1-4244-2509-9/09/$20.00 ©2009 IEEE

Computational Intelligence in Reliability and Maintainability Engineering

Marcia F. P. Salgado, Federal University of Minas Gerais Walmir M. Caminhas, PhD, Federal University of Minas Gerais Benjamim R. Menezes, PhD, Federal University of Minas Gerais

Key Words: computational intelligence, maintainability, reliability.

SUMMARY & CONCLUSIONS

In this paper the basics of reliability and maintainability modeling, prediction and optimization problems using stochastic models are briefly reviewed (for non-repairable and repairable systems). As an alternative to classical methods based on stochastic models, computational intelligence techniques such as neural networks and fuzzy systems as well as evolutionary computing, artificial immune systems and swarm intelligence are introduced. Classical methods, neural networks, evolutionary computing and immune algorithm are followed by examples demonstrating their applicability to reliability modeling, analysis and optimization. This is a fairly new research area and it has a great potential to support engineers on solving problems such as modeling, analysis and optimization of real-world industrial systems.

1 INTRODUCTION

In reliability and maintainability engineering there are always some questions to be answered using stochastic models (parametric or non-parametric models for non-repairable or repairable systems) as a traditional way of modeling and analyzing failure and repair data. Of course as data is available it is straightforward to build probabilistic models (choosing distributions and estimating their parameters) and use them to evaluate system performance and to make decisions (Ascher and Feingold, 1984), (Meeker and Escobar, 1998), (Kececioglu, 2002), (O’Connor, 2002), (Birolini, 2004). Most commercial packages such as the ones provided by Reliasoft, Relex or Minitab, or even Matlab®, have functionalities and toolboxes which help us on the practice of reliability and maintainability analyses using traditional approaches. The basic concepts of classical and alternative methods based on Computational Intelligence (CI) paradigms are reviewed followed by numerical examples of their application.

2 CLASSICAL MODELING AND ANALYSIS TECHNIQUES OF NON-REPAIRABLE AND REPAIRAPLE SYSTEMS

2.1 Non-repairable systems modeling and analysis

Non-repairable systems are the ones to which no maintenance actions rather than replacements are possible, i.e. for a non-repairable system a random variable time to failure

(TTF) is expected to follow some sort of probabilistic behavior that can be represented by probability density functions (pdf) such as Weibull and Exponential. In this case the failure pattern can be identified through classical modeling tools using many different commercial statistical packages. The steps to do such analyses are as follows: (1) Gather failure data; (2) Prepare and adjust the failure databases; (3) Use a statistical package for selecting the most applicable models which best fit the data available; (4) Estimate measures of interest such as the MTTF, confidence bounds as well as the estimated pdf parameters.

2.2 Non-repairable systems – Numerical Example

Failure data of a system was gathered and analyzed using Minitab® 15.1.1.0. Weibull distribution is more suitable for representing the time to failure data (correlation coefficient = 0.997). In this case the Weibull distribution was chosen to the reliability analysis (Figure 1) and its parameters are presented in Table 1.

Figure 1: Weibull distribution overview plot.

Table 1: Weibull distribution parameters (95% Confidence)

2.3 Repairable systems modeling and analysis

In section 2.1 the system was assumed non-repairable and classical reliability analysis was performed. In this section repairable system analysis is reviewed.

Parameter Estimate Lower Upper MTTF 25.0124 22.1730 28.2154 β 0.909790 0.841802 0.983270 α 23.9111 21.0043 27.2202

