FM – a pragmatic tool to model, analyse and predict complex behaviour of industrial systems

The Authors

Rajiv Kumar Sharma, Department of Mechanical Engineering, National Institute of Technology, Hamirpur, India

Dinesh Kumar, Department of Mechanical and Industrial Engineering, Indian Institute of Technology, Roorkee, India

Pradeep Kumar, Department of Mechanical and Industrial Engineering, Indian Institute of Technology, Roorkee, India

Abstract

Purpose – This paper aims to permit the system reliability analysts/managers/practitioners/engineers to analyze the system failure behavior using fuzzy methodology (FM)

Design/methodology/approach – In order to deal with both qualitative and quantitative information related to system performance the authors have adopted failure mode effect analysis (FMEA) and Petrinets (PNs), the well-known tools for reliability analysis, to build an integrated framework aimed at helping the reliability and maintenance managers in decision-making.

Findings – Using the proposed framework an industrial case from the paper mill is examined. From the results it is observed that the limitations associated with the traditional procedure of risk ranking in FMEA are efficiently modeled using fuzzy decision-making system (FDMS) based on FM. Also, the fuzzy synthesis of system failure and repair data helps to quantify the system behavior in a more realistic manner.

Originality/value – The simultaneous adoption of the proposed techniques to model, analyze and predict the uncertain behavior of an industrial system will not only help the reliability engineers/managers/practitioners to understand the behavioral dynamics of system but also to plan/adapt suitable maintenance practices to improve system reliability, availability and maintainability (RAM) aspects.

Article Type:

Research paper

Keyword(s):

System monitoring; Reliability management; Maintenance; Failure (mechanical); Paper industry.

Journal:

Engineering Computations: International Journal for Computer-Aided Engineering and Software

Volume:

Number:

Year:

2007

pp:

319-346

Copyright ©

Emerald Group Publishing Limited

ISSN:

0264-4401

1 Introduction

If everything went well and meets desired requirements, then there would be perhaps no failures but unfortunately failure is nearly an unavoidable phenomenon in mechanical systems/components. One can observe various kinds of failures in past under various circumstances such as nuclear explosions (Chernobyl nuclear disaster, 1986), Industrial plant leakages (Union carbide plant, Bhopal, 1984, Oil pipeline at Jesse Nigeria, 1998), aero plane crashes, and electrical network shutdowns, etc. which may be due to human error, poor maintenance, inadequate testing/inspection. With advances in technology and growing complexity of technological systems the job of reliability/system analyst has become more challenging. As they have to study, characterize measure and analyze the behavior of system using various qualitative and quantitative techniques (Cai, 1996; Modarres and Kaminsky, 1999; O'Connor, 2000; Vaurio, 2005; Aksu et al., 2006).

In reliability and maintainability studies a small number of researchers have seriously addressed the issue of handling uncertainties especially related with failure data (Fonseca and Knapp, 2001). The traditional analytical techniques (mathematical and statistical models) needs large amount of data, which is difficult to obtain because of numerous constraints such as rare events of component failure, human errors and economic considerations. Even if data are available, it is often inaccurate and thus, subjected to uncertainty, i.e. historical records can only represent the past behavior but may be unable to predict the future behavior of the equipment. Further age, adverse operating conditions and the vagaries of manufacturing processes affects each part/unit of system differently (Cizelj et al., 2001; Sergaki and Kalaitzakis, 2002). Therefore, it may be difficult or even impossible to establish rational database to accommodate all operating and environmental conditions. Though, virtually all the commercially available Computerized Maintenance Management System (CMMS) software packages offer data collection facilities but lack any decision analysis support for management. Hence, as shown in Table I (P _i, i=1 to 7 CMMS packages), a black hole exists in the column titled decision analysis (Shorrocks and Labib, 2000; Exton and Labib, 2002; Labib, 2003) because virtually no CMMS package offers decision support. Figure 1 shows how CMMS lacks in predictive maintenance data analysis and equipment failure analysis? (Swanson, 1997). According to Boznos (1998) most of companies are either dissatisfied or are neutral with the performance of CMMS. To counter the problem of uncertainty in decision analysis with respect to reliability and maintainability aspects both probabilistic and non-probabilistic methods are available in the literature. Based on the mature scientific theory, the probabilistic methods deals with uncertainty which is essentially random in nature but of an ordered kind. For instance, Bayesian methodology appeared in late 1970s is widely used in probabilistic risk assessment, an exercise aimed at estimating the probability and consequences of accidents for the facility/process under study (Siu and Kelly, 1998; Aven and Kvaløy, 2002). The non-probabilistic/inexact reasoning methods on the other hand study problems which are not probabilistic but cause uncertainty due to imprecision associated with the complexity of the systems as well as vagueness of human judgment. These methods are still developing and often use fuzzy sets, possibility theory and belief functions. For instance, in their work Sergaki and Kalaitzakis (2002) developed a fuzzy relational database model for manipulating the data required for criticality ranking of components in thermal powers plants. Liu et al. (2005), in their work proposed a framework for modeling, analyzing and synthesizing system safety of engineering systems on the basis of rule-based inference methodology using evidential reasoning. The framework has been applied to model system safety of an offshore and marine engineering system. From the literature studies it is observed that the field of inexact reasoning provides necessary help in coping with the imprecise and uncertain information pertaining to the system component/parts. In case of non-existence of precise data, approximate estimates of probabilities can be worked out.

