“…For uncertain risk analysis, scholars have not only investigated the concepts and corresponding theorems of indicators such as value-at-risk (Peng, 2013 ; Liu & Ralescu, 2017 ) and the risk index (Liu & Ralescu, 2014 ), but have also studied uncertain risk assessment (Zhou et al, 2017 ; Zhang et al, 2018a ), loss function risk measurement of uncertain systems (Peng, 2013 ), and the uncertain risk measure and comparison rules (Li & Peng, 2012 ), among others. Various systems, and the alpha most reliable paths, products and equipment in systems are all research objects in uncertain reliability analysis, and authors have discussed the mathematical models (Liu et al, 2015b ; Zu et al, 2018 ; Hu et al, 2019 ), reliability indicators and their calculation formulas (Zeng et al, 2017 ; Liu et al, 2018 ; Gao et al, 2018 ), and numerical algorithms or quantitative means (Wang et al, 2017 ; Zeng et al, 2018 ; Zhang et al, 2018b , 2019b ) of these objects. Fig.…”
Section: Discussion Of Development Historymentioning
Uncertainty theory is an area in axiomatic mathematics recently proposed by Professor Baoding Liu and aiming to deal with belief degrees. Retrieving 1004 journal articles from the Web of Science database between 2008 and 2019, and utilizing CiteSpace and Pajek software, we analyze the publications per year and by geographical distribution, productive scholars and their cooperation, key journals, highly cited articles and main paths of the field. In this way, seven key sub-fields of uncertainty theory and their research potential are derived. The results show the following: (1) The literature on uncertainty theory follows a linear growth trend, involves an extensive network of 1000 scholars worldwide and is published in 300 journals, indicating thus that uncertainty theory has become increasingly attractive, and its academic influence is gradually expanding. (2) Seven key sub-fields of uncertainty theory have clearly been identified, including the axiomatic system, uncertain programming, uncertain sets, uncertain logic, uncertain differential equations, uncertain risk analysis, and uncertain processes. Among them, uncertain differential equations and programming are the two main sub-fields with the largest numbers of published papers. Furthermore, for evaluating the research potential of sub-fields, maturity and recent attention indicators are calculated using the citations, total number of publications, quantity of most cited literature and half-life. Based on these indicators, uncertain processes shows the greatest development potential, and has remained a hot topic in recent years, being mainly concentrated on the uncertain renewal reward process, optimal control of discrete-time uncertain systems, and uncertain linear quadratic optimal control. Additionally, uncertain risk analysis is ranked second, and focuses on the analysis of expected losses, investment risk, and structural reliability of uncertain systems.
“…For uncertain risk analysis, scholars have not only investigated the concepts and corresponding theorems of indicators such as value-at-risk (Peng, 2013 ; Liu & Ralescu, 2017 ) and the risk index (Liu & Ralescu, 2014 ), but have also studied uncertain risk assessment (Zhou et al, 2017 ; Zhang et al, 2018a ), loss function risk measurement of uncertain systems (Peng, 2013 ), and the uncertain risk measure and comparison rules (Li & Peng, 2012 ), among others. Various systems, and the alpha most reliable paths, products and equipment in systems are all research objects in uncertain reliability analysis, and authors have discussed the mathematical models (Liu et al, 2015b ; Zu et al, 2018 ; Hu et al, 2019 ), reliability indicators and their calculation formulas (Zeng et al, 2017 ; Liu et al, 2018 ; Gao et al, 2018 ), and numerical algorithms or quantitative means (Wang et al, 2017 ; Zeng et al, 2018 ; Zhang et al, 2018b , 2019b ) of these objects. Fig.…”
Section: Discussion Of Development Historymentioning
Uncertainty theory is an area in axiomatic mathematics recently proposed by Professor Baoding Liu and aiming to deal with belief degrees. Retrieving 1004 journal articles from the Web of Science database between 2008 and 2019, and utilizing CiteSpace and Pajek software, we analyze the publications per year and by geographical distribution, productive scholars and their cooperation, key journals, highly cited articles and main paths of the field. In this way, seven key sub-fields of uncertainty theory and their research potential are derived. The results show the following: (1) The literature on uncertainty theory follows a linear growth trend, involves an extensive network of 1000 scholars worldwide and is published in 300 journals, indicating thus that uncertainty theory has become increasingly attractive, and its academic influence is gradually expanding. (2) Seven key sub-fields of uncertainty theory have clearly been identified, including the axiomatic system, uncertain programming, uncertain sets, uncertain logic, uncertain differential equations, uncertain risk analysis, and uncertain processes. Among them, uncertain differential equations and programming are the two main sub-fields with the largest numbers of published papers. Furthermore, for evaluating the research potential of sub-fields, maturity and recent attention indicators are calculated using the citations, total number of publications, quantity of most cited literature and half-life. Based on these indicators, uncertain processes shows the greatest development potential, and has remained a hot topic in recent years, being mainly concentrated on the uncertain renewal reward process, optimal control of discrete-time uncertain systems, and uncertain linear quadratic optimal control. Additionally, uncertain risk analysis is ranked second, and focuses on the analysis of expected losses, investment risk, and structural reliability of uncertain systems.
