Hybrid Uncertainty Research Articles

Fatigue reliability analysis for structures with hybrid uncertainties combining quadratic response surface and polynomial chaos expansion

This study attempted to propose a novel efficient and accurate uncertainty analysis method for structural fatigue life prediction with hybrid uncertainties. First, PCE was adopted to deal with independent normal random variables, where polynomial coefficients were expressed as quadratic functions of interval variables. In order to determine unknown coefficients, a SCC collocation technique was applied combining with LHS. Interval bounds of the corresponding mean value and standard variance were then deduced, after which a fatigue reliability analysis method was proposed. Finally, three numerical examples were implemented to validate the feasibility and efficiency of the proposed method.

Robust optimization of engineering structures involving hybrid probabilistic and interval uncertainties

A novel, yet practically feasible, robust optimization approach is proposed in this study for engineering structures involving hybrid uncertainties. Both stochastic and interval uncertain system parameters are incorporated within a single analysis-design computational scheme. The generalized beta distribution is adopted to model the bounded stochastic system uncertainties, which offers the benefit of evaluating the performance of objective function and constraints of the robust optimization. A multi-layered refining Latin hypercube sampling–based Monte Carlo simulation approach is proposed to assess the robustness of the objective function. Furthermore, a new concept, namely, the interval angular vector, is presented to evaluate the robust feasibility of the constraints of the optimization problem. In order to systematically solve the robust optimization problem, a new genetic algorithm is presented in this study which utilizes the order preference by similarity to ideal solution technique so the feasible design vectors can be sorted according to their distances to the negative ideal solution. The effectiveness and applicability of the proposed computational approach are demonstrated by one numeral example and two realistic complex engineering structures including the bucket linkage mechanism of an excavator and the upper beam of a high-speed punching machine.

Evaluation and selection of sustainable hydrogen production technology with hybrid uncertain sustainability indicators based on rough-fuzzy BWM-DEA

This study is aiming to propose a framework integrating new criteria system, rough-fuzzy best-worst method (BWM) and rough-fuzzy data envelopment analysis (DEA) for evaluation and selection of sustainable hydrogen production technologies (HPTs) with hybrid types of sustainability indicators (i.e. quantitative and qualitative) and hybrid decision uncertainties (i.e. intrapersonal linguistic vagueness and interpersonal preference randomness). The rough-fuzzy BWM is presented to determine the relative weights of sustainability criteria and the rough-fuzzy DEA is proposed for prioritizing the alternative HPTs with hybrid types of performance data (including crisp number, interval number and group linguistic terms). Such integration combines the collaborative capability of rough-fuzzy number in fully manipulating hybrid uncertainties, the merits of BWM in quickly searching optimal weights and the advantages of DEA in effectively obtaining efficiency with multiscale input and output data. The application of the proposed approach to five typical HPTs with 14 hybrid uncertain sustainability indicators show that the wind-based electrolysis has the most sustainability priority (0.480) for hydrogen production compared with coal gasification (0.035), steam methane reforming (0.056), biomass gasification (0.246) and photovoltaic-based electrolysis (0.182). Moreover, the comparisons with other methods demonstrate the validity and effectiveness of the proposed approach.

The design of a resilient and sustainable maximal covering closed-loop supply chain network under hybrid uncertainties: a case study in tire industry

This paper aims at designing a sustainable and resilient tire closed-loop supply chain network based on a real case study in Iran. To immune the network against disruptions, a new resiliency approach is proposed by extending efficient demand coverage plans. The developed model includes four objectives. The first objective minimizes the total costs of the network along with maximizing the coverage of customers’ demand. As a new concept, the second objective maximizes the operational reliability of facilities to extend a consistent and responsive tire supply chain. Also, to design a sustainable network, the third objective function minimizes CO2 emissions in the strategic and tactical planning horizons. Moreover, the fourth objective maximizes the social responsibility of the network via introducing new social factors. A new mixed fuzzy possibilistic flexible programming method is also proposed to handle constraints’ violations and parameters’ uncertainty. Outputs confirm the accurate performance of the extended model and ensure its applicability in the real-world case study.