Liao (2005) surveyed the development of various expert system methodologies over the last four decades. Among them fuzzy methodology (FM) has been widely applied in variety of areas such as fault diagnosis (Hauptmanns, 2004); structural reliability (Savoia, 2002); software reliability (Popstojanova and Trivedi, 2001; Wang et al., 2006); human reliability (Konstandinidou et al., 2006); safety and risk engineering (Sii et al., 2001; Guimarães and Lapa, 2007) and quality (Yang et al., 2003; Sharma et al., 2007a). In the words of Cai (1996) “Undoubtedly fuzzy methodology in system failure engineering is noticeable and growing area and is still lying in speculative research period and is premature”. In a hierarchical structure the reliability of a system is determined by the constituent sub-systems and reliability of each subsystem is, in turn, determined by the associated components and their possible failure modes. Therefore, to model, analyze and predict the system failure behavior a structured framework which makes use of information produced at lower level is required. Owing to its sound logic, effectiveness in quantifying the vagueness and imprecision in human judgment, the FM has been used as an effective tool in the study to synthesize the system information (obtained from operators experience or manufacturer's specification and expert opinions) with the help of fuzzy set principles. In the paper the author's presents an integrated framework to analyze the complex behavior of a system. The proposed framework makes use of both qualitative and qualitative techniques. In the qualitative framework first the root cause analysis (RCA) of the system has been done to list out potential failure causes using failure mode and effect analysis (FMEA). Using the selected experts, the values of failure of occurrence (O _f), likelihood of non-detection of failure (O _d), and severity (S) of failure of various components are ascertained and resulting risk priority number (RPN) is computed. Then, the problems associated with the traditional FMEA are handled by developing a decision support system based on FM. Further, to analyze the system behavior in quantitative terms, the failure rate and repair times associated with the system components are modeled using fuzzy synthesis of information. Various system parameters such as availability, mean time between failures (MTBF), reliability, expected number of failures (ENOF), etc. are quantified in terms of fuzzy, crisp and defuzzified values.

2 Failure analysis techniques

According to International Electro technical Vocabulary (IEV), failure is defined as “the termination of the ability of an item to perform a required function” (IEC 50 (191), 1990). Among various failure methods, which include reliability block diagrams (RBDs), Monte Carlo simulation (MCS), Markov Modeling (MM), FMEA, fault tree analysis (FTA) and Petri nets (PNs) (Ebeling, 2001; Adamyan and David, 2002, 2004). The section presents brief account of only those which are used in the study for analyzing the system behavior both in qualitative and quantitative terms.

