Fuzzy Bayesian reliability and availability analysis of production systems

Abstract

To have effective production planning and control, it is necessary to calculate the reliability and availability of a production system as a whole. Considering only machine reliability in the calculations would most likely result unmet due dates. In this study, a new modelling approach for determining the reliability and availability of a production system is proposed by considering all the components of the system and their hierarchy in the system structure. Components of a production system are defined as production processes; components of the processes are defined as sub-processes. In this hierarchical structure we could model all kinds of failures such as material and supply, management and personnel, and machine and equipment. In the analysis, a fuzzy Bayesian method is used to quantify the uncertainties in the production environment. The suggested modelling approach is illustrated on an example. In the example, also a separate reliability and availability analysis is conducted which only considered machine failures, and the results of both analyses are compared.

Introduction

Production starts with the decision to produce and continues until the finished product is complete. Any undesired stop in this duration can be defined as a failure of a production system. For example, production stoppage due to erroneous planning, late arrival of raw/semi finished materials or broken-down machines are different kinds of failures. Although many studies have examined reliability of individual components of a production system such as machines and humans, studies on reliability of a production system as a whole are limited in the literature. In terms of effective production planning and control, it is essential to compute the reliability of a production system especially if a company has high costs caused by unmet due dates.

Bral and Gardner (2002) proposed a method for determining production process availability by considering not only the machine but also other kinds of failures that stop the production process. They categorized the failures into two groups: process failures and machine failures. The process failures are defined as those which stop the production process such as the build-up of a metal shavings, swarf and turnings from the manufacturing operations. In the simulation model of the production system, a process failure of a machine is represented as a component of the machine. Our modelling approach is a generalization of Bral and Gardner (2002). Kumar and Huang (1993) modelled the reliability characteristics of a mine production system. They developed a simulation program to identify critical subsystems. They determined the production system’s appropriate boundaries, main subsystems and their related subsystems. The simulation program and analysis was used for preparation of mine production plans. Kusiak and Zakarian (1996) modelled and evaluated process reliability considering the reliability of activities of the process. They used integration definition methodology to evaluate the system’s reliability and provide industrial applications.

When modelling the production system one faces many different uncertainties. In general we can classify uncertainties as random and non-random. As stated in Singpurwalla and Booker (2004), random uncertainty is about outcomes of a random phenomenon and non-random uncertainty is about classification (also termed imprecision). In classification uncertainty we know the outcome, but we have difficulty placing it in a certain class. Probability theory and fuzzy set theory are effective tools to quantify random uncertainty and non-random uncertainty respectively.

Complex systems may have both kinds of uncertainty. Researchers have stated that probability theory can be used in concert with fuzzy set theory for the modelling of complex systems Zadeh, 1995, Barrett and Woodall, 1997, Ross et al., 2003, Singpurwalla and Booker, 2004. Bayesian statistics provide a natural framework combining random and non-random uncertainty so fuzzy Bayesian methods are developed for the solutions of the reliability problems. Sellers and Singpurwalla (2008) addressed the reliability of multistate systems with imprecise state classification. On the other hand, Gholizadeh et al., 2010, Gholizadeh et al., 2010, Huang et al., 2006, Taheri and Zarei, 2010, Viertl, 2009, Wu, 2004, Wu, 2006 addressed the problem of imprecise data (approximately recorded failure time durations and number of failures) in reliability studies. Chang, Chen, and Sun (2010) considered both fuzzy states and fuzzy information. Simon and Weber (2009) developed a Bayesian network model for determining reliability of the system with fuzzy states and imprecise probabilities.

The proposed model of a production system is solved using the fuzzy Bayesian reliability method developed by Wu, 2004, Wu, 2006. To utilize the imprecise data in the analysis, Wu, 2004, Wu, 2006 used fuzzy set theory and Bayesian statistics together. He assumed exponential lifetime distribution for each component of a system. He assigned a gamma prior distribution to the failure rate of the exponential distribution. Since exponential and gamma distributions are conjugate, posterior distribution of the failure rate parameter is also a gamma distribution. Then, using Mellin transform (Martz and Waller, 1991), a Bayesian point estimate of system reliability is computed from component reliabilities. Due to imprecise failure data, the failure rate is taken as a fuzzy random variable; also the parameters of the gamma distribution are taken as fuzzy real numbers. Using the definition of fuzzy random variables, an interval containing all the fuzzy Bayes point estimators of system reliability is defined. Membership function of the fuzzy Bayes point estimator is evaluated under resolution identity theorem.

The new modelling approach of a production system is introduced in Section 2. The reliability and availability analysis of a production system via fuzzy Bayesian method is represented in Section 3. In Section 4 the suggested modelling approach is illustrated on an example. Conclusions and future extensions of this work are given in Section 5.