Some effective control chart procedures for reliability monitoring

Reliability Engineering and System Safety 77 (2002) 143–150 www.elsevier.com/locate/ress

Some effective control chart procedures for reliability monitoring M. Xie*, T.N. Goh, P. Ranjan Department of Industrial and Systems Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore, Singapore 119260 Received 1 February 2002; accepted 29 March 2002

Abstract Control charts are widely used for process monitoring in the manufacturing industry. Little research is available on their use to monitor the failure process of components or systems, which is important for equipment performance monitoring. Some Shewhart control charts, especially those for the number of defects, can be used for monitoring the number of failures per fixed interval; however, they are not effective especially when the failure frequency becomes small. A recent control scheme based on the cumulative quantity between observations of defects has been proposed which can be easily adopted to monitor the failure process for exponentially distributed interfailure time. An investigation of its use for reliability monitoring is presented in this paper and the scheme can be easily extended to monitor inter-failure times that follow other distributions such as the Weibull distribution. Furthermore, the scheme is extended to the monitoring of time required to observe a fixed number of failures. The advantages of this scheme include the fact that the scheme does not require any subjective sample size, can be used for both high and low reliability items and can detect process improvement even in a high-reliability environment. q 2002 Published by Elsevier Science Ltd. Keywords: Reliability monitoring; Control charts; High reliable system; Gamma distribution; Improvement detection

1. Introduction Failure process monitoring is an important issue for complex or repairable systems. It is also a common problem for a fleet of systems such as equipment or vehicles of the same type in a company. Statistical control chart is a widely used process monitoring tool in the manufacturing industry; and it can be used in this type of failure process monitoring. This is usually done by plotting the number of failures or breakdowns per unit time such as week or month. Standard c-chart or u-chart, which is for the monitoring of the number of defects in a sample, can then be used for this purpose. However, this procedure requires a large number of failures and it is not appropriate for application to a highly reliable system. Fig. 1 is a typical example of periodic failure reports monitored with a c-chart. When there is an excessive number of failures, the chart will signal in such an out-of-control situations. Although the anticipated false alarm probability, the probability that the process is not changed when the plot shows an alarm, is 0.27% by a traditional chart, it could be much higher because when the number of failures is Poisson distributed, normal distribution which is used, is not a good approximation when the * Corresponding author. Fax: þ 65-6777-1434. E-mail address: [email protected] (M. Xie). 0951-8320/02/$ - see front matter q 2002 Published by Elsevier Science Ltd. PII: S 0 9 5 1 - 8 3 2 0 ( 0 2 ) 0 0 0 4 1 - 8

average number of failures is small. Moreover, the lower control limit (LCL) is usually set at zero, which is not useful because then process improvement cannot be detected. Chan et al. [1] recently proposed a procedure based on the monitoring of cumulative production quantity between the observations of two defects in a manufacturing process. This approach has shown to have a number of advantages: it does not involve the choice of a subjective sample size; it raises fewer false alarms; it can be used in any environment irrespective of whether the process is of high quality; and it can detect further process improvement. Since the quantity produced between the observations of two defects is related to the time between failures in reliability study, this approach can be readily adopted for process monitoring in reliability and maintenance. This will be investigated in this paper. As a brief review of related research, process monitoring of reliability related process characteristics has attracted some attention recently. Katter et al. [2], used the control chart to monitor the on-line welder condition. Steiner and Mackay [3] showed the use of control chart to detect process changed for censored data. Cassady et al. [4] introduced a combined control chart-preventive maintenance strategy. Haworth [5] showed how the multiple regression control charts could be used to manage software maintenance processes. Kopnov and Kanajev [6] discussed bearing

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parameter l is given by Fðt; lÞ ¼ 1 2 e2lt ;

Fig. 1. The traditional u-chart for the monitoring of number of failures per unit time.

degradation in a process and obtained an optimal control limit by defining the degradation process, considering the costs involved. These studies mainly follow the standard approach. The use of cumulative quality is a different and new approach, which is of particular advantage in reliability and maintenance engineering. We further extend the procedure to the case when cumulative time to the rth failure is used to monitor the failure process. The implementation and interpretations are provided and numerical examples are used to illustrate the application procedure. We also investigate some basic properties of the proposed scheme.