A fault tree is used to analyze the probabilities associated with the various failure causes and their effects on system performance. FTA starts by identifying a problem (an accident or an undesirable event) and all possible ways that the problem (failure) occurs. Since, 1960 the tool has been widely used for obtaining reliability information about the complex systems. The system failure analysis using fault tree methodology makes use of either qualitative or quantitative techniques. In quantitative techniques Monte-Carlo simulation and analytical solution approach is used to determine system reliability parameters where as in qualitative technique minimal cut set and path sets are used to determine system reliability parameters. Contrary to fault trees, Petri nets can more efficiently derive the minimal cut and path sets (Liu and Chiou, 1997; Adamyan and David, 2002). Also, the absorption property of PNs helps to simplify the Petri net model and determine minimal cut set and path sets by reorganizing the transitions which is possible as long as the firing time is not taken into consideration, i.e. transfer of tokens does not take place (static condition). Similar to fault tree, PNs makes use of digraph to describe cause and effect relationship between conditions and events. Petri nets have two types of nodes named place “P” and transition “T”. These nodes are connected by arcs “A”, i.e. arcs connect transitions to places or places to transitions. The basic symbols used in Petri net model (Figure 2) are defined as under:

○: Place, drawn as a circle.

–: Transition, drawn as a bar.

↑: Arc, drawn as an arrow, between places and transitions.

•: Token, drawn as a dot, contained in places.

Petri net, a directed bipartite graph is defined by a six-tuple (Peterson, 2000): Equation 1 where: T={t ₁,t ₂, … ,t _n}: a set of transitions, each transition representing an event or an action; P={p ₁,p ₂, … ,p _l}: a set of places, where a place is used to represent either the condition for the event or the consequences of the event; A⊆{T×P}∪{P×T}: a set of directed arcs that connect transitions to places and places to transitions; M ₀: the initial marking of the system that represents initial state of the system; I(t)={p|(p, t)∈A}: a set of input places of a transition t; and O(t)={p| (t,p)∈A}: a set of output places of a transition t.

It has two parts, i.e. static and dynamic. The static part consists of places (p), transitions (t) and arrows (A). While the dynamic part is related with marking of graph by tokens which are present, not present or evolves dynamically on firing of valid transitions. As shown in Figure 2(a), the static part, and Figure 2(b) the dynamic parts, i.e. before firing there is one token in each of input places P ₁ and P ₂ but no token in output place P ₃. Accordingly, the Petri net marking is M=(1, 1, and 0) and after firing of transition (based on enabling rules) the token moves from each of P ₁ and P ₂ to the output place P ₃.

From the literature studies it is observed that both Petri nets and fault tree methods are used for software reliability analysis (Kumar and Aggarwal, 1993), analysis of coherent fault trees (Hauptmanns, 2004) and fault diagnosis (Mustapha et al., 2004). Exclusively in the field of reliability engineering the application of Petri nets has been presented for reliability evaluation (Adamyan and David, 2002, 2004), Markov analysis (German, 2000; Aneziris and Papazoglou, 2004; Schoenig et al., 2006) and stochastic modeling (Ciardo et al., 1994; Sahner and Trivedi, 1996). In the paper, the authors has used the static part of PNs to model the quantitative behavior of system.

To diagnose the unreliable aspects of the system, RCA provides comprehensive classification of causes related to the various units comprising a system. The information obtained from RCA helps to establish a knowledge base for conducting FMEA Banking upon this information, FMEA is carried out by listing all the possible failure causes related to the system its components/parts. The tool has been used by reliability engineers to identify critical components/parts/functions whose failure will lead to undesirable outcomes such as production loss, injury or even an accident. It was developed at Grumman Aircraft Corporation in the 1950 and 1960s. Since, then, it has been extensively used as a powerful technique for system safety and reliability analysis of products and processes in wide range of industries – particularly aerospace, nuclear, automotive and medical (O'Connor, 2000; Ebeling, 2001; Xu et al., 2002; Bowles, 2003; Sharma et al., 2005b; Tay and Lim, 2006). The main objective of FMEA is to discover and prioritize the potential failure modes (by computing respective RPN), which pose a detrimental effect on the system and its performance. The results of the analysis help managers and engineers to identify the failure modes, their causes and correct them during the stages of design and production. The general procedure for carrying out FMEA process is shown by a flow chart (Figure 3).