A repairable system is defined as a system to which some sort of maintenance intervention might be taken in order to bring the system back to its normal operating condition after a failure has occurred. In this case, maintenance strategies as well as their quality are important information. Three different situations can be cited: (1) the system gets back to the same condition as it was when it was first put into operation (after the repair the system is ‘as good as new’, perfect repair); (2) the system gets back to the same condition as it was just before the failure (after the repair the systems is ‘as good as old’, minimal repair); (3) An imperfect repair is performed but the system condition is still degraded (imperfect maintenance better than minimal repair). Similarly to non-repairable systems, non-parametric or graphical methods as well as parametric models can be employed in repairable system identification (modeling and analysis). Non-parametric techniques are Duane plot, Mean Cumulative Function (MCF) plot and Total Time on Test (TTT) plot. Using these tools it is possible to have an idea about what kind of parametric model might be chosen from a list of models available. Details about how to build Duane, TTT or MCF plots can be found in (Rigdon, 2000). In terms of parametric models, minimal repairs can be modeled through non-homogeneous Poisson process (NHPP) models and Perfect repairs can be modeled through renewal process (RP) models or homogeneous Poisson process (HPP) which is a particular case of RP models with the time between failures following an exponential distribution (Rigdon, 2000). NHPP models might follow a Power law or Exponential law. In both cases deteriorating or improving conditions of a repairable system can be modeled depending on the estimates for the model parameters. HPP models are suitable only for representing stable conditions of the repairable system where the intensity of failures is constant. Imperfect repairs are modeled through a compromise between minimal repair and perfect repair models. A straight line on a Duane plot or TTT plot lying very close to a unit square diagonal indicates that NHPP Power Law Model is adequate. A straight line in a MCF plot indicates that system is stable (failures are remaining constant over time) so that HPP model is suitable. A concave down in a MCF plot indicates that the system reliability is improving (the time between failures is increasing over time) and a concave up indicates that the system reliability is deteriorating (the time between failures is decreasing over time). In such case NHPP models are suitable. Most real industrial systems are repairable what means that the modeling process must take into account the maintenance actions being performed and their impact to the system performance.

2.4 Repairable systems – Numerical Example

Failure and repair data for the system analyzed in section 2.1 was considered for the calculation of the time between failures (TBF). Mean cumulative function plot (MCF plot) was used to identify the system performance. In Figure 2 the MCF plot indicates that the system is improving. In this case a NHPP model should be used for modeling the system reliability. Using the Duane plot it can be seen (Figure 3) that

a Power law model is inadequate to model the system. In this case, an exponential law NHPP model was fitted (Figure 4). The exponential law NHPP model fitted can be used to predict the number of failures expected in a given time interval.

Figure 2: MCF plot for TBF (Repairable system analysis was

performed using Minitab® 15.1.1.0.)

Figure 3: Duane plot for TBF (Repairable system analysis

was performed using Minitab® 15.1.1.0.)

Exponential Law NHPP Model

ln (M(t)) = 0.0003*t + 3.4436

1

10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000TBF

ln (M

(TB

F))

Figure 4: Exponential Law NHPP model fit.

3 COMPUTATIONAL INTELLIGENCE TECHNIQUES IN RELIABILITY AND MAINTAINABILITY ENGINEERING

In the previous sections the basics of reliability modeling and analysis of non-repairable and repairable system were presented. As it was seen, most methodologies are based on the assumption that failure and repair data are well documented and available for quantitative modeling using classical methods.

Most recently alternative techniques to model and analyze complex systems are being used in many areas and in the reliability and maintainability engineering. Such techniques are known as Computational Intelligence (CI) techniques and are part of the so called research area Artificial Intelligence (AI). The term CI was coined in 1992 and it is used due to the fact these new techniques are inspired by the adaptive

mechanisms of nature which allow or facilitate intelligent behavior in complex and dynamic environment. These mechanisms include the paradigms originated from AI that have, in some sense, the ability to learn and adapt themselves to new situations, to generalize and make abstractions as well as discovery capabilities (Engelbrecht, 2002; Pedrycz and Gomide, 2007). The main paradigms of CI are Artificial Neural Networks (ANN), Evolutionary computing (EC), Swarm intelligence (SI), Fuzzy systems (FS) and Artificial Immune systems (AIS). They can be combined among themselves and with stochastic methods in order to develop more effective methods to solve difficult and complex engineering problems. In the next subsections the basics of EC, ANN and AIS are presented and numerical examples showing their applicability to reliability modeling, analysis and optimization are provided.