2. Control chart monitoring the time between failures 2.1. Distribution of time between failures For a system in normal operation, failures are random event caused by, for example, sudden increase of stress and human error. The failure occurrence process can usually be modelled by a homogeneous Poisson process with a certain intensity. Hence, our aim here is to monitor the failure process and detect any change of the intensity parameter. The procedure in Ref. [1] is based on the monitoring of the cumulative production quantity between observing two defects in a manufacturing process. This cumulative production quantity is an exponential random variable that describes the length till the occurrence of next defects in a Poisson process. When process failures can be described by a Poisson process, the time between failures will be exponential and the same procedure can be used in reliability monitoring. Here, we briefly describe the procedure for reliability monitoring. Since time is our preliminary concern, the control chart will be termed t-chart in this paper. This is in line with the traditional c-chart or u-chart, which our t-chart can be more suitable alternative. In fact, the notation also makes it easier for the extension to be discussed later. The distribution function of exponential distribution with

ð1Þ

t$0

The control limits for t-chart are defined in such a manner that the process is considered to be out of control when the time to observe exactly one failure is less than LCL, TL, or greater than upper control limit (UCL), TU. When the process is normal, there is a chance for this to happen and it is commonly known as false alarm. The traditional false alarm probability is set to be 0.27% although any other false alarm probability can be used. The actual acceptable false alarm probability should in fact depend on the actual product or process. Assuming an acceptable probability of false alarm of a, the control limits can be obtained as: TL ¼ l21 ln

1 1 2 a=2

and

TU ¼ l21 ln

2 a

ð2Þ

The median of the distribution is usually called the center line (CL), TC, and it can be computed as TC ¼ l21 ln 2 ¼ 0:693l21

ð3Þ

These control limits can then be utilized to monitor the failure times of components. After each failure, the time can be plotted on the chart. If the plotted point falls between the calculated control limits, it indicates that the process is in the state of statistical control and no action is warranted. If the point falls above the UCL, it indicates that the process average, or the failure occurrence rate, may have decreased which resulted in an increase in the time between failures. This is an important indication of possible process improvement. If this happens, the management should look for possible causes for this improvement and if the causes are discovered then action should be taken to maintain them. If the plotted point falls below the LCL, it indicates that the process average, or the failure occurrence rate, may have increased which resulted in a decrease in the failure time. This means that process may have deteriorated and thus actions should be taken to identify and remove them. In either case, the people involved can know when the reliability of the system is changed and by a proper follow up they can maintain and improve the reliability. Another advantage of using the control chart is that it informs the maintenance crew when to leave the process alone, thus saving time and resources. It can be noted here that the parameter l should normally be estimated with the data from the failure process. Since l is the parameter in the exponential distribution, any traditional estimator can be used and we omit this discussion here. 2.2. An example The procedure of a t-chart for failure process will be

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Table 1 Time between failures of a component Failure number

Time between failure (h)

Failure number

Time between failure (h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

30.02 1.44 22.47 1.36 3.43 13.2 5.15 3.83 21 12.97 0.47 6.23 3.39 9.11 2.18

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

15.53 25.72 2.79 1.92 4.13 70.47 17.07 3.99 176.06 81.07 2.27 15.63 120.78 30.81 34.19

illustrated with an example here. Table 1 shows the time between failures in hours. The first 20 values correspond to a historical l of 0.1. The last 10 points were simulated when the process average is shifted to 0.01, which means that the process has improved. In fact, Fig. 1 was plotted using this data assuming an observation interval of 15 h. That is, the number of failures in each 15-hour interval is used to plot the c-chart. Note that the LCL is set to zero since no observation can be negative. The control limits of t-chart for the same data set with a ¼ 0:0027 can be calculated as: lnð1 2 a=2Þ ¼ 0:0135; TL ¼ 2 l TU ¼ 2

lnða=2Þ ¼ 66:1 l

and Tc ¼ 2

lnð0:5Þ ¼ 6:9 l

The t-chart is shown in Fig. 2. It clearly gives an indication that the process average has shifted (improved). Again, this is a signal that was not captured by c-chart in Fig. 1 unless some complicated run rules are used.

