RELIABILITY

Dr. Ron LembkeSCM 352

Reliability Ability to perform its intended

function under a prescribed set of conditions Probability product will function when

activated Probability will function for a given

length of time

Measuring Probability Depends on whether components

are in series or in parallel

Series – one fails, everything fails

Measuring Probability Parallel: one fails, everything else

keeps going

Reliability Light bulbs have 90% chance of

working for 2 days. System operates if at least one

bulb is working What is the probability system

works?

Reliability Light bulbs have 90% chance of

working for 2 days. System operates if at least one

bulb is working What is the probability system

works?Pr = 0.9 * 0.9 * 0.9 = 0.72972.9% chance system works

Parallel

90% 80% 75%

Parallel 0.9 prob. first bulb works 0.1 * 0.8 First fails & 2 operates 0.1 * 0.2 * 0.75 1&2 fail, 3

operates =0.9 + 0.08 + 0.015 = 0.995

99.5% chance system works

90% 80% 75%

Parallel – Different Order

80% 75% 90%

Parallel 0.8 prob. first bulb works 0.2 * 0.75 First fails & 2 operates 0.2 * 0.25 * 0.90 1&2 fail, 3

operates =0.8 + 0.15 + 0.045 = 0.995

99.5% chance system works

Same thing!80% 75% 90%

Parallel – All 3 90% 0.9 prob. first bulb works 0.1 * 0.9 First fails & 2 operates 0.1 * 0.1 * 0.9 1&2 fail, 3 operates =0.9 + 0.09 + 0.009 = 0.999

99.9% chance system works

Practice

.95

.9

.75.80

.95

.9 .95

1: 0.95 * 0.95 2: Simplify

0.8 * 0.75 = 0.6 and 0.9 * 0.95 * 0.9 = 0.7695 Then 0.6 + 0.4 * 0.7695 = 0.6 + 0.3078 = 0.9078

Solutions

Practice

.9 .95

.95 .95

Solution 2 Simplify:

0.9 * 0.95 = 0.855 0.95 * 0.95 = 0.9025

Then 0.855 + 0.145 * 0.9025 = = 0.855 + 0.130863 = 0.985863

Practice

.90 .95

.9

.75.80

.9 .95

Simplify 0.9 * 0.95 = 0.855 0.8 * 0.75 = 0.6 0.9 * 0.95 * 0.9 = 0.7695

.855

.60

.7695

Simplify 0.6 + 0.4 * 0.7695 = 0.6 + 0.3078

=0.9078

.855

0.9078

0.855 + 0.145 * 0.9078 = 0.855+0.13631 =0.986631

These 3 are in parallel 0.855 + 0.145 * 0.6 +

0.145*0.4*.7695 =0.855 + 0.087 + 0.044631

= 0.986631

.855

.60

.7695

Lifetime Failure Rate 3 Distinct phases:

Infant Mortality Stability Wear-out

Failurerate

time, T

Exponential Distribution MTBF = mean time between failures Probability no failure before time T

Probability does not survive until time T = 1- f(T)

e = 2.718281828459045235360287471352662497757

MTBFTeTf /)(

Example Product fails, on average, after 100

hours. What is the probability it survives

at least 250 hours? T/MTBF = 250 / 100 = 2.5 e^-T/MTBF = 0.0821 Probability surviving 250 hrs =

0.0821 =8.21%

Normally Distributed Lifetimes Product failure due to wear-out

may follow Normal Distribution