This document discusses various measures of reliability in engineering. It defines key terms like quality, reliability, availability, and maintainability. It describes the bathtub curve with the three phases of failure rate over time. It also provides examples of how to calculate reliability using different failure distributions like normal, exponential, and Weibull distributions. Common measures of reliability discussed are mean time between failures (MTBF), mean time to failure (MTTF), and mean time to repair (MTTR).

1. RME-085
Total Quality Management
Topic: Reliability: Its evaluation and measures
By:
Dr. Vinod Kumar Yadav
Department of Mechanical Engineering
G. L. Bajaj Institute of Technology and Management
Greater Noida
Email: vinod.yadav@glbitm.org

2. Reliability Vs Quality:
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
Quality: Conformance to specification.
Reliability: How well the product maintains
its original level of quality over time.
Maintainability and Availability
Availability: Proportion of time for which
an item is not failed.
- More usefully unavailability (1 −
availability) describes the proportion of
time for which an item is failed.
Maintainability: The probability that a
failed item will be restored to operational
effectiveness within a given period of time
when the repair action is performed in
accordance with prescribed procedures.
Ways of improving Reliability
• By using proven, safe and simple designs.
• By proper testing before use
• By using redundant parts in high risk areas.
• By using components in parallel
• By using proven manufacturing methods.

3. Life-history curve (Bathtub curve)
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
Life-history curve (3-phases):
• Compares failure rate with time [1].
• Probability distributions (Fig. 2) are used to describe the three phases of bathtub curve.
Fig. 2 Failure rate as time function [1]
Fig. 1 Life history of a complex product for an infinite number of Items [1]

4. Life-history curve (Bathtub curve) contd.
Phase-II: Chance failure phase:
• Horizontal line - Rate of failure is constant (Random failures).
• The assumption of a constant failure rate is valid for most
products; however, some products may have a failure rate that
increases with time.
• Few products show a slight decrease, which means that the
product is actually improving over time.
• The exponential distribution and the Weibull distribution (Shape
parameters β = 1) are used to describe this phase of the life
history.
• When the curve increases or decreases, a Weibull shape parameter
greater or less than 1 can be used. Reliability studies and sampling
plans are, for the most part, concerned with the chance failure.
• Lower failure rate – better product.
Fig. 1 Life history of a complex product for an infinite number of Items [1]
I II III
Phase-I: Debugging phase or Burn-in phase or
infant-mortality phase:
• Marginal and short-life parts that cause a rapid
decrease in the failure rate.
• Weibull distribution (Shape parameters β < 1) is used
to describe the occurrence of failures.
• For some products, the debugging phase may be part
of the testing activity prior to shipment.
• For other products, this phase is usually covered by
the warranty period.
Phase-III: Wear-out phase
• Sharp rise in the failure rate
• Normal distribution best describes wear-out phase.
• Weibull distribution (Shape parameters β > 1) can be used

5. Normal Failure Analysis
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
• Although the normal curve is applicable to the wear-out phase, the
Weibull is usually used.
• Reliability can be determined using above formula
Table 1 can be used to find the area
under the curve to the left of time, t
, and is obtained from Appendix
Table 1, which is:
Table 1- Area under Normal curve (Adopted from Appendix a of Ref [1]
Rt = 1- P(t)
Problem 1: An ignition lighter has a
mean life of 1000 hours and standard
deviation of 500 hours. Calculate the
reliability at 1500 hours
Z =
𝑿− 𝜽
σ
Z =
𝟏𝟓𝟎𝟎−𝟏𝟎𝟎𝟎
𝟓𝟎𝟎
= 1.0
For Z = 1.0 P(t) from Table-1 is 0.8413
Rt at 1500 h = 1- P(t) = 1- 0.8413 = 0.1587 Or
15.87 %
Hence, 1587 out of 10000 lighters will last in
1500 h

6. Exponential Failure Analysis
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
• Exponential distribution and the Weibull distribution
(with β = 1) are used to describe the constant failure rate.
• Reliability can be determined using formula:
• t = time or cycles
• 𝜽 = mean life
Problem 2: Calculate the reliability of a product at (i) t =
50 h (ii) 60 h and (iii) 70 h. The mean life for a constant
failure rate was 60 h.
Ans:
• At 50 h Rt = 0.434
• At 60 h Rt = 0.367
• At 70 h Rt = 0.311
Rt = 𝒆−𝒕/𝜽
Rt with time
Weibull Failure Analysis
• Weibull distribution can be used for any of the
three phase of the bathtub curve (with β = 1, β < 1
or β > 1).
• If β = 1 Weibull approximates Exponential
distribution
• If β = 3.4 Weibull approximates Normal distribution
• β = Weibull slope (β and 𝜽 can be determined
graphically or analytically)
Problem 3: The failure pattern of a product follows
the Weibull distribution with slope 3.8 and mean life
𝜽 = 90 h. Determine its reliability at 100 h.
Ans: Rt = 0.224
Rt = 𝒆
−
𝟏𝟎𝟎
𝟗𝟎
𝟑
.
𝟖
Rt = 𝒆−(
𝒕
𝜽
)β
Power

7. Measures of Reliability
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
• Mean Time Between Failures (MTBF) - For repairable products – How
reliable a product is (measured in thousands or ten thousands of hours between failures).
• MTBF = Total operational time/ Number of failures
• MTBF = Available time / Number of failures
= (Planned time - Down Time) / Number of failures
• MTBF measures availability and reliability.
• Higher the value of MTBF, more reliable the product is.
• Mean Time To Failure (MTTF) - For non-repairable product : Maintenance
metric that measures the average amount of time a non-repairable component operates
before its failure. It’s the average life time of an asset that is irreparable.
• It’s the life time of any product or device.
• MTTF = Total operational time/Number of units under test.

8. Measures of Reliability contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU
Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
• Mean Time To Repair
• Refers to the amount of time required to repair a system and restore it to full functionality.
MTTR = Total maintenance time / Number of failures
MTTR = Down Time / No. of Failures
• MTTR measures Availability.
Availability = MTBF / (MTBF + MTTR) %
Reliability = e-(AT/MTBF)

9. References:
[1] Dale H. Besterfiled. A Text book on Quality Improvement. 9th Edition. Pearson (ISBN 10: 0-13-262441-9) pp: 169-184.

10. Thank you