“…The question is to obtain the specific analytical expressions for the steady-state probabilities distribution of the system states and for the SSP of the failure-free system operation, both in the general case and for some particular distribution types. We consider a stochastic process {v(t)}, t ≥ 0, defined on the set of states ε = {(0 * , 0) = 0, (1 * , 0) = 1, (0 * , 1) = 1, (1 * , 1) = 2}, where v(t) is the number of failed components of the system at time t. Using the Markov process [10] to describe the behavior of the system, we introduce a supplementary variable x(t) ∈ R 2 + -the time spent by moment t to repair the failed component and use the extended set of states E = ε × R 2 + . Thus, we get a two-dimensional process (v(t), x(t)) with an expanded phase space E = {(0 * , 0), (1 * , 0, x), (0 * , 1, x), (1 * , 1, x)}.…”
Section: Mathematical Model: Assumptions Notations Problem Settingmentioning
confidence: 99%
“…Repair after breakage was assumed imperfect. In [9,10], authors investigated a reliability model of repairable systems with stochastic lifetimes and uncertain repair times. They established, respectively, the mathematical models of the reliability of repairable series systems/parallel/series-parallel/parallel-series systems.…”
In the actual study, we carried out a reliability analysis of a repairable redundant data transmission system with the use of the elaborated mathematical and simulation model of a closed heterogeneous cold standby system. The system consists of one repair unit and two different data sources with an exponential cumulative distribution function (CDF) of their uptime and a general independent CDF of their repair time. We consider five special cases of the general independent CDF; including Gamma, Weibull-Gnedenko, Exponential, Lognormal and Pareto. We study the system-level reliability, defined as the steady-state probability (SSP) of failure-free system operation. The proposed analytical methodology made it possible to assess the reliability of the whole system in the event of failure of its components. Specific analytic expressions and asymptotic valuations are obtained for the steady-state probabilities of the system and the SSP of failure-free system operation. A simulation model of the system in cases where it is not workable to obtain expressions for the steady-state probabilities of the system in an explicit analytical form was considered, in particular for constructing the empirical system reliability function. The issue of sensitivity analysis of reliability characteristics of the considered system to the types of repair time distributions was also studied. The simulation modeling was done with the R statistics package.
“…Series-parallel systems [23], [24], parallel-series systems [25], [26], and k-out-of-n systems [27], [28] have been widely studied by scholars and practitioners, but the hybrid parallel systems do not get much attention from the researchers. As described in Figure 1, (a) and (b) in Figure 1 represent the basic structure of series-parallel systems and parallel-series systems, respectively.…”
Section: B System Description and Reliability Computationmentioning
For the selective maintenance planning (SMP), the decision makers are required to select components to be maintained as well as how to repair within a finite break between missions considering limited time, cost, and other resources. However, a few studies analyze the impact of team maintenance capability on SMP. In this paper, novel SMP models considering team maintenance capability and imperfect maintenance are introduced, which is ignored in current studies. In addition, a two-phase method integrating fuzzy Choquet integral based on λ-fuzzy measure and dynamic multi-objective artificial bee colony (DMABC) is proposed to optimize the SMP models. Fuzzy Choquet integral is used to deal with interaction among criterions and the DMABC is applied to obtain non-dominated solutions with different maintenance capability. Finally, the performance comparison experiment is made between the DMABC and NSGA-II, and the validity of the proposed algorithm has been demonstrated by the results.
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