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By describing the absorption, distribution, metabolism, and excretion of drugs, pharmacokinetics helps to find optimal therapies for patients and speed up drug development. In pharmacokinetics, dynamic uncertainties, such as fluctuations in hormone levels and external environment factors, are ubiquitous. Due to sparse clinical data and patient disease specificity, these uncertainties are mainly epistemic uncertainties rather than aleatory uncertainties, which cannot be handled well by probability theory based methods. Therefore, several pharmacokinetics based on uncertain differential equations under the framework of uncertainty theory were investigated, which all considered one compartment models. Noting that many drugs follow two compartment kinetics, this article proposes a two compartment pharmacokinetic model based on uncertain differential equations. Based on the proposed model, several essential pharmacokinetic parameters are investigated, which are important information required by regulatory bodies to approve drugs for public use. Estimation for unknown parameters in the model are given. Finally, a real data analysis using lidocaine drug concentration illustrates our methodology in details. [ABSTRACT FROM AUTHOR]

A general sampling problem can be described by an Ellsberg urn, which is a mathematical model that assumes that balls are randomly drawn from an urn with an uncertain numbers of colored balls. This means that the Ellsberg urn is essentially an intricate model with simultaneous randomness and epistemic uncertainty, and this is the core problem discussed in this paper. Since practical sampling is usually processed in an intricate environment, the solution for an equivalent mathematical problem is necessary. Suppose an Ellsberg urn contains three unknown numbers of colored balls (i.e., a two-degrees-of-freedom Ellsberg urn), and three balls are randomly drawn from the urn. Compared to the published papers, this paper first constructs a chance space with two-dimensional uncertainty space and three-dimensional probability space to rigorously calculate the color distributions for those drawn balls by uncertainty theory, probability theory, and chance theory. Moreover, it is interesting to find that all cases of the drawn balls are symmetric in such a specific situation of a sample problem with epistemic uncertainty. [ABSTRACT FROM AUTHOR]

The objective of this study is to present a novel method of level-2 uncertainty analysis in risk assessment by means of uncertainty theory. In the proposed method, aleatory uncertainty is characterized by probability distributions, whose parameters are affected by epistemic uncertainty. These parameters are described as uncertain variables. For monotone risk models, such as fault trees or event trees, the uncertainty is propagated analytically based on the operational rules of uncertain variables. For non-monotone risk models, we propose a simulation-based method for uncertainty propagation. Three indexes, i.e., average risk, value-at-risk and bounded value-at-risk, are defined for risk-informed decision making in the level-2 uncertainty setting. Two numerical studies and an application on a real example from literature are worked out to illustrate the developed method. A comparison is made to some commonly used uncertainty analysis methods, e.g., the ones based on probability theory and evidence theory. [ABSTRACT FROM AUTHOR]

Resilience, the ability of a system to withstand disruptions and quickly return to a normal state, is essential for a low-Earth-orbit satellite communication system (LEO-SCS), as its large number of satellites, may encounter various disturbances. As a developing system with asymmetrical and insufficient data, it is with both aleatory and epistemic uncertainties, which renders existing resilience measures inapplicable. This paper utilizes uncertainty theory to define belief instantaneous availability, based on which a new resilience measure for the LEO-SCS is proposed. As the resilience is mainly determined by the dynamic and uncertain characteristics of an LEO-SCS, an uncertain satellite network evolution model is developed to describe its operating patterns, and an evaluation method is proposed to estimate its resilience. To illustrate our proposed uncertain theory-based resilience evaluation method, an LEO-SCS case with 48 satellites is studied, and its resilience values under different backup strategies are compared to assist in system design decisions. [ABSTRACT FROM AUTHOR]

• Uncertain process is adopted for degradation modeling accounting for epistemic uncertainty. • A novel similarity based-uncertain weighted least squares estimation method is proposed. • A denoising method is proposed to deal with the noises caused by recovery phenomenon. • The proposed framework is applied to real lithium-ion battery degradation dataset for demonstration. Remaining useful life prediction based on degradation modeling is of great importance to condition-based maintenance, for which epistemic uncertainty due to the lack of sufficient knowledge needs to be characterized. For certain components, such as the batteries, the recovery phenomenon during degradation has to be considered, and the epistemic uncertainty associated with it is inevitable. This paper proposes a systematic method for degradation modeling and remaining useful life prediction based on uncertain process for degradation with recovery phenomenon. First, uncertain process is adopted for degradation modeling accounting for epistemic uncertainty. Then, a novel similarity based-uncertain weighted least squares estimation method is proposed to update the model parameters with real-time monitoring data. Afterwards, a denoising method is used to deal with the noises caused by recovery phenomenon. Finally, remaining useful life is calculated by uncertain simulation. A case study on real lithium-ion battery degradation dataset is performed to illustrate the effectiveness of the proposed method in comparison with traditional stochastic process. [ABSTRACT FROM AUTHOR]

