We adopt the first-order reliability method to approximate the component reliabilities and the inclusion-exclusion rule to approximate the system reliability. To solve the probabilistic optimization problem, we use a variant of the single loop method, which eliminates the need for an inner reliability analysis loop. The proposed method is amenable to implementation with existing deterministic topology optimization software, and hence suitable for practical applications. Designs obtained from component and system reliability-based topology optimization are compared to those obtained from traditional deterministic topology optimization and validated via Monte Carlo simulation.     

A stability investigation of a simulation- and reliability-based optimization

This study developed a reliability-based design optimization (RBDO) algorithm focusing on the ability of solving problems with nonlinear constraints or system reliability. In this case, a sampling technique is often adopted to evaluate the reliability analyses. However, simulation with an insufficient sample size often possesses statistical randomness resulting in an inaccurate sensitivity calculation. This may cause an unstable RBDO solution. The proposed approach used a set of deterministic variables, called auxiliary design points, to replace the random parameters. Thus, an RBDO is converted into a deterministic optimization (DO, α-problem). The DO and the analysis of finding the auxiliary design points (β-problem) are conducted iteratively until the solution converges. To maintain the stability of the RBDO solution with less computational cost, the proposed approach calculated the sensitivity of reliability (in the β-problem) with respect to the mean value of the pseudo-random parameters rather than the design variables. The stability of the proposed method was compared to that of the double-loop approach, and many factors, such as sample size, starting point and the parameters used in the optimization, were considered. The accuracy of the proposed method was confirmed using Monte Carlo simulation (MCS) with several linear and nonlinear numerical problems.     

Improved hybrid method as a robust tool for reliability-based design optimization

In the engineering problems, the randomness and the uncertainties of the distribution of the structural parameters are a crucial problem. In the case of reliability-based design optimization (RBDO), it is the objective to play a dominant role in the structural optimization problem introducing the reliability concept. The RBDO problem is often formulated as a minimization of the initial structural cost under constraints imposed on the values of elemental reliability indices corresponding to various limit states. The classical RBDO leads to high computing time and weak convergence, but a Hybrid Method (HM) has been proposed to overcome these two drawbacks. As the hybrid method successfully reduces the computing time, we can increase the number of variables by introducing the standard deviations as optimization variables to minimize the error values in the probabilistic model. The efficiency of the hybrid method has been demonstrated on static and dynamic cases with extension to the variability of the probabilistic model. In this paper, we propose a modification on the formulation of the hybrid method to improve the optimal solutions. The proposed method is called, Improved Hybrid Method (IHM). The main benefit of this method is to improve the structure performance by much more minimizing the objective function than the hybrid method. It is also shown to demonstrate the optimality conditions. The improved hybrid method is next applied to two numerical examples, with consideration of the standard deviations as optimization variables (for linear and nonlinear distributions). When integrating the improved hybrid method within the probabilistic model variability, we minimize the objective function more and more.     

Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model

Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.     

Optimizing an equilibrium supply chain network design problem by an improved hybrid biogeography based optimization algorithm

This paper presents a new equilibrium optimization method for supply chain network design (SCND) problem under uncertainty, where the uncertain transportation costs and customer demands are characterized by both probability and possibility distributions. We introduce cost risk level constraint and joint service level constraint in the proposed optimization model. When the random parameters follow normal distributions, we reduce the risk level constraint and the joint service level constraint into their equivalent credibility constraints. Furthermore, we employ a sequence of discrete possibility distributions to approximate continuous possibility distributions. To enhance solution efficiency, we introduce the dominance set and efficient valid inequalities into deterministic mixed-integer programming (MIP) model, and preprocess the valid inequalities to obtain a simplified nonlinear programming model. After that, a hybrid biogeography based optimization (BBO) algorithm incorporating new solution presentation and local search operations is designed to solve the simplified optimization model. Finally, we conduct some numerical experiments via an application example to demonstrate the effectiveness of the designed hybrid BBO.     

A new Chance-Variance optimization criterion for portfolio selection in uncertain decision systems

The Markowitz’s mean-variance (M-V) model has received widespread acceptance as a practical tool for portfolio optimization, and his seminal work has been widely extended in the literature. The aim of this article is to extend the M-V method in hybrid decision systems. We suggest a new Chance-Variance (C-V) criterion to model the returns characterized by fuzzy random variables. For this purpose, we develop two types of C-V models for portfolio selection problems in hybrid uncertain decision systems. Type I C-V model is to minimize the variance of total expected return rate subject to chance constraint; while type II C-V model is to maximize the chance of achieving a prescribed return level subject to variance constraint. Hence the two types of C-V models reflect investors’ different attitudes toward risk. The issues about the computation of variance and chance distribution are considered. For general fuzzy random returns, we suggest an approximation method of computing variance and chance distribution so that C-V models can be turned into their approximating models. When the returns are characterized by trapezoidal fuzzy random variables, we employ the variance and chance distribution formulas to turn C-V models into their equivalent stochastic programming problems. Since the equivalent stochastic programming problems include a number of probability distribution functions in their objective and constraint functions, conventional solution methods cannot be used to solve them directly. In this paper, we design a heuristic algorithm to solve them. The developed algorithm combines Monte Carlo (MC) method and particle swarm optimization (PSO) algorithm, in which MC method is used to compute probability distribution functions, and PSO algorithm is used to solve stochastic programming problems. Finally, we present one portfolio selection problem to demonstrate the developed modeling ideas and the effectiveness of the designed algorithm. We also compare the proposed C-V method with M-V one for our portfolio selection problem via numerical experiments.     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.