Fig. 2. A t-chart for the observed time between failures in Table 1.

2.3. Monitoring Weibull distributed time between failures The procedure can be easily generalized to other distributions. Here, we use the Weibull distribution to illustrate this. When items are replaced with a new one and the lifetime of each item follows the Weibull distribution with the distribution function "
 # t b FðtÞ ¼ 1 2 exp 2 ; t . 0; ð4Þ u The LCL, TL, and UCL, TU can be computed as  1=b 1 LCL ¼ u and lnð1 2 a=2Þ  1=b 1 UCL ¼ u lnða=2Þ

ð5Þ

Hence, a procedure could be easily developed by monitoring the time between failures with the above formula. When the time between failures is less than LCL, it is likely that there are assignable causes leading to significant process deterioration and it should be investigated. On the other hand, when the time between failures has exceeded the UCL, there are probably reasons that have lead to significant improvement. The policy or approach that had helped this should be maintained or even implemented for other similar systems or equipment. It should be noted that although similar ideas could be used for other distributions. The Weibull distribution is probably the most widely used one and it is very flexible for modeling increasing or decreasing failure rate. Sun et al. [7] used the Weibull distribution to model the time to failure of a hard disk drive and came out with a failure percent control chart. On the other hand, when systems undergo regular maintenance or replacement, the actual failure process can be modeled with a Poisson process [8]. This is the case because maintenance actions will prevent the system to

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enter the wear-out region quickly and the common situation such failure process is monitored. Another important justification for Poisson process when monitoring failure process of a large number of systems is that this can be considered as a superposition of a number of failure processes one for each system. It is shown in Hoopen and Reuver (1966) [9] that under some general conditions, the superposition of random sequence of events approaches a Poisson process. Hence, we will focus on the use of exponentially distributed lifetime here.

3.1. The distribution of Tr To monitor the system based on the time between the occurrences of r failures, we need a distribution to model the cumulative time till the rth failure, Tr : It is well known that the sum of r exponentially distributed random variables is the Erlang distribution. An Erlang random variable is defined as the length until the occurrence of r defects (failures) in a Poisson process. Then the probability density function of Tr is given as: ð6Þ

The cumulative Erlang distribution is Fðtr ; r; lÞ ¼ 1 2

rX 21 k¼0

ðltr Þk exp{ 2 ltr } k!

FðUCLr ; r; lÞ ¼ 1 2

rX 21

e2lUCLr

k¼0 rX 21

e2lCLr

k¼0

Monitoring the failure occurrence process using the tchart is straightforward. However, since the decision is based on only one observation, it may cause many false alarms or it is insensitive to process shift if the control limits are wide (with small value of a ). To deal with this problem, we can monitor using the time between r failures. Denote the time to observing the rth failure by tr ; a tr-chart is proposed and studied here. A practical scenario is that when the reliability of a complex system is to be monitored and the failure of any components or incident is reported, the occurrence process can be modeled by a Poisson process. In fact, when components are replaced, we have a superposition of renewal processes. As mentioned earlier, the superimposed process can be approximated by a Poisson process, and hence providing an important justification for the use of our model.

lr trr21 exp{ 2 ltr } ðr 2 1Þ!

To calculate the control limits of the tr-chart, the exact probability limits will be used. If a is the accepted false alarm risk then the upper control limit, UCLr, the center line, CLr, and the lower control limit, LCLr, can be easily calculated by using Eq. (7) in the following manner:

FðCLr ; r; lÞ ¼ 1 2

3. Monitoring time between r failures

f ðtr ; r; lÞ ¼

3.2. Control limits of tr-chart

ð7Þ

This Tr can then be used to model the time to the rth failure in a Poisson process. We propose another chart; henceforth known as the tr-chart, to monitor Tr : It should be noted that for r ¼ 1; both the Gamma distributions reduce to the exponential distribution. Hence, the t-chart is the same as our t1-chart.

ðlUCLr Þk ¼ 1 2 a=2 ð8Þ k!