Risk-based design optimization under hybrid uncertainties

The rapidly changing requirements of engineering optimization problems require unprecedented levels of compatibility to integrate diverse uncertainty information to search optimum among design region. The sophisticated optimization methods tackling uncertainty involve reliability-based design optimization and robust design optimization. In this paper, a novel alternative approach called risk-based design optimization (RiDO) has been proposed to counterpoise design results and costs under hybrid uncertainties. In this approach, the conditional value at risk (CVaR) is adopted for quantification of the hybrid uncertainties. Then, a CVaR estimation method based on Monte Carlo simulation (MCS) scenario generation approach is derived to measure the risk levels of the objective and constraint functions. The RiDO under hybrid uncertainties is established and leveraged to determine the optimal scheme which satisfies the risk requirement. Three examples with different calculation complexity are provided to verify the developed approach.

Cable Angle and Minimum Resultant Force Response Analysis of Lower Limb Traction Device for Rehabilitation Robot With Interval Parameters

Abstract This paper proposes a hybrid uncertainty analysis method (HUAM) based on the first-order interval perturbation method (FIPM) and Monte Carlo method (MCM) for minimum resultant force response analysis of the lower limb traction device (LLTD) of a hybrid-driven parallel waist rehabilitation robot (HDPWRR) with interval parameters. Based on the analysis of cable angles by using the interval algorithm, the problem of non-uniqueness of the force solution in redundant constraint mechanisms is solved. The force response domain prediction with interval parameters on rehabilitation patients is estimated by using the HUAM which combining the first-order interval perturbation technique with direct Monte Carlo method in different stages, and it reduces the calculation amount. First, the kinematic and static models of the LLTD with deterministic information are established according to its work principle. Then, the interval matrices with interval parameters are calculated by using the FIPM and the response of cable angles is combined with the static model. Third, by numerical examples, the accuracy and efficiency of the HUAM for solving the force response domain problem with interval parameters are verified. The bounds of cable angle response domain of the interval LLTD model are determined. Finally, the minimum resultant force response domain prediction with interval parameters on rehabilitation patients is estimated by combining the FIPM and MCM.

Slope hybrid reliability analysis considering the uncertainty of probability-interval using three-parameter Weibull distribution

A reliability model is proposed to solve the problem of hybrid uncertainty with both random and interval variables in slope engineering. A hybrid uncertainty model based on the dimension reduction method and Taylor expansion is constructed to approximate the limit state function. Using the polynomial theorem and variable transformation method, the origin and center moments’ interval of the limit state function are calculated. Moment information is applied to the expansion of a three-parameter Weibull distribution, and the cumulative distribution function and probability density function of limit state function are determined. As a result, the failure probability interval of the slope is calculated. The interval uncertainty problem is transformed into an interval certainty problem using Taylor expansion without solving for the statistical moment of limit state function using multiple integrals and iteratively searching for the most probable failure points. The numerical results from two slopes show that the proposed method is effective and feasible.

A global sensitivity analysis approach for multiple failure modes based on convex-probability hybrid uncertainty

PurposeThis paper aims to develop a method for evaluating the failure probability and global sensitivity of multiple failure modes based on convex-probability hybrid uncertainty.Design/methodology/approachThe uncertainty information of the input variable is considered as convex-probability hybrid uncertainty. Moment-independent variable global sensitivity index based on the system failure probability is proposed to quantify the effect of the input variable on the system failure probability. Two-mode sensitivity indices are adopted to characterize the effect of each failure mode on the system failure probability. The method based on active learning Kriging (ALK) model with a truncated candidate regions (TCR) is adopted to evaluate the systems failure probability, as well as sensitivity index and this method is termed as ALK-TCR.FindingsThe results of five examples demonstrate the effectiveness of the sensitivity index and the efficiency of the ALK-TCR method in solving the problem of multiple failure modes based on the convex-probability hybrid uncertainty.Originality/valueConvex-probability hybrid uncertainty is considered on system reliability analysis. Moment-independent variable sensitivity index based on the system failure probability is proposed. Mode sensitivity indices are extended to hybrid uncertain reliability model. An effective global sensitivity analysis approach is developed for the multiple failure modes based on convex-probability hybrid uncertainty.

