Reliability analysis of repairable systems using Petri nets and vague Lambda-Tau methodology

Abstract

The main objective of the paper is to developed a methodology, named as vague Lambda-Tau, for reliability analysis of repairable systems. Petri net tool is applied to represent the asynchronous and concurrent processing of the system instead of fault tree analysis. To enhance the relevance of the reliability study, vague set theory is used for representing the failure rate and repair times instead of classical(crisp) or fuzzy set theory because vague sets are characterized by a truth membership function and false membership functions (non-membership functions) so that sum of both values is less than 1. The proposed methodology involves qualitative modeling using PN and quantitative analysis using Lambda-Tau method of solution with the basic events represented by intuitionistic fuzzy numbers of triangular membership functions. Sensitivity analysis has also been performed and the effects on system MTBF are addressed. The methodology improves the shortcomings of the existing probabilistic approaches and gives a better understanding of the system behavior through its graphical representation. The washing unit of a paper mill situated in a northern part of India, producing approximately 200 ton of paper per day, has been considered to demonstrate the proposed approach. The results may be helpful for the plant personnel for analyzing the systems' behavior and to improve their performance by adopting suitable maintenance strategies.

Introduction

The majority of industrial systems are repairable and consist of several subsystems. Each subsystem is composed of numerous complex components and the probability that the system survive depends directly on each of its constituent components. With the advances in technology and growing complexity of systems and insistence on product quality, the importance of reliability and maintainability in industries has become prime. However, since plant managers and engineers are faced with important preventive maintenance decisions and these affect the plant performance and reliability, therefore reliability study and maintenance policy play a crucial role in a process operation. The understanding of this role requires an attempt to study, characterize, measure and analyze the systems' behavior by eliminating or reducing the likelihood of failures and thus increasing their designed life and operational availability.

In order to measure the performance/behavior of the system, several techniques are available in the literature. Some widely used are event tree, fault tree analysis (FTA), reliability block diagrams(RBDs), Petri nets (PNs), and Markovian approach etc. [1], [2], [3], [4], [5], [6]. Both PNs and FTA recognized as a powerful tool for estimating the reliability of large scaled systems, where system success or failure is described by the state of the top event. The probability of a top event is a function of the failure probability of a primary event, whose data are collected either from the available historical data or raw data provided by the experts. Further, the data obtained from the past history of any industrial system are imprecise, incomplete, vague and conflicting. If the data are used as such in the calculations, the results will be highly uncertain. Thus, the probabilistic approach to the conventional reliability analysis is inadequate to account for such built-in uncertainties in the data. On the other hand, fuzzy methodology can deal with imprecise, uncertain dependent information related to system performance and provides a better, consistent and mathematically more sound method for handling uncertainties in data than conventional methods, such as Markov process, Bayesian statistics etc. The concept of fuzzy set theory and fuzzy arithmetic has been used in the evaluation of the reliability of the system by the various researchers [1], [7], [8], [9], [10].

Furthermore, if analysis has been done by using some suitable techniques listed above then any reliability index alone, as discussed by the above researchers, is inadequate to give deeper idea about such type of systems' behavior because a lot of factors exist which overall influence the systems' performance and consequently their behavior. To remove this, [1] proposed a new methodology, named as Fuzzy Lambda-Tau Methodology (FLTM), by making use of Petri nets instead of fault trees. Fuzzy set theory was used to represent failure and repair data and analyzed the behavior of the system by using various reliability indices (measures). These indices include failure rate, repair time, mean time between failures (MTBF), expected number of failures (ENOF), availability and reliability of the system which gave more sound idea about the behavior of the system. Based on that, behavior analysis of some complex repairable industrial system are analyzed by [3], [5], [11], [12], [13]. The basic expressions for the failure rate $(\lambda )$ and repair time ($\tau$) associated with the logical AND- and OR-gates used by them is summarized in Table 1.

From the above literature, it has been shown that fuzzy set theory is a useful tool to handle such situations by defining the fuzzy set $\tilde{A}$ which accommodate the various degree of membership on the real continuous interval [0, 1] by the membership function ${\mu }_{\tilde{A}}\in [\mathrm{0,1}]$ where ${\mu }_{\tilde{A}}$ is the degree of membership of element x in fuzzy set $\tilde{A}$. After the introduction of the concept of fuzzy sets by Zadeh in 1965 several researches were conducted on the extensions of the notion of fuzzy sets. Among these extensions the one that has drawn the attention of many researches during the last decades is the theory of intuitionistic fuzzy sets (IFSs) introduced by Attanassov in 1986 [14]. Gau and Buehrer [15] extended the idea of fuzzy sets by vague sets. IFS add an extra degree to the usual fuzzy sets in order to model hesitation and uncertainty about the membership degree of belonging. In fuzzy set theory the hesitation degree (or non-membership degree) of an element of the universe is implicity defined as one minus the membership degree, and hence it is fixed. In IFS theory the hesitation degree is somehow independent. Moreover in fuzzy set, the degree of acceptance is considered only but IFS is characterized by a membership function (acceptance) and a non-membership function (rejection) so that the sum of both values is less than one [16]. Chen [17] proposed a new method for analyzing the fuzzy system reliability based on vague sets. Kumar et al. [18] and Mahapatra and Roy [19] presented a method for fuzzy system reliability analysis using idea of interval valued vague sets and intuitionistic fuzzy numbers respectively. Taheri and Zarei [20] investigated the Bayesian system reliability assessment in vague environment. Kumar and Yadav [21] had analyzed the fuzzy system reliability using different types of intuitionistic fuzzy numbers (IFNs) instead of the classical probability distribution for the components. Functions of the IFNs are calculated to construct the membership function and non-membership function of the fuzzy reliability via non-linear programming problem using commercial package GINO. Also, they proposed a weakest t-norm based intuitionistic fuzzy fault tree model for analyzing the system reliability [22]. Verma et al. [23] had analyzed the behavior of the compressor system in a vague environment for series system only.

The present paper proposed a new methodology, named as vague Lambda-Tau methodology (VLTM), for analyzing the behavior of the complex repairable industrial system by utilizing vague, uncertain and imprecise data. In FLTM, the highest level of confidence of domain experts is assumed to be 1. But in real life situation, it lies between [0, 1] according to expert's knowledge. Keeping this point in view, VLTM has been proposed in which the effects of failures and course of action on the system performance have been analyzed. To remove the uncertainty in the available/collected data, intuitionistic fuzzy numbers are developed using fuzzy possibility theory. PN has been used for modeling and analysis of complex industrial systems and process due to its ability to model the dynamic of the system. Desrochers and Al-Jaar [24], Liu et al. [25] demonstrate that PN modeling was superior to traditional Markov chain modeling and FTA, as they provide a powerful formalism to model various classes of discrete events and may be used for qualitative and quantitative purposes simultaneously. Keeping this point in view, the interactions among the various units of system are modeled using PNs and different cut sets are obtained by using matrix method [25]. To strengthen the analysis various reliability parameters such as failure rate, repair time, MTBF, unreliability, availability, reliability, maintainability and ENOF are computed in the form of vague membership functions, which improve the shortcoming/drawbacks of existing fuzzy and probabilistic approach and give a better understanding of the system behavior through its graphical representation. The proposed approach has been illustrated through a case study of washing unit of a paper mill, a complex repairable industrial system situated in the northern part of India. Sensitivity analysis on the system MTBF has also been addressed. The obtained results will help the management for reallocating the resources to achieve the targeted goal of higher profit.