Volume 6, Issue 1

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A Model to Estimate Restoration Factors

[Editor's Note: In the online version of this article, we have corrected "Table 1: Analysis Results Comparison" used in the printed edition of Volume 6, Issue 1. In the "Proposed Model" column, the values for Beta and Lambda were switched.]

In repairable system maintainability and availability analysis using the BlockSim software, the user is given the option to specify a restoration factor (repair effectiveness) that describes the percentage to which a component will be restored after the performance of the maintenance action. This provides the ability to model maintenance using "used parts" or imperfect maintenance. The commonly used models for analyzing repairable systems data are perfect renewal processes (PRP), corresponding to perfect repairs, and nonhomogeneous Poisson processes (NHPP), corresponding to minimal repairs. However, most repair activities may realistically not result in such two extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance).

In this article, we explore a model that can analyze complex repairable systems with various degrees of repair. A general likelihood function formulation for single and multiple repairable systems is presented for estimation of the new model parameters. The practical use of the proposed statistical inference is demonstrated by two examples, and the results show that the proposed method is a promising and efficient approach with the potential to become useful in industry. The proposed model will be available in upcoming ReliaSoft products.

Introduction
In BlockSim, the restoration factor can be a number from 0 to 1, where 1 indicates that the block (component) is "as good as new" after the maintenance action (i.e., with a new age of 0 ) and 0 indicates that the block has not been improved at all by the maintenance action (i.e., with an age the same as the age before the maintenance was performed). A restoration factor of .5 implies a 50% improvement to the block and sets the age of the block to 50% of the age of the block at the time of the maintenance action. A restoration factor of .75 implies a 75% improvement to the block and sets the age of the block to 25% of the age of the block at the time of the maintenance action.

Repairable systems receive repair/maintenance actions that restore system components when they fail. These actions change the overall makeup of the system and affect the system behavior differently due to different maintenance approaches. Basically, there are two major categories: corrective maintenance or preventive maintenance. Each can be classified in terms of the degree to which the operating condition of an item is restored by maintenance in the following way [1][2]:

  • Perfect repair or maintenance: a maintenance action that restores the system operating condition to be "as good as new."
  • Minimal repair or maintenance: a maintenance action that restores the system operating state to be "as bad as old."
  • Imperfect repair or maintenance: a maintenance action that restores the system operating state to be somewhere between as good as new and as bad as old.
  • Worse repair or maintenance: a maintenance action that makes the operating condition worse than that just prior to failure.
  • Worst repair or maintenance: a maintenance action that makes the system fail or break down undeliberately.

In this article, a model is proposed to analyze complex repairable systems and to estimate the restoration factor. A general likelihood function formulation for single and multiple systems with time truncated data and failure truncated data is presented for the estimation of the parameters.

Assumptions

  • The Time to First Failure (TTFF) distribution is known and can be estimated from the available data.
  • The repair time is assumed to be negligible so that the failures can be viewed as point processes.

Proposed Model and Maximum Likelihood Estimates
The proposed model [3][4] assumes that repairs fix all of the wearout and damage accumulated up to the current time. Consider a repairable system, observed from time t = 0. Denote by t1, t2, ... the successive failure times and let the times between failures be denoted by x1, x2, .... Thus we have:

equation

where for convenience we define t0 = 0. The sequence t1, t2, ... of failure times and the sequence x1, x2, ... of inter-arrival times thus contain exactly the same information about a particular realization of the process.

A Repairable System Process

Previous research assumes that the nth repair can remove the damage incurred only during the time between the (n-1)th and nth failures. In practice, not only does the nth repair depend on the (n-1)th repair, but it also depends on all previous repairs. We assume that the repair action could remove all damage accumulated up to the nth failure; accordingly, the virtual age after the nth repair becomes:

equation

where q is the degree of the nth repair (restoration factor) where 0 <= q <= 1, thus:

equation

If the system has the virtual age Vn-1 = y immediately after the (n - 1)th repair, the nth failure time X is distributed according to the following cumulative distribution function (cdf ):

equation

In this model, q = 0 corresponds to a perfect repair (PRP, as good as new) while q = 1 indicates a minimal repair (NHPP, as bad as old). The case of 0 < q < 1 corresponds to an imperfect repair (better than old but worse than new) while q > 1 indicates worse repair (worse than old). The case of q < 0 suggests a system restored to a condition of better than new. Physically speaking, therefore, q can be an index for repair effectiveness.

Maximum Likelihood Estimation of the Parameters
A Monte Carlo approach proposed by Kaminskiy and Kivtsov [5] provides a simulation method for statistical estimation of the proposed model, and it has been used in the automotive industry. However, this approach needs to estimate the distribution of the TTFF from a large amount of data and requires a very long time to estimate the parameters. For these reasons, using MLE to estimate the proposed model's parameters is preferable.

Single Repairable System
A maximum likelihood estimation method is possible for cases in which there is reasonably enough data available. Let t1, t2, ... tn be the inter-arrival time between failure i - 1 and i, assuming a Weibull distribution for TTFF, the nth failure time is distributed according to the following cdf:

equation

Thus the conditional probability density function (pdf ) of ti is:

equation

Where ti > ti-1. The corresponding likelihood is:

equation

Taking the natural log on both sides gives:

equation

Where vi can be obtained by Eqn. (1).