Sustainable allocation of water resources in water-scarcity conditions using robust fuzzy stochastic programming

Economic, social, and demographic development has globally led to an increased demand for water, consequently forcing some countries to face the shortage of water resources. In Iran, the problem of water scarcity is such a serious challenge that the sustainable allocation of water resources is sought after so as to help deal with this crisis. To this aim, the present paper proposes a new Robust Fuzzy Stochastic Programming model for the allocation of water resources. For the first time in the literature of stochastic water allocation problem, we applied the sustainability index to the optimization model and considered different levels of groundwater in an allocation system. Another feature of this model is its capability in considering a hybrid uncertainty in the parameters of the model, i.e. fuzzy stochastic scenarios, in a robust fuzzy stochastic programming model. Our proposed model considers three objective functions, namely maximizing the profits of different consumption sectors, minimizing the loss of different sectors caused by water shortage in different scenarios, and minimizing the allocation of salt water from ground water. The model has been applied in Mashhad city, northeastern Iran, which has long had water scarcity as an issue. We studied the allocation of water from surface, ground, and recycled wastewater resources to the three sectors of agricultural, municipal, and industrial; the results indicated that water allocation patterns and water shortages were affected by different scenarios. In fact, the municipal and agricultural sectors did not face water shortage in all scenarios, whereas the industrial sector encountered water scarcity in a drought year. Moreover, the assessment of optimality and feasibility robustness of the model indicated that the proposed model is robust.

A composite-projection-outline-based approximation method for system reliability analysis with hybrid uncertainties

This paper investigates the system reliability analysis under random and interval variables (SRA-RI). When performance functions in SRA-RI are replaced by surrogate models, it is found that the composite projection outlines on the composite limit-state surface should be well approximated so as to accurately assess system failure probability bounds. Then a composite-projection-outline-based active learning Kriging (CPOK) method is proposed in this paper. To refine the approximated composite projection outlines, three system learning functions are defined in CPOK for parallel, series and mixed systems, respectively. Based on these three functions, new points around the composite projection outlines are sequentially selected and used for the update of Kriging models. Meanwhile, prediction uncertainties of Kriging models are quantified and used for terminating the update of Kriging models. Finally, system failure probability bounds are evaluated by Monte Carlo simulation. The advantages of CPOK are validated by four numerical examples and a piezoelectric energy harvester example.

An extended stochastic pseudo‐spectral Galerkin finite element method (XS‐PS‐GFEM) for elliptic equations with hybrid uncertainties

SummaryNouy and Clement introduced the stochastic extended finite element method to solve linear elasticity problem defined on random domain. The material properties and boundary conditions were assumed to be deterministic. In this work, we extend this framework to account for multiple independent input uncertainties, namely, material, geometry, and external force uncertainties. The stochastic field is represented using the polynomial chaos expansion. The challenge in numerical integration over multidimensional probabilistic space is addressed using the pseudo‐spectral Galerkin method. Thereafter, a sensitivity analysis based on Sobol indices using the derived stochastic extended Finite Element Method solution is presented. The efficiency and accuracy of the proposed novel framework against conventional Monte Carlo methods is elucidated in detail for a few one and two dimensional problems.

A class of bi-level vector extremum models with hybrid uncertain coefficients and its application to supply chain planning problems

Decision-making optimization for supply chain planning under uncertain environment is of vital importance since it determines the reliability and efficiency of whole system. In this paper, a bi-level supply chain planning model is established in which the core company is the leader and the suppliers and retailers are the followers in the hierarchal process. Considering the hybrid uncertainty of randomness and fuzziness in the bi-level vector extremum models, a class of linear combination between fuzzy coefficients and decision variables is studied and its crisp equivalent transformation under the given possibilistic level is proposed. Furthermore, the fuzzy decision is applied to separate the two-planner Stakelberg-Nash equilibrium and the Genetic Algorithm (GA) is developed to solve the bi-level multi-objective model with hybrid variables. A practical case is provided to illustrate robustness and practicality of the proposed methodology, and algorithm comparison is then given to prove the efficiency of the GA. The computational results indicate that the proposed model and techniques can provide appropriate tools to tackle the other supply chain planning problems with hybrid variable in uncertain decision-making environment.