The main disadvantage of RPN approach is that various sets of input terms, i.e. O _f, S and O _d, may produce an identical value, however, the risk implication may be totally different which may result in high-risk events may go unnoticed. For instance, consider two different events having values of O _f=3, S=4, O _d=5 and O _f=1, S=10, O _d=6, respectively. Both these events will have a total RPN value of 60 however; the risk implications of these two events may not necessarily be the same which may result in high-risk events may go unnoticed. The other disadvantage of the RPN ranking is that it neglects the relative importance among O _f, S and O _d. The three factors are assumed to have the same importance but in real practical applications the relative importance among the factors exists. For instance, a failure mode with a very high severity, low rate of occurrence, and moderate detectability (say 9, 3, and 5, respectively) may have a lower RPN (135) than one with all parameters moderate (say 5, 6, and 6 yielding an RPN of 180) even though it should have a higher priority for corrective action. To address these disadvantages a computer-based intelligent fuzzy decision support system is designed to prioritize the failure causes.

3 Basic notions on fuzzy approach

The section presents only those basic concepts related to fuzzy set theory, which are helpful for analyzing system behavior and developing a decision support system for FMEA (Zadeh, 1996; Zimmermann, 1996; Kokso, 1999; Ross, 2000; Tanaka, 2001).

3.1 Crisp versus fuzzy sets

Crisp (classical) sets contain objects that satisfy precise properties of membership functions (MF). Only two possibilities whether an element belongs to, or not belongs to a set exist. A crisp set “A” can be represented by a characteristic function M _A/ u={0, 1}: Equation 2 where: U: universe of discourse; X: element of U, A: crisp set and M: characteristic function.

On the other hand fuzzy sets contain objects that satisfy imprecise properties of MFs, i.e. membership of an object in a fuzzy set can be partial. Contrary to classical sets, fuzzy sets accommodate various degree of membership on continuous interval [0, 1] where “0” conforms to no membership and “1” conforms to full membership. Mathematically defined by equation (2): Equation 3 where: μ _A˜(x): degree of membership of element x in fuzzy set A˜.

Larger the μ _A˜(x), stronger is the degree of belongingness for x in A˜. If μ _A˜(x)=1, the element x (with 100 percent certainty) completely belongs to fuzzy set A. otherwise if μ _A˜(x)=0, the element x does not belongs to A. Values between “0” and “1” mean intermediate values of certainty.

3.2 Membership functions, α cuts and linguistic variables

A MF is a curve that defines how each point in the input space is mapped to a membership value (partial truth) between 0 and 1. Various types of MFs such as triangular, trapezoidal, γ and rectangular can be used for reliability analysis. However, triangular membership functions (TMFs) are widely used for calculating and interpreting reliability data because of their simplicity and understandability (Yadav et al., 2003; Bai and Asgarpoor, 2004). For instance, imprecise or incomplete information such as low/high failure rate, i.e. about 5 or between 6 and 8 is well represented by TMF. In the study, TMF is used as it reflects the dispersion of the data adequately.

To resolve fuzzy sets in terms of constituent crisp sets, the concept of α cuts is used. They are indispensable in performing arithmetic operations with fuzzy sets. The α cut of a fuzzy set M˜ denoted by M˜ α, is the crisp set comprised of all elements x of universe of discourse for which membership of greater than or equal to α that is: Equation 4 where α is a parameter in the range 0≤x≤1 with introduction of α cuts, M˜ is defined as: Equation 5 The confidence interval defined by a cut is written as (equation (3)) (Figure 4). The confidence interval defined by a cut is written as (equation (3)): Equation 6 Moreover, when an event is imprecisely or vaguely defined, the experts would simply say that the possibility of occurrence of a given event is “low” “high” and “fairly high”. To estimate such subjective events linguistic expressions are used. The analyst can use linguistic variables to assess and compute the events using well-defined fuzzy MFs. A linguistic variable is characterized by (X, T, U, M) where:

X: the linguistic variable, for example, X is the FL of an item.

T: the set of linguistic values that X can take, for example, T={Very low, Low, Average, Frequent, Highly frequent}.

U: the actual physical domain in which the linguistic variable X takes its quantitative (crisp) values, for example, U=[F _{Very low},F _{Highly frequent}].

M: semantic rule that relates each linguistic value in T with a fuzzy set in U, for example, M relates “very low” “low” “highly frequent” with the specific MF.

In the study, the linguistic terms such as remote, low, moderate, high and very high are used to represent the probability of failure of occurrence, non-detectability and severity related with the failure causes in FMEA.

3.3 Fuzzy rule base and inference system

The rule base describes the criticality level or riskiness of the system for each combination of input variables. Rules are formulated in linguistic terms using two approaches:

expert knowledge and expertise; and
fuzzy model of the process.