Epistemic uncertainties, caused by data asymmetry and deficiencies, exist in resilience evaluation. Especially in the system design process, it is difficult to obtain enough data for system resilience evaluation and improvement. Mathematics methods, such as evidence theory and Bayesian theory, have been used in the resilience evaluation for systems with epistemic uncertainty. However, these methods are based on subjective information and may lead to an interval expansion problem in the calculation. Therefore, the problem of how to quantify epistemic uncertainty in the resilience evaluation is not well solved. In this paper, we propose a new resilience measure based on uncertainty theory, a new branch of mathematics that is viewed as appropriate for modeling epistemic uncertainty. In our method, resilience is defined as an uncertainty measure that is the belief degree of a system's behavior after disruptions that can achieve the predetermined goal. Then, a resilience evaluation method is provided based on the operation law in uncertainty theory. To design a resilient system, an uncertain programming model is given, and a genetic algorithm is applied to find an optimal design to develop a resilient system with the minimal cost. Finally, road networks are used as a case study. The results show that our method can effectively reduce cost and ensure network resilience. [ABSTRACT FROM AUTHOR]

• This paper proposes a new uncertain accelerated degradation model. • The model considers the epistemic uncertainties in time and unit dimensions. • This paper presents the uncertain statistical method for parameter estimations. • The model is suitable for limited data sizes in time and unit dimensions. Accelerated degradation testing (ADT) has been widely used for the reliability and lifetime evaluations of highly reliable products. Generally speaking, the obtained ADT data are composed of two parts - the deterministic degradation trend and uncertainties. When only limited ADT data can be obtained, there will be a lack of knowledge on recognizing the population and lead to epistemic uncertainties. However, current accelerated degradation models fail to clearly distinguish and properly quantify the epistemic uncertainties in time and unit dimensions. Motivated by this problem, this paper constructs a new uncertain accelerated degradation model based on uncertainty theory and belief reliability theory, and presents the uncertain statistical method for parameter estimations. An application case is used to illustrate the proposed methodology and conduct discussions for the deterministic degradation trend and uncertainty analyses of the proposed methodology and the analyses of the proposed methodology to the data sizes in time and unit dimensions. Results show that under limited data sizes in time and unit dimensions, the proposed model can clearly distinguish and properly quantify the epistemic uncertainties in time and unit dimensions and is more suitable for the ADT modeling and analysis than the other two widely used accelerated degradation models. [ABSTRACT FROM AUTHOR]

For newly developing systems, the determination of the reliability index is an important task in the system demonstration process that determines both the system design goal and the verification requirement. However, due to the insufficient data at the beginning of the system life cycle, the problem of epistemic uncertainty cannot be ignored. Researchers have proposed several belief reliability metrics and analysis methods to quantify the epistemic uncertainty of a system, but to the best knowledge of the authors, no relevant studies have been performed on system reliability index determination. In this paper, we propose a belief reliability index determination method based on group decision‐making. Using a similar system‐based reliability index determination method, this method comprehensively considers the demands of users and the capabilities of contractors. In addition, the Delphi method and Hadamard multiplication convex combination are combined to effectively solve the epistemic uncertainty quantification problem. The Delphi method is used to find the conflicts caused by multi‐expert scoring, and the Hadamard multiplication convex combination can integrate the scoring matrix of all experts, which makes the results more reasonable. Finally, a Low‐Earth‐orbit (LEO) satellite communication system is used to verify the effectiveness of our belief reliability index determination method. [ABSTRACT FROM AUTHOR]

