INTRODUCTION
Effective maintenance management is essential and critical as a way to reduce
the adverse effect of equipment failures and to maximize equipment availability.
The increase in equipment availability means higher productivity which translates
into higher profitability provided that the maintenance optimization does include
the cost reduction factor. This has lead to increase research interest in the
subject of optimizing maintenance management. It is estimated that 15 to 45%
of total production cost are attributed to maintenance cost with 30% of total
manpower involvement (AlNajjar and Alsyouf, 2003). The
total cost of maintenance will be significant; however, the consequence of an
inefficient maintenance management is far beyond the direct cost of maintenance
where, in most cases, it is not easily quantifiable. The maintenance’s
high cost and low efficiency is one of the last costs saving frontier for companies
to improve profitability (AlNajjar and Alsyouf, 2003).
One of the ways to improve maintenance efficiency is to have a better prediction
of the next failure occurrence so that all the resources involved can be optimized.
There are a number of models available to predict reliability and subsequently
equipment failure, however, there are obvious gaps observed between researchers
and practitioners of maintenance (Louit et al.,
2009). This is due to two main challenges, one, most models are based on
specific assumption that rarely represent actual operating conditions and two,
the scarcity of failure and maintenance data making statistical inferences inaccurate.
Various failure data sources have also been used to predict system reliability
as presented by Peiravi (2009). However, this approach
might introduce error due to different assumption of operating condition represented
by the failure data.
The current research focuses on the development of reliability assessment model
for repairable equipment by utilizing equipment performance data instead of
failure data. A repairable system is defined as a system which can be restored
to satisfactory working condition by repairing or replacing the damaged components
that caused the failure to occur other than replacing the whole system (Feingold
and Ascher, 1984).
The model was developed by utilizing system performance data which can either
be discrete or continuous. The performance data is defined as set of data that
are used to monitor the system output performance and is assumed to be directly
representing the health of a system. The data could be in the form of hourly
flow rates from a pump or amount of cooling in an hour for an air conditioning
system or other forms of measurement that can correlate to the condition of
the system. It is in a way analogous to the measurement of blood compositions
to indicate the state of person with respect to cancer progression (Gibson,
2008). In a lot of cases, these data are commonly available and closely
monitored as they are directly linked to total plant production output and thus
to revenue or income of an organization (Roberts and Barringer,
2001). For instance, the flow rate of a critical pump in a gas production
system is measured and monitored continuously with a calibrated flow meter.
Similarly, for a district cooling plant, the output from chillers in terms of
Refrigeration Ton hour (RTh) is also continuously monitored. The idea behind
this research is to use the abundantly available performance data to relate
to the system’s degradation and to predict the system reliability particularly
in cases where the failure data is limited or unavailable.
The use of performance data in reliability analysis is based on the following
assumptions:
• 
The system is assumed to be in operation if the performance
is above the minimum required level. If the performance falls below the
required level, the system is assumed to be in failed state 
• 
The performance level during standby is excluded from analysis as the
system is not considered operational, thus assumed no degradation in performance.
Standby time is identified based on production or maintenance records and
can be either in warm standby or cold standby 
• 
The changes in performance level and thus the system’s states occur
purely at random with no external intervention except for repairs and maintenance.
Changes to the performance level due to manual adjustments are excluded
from the analysis 
• 
The performance level as random variable is assumed to be Independent
and Identically Distributed (IID) from one interval to another. This means
that the level of system performance at one interval would not influence
the performance level at the next interval and they are both of same distribution 
MATERIALS AND METHODS
The proposed steps in performing the analysis are as shown in Fig.
1 which is based on the method proposed by Louit et
al. (2009) and is summarized as follows:
• 
Acquire performance data for the selected system. The data
chosen has to be from a calibrated metrology and would be able to represent
overall system performance 
• 
Define the minimum performance level where any performance falls below
this level is considered functionally failed. From the definition, Time
to Functional Failure (TTFF) can be calculated 
• 
Arrange the TTFF chronologically. Plot cumulative failures versus time.
Observe the trend 
• 
Perform independent test. This test is to confirm whether the there is
any influence from one functional failure to the next. The test is done
by plotting T_{n} (current failure) versus T_{n1} (previous
failure) and observe whether there is any clustering or trending indicating
dependent sample. Model for dependent TTFF is not part of the scope of study 
• 
Perform Mann test (NIST, 2011). The null hypothesis
for this nonparametric test is that the TTFF can be assumed as Renewal
Process (RP) and thus if this hypothesis is accepted, the TTFF can be analyzed
using distribution fitting. The alternative hypothesis is a monotonic trend
which will invalidate the analysis using distribution fitting. The test
statistic, M, is calculated based on the reverse arrangement of the data.
This means that if T_{1}, T_{2}, …, T_{n},
were the n failure data, then the reverse arrangement occurs whenever T_{i}<T_{j}
for each i<j. Therefore, in general: 
where I(.) is an indicator variables taking the value of 1 whenever the condition
is met.