Oftenly expressed in “If-Then” form, where, If: an antecedent which is compared to the inputs and Then: a consequent, which is the result/output.

For instance, the format of rules is defined as: Equation 7 In practical applications the fuzziness of the antecedents eliminates the need for precise match with the inputs. All the rules that have any truth in their antecedent will fire and contributes towards the fuzzy conclusion set. Each rule is fired to a degree that is function of the degree to which its antecedent matches the input. This imprecise matching provides a basis for interpolation between possible input states and serves to minimize the number of rules needed to describe the input-output relation (Bowles and Pelaez, 1995).

By using the inference mechanism an output fuzzy set is obtained from the rules and the input variables. Figure 5, shows the schematic representation of the fuzzy reasoning mechanism with two rules. First, the numerical input variables (occurrence, severity) are fuzzified using appropriate MFs. Then, the min operator is used for the conjunction and for the implication operations. The outputs (individual fuzzy sets) are aggregated by using the max operator and finally, the aggregated output is defuzzified to obtain a crisp value.

The two most common types of inference systems frequently used are:

the max-min inference; and
the max-prod inference method.

Examples of t-norms are the minimum, oftenly called “Mamdani implication” and the product, called the Larsen implication. In the study Mamdani's max-min inference method is used. The min operator is used for the conjunction of the rule and for the implication function and the max operator is used for the aggregation of the fuzzy sets. The compositional rule of inference proposed by Zadeh (1975) results in equation (5): Equation 8 where, β _k=minα _i,k[α _i,k=sup min (μ _A ^′(x _i),μ _Ai,k(x)) ]

3.4 Defuzzification

In order to obtain a crisp result from fuzzy output, deffuzification is carried out. In the literature various techniques for defuzzification such as centroid, bisector, middle of the max, weighted average exists. The criterion's for their selection are:

disambiguity (result in unique value);
plausibility (lie approximately in the middle of the area); and
computational simplicity (Kokso, 1999; Ross, 2000; Zimmermann, 1996).

In the study, the centroid method is used for defuzzification in both qualitative and quantitative analysis as it gives mean value of the parameters. Mathematically, represented as (equation (6)): Equation 9 where, B′ is the output fuzzy set, and μ _B′ i is the MF.

4 An example

As an example a case from process industry (paper mill) situated in northern part of India (producing 180 tons of paper per day) is taken to discuss the failure behavior of system using the failure analysis techniques discussed in Section 2. There are many functional units in a paper mill such as feeding, pulp preparation, pulp washing, screening, bleaching and preparation of paper. The present analysis is based on the study of washing system. The schematic diagram of the system is shown in Figure 6. The system consists of four main subsystems defined as:

Filter [SS₁]: filter is employed to drain black liquor from the cooked pulp.
Cleaners [SS₂]: cleaners has three units in parallel. Here, water is mixed with pulp to cleanse by centrifugal action. Failure of any one will reduce the efficiency of the system, which reduces the quality of paper.
Screen [SS₃]: screener has two units in series. These are used to remove oversized, uncooked and odd shaped fibers from pulp through straining action. Failure of any one unit will cause system to fail.
Decker [SS₄]: decker has two units in parallel. Complete failure of decker occurs when both fail.

4.1 Qualitative framework

The unreliable aspects of the washing system (Figure 6) are determined by comprehensive classification of failure causes pertaining to the system components (deckers, filters, screeners and cleaners) using RCA as shown in Figure 7. Then, FMEA of the system is carried out by listing all the possible failure causes, their function and effect on system. In brief the methodology used to compute the scores related to failure of occurrence (O _f), likelihood of non-detection of failure (O _d), and severity (S) of failure of various components in FMEA is discussed as under (Sharma et al., 2005b).

4.1.1 Probability of occurrence of failure (O _f)

Probability of occurrence of failure is evaluated as a function of MTBF. The data related to MTBF of components is obtained from previous historical records, maintenance log-books and is then integrated with the experience of maintenance personnel. For instance, if MTBF of component is between two and four months then probability of occurrence of failure is high (occurrence rate 0.5-1 percent) with the score ranging between seven and eight. Table II presents the linguistic assessment of probability of failure occurrence with corresponding MTBF and scores assigned.