• Uncertainty theory is used to express the epistemic uncertainty. • Uncertainty expressions considering analyst's knowledge stages are developed. • Propagations methods corresponding to knowledge stages are developed. • A knowledge-increasing process is simulated based on the proposed methods. Uncertainty analysis plays a significant role in risk assessment, which consists of two tasks: uncertainty expressions of input variables in the model and their propagations through the model built. We aim to provide, in fault tree analysis context, suitable methods of expression and propagation of uncertainty corresponding to different stages of knowledge that the risk analyst own, where frequentist probability is used to express the aleatory uncertainty and uncertainty theory is used to represent the epistemic uncertainty. To do so, we divide the analyst's knowledge state into five different stages, and develop the correct expression of uncertainty corresponding to each stage, where different combinations of probability and uncertainty are considered. Methods of propagation of these uncertainties through fault trees are further developed, where we introduce probability distributions, uncertainty distributions, newly-developed level-2 distributions, and the varying time t into the operational law for Boolean uncertain random system to better address the needs of practical risk assessments. A case study is conducted to show the differences in the propagation methods corresponding to various knowledge stages, and the results highlight that the proposed methods are effective and could deliver clear messages to decision makers. [ABSTRACT FROM AUTHOR]

This paper copes with a joint Location-Allocation-Inventory problem in a three-echelon base-level spare part support system with epistemic uncertainty in uncertain demands of bases. The aim of the paper is to propose an optimization model under the uncertainty theory to minimize the total cost, which integrates crucial characterizations of the inventory control decisions and the location-allocation scheme arrangement under a periodic review order-up-to-S (T, S) policy. Uncertainty theory is introduced in this paper to characterize epistemic uncertainty, where demands are treated as uncertain variables and stockout loss is represented by value-at-risk in uncertain measurement. To solve the original uncertain optimization model, an equivalent deterministic model is derived and addressed by an improved bilevel genetic algorithm. Moreover, the proposed models and algorithm are encoded into numerical examples for supply chain programming. The results highlight the applicability of the model and the algorithm's effectiveness in approaching the optimal solution compared with traditional genetic algorithm. Sensitivity analyses are further made for the impacts of review time and inventory capacity on different cost components. [ABSTRACT FROM AUTHOR]

Due to economic and technical reasons, in accelerated degradation testing (ADT) there are usually limited ADT data, which embody lots of epistemic uncertainties that cannot be depicted well by probability theory. Noting these facts, several uncertain processes based accelerated degradation models (UADMs) are proposed under the framework of uncertainty theory. However, these UADMs described uncertainties from a macro perspective which ignored the characteristic of performance degradation in small time scales and failed to describe the essence of nonlinear degradation. What is more, to estimate unknown parameters, these models used empirical distribution functions, which may have a great impact on the estimation results with limited samples. Motivated by these problems, based on uncertainty theory this paper builds up an uncertain differential equation based accelerated degradation model (UDEADM), which considers the dynamic cumulative change process of performance in small space–time scale, and explains the essence of nonlinear for uncertainties in degradation trend. Unknown parameters are estimated using uncertain generalized moment estimations. Furthermore, reliability and lifetime are analyzed under the normal operating condition based on belief reliability theory. A simulation study and a real data analysis are conducted to illustrate the effectiveness and advantages of the proposed methodology in details. • An uncertain differential equation based accelerated degradation model is proposed. • Dynamic cumulative change of performance in small space–time scale is considered. • Unknown parameters are estimated by uncertain generalized moment estimations. [ABSTRACT FROM AUTHOR]

Software reliability plays an important role in modern society. To evaluate software reliability, software reliability growth models (SRGMs) investigate the number of software faults in the testing phase. Obviously, testing progresses are inevitably influenced by dynamic indeterministic fluctuations such as the testing effort expenditure, testing efficiency and skill, testing method, and strategy. To model these dynamic fluctuations, several probability theory-based SRGMs are proposed. However, probability theory is suitable for dealing with aleatory uncertainty, but fails to deal with epistemic uncertainty widely existing in software faults. Therefore, this article considers software reliability from a new perspective under the framework of uncertainty theory, which is a new mathematical system different from probability theory, and proposes a software belief reliability growth model (SBRGM) based on uncertain differential equations for the first time. Based on this SBRGM, properties of essential software reliability metrics are investigated under belief reliability theory, which is a brand-new reliability theory. Parameter estimations for unknown parameters in SBRGM are presented. Furthermore, some numerical examples and real data analyses illustrate our methodology in detail, and show that it performs better than several famous probability-based SRGMs in terms of fitting ability and prediction ability. Finally, an optimal software release policy is discussed. [ABSTRACT FROM AUTHOR]

