A fuzzy universal generating function-based method for the reliability evaluation of series systems with performance sharing between adjacent units under parametric uncertainty
Introduction
The performance sharing mechanism is a kind of redundancy of multi-state systems (MSSs). In MSSs with performance sharing, each multi-state unit has a random performance and a random demand. If the performance of unit i exceeds its individual demand (), its surplus performance can be shared with any other unit j that experiences performance deficiency (). The system reliability in this case is defined to be the ability to meet the required performance level of the system [1].
The type of performance sharing was first studied by Lisnianski and Ding [2], where a main unit and a reserve unit were considered. The surplus performance can only be transmitted from the reserve unit to the main unit if the main unit suffers performance deficiency. A universal generating function (UGF)-based method was proposed to evaluate the reliability of such system. Levitin [1] proposed the formal notion of performance sharing mechanism and the “common bus” performance sharing model in which the surplus performance can be redistributed from any unit to any other unit through a common bus, and the only limitation on the performance transmission was imposed by the limited transmission capacity of the common bus. A UGF-based method was proposed to evaluate the reliability of MSSs with common bus performance sharing. Yu et al. [3] studied the repairable MSSs with common bus performance sharing, and proposed a method based on the combination of the stochastic process method and the UGF technique to evaluate the instantaneous availability of such systems. Xiao and Peng [4] proposed a UGF-based method to evaluate the availability of series-parallel systems with common bus performance sharing, and used the genetic algorithm to optimize the allocation and maintenance of the units in such systems. Xiao et al. [5], [6] studied the effect of the load on the performance and failure rates of the units, and proposed a UGF-based method to evaluate the load dependent series-parallel systems with common bus performance sharing. Peng et al. [7] extended the common bus model by allowing units to be in two performance sharing groups, and proposed a UGF-based method to evaluate the reliability of systems with two common bus performance sharing groups. Peng et al. [8] proposed a UGF-based method to evaluate the reliability of a series system with a common bus performance sharing group of limited size, and developed a dynamic connection strategy to optimize the system reliability. Zhai et al. [9] studied the defense and attack strategies for a system with a common bus performance sharing mechanism that is subject to intentional attacks. Yu et al. [10] developed a recursive algorithm to evaluate the reliability of phased-mission common bus systems with common cause failures. Zhao et al. [11] proposed a method based on the combination of the stochastic process method and the UGF technique to evaluate the reliability of k-out-of-n: G system with common bus performance sharing. The previous studies assume that the MSS with performance sharing works with a common bus, in which the total surplus performance is redistributed over the whole system. Wang et al. [12] thought this assumption was not appropriate for all practical systems, and proposed the performance sharing model without the assumption of the common bus. In their performance sharing model, the surplus performance of a unit was first transmitted to its adjacent units, no matter they experience deficiency or not, and then the remaining surplus performance is further transmitted to non-adjacent units. A UGF-based method was proposed to evaluate the reliability of such systems. Qiu and Ming [13] proposed a UGF-based method to evaluate the reliability of systems with common bus performance sharing considering transmission loss.
From the existing studies of MSSs with performance sharing, it is found that the surplus performance is assumed to be redistributed over the whole system directly (common bus performance sharing model) or indirectly (performance sharing model), and the performance/demand levels and corresponding state probabilities are assumed to be precise values. Because these two assumptions are not always true, there are two issues that have not been considered in the existing studies: i) in some real-world systems, the surplus performance of a unit can only be shared with its adjacent units which experience deficiency, and cannot be shared with non-adjacent units; ii) it is sometimes difficult to estimate the precise state probabilities due to the inaccuracy and insufficiency of data. This motivates us to propose a new method for the reliability evaluation of MSSs with performance sharing between adjacent units under parametric uncertainty related to the state probabilities.