3.1 Artificial Neural Networks – Basic concepts

An artificial neural network (ANN) is a model with a large number of highly interconnected neurons which can be used to map a set of inputs into one or more outputs based on a set of weights and activation functions. There are many different ANN architectures which can be modeled in according to their performance and robustness which are highly dependent on the complexity of the system being analyzed. ANN’s can be applied to reliability engineering problems such as probability density functions approximation as well as to reliability and maintainability analyses and optimization based on failure and repair data. Once the ANN architecture is chosen, it must be trained in order to update its weights according to a set of inputs and desired output(s). This training process can be implemented through different optimization algorithms such as the classical back-propagation based on the gradient descent method. The objective function in this case is the ANN output mean squared error (MSE) which is calculated comparing the target values to the ANN output along the training process.

3.2 Artificial Neural Networks – Numerical example

In order to show the applicability of ANN to reliability problems, the probability density function (PDF) of a system (assumed non-repairable) is estimated using a classical method and then compared to the output of a trained ANN. Using an ANN in this way might be advantageous due to the practical limitations involved when arbitrary pdf’s are to be chosen. To give an example, let say historical failure data of a component it is available and its performance for different mission times is to be estimated. In this case a multilayer feed-forward neural network (MLP) can be easily implemented and used to approximate the reliability function. In this paper a MLP was implemented using Matlab Neural Network toolbox. To train the ANN, the expected reliability values (Target) were presented to the network as well as the time to failures (TTF). To obtain the target values a Weibull pdf was fitted to the data and its parameters were estimated (scale parameter η = 28.1844 hours, and shape parameter β = 0.860621). There were 309 samples and 70% of them (217) were used to train

the ANN, and the other 30% were used to validate (15%, 46) and test (15%, 46) the network. To evaluate the ANN performance the mean squared error (MSE) comparing the Target to ANN output was evaluated for the training, validation and test data. In Figure 5 the Reliability function estimated using statistical toolbox is compared to the Reliability function approximated through the ANN. There are many other possible applications of ANN in reliability problems.

Figure 5: Reliability function approximation.

3.3 Evolutionary computing – Basic concepts

Evolutionary computing techniques are based on the natural evolution mechanisms in which the core concept is the survival of the individuals more adapted to the environment or operating context. In the natural evolution the survival capability is reached through a reproduction process when new individuals are generated from their parents and have genes as well as the best characteristics of all. Individuals with bad characteristics do not survive and the good ones have more chances of keeping its characteristics in the population. In evolutionary computing a population of individuals are generated and evolved through many generations in what individuals compete in order to reproduce themselves. An individual is generated from the genes of its parents through a crossover process and it is eventually subjected to a mutation process. After each generation the best individuals are chosen to be in the next generation what it is called ‘elitism’. Engelbrecht (2002) cites the classes of algorithms that are part of what is named evolutionary computing: (1) genetic algorithms which model the evolutionary process; (2) genetic programming which is based on genetic algorithms but the individuals are programs represented in trees; (3) evolutionary programming is the result of the adaptive behavior simulation of the evolutionary process; (4) evolutionary strategies model the parameters that control the evolutionary process; (5) differential evolution is similar to genetic algorithms but the reproduction process is slightly different; (6) cultural evolution models the evolution of the culture of a population influencing also the culture of the genetic evolution; (7) Co-evolution is a evolution process where at the beginning the individuals are not adapted but might evolve through a cooperation and competing process with the other individuals. According to Engelbrecht (2002), evolutionary computing has been successfully applied to many real-world problems such as data-mining, optimization, clustering, programming and

approximation of time series. More details regarding the design and application of

evolutionary computing can be found in Ashlock (2005), Menon (2004) and Jong (1997). Levitin (2007) presents many examples of evolutionary computing application to reliability engineering problems.