4.1.2 Probability of non-detection of failures (O _d)

The chance of detecting a failure cause or mechanism depends on various factors such as ability of operator or maintenance personnel to detect failure through naked eye or by periodical inspection or with the help of machine diagnostic aids such as automatic controls, alarms and sensors. For instance, probability of non-detection of failure of a component through naked eye is say, 0-5 percent is ranked 1 with non-detectability remote.

4.1.3 Severity of failure(S)

Severity of failure is assessed by the possible outcome of failure effect on the system performance. If the effect is not noticed and has no effect on performance than the severity is termed as remote. The slight deterioration in system performance, regards severity as moderate and significant deterioration regards severity as high. The effect is said to be very high when non-conforming products are produced or there is appreciable production loss. For instance, if MTTR of facility/component in washing unit is less, say lies between 1/4 and 1/5 h, than effect may be regarded as remote. If external intervention is required for repairs, or MTTR exceeds 1/2 days and there is appreciable deterioration in the quality of the pulp than effect may be regarded as high and if system degrades resulting in line shut down/production stoppage than the severity may be regarded as very high. Apart from the parameters listed in Table II the economics (associated with maintenance, spares and manpower) and safety aspects can also be used for determining severity effect.

Table III presents the detailed FMEA analysis for the system. The numerical values of FMEA parameters, i.e. O _f, S and O _d are obtained by using the discussed methodology. Then RPN number for each failure cause is evaluated by multiplying the factor scores, obtained from the selected experts.

From Table III it is observed that for deckers, WC₄₈ and WC₄₉ represented by different sets of linguistic terms produce an identical RPN, i.e. 336, however, the risk implication for both the causes may be totally different. For pumps, the causes WC₅₄ and WC₅₇ represented by same linguistic terms produce different RPN and are ranked 2nd and 1st, respectively, which could be misleading and so on. The above listed limitations of traditional FMEA are addressed by using FM. A decision support system based on fuzzy set theory is developed using MATLAB Toolbox 6.3. The basic system architecture of the proposed system as shown in Figure 8 consists of three main modules, i.e. knowledge base module and user input/output interface module. The input parameters, i.e. O _f, S and O _d, used in FMEA, are fuzzified using appropriate MFs to determine degree of membership in each input class. For the output variable, riskiness/priority level (Figure 9 (a) and (b)) both triangular and trapezoidal MFs are used. Multiple experts with different degree of competencies “C” are used to construct the MF. The resulting fuzzy inputs are evaluated in fuzzy inference engine, which makes use of well-defined rule base. In the study, based on the MFs of three input variables O _f, S, O _d with, five fuzzy sets in each, a total of 125 rules can be generated. Table IV shows the nature of rules. However, these rules are combined (wherever possible). For example, the three rules as shown in Table IV can be combined and read as: If O _f is High and S is Moderate and O _d is High Then Risk Priority is Moderate-High or any other combination of the three linguistic terms assigned to O _f, S and O _d.

Finally, to express the riskiness/criticality level of the failures so that corrective or remedial actions can be prioritized accordingly, defuzzification is done using centroid method to obtain crisp ranking from the fuzzy conclusion set. Figure 10 shows the fuzzy inference outputs for two failure causes WC₃₃ and WC₄₅.The comparison of the results obtained through traditional and fuzzy approach is presented in Table V, respectively.

4.1.4 Discussion

From the comparative results presented in Table V, for cleaners, it is evident that in traditional FMEA, events with same linguistic terms produce different RPN but identical ranking is produced with help of fuzzy decision-making system (FDMS). For instance, in case of cleaners causes WC₂₂ and WC₂₄, where O _f, S and O _d are described by same linguistic terms, i.e. moderate, moderate, high, respectively, the defuzzified output is 0.617. This entails that these two causes should be given the same priority for attention. The RPN method, however, produces an output of 288 and 240 for these causes and ranks them at third and fourth place, respectively. This means that WC₂₂ has the highest priority than WC₂₄, which could be misleading.

