Mathematical Problems in Engineering

Volume 2018, Article ID 6524629, 8 pages

https://doi.org/10.1155/2018/6524629

## Dynamic Fuzzy Reliability Analysis of Multistate Systems Based on Universal Generating Function

^{1}School of Mechanical Engineering, Liaoning Shihua University, Liaoning 113001, China^{2}School of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China^{3}School of Mechanical Engineering and Automation, Northeastern University, Liaoning 110819, China^{4}Department of Industrial Design, Liaoning Shihua University, Liaoning 113001, China

Correspondence should be addressed to Peng Gao; nc.ude.upnl@gnepoag

Received 24 May 2017; Revised 22 March 2018; Accepted 4 April 2018; Published 10 May 2018

Academic Editor: Gen Q. Xu

Copyright © 2018 Peng Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Considering the fuzziness of load, strength, operational states, and state probability, reliability models of multistate systems are developed based on universal generating function (UGF). The fuzzy UGF of load and the fuzzy UGF of strength are proposed in this paper, which are used to derive the fuzzy component UGF and the fuzzy system UGF. By defining the decomposition operator and the inner product operator, failure dependence and effects of multiple load applications are taken into account in the established reliability models. Moreover, dynamic fuzzy reliability models of multistate systems are constructed considering the strength degradation of components. The results show that failure dependence and the effects of multiple load applications have significant impacts on system reliability, which considerably decrease system reliability and increase the fuzziness of system reliability under low performance requirements. Besides, in the dynamic reliability analysis of multistate systems, strength degradation dependence could lead to large computational error.

#### 1. Introduction

Reliability is an important index in quality evaluation [1]. Traditional hypothesis of binary state about technological systems cannot always be satisfied in practice. Partial failure or degradation of systems could be encountered. In this case, it is insufficient to define the states of the systems by working normally or failing to work. Thus, reliability analysis of multistate systems has become more and more attractive. The universal generating function (UGF) is important and effective mathematical means to calculate the reliability of multistate systems. Compared with other methods, the UGF can clearly present the mapping relation between the working states of the systems or the components and the corresponding state probabilities. Besides, high computational efficiency is another important advantage of the UGF. Hence, the UGF has been widely used in reliability analysis of multistate systems such as electric power systems, electronic systems, and mechanical systems.

In the last few decades, a great deal of innovative research has been carried out on the reliability analysis of multistate systems based the UGF technology. For instance, a new expression was proposed for the UGF counting (nonsingular) walks with small steps in the quarter plane in terms of infinite series by Kurkova and Raschel [2]. Besides, three cases (an algebraic case, a transcendental D-finite case, and an infinite group model) are used to illustrate the expression of the generating function in the literature. Dherin proved that this UGF converges for analytical Poisson structures and shows that the induced local symplectic groupoid coincides with the phase space of Karasev–Maslov [3]. The method for probabilistic production simulation of wind power systems was developed based on universal generating function (UGF) by Jin et al. which completes the production simulation with the chronological wind power and load demand [4]. Farsi developed a new method for reliability evaluation via the UGF for a solar array mechanism [5]. Levitin provided a detailed discussion on the origin and the algorithm of UGF [6]. Moreover, a series of reliability models were given for various typical systems (such as series system, parallel system, and voting system), which laid the foundation for reliability analysis of multistate systems by using the method of UGF. In addition, the problem of reliability-based optimization of multistate system can be addressed by combining the UGF with various optimization algorithms such as the genetic algorithm, and ant colony algorithm. [7, 8]. Ding and Lisnianski proposed the fuzzy UGF technique to analyze the reliability of a series-parallel system [9].

For many technology systems, the system states are closely related to the working load [10, 11]. Meanwhile, the failure of these systems always occurs in the case where the strength exceeds the working load. Further investigation of load effects on multistate systems should be carried out. In addition, to explore the randomness of the variables in the UGF of multistate systems and components, a large amount of samples is required. In some situations, samples are insufficient, some uncertain information is based on experience and judgment, or experimental data is obtained under the unknown or nonconstant reproduction conditions [12–15]. In this case, fuzzy mathematics theory can be used as an alternative method to analyze the system reliability [16–18]. In practice, state degradation is always encountered, such as the degradation in the failure modes of fatigue, abrasion, or corrosion, which makes the system states different at different time instants. The system states and the corresponding possibility that these states show obvious dynamic characteristics. In order to describe the change of system states, state possibility and system reliability with time and dynamic fuzzy reliability models of multistate systems are developed in this paper. Moreover, in the models, the factors of both load and strength are taken into consideration in component reliability derivation, which facilitates the quantitative description of the influences of the failure dependence on system reliability.

#### 2. Load UGF and Strength UGF

As a matter of fact, it is important to obtain the fuzzy component states and corresponding fuzzy probabilities. The concept of the component states and the corresponding probabilities in this paper are similar to that defined in [6, 10]. However, the uncertainty in the component states and the corresponding probabilities at each load application is described by the membership degree. The detailed method to obtain the membership degree of a fuzzy variable can be referred to in [19].

The UGF establishes the mapping relationship between the states of the components or the systems and the occurrence probability of the corresponding states in the form of polynomial. Assume that the component in a system has states and the system has states. and represent the th state of the component and the corresponding probability, respectively. and are the state of the th state of the system and the corresponding probability, respectively. Then the UGF of the component and the UGF of the system can be expressed as

The operation between the UGF is carried out via different operators as follows:

As mentioned above, we will be mainly concentrated on the fuzzy reliability of multistate systems in this paper. The fuzzy UGF can be expressed as [9]where stand for the fuzzy state probability and fuzzy working state, respectively. For computational convenience, the triangular fuzzy number is always used to model the fuzzy variables in reliability models. The triangular fuzzy number can be expressed as with the membership function shown in Figure 1 and