## 2019

STATISTICS ROUNDTABLE

# Reliability Assessment By Use-Rate Acceleration

**by Necip Doganaksoy, Gerald J. Hahn and William Q.
Meeker**

Statistical evidence is often needed to show that a proposed product meets or exceeds its reliability goals. Many times, such evidence must be obtained in a compressed time period. For instance, the manufacturer of a newly designed washing machine might want its product to operate failure-free for five years or more, but might have only six months to prove this.

For products that are not used continuously, the desired validation can often be achieved by use-rate acceleration—that is, by running the product more frequently than under normal usage.

For example, by operating a toaster 100 times daily, you can simulate 10 years of operation in about 73 days, assuming a twice-a-day use rate by purchasers. Accelerated use-rate testing might be appropriate in testing other products such as photocopiers, printers, bicycles and laptop computers.

Such testing assumes the increased cycling rate will excite the failure modes seen in normal operations; for instance, failures that result directly from product operation and not, say, chemical change over time. This assumption is reasonable for many, but not all, failure modes. Failures due to corrosion provide a counter example; these are likely to be dependent on elapsed time, rather than usage.

Accelerated use-rate testing might be performed on the component, subsystem or system level. It requires selecting appropriate units for testing and establishing a plan that provides useful data for statistical analysis.

### Washing Machine Motor Example

A new model motor had been built for use in washing
machines.^{1} The motor was completely redesigned to
reduce noise and improve reliability. Skilled design engineers
used top quality materials and state-of-the-art methods to
correct reliability problems on previous designs. They also
performed short highly accelerated life tests (HALTs), subjecting
components and a few prototype motors to intensive temperature
cycling, vibration and overvoltage conditions to discover,
understand and remove potential failure modes.

The new motor’s reliability was required to be at least 97% after 10 years of operation. Some motor bearings could wear out, causing failures, but this was felt to be unlikely during the first 10 years.

The engineers were confident they had developed a highly reliable motor. But had they? Would 97% of the motors last 10 years? In light of the extensive design changes, experience and engineering judgment provided only baseline estimates. To really find out, the engineers had six months before product release to conduct an appropriate life test on new motors at conditions that simulated customer use.

The testing was intended to quickly identify and eliminate any remaining reliability problems. It also was meant to demonstrate, with 95% statistical confidence, that the desired reliability goal would be met.

### Testing Strategy

How could we obtain the equivalent of 10 years of field experience in six months? We did it by running a sample of motors continuously at stresses that simulated their operation in washing machines and shutting down for only brief cooldowns between periods of continuous running. (Such cooldowns were to activate potential failure modes due to temperature cycling.

This strategy allowed us to run 24 cycles daily, simulating
3.5 years

of field operation in each month of testing, assuming a use rate
of four washes per week. The testing was conducted on prototype
motors using special equipment that subjected them to mechanical
loads simulating those encountered during a typical washing
cycle. The underlying assumption that failures depend on motor
running time and shutdowns—independent of elapsed
time—seemed reasonable from engineering considerations.

### The Test Plan

Statistical evaluations^{2} showed that testing 66
motors for six months would provide an appropriate sample size,
balancing precision of information gained against cost. The
motors used for the test needed to reflect, as closely as
possible, the variability expected in large scale production.
Thus, the 66 motors were built at three different times, using
multiple lots of materials for each of the key assembled parts.
Detailed records of the motors’ histories and test
performance were maintained.

**Results after one month:** To everybody’s
surprise and consternation, after one month of testing (728 test
cycles or 3.5 years of customer use), four of the 66 motors had
failed. The resulting estimated reliability of 94%—(62/66)
x 100—in 3.5 years was far short of the goal of 97% in 10
years. (This shows how it is easier to prove inability to meet
reliability goals than to demonstrate high reliability.)

Physical evaluation indicated that a manufacturing defect was the root cause of the four failures. In fact, all these failures occurred during the first week of testing. Fortunately, it was easy for manufacturing to fix the process to avoid such failures on future motors, and the test was continued with the four failed motors replaced by new ones.

**Results after three months:** After three
months of testing (2,184 test cycles or 10.5 years of customer
use) four more failures occurred. One bearing failure took place
at 1,989 cycles, or the equivalent of 9.6 years of customer
use.

There were three additional failures (at 4.5, 6.7 and 7.5 years) due to malfunction of a plastic part. This failure mode also needed to be eliminated to meet the reliability goal. The design team felt confident it could correct this malfunction by changing the geometry of the plastic part. A separate program was initiated to validate the resulting design fix.

Assuming the other two identified failure modes (manufacturing defect and plastic part malfunction) are successfully corrected, only the bearing failure mode remained applicable in assessing the reliability of future motors. This resulted in an estimated 10.5-year reliability of 98.3% (58 out of 59 surviving motors), which exceeded the 10-year reliability goal of 97%.

However, this estimate, based upon testing a relatively small
sample of motors, was subject to much statistical uncertainty. To
account for this, a lower 95% confidence bound on reliability
(based on the binomial distribution^{3}) was calculated
as 92.2%. Generally speaking, this meant we could claim with 95%
confidence that the 10.5-year motor reliability was at least
92.2%. This value was appreciably less than the 10-year goal of
97%.