ðlCLr Þk ¼ 0:5 k!

ð9Þ

and FðLCLr ; r; lÞ ¼ 1 2

r21 X

e2lLCLr

k¼0

ðlLCLr Þk ¼ a=2 k!

ð10Þ

The control limits can be easily calculated using some mathematical or statistical software such as Mathematica. Table 2 shows the calculated control limits for some trcharts with the false alarm risk, a ¼ 0:0027: It should be noted that UCLr and LCLr appear in Eqs. (8) – (10) in product with l, the control limits are inversely proportional to l. That is, when l is increased by a factor, the limits will be decreased by the same factor. The decision-making procedure for the tr-chart remains same as the t-chart. A point plotted below the LCL signifies deterioration of process and warrants corrective action, while a point plotted above the UCL denoted process improvement and action should be taken to identify the cause of improvement and to maintain it. From reliability point of view, a point plotted below the control limit points out that the time between failures may have decreased while a point plotted above the control limit indicates that the time between failures may have increased. 3.3. An illustrative example Table 3 shows a set of failure time data. The first 30 times were simulated with the process average, l, of 0.001 and the second 30 times were simulated with the process average changed to l ¼ 0:003: The accepted false alarm risk is a ¼ 0:0027: Fig. 3 shows the t-chart for the data in Table 4. It can be seen that the t-chart fails to raise an alarm. The control limits of the t-chart are UCL ¼ 6607:7 and LCL ¼ 1:4: Since t-chart makes use of a single observation in decision making, a t3-chart could be used if more observations are to be taken into consideration in an easy way. The data shown in Table 3 is converted into the data of Table 4 which shows the cumulative time to failure between every three failures, i.e. T3. The performance of t-chart can be compared with the performance of t3-chart by using the data of Table 4. The control limits of the t3-charts can be calculated by solving

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Table 2 Some control limits of some tr charts with a ¼ 0:0027

l

UCL

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

t1 66 071.4 33 036.9 22 025 16 518.9 13 215.2 11 012.7 9439.5 8259.6 7341.8 6607.7

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

t3 108 54 36 27 21 18 15 13 12 10

CL

695 347.6 230.1 173.8 738.4 115.9 527.9 586.8 077 869.3

LCL

6931.5 3465.7 2310.5 1732.9 1386.3 1155.3 990.2 866.4 770.2 693.1

13.5 6.8 4.5 3.4 2.7 2.3 1.9 1.7 1.5 1.4

26 740.6 13 370.3 8913.5 6685.2 5348.1 4456.8 3820.1 3342.6 2971.2 2674.1

2116.9 1058.4 705.6 529.2 423.4 352.8 302.4 264.7 235.2 211.7

UCL

CL

t2 88 999.2 44 500 29 667.3 22 250.5 17 800.4 14 833.7 12 714.6 11 125.3 9888.3 8899.9

16 783.5 8391.7 5594.5 4195.9 3356.7 2797.2 2397.6 2097.9 1864.8 1678.4

528.8 264.5 176.3 132.2 105.8 88.2 75.6 66.1 58.8 52.9

t4 126 63 42 31 25 21 18 15 14 12

36 720.6 18 360.3 12 240.2 9180.2 7344.1 6120.1 5245.8 4590.1 4080.1 3672.1

4653 2326.5 1551 1163.2 930.6 775.5 664.7 581.6 517.1 465.3

805 402.1 268.2 700.2 360.9 132.8 114.9 850.3 089.4 680.5

LCL

the following equations: Table 3 Failure time data of the components Failure number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Time 1065.55 535.8 540.53 716.2 2525.43 1264.18 479.44 1783.22 473.67 2265.42 2191.75 1097.26 597.59 971.16 3157.29 2932.96 987.67 1816.18 117.21 190.65 943.99 1084.48 2306.54 6.56 3111.51 283.86 659.39 683.48 36.14 754.16

12

2 X

e20:001UCL3

k¼0

Failure number 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Time 35.85 362.8 357.85 334.48 80.13 1939 77.88 4.03 98.67 17.19 289.79 63.99 2.46 697.68 1167.33 239.66 93.78 680.45 4.83 102.91 479.05 156.67 1286.24 443.97 360.03 414.66 128.9 36.1 197.31 418.12

12

2 X k¼0

12

2 X

e20:001CL3

ð0:001UCL3 Þk ¼ 0:99865; k!