Fig. 1: 
Proposed process flow for the TTFF (time to functional failure)
data to apply the appropriate model 
As M can be approximated by normal distribution, the calculated M value is
then compared to the standard value to decide whether the null hypothesis can
be accepted. The equation in relating M to z factor is as shown in Eq.
2:
where, r is the number of repair (or failure).
• 
Perform Laplace test which has the null hypothesis of homogenous
Poisson process versus an alternative hypothesis of nonhomogenous Poisson
process. If the hypothesis is accepted, then the times to failures are assumed
to be independent and identically (IID) exponentially distributed. The principle
behind the test is to compare the mean value of time to failures in an interval
with the midpoint of the interval. If the mean deviates from the midpoint,
then the data exhibit trending and thus cannot be assumed to be IID. The
test statistical value, L, follows a standard normal distribution and can
be calculated as below: 
where, T_{j} is the time to jth failure and [a,b] is the interval of
the observation.
on the other hand is given by:
• 
Apply the appropriate model based on both graphical test and
statistical tests 
Case study
System description: The system chosen to illustrate the applicability of
the model is one of the gas powered turbines operating in a gas district cooling.
The capacity of the turbine is 4.2 MW. The daily system performance is based
on peak hour demand between 8 am to 5 pm during weekdays. A total of 1290 days
of data were gathered and used in the analysis. Figure 2 shows
the plot system performance data versus days of production. The minimum production
limit is set based on research work done by Majid and Nasir
(2011) on a similarly configured system. The system is assumed to experience
functional failure whenever the system performance falls below the minimum production
limit. Based on this assumption, the TTFF is shown in Table 1.

Fig. 2: 
Production output (KW) versus days of production used to calculate
the TTFF (time to functional failure) 
Table 1: 
TTFF (time to functional failure) Data calculated based on
the time the system falls below the minimum production limit 

RESULTS AND DISCUSSION
The results provided in the following sections prove the possibility of using
performance data in reliability analysis in cases where failure data is not
available. This is evident from statistical analysis comparing actual function
failures to the predicted failures. However, each TTFF needs be tested according
to the proposed process flow so that appropriate analysis can be applied.
Test for independence: Test for independence was performed by using
serial correlation test (Kumar and Klefjso, 1992) which
will detect the presence of dependent data. In this test, the TTFF data was
plotted against a one lag time data as shown in Table 2 and
Fig. 3. Since the data was randomly scattered based on Fig.
3, it can be concluded that one failure to the next was independent. In
other words, the current failure did not have any influence over immediate subsequent
failure.
Trend test: The second step in the analysis is to perform a graphical
trend test based on the plot of cumulative time versus number of failure of
the data in Table 1.