Although the three-month results appeared favorable, assuming successful fixes of the two identified problems, continued testing was required to narrow the gap between the estimated reliability and the lower confidence bound to a level that would demonstrate the desired reliability statistically.

**Results after six months:** After six months of
testing (4,368 test cycles; 21 years of customer use), seven
bearing failures and four additional plastic part failures
occurred, as shown in Table 1. Note the seven failures due to the
plastic part malfunction are taken to be censored observations in
this tabulation (for example, all that we know about their
failure times is that these exceeded their observed survival
times), under the assumption that this malfunction would not
occur on future motors.

### Analysis of Results After Six Months

Weibull and lognormal distributions were fitted to the
data.^{4,5,6} Both models are frequently used in such
applications and seemed reasonable on both theoretical and
empirical grounds. The findings from the two analyses were
similar. Therefore, only the Weibull distribution results are
shown in Figure 1.

The solid line in Figure 1 is the Weibull distribution
estimate (using maximum likelihood) of the percentage of devices
failing as a function of years in service. The dotted curve
provides approximate upper 95% confidence bounds on these failure
probabilities. Even though Figure 1 shows only the seven bearing
failures, the line was fitted taking all data into
consideration.^{7,8}

The estimated 10-year reliability was now 99.4% (failure probability of 0.006), with a 95% lower confidence bound of 96% (failure probability of 0.04). The plotted points in Figure 1 (p. 75) scatter around a straight line, supporting the Weibull distribution model assumption within the range of the data.

The 95% lower confidence bound of 96% on 10-year reliability just missed providing the desired demonstration of 97%. But the 97% demonstration can be made with 92% confidence—and this was judged sufficient for production start-up.

### Further Evaluation and Testing

All failed motors, plus a sample of the unfailed ones, were taken apart and evaluated to obtain information to improve future product reliability.

Ten surviving motors were selected randomly for another 4,000 cycles of testing to obtain more precise reliability estimates. In addition, 25 units built during a one-month period and 25 units randomly sampled from the first week of high volume production—all incorporating the two fixes—were tested for varying times. The results confirmed that the earlier problems had been successfully resolved.

Finally, to check for possible new process problems, five motors are selected randomly each week from production. Four are tested for one week and one for three months.

### Other Situations

In this washing machine example, 10 years of normal operation was simulated in three months by accelerating the use rate. For products that operate continuously, such as refrigerators and power generation equipment, this type of accelerated test is not possible. In such cases, testing might be conducted under more severe environments, such as higher temperature or humidity, to accelerate the physical or chemical degradation process that causes certain failures, such as the weakening of an adhesive bond. Accelerated testing also might involve exposing the units to increasing amounts of stress, such as voltage or pressure.

Statistical concepts can be used in both cases to develop a
life test of one or more of the accelerating variables at various
conditions. The resulting data are used to predict time to
failure at normal operating conditions, with statistical
confidence bounds, based on a physically appropriate mathematical
model that relates the accelerating variable to time to
failure.^{9}

### REFERENCES

- Necip Doganaksoy, Gerald J. Hahn and William Q. Meeker, “Assuring Product Reliability and Safety,” Statistics: A Guide to the Unknown, fourth edition, Duxbury Press, 2006.
- William Q. Meeker, Gerald J. Hahn and Necip Doganaksoy, “Planning Reliability Assess-ment,” Quality Progress, June 2005, pp. 90-93.
- Gerald J. Hahn and William Q. Meeker, Statistical Intervals: A Guide for Practitioners, Wiley, 1991, section 6.2.
- Necip Doganaksoy, Gerald J. Hahn and William Q. Meeker,
“Product Life Data Analysis: A Case Study,” Quality
Progress, June 2000,

pp. 115-121. - William Q. Meeker and Luis A. Escobar, Statistical Methods for Reliability Data, Wiley, 1998.
- Wayne B. Nelson, Applied Life Data Analysis, Wiley, 1982.
- Necip Doganaksoy, William Q. Meeker and Gerald J. Hahn, “Divide and Conquer: Reliability Analysis by Individual Failure Modes,” Quality Progress, June 2002, pp. 47-52.
- Meeker and Escobar, Statistical Methods for Reliability Data, see reference 5, section 6.4.
- Gerald J. Hahn, William Q. Meeker and Necip Doganaksoy, “Speedier Reliability Analysis,” Quality Progress, June 2003, pp. 58-64.

**NECIP DOGANAKSOY** is a statistician and Six
Sigma Master Black Belt at the GE Global Research Center in
Schenectady, NY. He has a doctorate in administrative and
engineering systems from Union College in Sche-nectady.
Doganaksoy is a senior member of ASQ and a fellow of the American
Statistical Assn.

**GERALD J. HAHN** is a retired
manager of statistics at the GE Global Research Center in
Sche-nectady, NY. He has a doctorate in statistics and operations
research from Rensselaer Polytechnic Institute in Troy, NY, where
he is also an adjunct faculty member. Hahn is a fellow of ASQ and
the American Statistical Assn.

**WILLIAM Q. MEEKER** is professor of statistics
and distinguished professor of liberal arts and sciences at Iowa
State University in Ames, IA. He has a doctorate in
administrative and engineering systems from Union College in
Schenectady, NY. Meeker is a fellow of ASQ and the American
Statistical Assn.

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