A dependency bounds analysis method for reliability assessment of complex system with hybrid uncertainty

In reliability assessment, a difficulty is to handle a complex system with hybrid uncertainty (aleatory and epistemic uncertainty) and dependency problem. Probability-box is a general model to represent hybrid uncertainty. Arithmetic rules on the structure are mostly used between independent random variables. However, in practice, dependency problems are also common in reliability assessment. In addition, in most real applications, there is some prior information on the dependency of components, but the available information may be not enough to determine dependent parameters. The issue is named non-deterministic dependency problem in the paper. Affine arithmetic is hence used to produce dependent interval estimates. The arithmetic sometimes has a better effect than probability-box arithmetic (interval arithmetic) in dealing with dependency problem. Bayesian network is a commonly used model in reliability assessment. Under Bayesian network framework, this paper proposes a dependency bounds analysis method that combines affine arithmetic and probability-box method to handle hybrid uncertainty and non-deterministic dependency. For the sake of illustration, this method is applied to two real systems. To show the advantages of the proposed method, the proposed method is compared with the Frechet inequalities and 2-stage Monte Carlo method in the second case study.

Reliability-based analysis method of fluid dynamics for turbomachinery with interval distribution parameters

Machining level and manufacturing accuracy have an important influence on the reliability of turbomachinery flow physics and dynamic performance. This article proposes a hybrid uncertainty analysis method for fluid dynamics problems with manufacturing constraints. The parameters with sufficient and insufficient experimental samples are processed by the stochastic method and interval method, respectively, which can quickly solve the difficulties of epistemic uncertainty modelling caused by the lack of experimental samples. On this basis, an efficient iterative algorithm is proposed to solve the hybrid reliability model based on the update response surface, and the interval of the failure probability of fluid dynamics for turbomachinery is calculated. Two examples of cooling axial flow and centrifugal turbomachinery are given to verify the validity of the proposed method. The results show that the reliability of the turbomachinery flow performance can be significantly improved by increasing the uncertainty level of key parameters.

Multi-Objective Robust Optimization of a Dual-Flux-Modulator Magnetic Geared Machine With Hybrid Uncertainties

In order to improve the robustness and torque performance of a dual-flux-modulator magnetic-geared machine (DFM-MGM) considering simultaneously both random and interval uncertainties of design parameters, a multi-objective robust optimization (MORO) method with multiple Monte Carlo simulations (MCSs) is proposed. In this method, the multiple MCSs are adopted to evaluate the effects of parametric hybrid uncertainties on the robustness of optimization results. To build a MORO model of the DFM-MGM, the three-dimensional finite element model is established firstly and then validated by the experiment. Through a parametric study, it is found that five dimensional parameters of the permanent magnets (PMs) and stator have more significant effects on the stall torque (ST) and ST per PM volume (STPPV). Finally, a multi-objective particle swarm optimization algorithm with surrogate models, sigma criteria design and multiple MCSs method is implemented to solve the MORO problem. Both the average standard deviations and standard deviation differences of the ST and STPPV are used to deal with hybrid uncertainties during MORO. The optimized DFM-MGM by MORO has a STPPV 6.3% higher than that of the initial design under the same ST constraint. Moreover, the average standard deviations and standard deviation differences obtained by MORO are much smaller than those achieved by the deterministic optimization, indicating that the robustness of optimal results can also be significantly improved by the MORO.

Toward a coordination of inventory and distribution schedules for blood in disasters

The need for efficient blood supply is of more significance in the event of disasters, when there is a lack of coordination between distribution and inventory management. The recent earthquake in Kermanshah province in Iran is among such cases that confirmed the need for coordinating such schedules. In this respect, a two-stage stochastic programming (SP) approach is presented for planning supply of blood after disasters that can assist in inventory decisions under hybrid uncertainty, minimizing the shortage and wastages. The uncertainty stems from imprecise parameters and scenario variability, and a robust-fuzzy-stochastic programming (RFSP) approach is devised to hedge against the uncertainty. The perishability of blood, the substitutability of blood groups, and the age-based characteristic of blood are taken into account to make the model more reliable. The compromise programming is applied to solve the multi-objective model. The results illustrate that the RFSP model can make a reasonable trade-off between mean value, feasibility robustness, and optimality robustness, which results in a robust and reliable solution under disastrous conditions.