Fig. 3: 
Result of dependency test for TTFF (time to functional failure)
which shows no correlation, thus failures are independent 

Fig. 4: 
Graphical trend test for TTFF (time to functional failure)
which shows good fit to the straight line indicating no trend in the failure
data 
The result is shown in Fig. 4 which indicates a good fit
for linear regression (R^{2} = 0.95). This means that the data did not
indicate any trend and thus can be assumed to be identically distributed.
Analytical trend test: To further validate the graphical test of IID,
two analytical tests were conducted namely Mann test and Laplace test. Both
tests were performed based on 95% confidence limits and the results are shown
in Table 3. Since the calculated values for both tests fall
between the z_{cr} values at 95% confidence interval, the data can be
assumed to be identically distributed.
Model assumptions: Based on both graphical and analytical tests, the
TTFF data can be safely assumed to be IID and exponentially distributed. As
such, the system failure rate can be estimated by fitting exponential distribution
to the TTFF data. The Probability Distribution Function (PDF) is shown in Fig.
5. Rank regression method was subsequently used to estimate the failure
rate, λ = 0.0097 failure day^{1}.

Fig. 5: 
PDF plot for TTFF (time to functional failure) which show
a good fit to exponential distribution 
Table 2: 
TTFF (time to functional failure) and lagtime TTFF used to
test for independence between two subsequent failures 

Table 3: 
Mann and Laplace test results showing no trend in failure
data at 95% confidence interval 

With the failure rate, the reliability function for the system with respect
to time can be represented as:
The plot of reliability function is shown in Fig. 6.
The reliability plots shows that system reliability is approaching zero at
640 days (or the probability of failure equals 1). This means that, there will
be at least 1 system failure when the system operates continuously for 640 days.
As mentioned above, the failure is when the system fails to meet the minimum
requirement and not the total system failure. This result can be used to schedule
a shutdown for planned maintenance.

Fig. 6: 
Reliability versus time plot showing the expected functional
failure to occur at 640 days 

Fig. 7: 
Effect to reliability as demand changes. As the demand increases,
the probability of not meeting the demand also increases 
The timing for the shutdown can also be based on system reliability target
as required by the organization. For example, a preventive maintenance should
be performed when the reliability figure falls below 20% which in this particular
case, between 5 to 6 months of operation time. Bear in mind that the reliability
is for the whole system with all modes of failure. Further analysis using the
maintenance data should be performed to understand failure modes so that proper
preventive action can be done during the planned shutdown.
The analysis can be further extended to look at the impact to reliability as
demand or the minimum requirement changes. Obviously, the higher the demand,
the more likely the system will not be able to meet the requirement. Figure
7 shows the changes in reliability as requirement changes.
Model validation: The proposed approach was validated by comparing the
predicted failure data to the actual system failure.

Fig. 8: 
Comparison between predicted failure and the actual occurrence
of functional failures 

Fig. 9: 
Statistical comparison between actual failures and predicted
failure 
The predicted failure is based on mean time between failures which is the reciprocal
of failure rate. The result shown in Fig. 8 shows the predicted
failure time based on mean time between failures compared with the actual occurrence
of failure. A statistical comparison using Dunnett’s test (Walpole
et al., 2007) shows statistical equivalent proving the validity of
the approach as shown in Fig. 9.
CONCLUSION
This study presents an alternative approach in assessing system reliability
based on abundantly available system performance data instead of failure data.
Based on the approach, the reliability result is actually measuring the probability
that the system is able to meet the minimum requirement for a given operating
condition for a stated period of time. The result is valuable in planning a
system shutdown depending on the organization’s reliability target.
However, this approach is only valid for a system level reliability encompassing
all modes of failures. As such, other information such as component failures
or types of failure are not apparent. Thus, maintenance data and condition monitoring
data are still essential in assessing the component level of reliability performance.
ACKNOWLEDGMENT
The authors wish to thank Universiti Teknologi PETRONAS for providing the necessary
support for this research.