Similarly, in case of deckers, the causes WC₄₁ and WC₄₂ described with same linguistic terms, i.e moderate, moderate, moderate produce different RPN and are assigned different priorities but fuzzy method produces same output and same priorities for them. Also, causes WC₄₈ and WC₄₉, where O _f, S and O _d are described by different linguistic terms, i.e. high, moderate, high and moderate, high, high, respectively, the FMEA output/rank is same, i.e. 336/2nd but FDMS differentiates them and ranks WC₄₈ at 3rd place and WC₄₉ at 2nd place. For pumps, the causes WC₅₄ and WC₅₇ represented by same linguistic terms produce different RPN using traditional FMEA and are ranked 3rd and 1st, respectively, which could be misleading. On the other hand fuzzy approach produce same output/rank.

4.2 Quantitative analysis

In order to measure and analyze the behavior of system quantification of various system parameters (such as repair time (τ), failure rate (λ), MTBF, Av. and ENOF) is essential for managerial decision-making with respect to maintenance and manpower planning. In this framework, the Petrinet model of the system is obtained first from its equivalent fault tree model (Figure 11(a) and (b)) and then with the help of fuzzy synthesis of system information the fuzzy, crisp and defuzzified results are obtained (based on the steps shown in Figure 12) to analyze the system behavior.

In brief the steps are discussed as:

Under the information extraction phase, the data related to failure rate [λ _i] and repair time [τ _i] of the components (filter, cleaner, screener and deckers) is collected from present/historical records of a paper mill and is integrated with expertise of maintenance personnel as presented in Table VI(a) (Kumar, 1991; Sharma et al., 2005c).
To account for imprecision and uncertainties in data, the crisp input data of λ and τ is converted to fuzzy numbers using TMF, with ±15 percent spread on crisp value (as shown in the Figure 13 for the first component, i.e. filter). To obtain fuzzy probabilities values, the fuzzy transition expressions for λ and τ are obtained by using the extension principle coupled with an α cut and interval arithmetic operations on conventional AND/OR expressions (Singh and Dhillion, 1991). For instance, AND transition expressions for λ and τ are represented by equations (i) and (ii) in the Table VI(b).
After knowing the input fuzzy triangular numbers for all the components shown in Petrinet model the corresponding fuzzy values of λ and τ for the system at different confidence levels are determined using fuzzy transition expressions. To analyze the behavior of system in quantitative terms various parameters of interest such as system availability, system reliability, ENOF, and MTBF are computed at different α cuts. The summary of the fuzzy reliability parameters, for each confidence factor, i.e. α level, ranging from 0 to 1, with increments of 0.1, is presented in Table VII(a-c) with left and right spreads. Depending upon the value of α, the analyst predicts the measures for the system. The graphical results (Figure 14) shows that if uncertainty in input data are described by means of triangular fuzzy numbers, then the possibility distribution of failure rate and repair time is a distorted triangle because after applying the fuzzy mathematics, the linear sides of triangle changes to parabolic one (Sittithumwat et al., 2004; Sharma et al., 2005c).
In order to make decisions with respect to maintenance actions it is necessary to convert fuzzy output into a crisp value. By using centroid method (as discussed in Section 3) defuzzification is carried out. The defuzzified values for the respective system parameters are calculated at different spreads. Table VIII presents both crisp and defuzzified values for the unit.

4.3 Behavior analysis

To analyze and predict the behavior of system in quantitative terms, the crisp and defuzzified values of system parameters are calculated at ±15,±25, and ±60 percent spread as the managers are always confronted with the maintenance decisions with respect to the systems and their components/parts. From the Table VIII, it is observed that defuzzified value changes with change in percentage-spread. For instance, repair time first increases by 5.28 percent when spread changes from ±15 to ±25 percent and further by 14.20 percent when spread changes from ±25 to ±60 percent. Similarly, for failure rate and ENOF, with increase in spread, increase in defuzzified values, is observed. On the other hand, at the same time for MTBF a decrease of 0.34 percent when spread changes from ±15 to ±25 percent and further by 2.836 percent when spread changes from ±25 to ±60 percent is observed. Similarly, for availability and reliability decrease in defuzzified values with increase in spread is observed. The computation of various system parameters (failure rate, repair time, ENOF, MTBF, availability and reliability) at different spreads helps the maintenance managers to understand the dynamics of system behavior. From the above discussions it is inferred that the maintenance action for the system should be based on defuzzified MTBF rather than on crisp value because with the reduced MTBF values a safe interval between maintenance actions can be established and inspections (continuous or periodic) can be conducted to monitor the condition or status of various equipments constituting the system before it reaches the crisp value. It can also be observed that with increase in repair time the availability goes on decreasing.