ð0:001CL3 Þk ¼ 0:5; k!

e20:001LCL3

k¼0

ð0:001LCL3 Þk ¼ 0:00135 k!

Solving the above equations, the control limits of the t3-chart are obtained as UCL3 ¼ 10 869:3;

CL3 ¼ 2674:1;

LCL3 ¼ 211:7

The t3-chart is shown in Fig. 4. As can be seen from the figure, the t3-chart raises an alarm signifying process deterioration from l ¼ 0:001 to l ¼ 0:003: This can be

Fig. 3. The t-chart for the data in Table 5 and no alarm is raised.

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Table 4 Cumulative failure time between every three failures Observation number

Time to the accumulation of three failures

Observation number

Time to the accumulation of three failures

1 2 3 4 5 6 7 8 9 10

2141.88 4505.81 2736.33 5554.43 4726.04 5736.81 1251.85 3397.58 4054.76 1473.78

11 12 13 14 15 16 17 18 19 20

756.5 2353.61 180.58 370.97 1867.47 1013.89 586.79 1886.88 903.59 651.53

compared with the t-chart shown in Fig. 3, which fails to raise an alarm. Hence, by accumulating three failure times, the sensitivity of monitoring to detect process change can be improved compared with t-chart. On the other hand, it is a subjective issue what r should be used in tr-chart. Usually r should not be too large, as it may need to accumulate a long time before a decision is made. In Section 4, we investigate the properties for tr-chart for reliability monitoring and discuss the issue of chart sensitivity and implementation.

4. Some statistical properties of tr-chart One of the most frequently used terms associated with the control charts for process monitoring is the average run length (ARL) which is generally defined as the average number of points that must be plotted on the control chart before a point indicates an out of control situation. A good control chart should have a large ARL when the process is in control and small ARL when the process shifts away from the target. This means that when the process runs in control, the control chart should raise few false alarms while on the other hand, when the process runs out of control the control chart should raise frequent false alarms to indicate the shift in the process parameters. Due to these reasons, it becomes important for any type of control chart to exhibit desirable ARL property.

Assume that the probability for the time Tr falling within the control limits of the t3-chart can be denoted by br : Then, br can be represented as:

br ¼ FðUCLr ; r; lÞ 2 FðLCLr ; r; lÞ

ð11Þ

where Fðt; r; lÞ is the distribution of Tr : Using Eq. (7), the probability that the points do not fall between the control limits which is represented as ð1 2 br Þ can be obtained as 1 2 br ¼ 1 " # rX 21 rX 21 k k 2lLCLr ðlLCLr Þ 2lUCLr ðlUCLr Þ 2 e e 2 k! k! k¼0 k¼0 ð12Þ On an average, only one out of 1=ð1 2 br Þ points falls outside the control limits. If the process failure occurrence rate is l, on an average, r defects will occur for r=l (the mean of the Erlang distribution) items inspected. The average run length of the t3-chart (ARLr) can then be represented as: ARLr ¼ EðTÞ

1 ð1 2 br Þ

ð13Þ

Substituting the values of EðTÞ and ð1 2 br Þ; Eq. (14) can be written as: ARLr ¼ £

"

l 1þ

r rX 21 k¼0

e

2lUCLr

rX 21 ðlUCLr Þk ðlLCLr Þk 2 e2lLCLr k! k! k¼0

#

ð14Þ

Fig. 4. The t3-chart for the data in Table 4.