• Dataset discrepancies and epistemic uncertainties are considered in the integration. • The evaluation index system for an ADT dataset is constructed. • Relative qualities of multi-source ADT datasets are quantified. • A statistical method is given for dataset discrepancies and epistemic uncertainties. • Uncertainties in the degradation evaluation are notably decreased. Degradation data collected from a single accelerated degradation testing (ADT) are usually non-sufficient, which causes a lack of knowledge and leads to epistemic uncertainties. So multi-source ADT datasets are usually integrated for degradation analysis and reliability evaluations. Nevertheless, the discrepancies among multi-source ADT datasets are ignored in current integration methods, which cannot make full use of all information in the integration process. To address these problems, an integration method of multi-source ADT datasets is developed for degradation analysis and reliability evaluations. Firstly, the evaluation index system for an ADT dataset is constructed. Then, the relative qualities of ADT datasets are computed through an additive model and a weight formula. After that, the degradation process is described by an uncertain accelerated degradation model to quantify epistemic uncertainties, and the related statistical analysis is given through a proposed uncertain least weighted squared error to consider dataset discrepancies. A simulation study and an application case are utilized to illustrate the proposed method. Results show that the proposed method can well capture the discrepancies among multi-source ADT datasets and make a quantitative comparison; additionally, it can notably decrease uncertainties in the degradation analysis. [ABSTRACT FROM AUTHOR]

• URADM is proposed to integrate aleatory and epistemic uncertainties from multiple dimensions in ADTs. • Aleatory and epistemic uncertainties are divided and quantified separately in the URADM. • Integration of aleatory and epistemic uncertainties is measured through chance theory. • A dynamic uncertain random simulation algorithm is proposed to approximate reliability. • More accurate and stable degradation predictions are achieved under multiple situations. Accelerated degradation testing (ADT) is commonly employed for degradation analysis and reliability evaluations. Due to limitations in practical ADTs, aleatory and epistemic uncertainties are simultaneously embodied in time, unit, and stress dimensions but they are not considered comprehensively and integrated scientifically currently. To address such problems, we focus on the most common situation in ADTs that test time is relatively long (reflecting aleatory uncertainties in time dimension) while the numbers of test units and stress levels are quite small (reflecting epistemic uncertainties in unit and stress dimensions), and establish an uncertain random accelerated degradation model (URADM). In the URADM, aleatory and epistemic uncertainties are quantified using probability theory and uncertainty theory, respectively, and they are integrated using chance theory. Then, first passage time and reliability evaluations are derived. Next, a three-step uncertain random statistical method is presented for parameter estimations, where aleatory and epistemic uncertainties are divided carefully and quantified separately. A simulation study and a practical case are conducted to show the effectiveness of the URADM. Results reveal that the URADM can not only predict deterministic degradation trends with high accuracy as existing methods do, but contribute degradation boundaries well covering ADT data with narrower boundaries and higher stability. [ABSTRACT FROM AUTHOR]

• The epistemic uncertainty of trigger effect in reliability evaluation is analyzed. • Uncertain theory is applied to quantify the PoF model-form uncertainty. • The uncertain degradation model of failure mechanism is built. • Belief reliability of the system with failure trigger effect is deduced. • The advantages of the new method are discovered in the engineering case. Reliability evaluation of system with dependent failure processes is a problem that draws more research attention. It is essential to quantify the epistemic uncertainty in the system for high evaluation accuracy. Trigger is one of the typical dependent failure processes. Trigger effect indicates that some events occur because of the trigger of other events. This paper studies the epistemic uncertainties in the reliability evaluation of the system with trigger effect based on uncertainty theory. The model-form uncertainty of physics of failure (PoF) model, the failure process uncertainty due to lack of knowledge and the propagation of them in the system are quantified with Belief Adjustment Factors Method (BAFM) and the Liu process. System belief reliability is evaluated by decoupling the correlations between the trigger failure mechanism and triggered failure mechanisms. A conductive slip ring in electromechanical orientation device is introduced as a case. The trigger effect and system belief reliability are analyzed. Results show that, compared with probability method, evaluation values with uncertainty method are more conservative and stable in the case.. [ABSTRACT FROM AUTHOR]