5 Conclusion

Owing to its sound logic, effectiveness in quantifying the vagueness and imprecision in human judgment, the FM is successfully used to evaluate and assess the system failure behavior. The in-depth analysis of system with the help of RCA helps to identify the various failure modes and their causes, the information on which FMEA for the system can be build. The use of FDMS not only addresses the seriously debated disadvantages associated with traditional procedure for conducting FMEA but also integrates expert judgment, experience and expertise in more flexible and realistic manner. For example, as evident from the comparative results in Table V that a decision support system based on fuzzy set theory combine severity, occurrence and non-detectability (the three input parameters in FMEA) in more flexible manner as compared to traditional FMEA.

Further, the quantitative analysis conducted to analyze and predict the system behavior helps the maintenance managers to understand the behavioral dynamics of the system. For instance, the triangular fuzzy graphical representations of system parameters (Figure 14) provides a general idea to the maintenance managers and helps them to predict the reliability measure(s) for the system (depending upon the value of confidence factor, α).The dispersion represented with the TMFs takes care of inherent variation in human performance, vagueness in system performance due to age and adverse operating conditions. Thus, it becomes intitutive for the engineers to arrive at decisions.

It is concluded from the study that the application of inexact reasoning methods (particularly the FM) in system failure engineering help the system/reliability analysts to:

analyze failure behavior of industrial systems in more realistic manner as they often makes use of subjective judgments, uncertain data and approximate system models;
model and predict the behavior of industrial systems in more consistent manner; and
plan suitable maintenance practices/strategies for improving system performance.

6 Managerial implications

The important managerial implication of the unified approach presented in the study is to enable the managers/practitioners/researchers to define, measure, and analyze the imprecise information associated with reliability and maintainability aspects of the production systems. Certain other implications are summarized as under.

The FM in system failure engineering:

helps to handle ambiguous, qualitative or imprecise information as well as quantitative data used for reliability assessment of systems in consistent manner;
helps to combine severity, occurrence and non-detectability, the three input parameters (in FMEA) in more flexible manner to obtain RPN;
helps to plan suitable maintenance practices/strategies for improving system performance and there by reducing operation and maintenance (O&M) costs; and
can be extended to evaluate system behavior in other process industries (petroleum, food processing, chemical, sugar, etc.).

Though, the results still depends upon the expert's judgment (especially when building a knowledge base for fuzzy inference system in FMEA) and the quality of the information obtained from various sources, as with any modeling framework one has to exercise great care to ensure that the data and inputs presented to the method are of good quality without which the results could be biased.

Equation 1

Equation 2

Equation 3

Equation 4

Equation 5

Equation 6

Equation 7

Equation 8

Equation 9

Figure 1Applications of CMMS modules

Figure 2(a) Static (b) dynamic Petrinets

Figure 3Failure mode effect analysis

Figure 4 α cut of a fuzzy set

Figure 5Fuzzy reasoning mechanism

Figure 6Washing system

Figure 7Root cause analysis (washing system)

Figure 8Architecture of fuzzy decision making system (FDMS)

Figure 9Fuzzy representation of MFs (a) O _s, S, and O _d (b) risk priority

Figure 10FIS outputs for WC₃₃ and WC₄₅

Figure 11Washing system (a) fault tree model (b) Petrinet model

Figure 12Framework of quantitative analysis

Figure 13Input fuzzy triangular number representation (filter)

Figure 14Fuzzy representations of system parameters (a) repair time and failure rate; (b) reliability and availability; and (c) meantime between failures and expected number of failures

Table ICommon CMMS packages

Table IIScale used to measure (probability of failure occurrence, severity and detection)

Table IIIDetailed FMEA (washing system)

Table IVIf-Then rules

Table VComparative results (traditional and fuzzy ranking)

Table VI(a) System data and (b) fuzzy expressions

Table VII(a-c) Computed system parameters

Table VIIIDefuzzified values at different spreads

References

Adamyan, A., David, He. (2002), "Analysis of sequential failures for assessment of reliability and safety of manufacturing systems", Reliability Engineering & System Safety, Vol. 76 pp.227-36.