The UGF technique is widely used in the modeling of MSSs with performance sharing. In the existing UGF-based methods, the state probabilities associated with the performance/demand levels are all precise values. However, due to the inaccuracy and insufficiency of data, it is sometimes difficult to estimate the precise state probabilities, and the conventional UGF technique is not applicable in the imprecise case. There are some extensions of the UGF technique for imprecise cases in the literature. Ding and Lisnianski [14] proposed the fuzzy UGF (FUGF) technique based on the conventional UGF technique and fuzzy set theory, and evaluated the reliability of fuzzy MSSs when the performance rates and corresponding probabilities were fuzzy values. Liu et al. [15] proposed a fuzzy Markov process and fuzzy Markov reward process to establish the dynamic state probabilities and fuzzy reward, and a parametric programming algorithm to obtain the probability and reward membership functions. A FUGF technique was proposed to evaluate the dynamic fuzzy reliability of MSSs. Ding et al. [16] used the fuzzy recursive method and the FUGF technique to evaluate the approximate reliability of multi-state weighted k-out-of-n systems. Li et al. [17] proposed the interval UGF based on the interval arithmetic and the conventional UGF technique to estimate the interval-valued reliability of MSSs. Destercke and Sallak [18] proposed the belief UGF technique based on the belief functions theory and the conventional UGF technique to evaluate the availability of MSSs under epistemic uncertainty. Li et al. [19] introduced random fuzzy variables to represent aleatory and epistemic uncertainties, and proposed the hybrid UGF technique to characterize random fuzzy variables and evaluate the system availability p-boxes under uncertainty. Mi et al. [20] proposed a belief UGF technique based on the belief functions theory and the conventional UGF technique to evaluate the reliability of MSSs with epistemic uncertainty and common cause failures. Qiu and Ming [21] proposed a belief UGF-based method to evaluate the reliability of systems with performance sharing under epistemic uncertainty.
Fuzzy set theory is a useful tool for the reliability and risk analysis of systems under uncertainty [14], [22], [23], [24]. Karimi and Hüllermeier [25] used fuzzy set theory to assess the risk of natural disaster when statistical data and/or physical knowledge are insufficient for probabilistic analysis. Zio et al. [26] developed a fuzzy expert system for human reliability analysis to elicitate factors influencing conditional human error for two dependence successive operator actions in a nuclear power plant accident. Purba et al. [27] developed a fuzzy reliability algorithm to generate basic event failure probabilities without reliance on quantitative historical failure data through qualitative data processing. In the FUGF technique [14], the state probabilities or performance/demand levels of the system units can be fuzzy values to represent the uncertainty. In this work, a FUGF-based method is proposed to evaluate the reliability of MSSs with performance sharing between adjacent units under parametric uncertainty related to the state probabilities, and some fuzzy composition operators are introduced for the calculation.
The paper is organized as follows. Section 2 presents briefly the FUGF technique. Section 3 depicts the architecture of the fuzzy multi-state series system with performance sharing between adjacent units under parametric uncertainty. Section 4 presents the proposed FUGF-based reliability evaluation method. Section 5 gives an illustrative example to validate the proposed method. Section 6 concludes this paper and gives the future perspective.
Section snippets
UGF technique
In a MSS consisting of n multi-state units, any unit i can have different states representing different performance levels, denoted by the set . is the performance level of unit i at state j. The performance of unit i is a random variable taking its values from . The probabilities associated with different performance levels of unit i are denoted by the set , where . The mapping is called the probability mass
System model
Fig. 1 is the architecture of a fuzzy multi-state series system with performance sharing between adjacent units under parametric uncertainty. Each unit must first meet its individual demand. If the individual demand is satisfied and some surplus performance exists, this performance can be shared with other units that experience performance deficiency. The performance transmitted through the bus is limited by the transmission capacity of the bus. In this work, an important assumption of the
FUGF-based reliability evaluation method
In this section, a FUGF-based method is proposed to evaluate the fuzzy reliability of series systems with performance sharing between adjacent units under parametric uncertainty related to the state probabilities of the performance/demand levels of units.
The system consists of n units connected in series. Any unit i has different performance levels with the corresponding fuzzy pmf and , and different demand levels with the corresponding
Illustrative example
In this section, the proposed reliability evaluation method is applied to an illustrative example considering two different cases:
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Case 1: no parametric uncertainty is taken into account. In this case, all state probabilities are precise values. Precise state probabilities can be regarded as a special case of fuzzy state probabilities, therefore, the proposed FUGF-based method can be applied in this case.
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Case 2: the parametric uncertainty related to state probabilities is taken into account. In
Conclusion
This paper studies a generalized multi-state series system with performance sharing between adjacent units under parametric uncertainty, and proposes a FUGF-based method to evaluate the fuzzy reliability of such systems. Rather than the common bus performance sharing model and the performance sharing model, the model of performance sharing between adjacent units is discussed in this work. In the model of performance sharing between adjacent units, the surplus performance of a unit can only be
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is supported by National Natural Science Foundation of China (Grant No. 51805326 and 71632008) and National Major Science and Technology Projects of China (Grant No. 2017-I-0007-0008). The authors are thankful to the editor and anonymous reviewers for their valuable comments that help us improve the quality of the paper.
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