Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 156059, 11 pages

http://dx.doi.org/10.1155/2015/156059

## Fuzzy Availability Assessment for a Repairable Multistate Series-Parallel System

^{1}College of Science, Yanshan University, Qinhuangdao 066004, China^{2}School of Economics and Management, Yanshan University, Qinhuangdao 066004, China

Received 27 January 2015; Revised 12 April 2015; Accepted 15 April 2015

Academic Editor: Victor S. Kozyakin

Copyright © 2015 Linmin Hu 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

This paper considers a repairable multistate series-parallel system (RMSSPS) with fuzzy parameters. It is assumed that the system components are independent, and their state transition rates and performance rates are fuzzy values. The fuzzy universal generating function technique is adopted to determine fuzzy state probability and fuzzy performance rate of the system. On the basis of *α*-cut approach and the extension principle, parametric programming technique is employed to obtain the *α*-cuts of some indices for the system. The system fuzzy availability is defined as the ability of the system to satisfy fuzzy consumer demand. A special assessment approach is developed for evaluating the fuzzy steady-state availability of the system with the fuzzy demand. A flow transmission system with three components is presented to demonstrate the validity of the proposed method.

#### 1. Introduction

The study of repairable systems is an important topic in engineering systems. System availability is very good evaluation for performance of repairable systems and occupies an increasingly important issue in power plants, manufacturing systems, industrial systems, and transportation systems, and so forth.

In the real-world problems, many repairable systems are designed to perform their intended tasks in a given environment. One type of these repairable systems is repairable multistate system (RMSS). The RMSS is able to perform its task with various distinguished levels of efficiency usually referred to as performance rates. Since the number of RMSS states increases very rapidly with the increase in the number of its components, the universal generating function (UGF) technique was introduced and proved to be efficient in evaluating the reliability of the multistate systems [1–4].

The repairable multistate series-parallel system (RMSSPS) model is frequently used in practice and has been extensively studied for many years. The conventional study for the RMSSPS considers the assumptions of the exact state transition rate and performance rate for each system component. However, in many engineering applications it is very difficult to obtain accurate and sufficient data to estimate the precise values of the state transition rate and performance rate for each system component. For this reason, the concept of fuzzy reliability has been introduced and developed by several authors [5–9].

Fuzzy set theory proposed by Zadeh [10] is a very good approach to deal with fuzzy uncertainty and has gained successful applications in various fields. It provides useful tools to investigate and analyze imprecision phenomena in queuing systems [11], rock engineering classification systems [12], transport systems [13], manufacturing systems [14], supply chain problems [15], and various optimization problems [16–18]. Wong and Lai [19] provided a survey of applications of the fuzzy set theory technique in production and operations management and pointed out that nearly every application is potentially able to realize some of the benefits of fuzzy set theory. Furthermore, fuzzy set theory also provides useful methodology to analyze the reliability in uncertain systems. It can deal with the problem of lacking of inaccuracy or fluctuation data for system components in reliability analysis of some realistic engineering systems. Thus, it is necessary to introduce fuzzy set theory into the reliability theory to deal with reliability of the system with uncertain parameters. The theory of fuzzy reliability has been developed on the basis of fuzzy set theory. Many research works on the application of fuzzy set theory to problems in reliability or availability of systems were presented in [20–23], and a systemic review on fuzzy reliability of systems with binary-state was provided by Cai [24].

Recently, fuzzy reliability research has focused on reliability evaluation of fuzzy multistate systems. Ding et al. [25, 26] proposed firstly the concept of fuzzy multistate system and assessed the fuzzy reliability of fuzzy multistate systems with fuzzy demand using fuzzy UGF method. Liu et al. [27, 28] investigated the dynamic fuzzy state probabilities, fuzzy performance rates, and fuzzy availability for fuzzy unrepairable multistate elements and fuzzy unrepairable multistate system according to the parametric programming technique and the extension principle. Bamrungsetthapong and Pongpullponsak [29] discussed fuzzy confidence interval for the fuzzy reliability of a RMSSPS with a fuzzy failure rate and a fuzzy repair rate and considered the performance of fuzzy confidence interval based on the coverage probability and the expected length.

Steady-state availability of a repairable system, as a system performance measure, is the probability that the system is performing satisfactorily over a reasonable period of time [30]. The RMSS availability is defined as the ability of the system to satisfy consumer demand, which is equal to the sum of the probabilities of occurrence of system states in which the system performance rates satisfy consumer demand. Comparable work on the steady-state availability assessment for the RMSSPS with fuzzy state transition rate and fuzzy performance rate is rarely found in the literature. This motivates us to develop the fuzzy steady-state availability assessment for a RMSSPS with fuzzy parameters. According to [4], in this paper, we assumed that the fuzzy performance rate of each multistate parallel subsystem is the sum of the fuzzy performance rates of its all components, and the fuzzy performance rate of the entire RMSSPS is the minimum of the fuzzy performance rates of all parallel subsystems. The purpose of this study is to utilize the *α*-cut approach, the extension principle, and parametric programming technique to determine the fuzzy state probability and fuzzy performance rate of the RMSSPS and to evaluate the fuzzy steady-state availability of the system with the fuzzy consumer demand.

The rest of the paper is organized as follows. Section 2 introduces the description of the system considered here and analyzes the fuzzy state probability of multistate component. The fuzzy state probabilities and fuzzy performance rates of the repairable parallel subsystem and the RMSSPS are presented in Sections 3 and 4, respectively. The fuzzy availability assessment method for the RMSSPS is given in Section 5. An illustrative example is presented in Section 6. Finally, Section 7 gives conclusions.

#### 2. RMSSPS with Fuzzy Parameters

The RMSSPS we considered here is composed of subsystems connected in series, and each subsystem consists of components in parallel, . The structure of a RMSSPS is shown in Figure 1.