• The uncertainty theory-based reliability evaluation is proposed to analyze the limited and censored time-to-failure data. • Two types of life testing censoring and two types of time-to-failure data are considered. • The proposed method can quantify epistemic uncertainties and provide stable and accurate mean time to failure results. Limited and censored time-to-failure data are common in practical life testing, which may lead to a lack of knowledge, i.e., epistemic uncertainties, and makes it difficult to evaluate product reliability accurately. Under this situation, the large-sample based probability theory is not appropriate anymore. To deal with such problem, in this paper, the uncertainty theory based reliability evaluation is proposed to cope with the limited and censored time-to-failure data, where two types of life testing censoring (type-I and type-II censoring) and two types of time-to-failure data (precise and interval data) are considered. Firstly, the lifetime model and reliability evaluations based on uncertainty theory are presented. Then, the uncertain statistics method is presented for each combination of censoring types and data types in life testing, which includes four steps: order statistics construction, belief degree construction according to the Laplace principle of indifference, parameter estimation based on the uncertain principle of least squares, and uncertain hypothesis testing. Two simulation studies and two practical cases with comprehensive and in-depth discussion are conducted to illustrate the practicability and effectiveness of the proposed method. The results show that the proposed method can quantify epistemic uncertainties properly and provide more stable and accurate mean time to failure results under limited and censored time-to-failure data compared with probability and Bayesian probability methods. [ABSTRACT FROM AUTHOR]

• The hybrid uncertainty includes the aleatory uncertainty and epistemic uncertainty is analyzed in DCFP. • The uncertainty random renewal reward process and arithmetic Liu process are used to describe the shock and degradation process. • The analytic formulas of system reliability models considering three types of DCFPs are derived. • A combinational algorithm is proposed to calculate system reliability. • The proposed method is proved to be an appropriate method to qualify the hybrid uncertainty in the case. Dependent competing failure process (DCFP) has been widely studied in recent years. When evaluating the reliability of DCFP system, the aleatory uncertainty due to randomness and the epistemic uncertainty due to lack of data and knowledge should be considered. In this paper, the hybrid uncertainty of DCFP system is analyzed and quantified with chance theory. The uncertainty random renewal reward process and the arithmetic Liu process are used to describe the shock and degradation process. The quantification of uncertainty propagation is realized by decoupling DCFP from the physical correlation aspect. Reliability of DCFP system considering two kinds of unilateral dependencies and mutual dependency is modeled and a combinational algorithm is proposed to solve it. Finally, the DCFP effect in control system of an aircraft is studied as a case, aleatory uncertainty and epistemic uncertainty are modeled, and the system reliability is evaluated. Meanwhile, the effects of key parameters on system reliability are discussed. Results show that the chance theory is an appropriate method to qualify the hybrid uncertainty for engineering systems when data is insufficient. The mutual dependency of soft failure and hard failure will reduce the system reliability. The smaller the parameter of hard failure, the greater the influence of mutual dependency on the results. [ABSTRACT FROM AUTHOR]

• A general ADT modeling framework considering uncertainties are presented. • A 3-Dimensional ADT model and a practical application procedure are developed. • A bidirectional Liu process is utilized to describe the epistemic uncertainty. • A new statistical method is presented for parameter estimations. • Cases show the accuracy and practicability of the proposed model and framework. Accelerated degradation testing (ADT) has been widely used to identify the performance degradation law so as to evaluate the reliability and lifetime. During this process, there are aleatory and epistemic uncertainties in the unit, time and stress dimension. However, the current studies do not have a comprehensive modeling framework and a general ADT model that fully considers and properly quantifies all sources and types of uncertainties. Aiming at these problems, this paper proposes a 3-Dimesional modeling framework and a general ADT model, and gives a new uncertain ADT model considering epistemic uncertainties in the unit, and stress dimension. Furthermore, considering the weights of observed degradation data under different stress levels, this paper proposes a new parameter estimation method, uncertain weighted least squares, to well predict degradation law and control uncertainties. Finally, a microwave case, a rubber seal case and two simulation cases are conducted to show the practicability of the proposed framework and model. The results show that the proposed framework can guide the construction of different models according to different situations; and compared with current methods, the proposed model are more appropriate for performing ADT model and quantifying uncertainties when uncertainty in the stress dimension cannot be ignored. [ABSTRACT FROM AUTHOR]