In this paper a genetic algorithm is used to optimize the performance of three generic systems in terms of their reliability through redundancy and reliability allocation subject to cost, weight and volume constraints. This is a well known design problem and typically very difficult to solve using exact optimization algorithms. Many papers have treated this problem using different structures (models) and methods. In Levitin (2007) the most recent applications of evolutionary computing in reliability optimization are presented. A detailed introduction to system reliability and reliability optimization can be found in Kuo et al. (2001). An overview of employed methods for solving many reliability optimization problems since 1970's is presented by Kuo (2000) providing a broad bibliographic review on the subject.

A genetic algorithm (GA) is based on the Darwin evolutionary theory, where a population of individuals are subjected to mutations and selected through the generations according to their degree of adaptability to the environment. It is expected populations even more adapted to the environment through the application of crossover, mutation and selection operators. This algorithm allows a good local exploration among best individuals.

3.4 Evolutionary computing – Numerical example

A GA was implemented to evolve 50 individuals through an evolution process during 500 generations with biased crossover of 80% probability, with mutation of 5% probability and selection being done through ranking strategy applied to the whole population. The system optimization in regards to reliability is associated to decide about which solution to choose in order to meet the requisites in terms of technology and other specified aspects. In Levitin (2007) five formulations for the reliability optimization problem are cited. In this paper the formulation named Traditional was adopted which has been adopted by Chen (2006) as well. The formulation is presented as follows. Let m be the number of subsystems, [ ] m 1,0∈r the vector of components reliability of each subsystem and m Ν∈n a vector with the number redundancies for each subsystem. The reliability of each subsystem and the constraint functions of volume, cost and weight are defined by equations 1 to 4, respectively:

( ) ( )( ) ( )

( ) ( ) ( )

( )

2 21

1

/42

1

/43

1

, 1 (1 ) , 1,..., 1

2

1000 3 ln

i

i

i

i

ni i i i

m

i i ii

mn

i ii i

mn

i ii

R r n r i m

g V w v n

g C n e r

g W w n e

β

α

=

=

=

= − − =

= −

⎛ ⎞= − − +⎜ ⎟

⎝ ⎠

= −

∑

∑

∑

r, n

r, n

r, n ( ) 4

The termsα and β are parameters representing physical characteristics of the components, r is a vector of subsystems’ reliability, n is a vector in which the elements

represent the number of redundancies in each subsystem, andV , C and W are targets for the volume, the cost and the weight of the system being optimized.

The reliability function of a generic complex (bridge) system as presented in Figure 6, is given by

( )

( )

3 1 2 3 4 1 4 5

2 3 5 1 2 3 4 1 2 3 5

1 2 4 5 1 3 4 5 2 3 4 5 1 2 3 4 5

,

2 7

sR R R R R R R RR R R R R R R R R R R

R R R R R R R R R R R R R R R R R

= + ++ − −− − − +

r n

Figure 6: A generic complex system (bridge).

Reliability-redundancy allocation problems are typically defined as non-linear integer programming (Gen, 2006). In this paper, complex design of five subsystems as presented by Chen (2006) was chosen as a test problem. The values for the model parameters are given in the table 2.

Table 2: Parameters for complex system.

A statistical analysis was performed for 200 trials of the GA, as illustrated in Figure 7. The best results are showed in Table 3. The problem was evaluated 25,000 times in each simulation of the GA.

Figure 7: GA statistical analysis - Complex System

optimization results.

n r ( )nr,sR ( )nr,g

(3,3,3,3,1)

0,8070836 0,8779927 0,8628981 0,6965346 0,7823552

0,9998863 18

0,0740375 4,2647698

Table 3: GA best optimization results – Complex System.

Reliability allocation problems of series and series-parallel systems were also tested using the GA. The results can be found in Salgado et al. (2007). A more realistic approach

was implemented using multi-objective optimization techniques and can be found in Salgado et al. (2008).