Multiple Systems
Suppose there are k systems:

equation

Taking the natural log on both sides gives:

equation

There are three parameters (q, λ and β) that need to be estimated. However, there is no closed form mathematical solution. A numerical algorithm has been developed to solve both the single repairable system and multiple repairable systems.

Grouped Data
The grouped data type is used for tests where the exact failure times are unknown and only the number of failures within a time interval is known (e.g., inspection data). As an example, this data type would be applicable when multiple units are run and the test units are inspected after predetermined time intervals and the number of failed units is recorded. The failed units are then subsequently repaired and put back into the test or removed.

The likelihood function is as follows:

equation

Taking the natural log on both sides gives:

equation

In order to estimate the unknown parameters, we use a numerical method to maximize the log likelihood function like the single repairable system and the multiple repairable systems.

Example 1
To illustrate the general application of this model, consider a system tested for T = 395.2 hours with the 56 failure times given in Table 1. The first failure was recorded at .7 hours into the test, the second failure was recorded 3 hours later at 3.7. The last failure occurred at 395.2 hours into the test and the system was removed from the test.

Table 1: Failure Data for a Repairable System [6]
Table 1: Failure Data for a Repairable System [6]

This data set is failure truncated. Based on this data set, different ML estimates of λ, β and repair degree q can be calculated corresponding to different model assumptions. The results of the ML estimates are shown in Table 2. For the PRP column, we assume the repair activities are perfect repairs and the failure intensity is as good as new. Thus, we can obtain the λ, β estimates and LKV using the Weibull++ 6 software. For the NHPP column, we assume that the repair actions restore the system operating state to be as bad as old. λ and β can be estimated using the RGA 6 software, which provides an excellent tool to analyze this type of repairable system. For the Proposed Model column, we use MLE to obtain λ, β and q estimates.


Table 1: Analysis Results Comparison

Table 2 shows that, based on LKV, the proposed model is the best fit for this data set. The q parameter is estimated to be 0.93156903. This value can be used as the restoration factor in BlockSim.

Figure 1 shows the cumulative number of failures. Cumulative number of failures depends on the virtual time. This plot can also be used to estimate the cumulative number of failures for future times.

Cumulative Number of Failures and Two-Sided 90% Confidence Bounds
Figure 1: Cumulative Number of Failures and Two-Sided 90% Confidence Bounds

Example 2
Suppose K = 6 systems are observed during [0, Ti ], i = 1, ..., k. That is, the data are time truncated with T1 = 8760, T2 = 5000, T3 = 6200, T4 = 1300, T5 = 2650 and T6 = 500. Failure data are given in Table 3.

Table 3: Failure Data for Repairable Systems
Table 3: Failure Data for Repairable Systems

This failure data set is from multiple repairable systems. We can estimate λ, β, q and the cumulative number of failures utilizing the data from all six systems in Table 3. Table 4 shows the results of ML estimates based on different models.


Table 4: Analysis Results Comparison

From Table 4, based on LKV, the proposed model is the best fit for this data set. The estimated q value, 0.55215907, can be used as the restoration factor in BlockSim. Figure 2 shows the cumulative number of failures and can be used to estimate the cumulative number of failures for future times. We can see that the proposed model fits these multiple repairable systems very well and provides promising results.

Cumulative Number of Failures vs. Time
Figure 2: Cumulative Number of Failures vs. Time

Conclusion
In this article, we explored a model based on the Weibull distribution for representing the reliability of complex repairable systems. The emphasis has been on solving problems with different types of data through model fitting and validation. A systematic MLE method is proposed for the parameters of the proposed model (including the restoration factor), by assuming values of the repair effectiveness (restoration factor) parameter of 0 and 1, the traditional ML estimators for NHPP and PRP can be obtained. Examples and procedures specifically illustrating these methods were given for two real world situations. The proposed method provides excellent predictions with the potential of becoming very useful in practice. Look forward for this analysis in upcoming ReliaSoft products.

References
1. Mettas, A. and Zhao, W. Modeling and Analysis of Complex Repairable Systems, Technique Report, ReliaSoft Corporation, 2004.

2. Wang, H. and Pham, H. "Some maintenance models and availability with imperfect maintenance in production systems." Annals of Operations Research, 91: 305-318, 1999.

3. Kijima, M. and Sumita, N. "A useful generalization of renewal theory: counting process governed by non-negative Markovian increments." Journal of Applied Probability, 23, 71-88, 1986.

4. Kijima, M. "Some results for repairable systems with general repair," Journal of Applied Probability, 20, 851-859, 1989.

5. Kaminskiy, M. and Krivtsov, V. "A Monte Carlo approach to repairable system reliability analysis." Probabilistic Safety Assessment and Management, New York: Springer; p. 1063-1068, 1998.

6. ReliaSoft Corporation, Reliability Growth and Repairable System Data Analysis Reference, 2004.

End Article

 

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