The UCLr and LCLr can be calculated as before. Some ARLr values for l0 ¼ 0:001 (process average) a ¼ 0:0027 are shown in Table 5. As it is evident from the table, when the process is in control, the ARL value is quite large and when the process average shifts the ARL value decreases, thus proving that the t3-chart has a desirable ARL property. Fig. 5 shows some ARLr curves for l0 ¼ 0:001 and a ¼ 0:0027:

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Table 5 Some ARL values for t3-chart with lo ¼ 0:001 and a ¼ 0:0027

l

ARL-1

ARL-2

ARL-3

ARL-4

0.00001 0.00002 0.00005 0.00008 0.0001 0.0002 0.0005 0.0008 0.001 0.002 0.003 0.004 0.005 0.008 0.01 0.02 0.05 0.08 0.1

106 829 57 062.4 27 827.4 21 203.3 19 357.7 18 726.8 53 451.5 203 557 370 495 185 310 82 471.3 46 421.4 29 729.7 11 636.7 7457.5 1877 306.4 122.1 79.2

200 749 101 428 43 196.7 29 761 25 767.9 21 327.5 62 504.3 336 113 740 622 191 772 58 841.3 25 700.3 13 621.1 3685 2018.7 350.3 54 27.1 20.7

300 059 150 219 61 093.5 39 806.2 33 221.3 23 819.2 64 736.5 432 901 1 110 930 162 343 37 421.5 13 791.3 6567.1 1555.5 845.2 188.9 60.1 37.5 30

400 004 200 028 80 327.5 51 014.5 41 665.5 26 672.5 64 815.6 505 458 1 481 480 133 114 25 032.6 8421.8 3882.3 979.4 585.8 203.5 80 50 40

From Table 5 it can be seen that ARLr increases when l , 0:0002; which is due to the effect of the term r=l on ARLr. When l becomes small (approaches zero), it tends to increase the ARL. In addition, 1=ð1 2 br Þ tends to decrease the ARL when the process improves but its effect is less dominant as compared to the other effect for small values of l. As a result, when the process improves (i.e. l decreases), the ARLr first decreases and then increases. From Fig. 5 it can also be seen that as l increases, the ARLr drops more sharply than the traditional t-chart and thus the t3-charts (for r . 1) are more sensitive to process deterioration than the t-chart. Chan et al. [1] had shown the ARL property of t-chart with only LCL, which means that only the process deterioration was of interest. Fig. 6 shows the ARLr curves of some t3-charts having only a LCL with l0 ¼ 0:001 and a ¼ 0:00135:

5. Conclusions Until now, statistical control charts have been mostly

Fig. 6. Some ARL curves of t3-charts (with only a LCL) with l0 ¼ 0:001 and a ¼ 0:00135:

used to monitor production processes. Although reliability monitoring, especially that for complex equipment or fleet of systems is an important subject, little study has been carried out on the applications of traditional control chart for defects such as the c-chart or u-chart. In fact, they might not be suitable unless the number of failures per monitoring interval is large. If the time interval itself is long, such as months or quarters, deteriorating systems will not be detected quickly. In this paper, we have studied the use of control charting technique to monitor the failure of components. To deal with the problems suffered by conventional quality control charts, the monitoring procedure specifically based on exponential distribution can be used. Extension of this to other lifetime distribution is straightforward although there are usually simple justifications for using exponential distribution, especially for maintained system and fleet of systems. The procedure can also be extended to general nonrepairable system when non-homogeneous Poisson process has to be used as well. For this type of processes, a timedependent intensity function is needed and the data can be transformed to another time-scale so that the process becomes Poisson process. On the other hand, for that non-homogeneous Poisson process, what is usually more important is to predict the trend rather than monitoring the failure process itself. In this paper, a new procedure based on the monitoring of time to observe r failures is also proposed and it can be more appropriate for reliability monitoring. It differs from the tchart in the sense that it plots the cumulative time until observing r failures. This procedure is useful and more sensitive when compared with the t-chart although it will wait until r failures for a decision. Statistical properties of this procedure are investigated. Also notable is the fact that with the traditional c-chart and u-chart, when the lower limit is commonly set to zero, further improvement cannot be detected since no defect count can be below that. The new procedure is able to detect process improvement as well as deterioration.

Acknowledgments Fig. 5. Some ARL curves of t3-charts with l0 ¼ 0:001 and a ¼ 0:0027:

Part of this research was supported by the National

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M. Xie et al. / Reliability Engineering and System Safety 77 (2002) 143–150

University of Singapore under the research grant RP3992679 for a project entitled ‘Reliability Analysis of Engineering Systems’.

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