3.5 Artificial Immune Systems – Basic concepts

Recently a new class of algorithms inspired by immune system mechanisms has been applied to different failure diagnosis and pattern recognition problems (de Castro and Zuben, 2002; de Castro Silva, 2003; de Castro e Silva, 2008). This paper shows an example of application to solve the same reliability optimization problems presented in section 3.2.2. An immune algorithm (IA) based on ‘clonal’ principle which is well explained in Castro Silva (2003) was implemented. The immune system has many properties and functioning mechanisms that inspire the development of algorithms which can be used to solve many complex engineering problems.

3.6 Artificial Immune Systems – Numerical example

The IA was adjusted to a cloning rate of 50%, a cloning factor of β = 0.6, a decay maturation ratio of ρ = 1 applied to an initial population of 50 antibodies maturated during 500 generations.

A statistical analysis was performed for 200 trials of the IA, as illustrated in Figure 8. The best results are showed in Table 4. The problem was evaluated 25,000 times in each simulation of the IA.

Figure 8: IA statistical analysis - Complex System

optimization results.

n r ( )nr,sR ( )nr,g

(3,3,2,4,1)

0,8257889 0,8626022 0,9127624 0,6430090 0,7251948

0,9998890 5

0,0411987 1,5604663

Table 4: IA best optimization results – Complex System.

Reliability allocation problems of series and series-parallel systems were also tested using the IA. The results can be found in Salgado et al. (2007). A more realistic approach was implemented using multi-objective optimization techniques and can be found in Salgado et al. (2008).

4 FINAL CONSIDERATIONS

In this paper an introduction to computational intelligence techniques and their applicability to reliability engineering problems were demonstrated through numerical examples.

The study of optimization techniques is extremely important once the searches for feasible solutions which maximize the performance and minimize the costs associated with a system are intrinsic goals of reliability and maintainability engineers. Reliability optimization problems are combinatorial what make difficult to effectively solve them in polynomial-time by any of exact or deterministic methods. Stochastic optimization techniques are more suitable in these cases, what motivates the use of Computational Intelligence techniques such as evolutionary computing, artificial immune systems and swarm intelligence.

REFERENCES

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2. D. Ashlock, (2005). Evolutionary Computation for Modeling and Optimization, Springer Verlag.

3. A. Birolini (2004). Reliability Engineering: Theory and Practice, 4 edn, Springer-Verlag.

4. K. D. Jong (1997). The handbook of Evolutionary Computation, IOP Publishing Ltd and Oxford University Press.

5. D. Kececioglu (2002). Reliability Engineering Handbook, Vol. 1-2, DEStech publications.

6. J. Kennedy and R. C. Eberhart (2001). Swarm Intelligence, Morgan Kauffman.

7. G. Levitin, (2007). Computational Intelligence in Reliability Engineering: Evolutionary Techniques in Reliability Analysis and Optimization and New Meta-heuristics, Neural and Fuzzy Techniques in Reliability, Vol. 39-40, Springer-Verlag.

8. W. Meeker and L. Escobar (1998). Statistical Methods for Reliability Data, John Wiley.

9. A. Menon (2004). Frontiers of Evolutionary Computation, Kluweer Academics Publishers.

10. P. O’Connor (2002). Practical Reliability Engineering, 4th edn, John Wiley.

11. V. Palade, C. D. Bocaniala and L. Jain (2006). Computational Intelligence in Fault Diagnosis, Springer-Verlag.

12. S. Sandri, D. Dubois and H. Kalfsbeek (1990). Elicitation, assessment and pooling of expert judgments using possibility theory.

13. W. Pedrycz, F. Gomide (2007) Fuzzy Systems Engineering: Toward Human-Centric Computing. Wiley-IEEE Press, 526pp.

14. S. E. Rigdon and A. P. Basu (2000). Statistical Methods for the Reliability of Repairable Systems. John Wiley. 224pp.