Coupled fuzzy-interval model and method for structural response analysis with non-probabilistic hybrid uncertainties

In many engineering practices with uncertainty, the non-probabilistic methods play an increasing important role. To overcome the limitation of traditional single-uncertainty modeling methods in handling coupled uncertain problems, this paper develops a more general hybrid uncertainty analysis framework. The non-probabilistic hybrid uncertainties are expressed as coupled fuzzy-interval variables, where the bounds of interval are interpreted as fuzzy sets instead of deterministic values. By means of the cut-set strategy and decomposition theorem in fuzzy set theory, the hybrid uncertain problem is transformed into a series of dual-interval problems. The conservative and radical extreme-value predictions in different variable subspaces are adopted to characterize the coupled uncertainty in output response. To further improve the computational efficiency of extreme-value prediction, an adaptive response surface model using radial basis function is proposed, where the potential global optimum points are introduced as the new sample points in the sequential sampling process. Finally, the effectiveness of the proposed method on dealing with non-probabilistic hybrid uncertainties is validated by two examples.

Niedokładna analiza czułościowa niezawodności systemu w oparciu o sieć bayesowską i pole prawdopodobieństwa (p-box)

Sensitivity analysis measures how changes in system inputs affect outputs. Previously, a large amount of sensitivity analysis research was relevant to the precise probability that is regarded as an ideal condition of engineering. Due to insufficient test samples and the low accuracy of test data, system reliability with hybrid uncertainty is difficult to be described as a precise value. As a profusion of highly integrated electromechanical equipment is applied in modern life, it is impossible to apply sufficient resources to eliminate the stochastic property of every component, which necessitates the identification of highly sensitive components to efficiently reduce imprecision. Hence, based on the theory of imprecise probability, imprecise sensitivity analysis has become a popular research topic in the last decade. In this paper, a method for uncertain system reliability and imprecise sensitivity analysis is proposed based on a Bayesian network, a probability box and the pinching method. The feasibility and accuracy of the combined method are fully verified through the evaluation and analysis of a numerical example and a case study of an electromechanical system, and the highly sensitive components that heavily influence the imprecision of system outputs are accurately identified.

Hybrid uncertainty analysis of functionally graded plates via multiple-imprecise-random-field modelling of uncertain material properties

This paper presents the mixed probabilistic/non-probabilistic uncertainty problem regarding the hybrid uncertainty propagation on physical model of functionally graded plates through dissimilarly uncertain characterisation of material properties in the presence of the multiple-imprecise-random-field modelling. The spectral stochastic finite element framework is extended to perform mixed uncertainty analysis of functionally graded material whereby the spatial variation of elastic modulus of the plate is uniquely characterised as the multiple-imprecise-random-field modelling more than a purely homogeneous random field. This characterisation has originated in heeding the construction of physical models and the significant influences of separately uncertain characteristics propagated on the physical model through different material ingredients are examined to make physical models more reliable and realistic. Mechanics and mathematics intrusive-formulations are then established to explicitly produce the uncertain-but-bounded statistical characteristics of responses, namely interval mean value, interval standard deviation. Bounded surfaces of statistical moments and bounding distribution functions are vividly depicted for demonstration. The feasibility and effectiveness of the presented work are illustrated via test cases of numerical examples.

New reliability modeling methods for structural systems with hybrid uncertainty

AbstractThe present study investigates the hybrid reliability modeling of structures in which the inputs contain both random variables and interval variables. Hybrid uncertainty is divided into three categories, including random variables mixed with random variables, interval variables mixed with interval variable, and random variables mixed with interval variables. In order to perform the reliability analysis of structural systems, first, the Bayes method is proposed in the present study to obtain distribution parameters of random variables. Moreover, the self‐sample method is introduced to obtain the interval boundaries based on the least available measuring data. Then, the reliability models are established for three situations and the reliability indices are defined and derived accordingly. The abovementioned three types of reliability indices outline the general situation of structural systems. Finally, the specific calculation process is described in details through different examples. Furthermore, the accuracy and efficiency of the proposed method is discussed by comparing the results obtained from the Monte Carlo simulation and those of other methods. The obtained results indicate that the performance of the proposed model in solving reliability modeling problems is better.