15. A. P. Engelbrecht (2003). Computational Intelligence: An Introduction. Wiley. 310 pp.

16. T.-C. Chen (2006). Ia based approach for reliability redundancy allocation problems, Elsevier Applied Mathematics and Computation 182: 1556-1567.

17. L. N. de Castro and F. J. V. Zuben (2002).Learning and optimization using the clonal selection principle, IEEE

Transactions on Evolutionary Computation, N.3 6: 239-251.

18. W. Kuo and V. R. Prasad (2000). An annotated overview of system-reliability optimization. IEEE Transactions on Reliability N.2 49: 176-187.

19. W. Kuo, V. R. Prasad, F. A. Tillman and C.L. Hwang (2001). Optimal Reliability Design: Fundamentals and Applications, 1 edn, Cambridge University Press.

20. M. Gen and Yun, Y. (2006). Soft computing approach for reliability optimization: State-of-the-art survey, Reliability Engineering and System Safety 91: 1008-1026.

21. M. F. P. Salgado, A. C. Lisboa, R. R. Saldanha, W. M. Caminhas, B. R. Menezes (2007). Application of evolutionary computing in system reliability optimization. In: Proceedings of I Computational Intelligence Brazilian Symposium of Neural Network Brazilian Society (SBRN). UFSC, V.1: 1-7. (In Portuguese).

22. M. F. P. Salgado, A. C. Lisboa, R. R. Saldanha, W. M. Caminhas, B. R. Menezes (2008). Application of evolutionary computing in system reliability optimization. Learning and Nonlinear Models – Brazilian Society of Neural Network Magazine. Accepted in October, 2008. (In Portuguese).

23. M. F. P. Salgado (2008). Application of Optimization Techniques on Reliability Engineering. Master Thesis - Electrical Engineering Post-graduation Program of Federal University of Minas Gerais, Brazil. (In Portuguese).

BIOGRAPHIES

Marcia F. P. Salgado, PhD Student. Electrical Engineering – Computational Intelligence Research Group, Federal University of Minas Gerais (UFMG) Antônio Carlos Avenue, 6627 - Pampulha - Belo Horizonte, MG, 31270-901 Brazil.

E-mail: marcia.platilha@cpdee.ufmg.br

Marcia F. P. Salgado is currently a PhD student at UFMG. Her research is focused on the development and applications of computational intelligence techniques to reliability engineering problems with the support of the National Council for Scientific and Technological Development (CNPq). She is also a reliability consultant and has been working in projects for different Steel, Mining and Refining companies in Brazil and abroad. She earned her bachelors and masters degrees in Electrical Engineering at UFMG, Brazil.

Walmir Matos Caminhas, PhD. Electrical Engineering – Computational Intelligence Research Group, Federal University of Minas Gerais (UFMG) Antônio Carlos Avenue, 6627 - Pampulha - Belo Horizonte, MG, 31270-901 Brazil.

Walmir M. Caminhas is an associated Professor at UFMG. He has experience in electrical engineering with emphasis on process control working mainly on the following subjects: computational intelligence (neural networks, fuzzy systems and artificial immune systems) and failure diagnosis in dynamical systems. He earned his bachelors and masters degrees in Electrical Engineering at UFMG and his PhD degree at State University of Campinas, Brazil (Unicamp).

Benjamim Rodrigues de Menezes, PhD. Electrical Engineering – Computational Intelligence Research Group, Federal University of Minas Gerais (UFMG) Antônio Carlos Avenue, 6627 - Pampulha - Belo Horizonte, MG, 31270-901 Brazil.

Benjamim R. de Menezes is a Professor at UFMG. He has experience in electrical engineering with emphasis on process control and system reliability working mainly on the following subjects: sliding mode control, failure diagnosis and industrial processes reliability analysis. He has been involved in many R&D technological projects. He earned his bachelors degree in Electrical Engineering at UFMG, his masters degree at Federal University of Rio de Janeiro (UFRJ) and his PhD degree at Institut National Polytechnique de Lorraine, France.