Chapter 2: Mathematics for Maintenance

1
Chapter 2 Mathematics for Maintenance

• Overview
• Basic Statistics
• Derivative and Integral Calculations
• Laplace Transforms and Differential Equations
• Basic Economics

2
Basic Statistics Probability and Distributions

• Random events and probability
• An event is a particular outcome or a set of
  outcomes, of a particular trial or experiment
• A random event E will occur with some probability
  denoted by P(E) where 0 P(E) 1
• The collection of all possible outcomes (events)
  relative a random process is called the sample
  space set S, where S E1,E2, En and P(S)1
• If two or more classes of outcomes are all
  non-overlapping, then these events are mutually
  exclusive
• Two or more classes of events are independent if
  the probabilities of occurrence of their outcomes
  do not depend on each other.

3
Probability and Distributions

• Rules of probability
• Every event A has associated with it
  complementary event and P(A) 1 P( )
• Union probability P(A?B) P(A) P(B) P(A?B).
  If A and B are mutually exclusive then P(A?B)
  P(A) P(B)
• Intersection probability P(A?B) P(A)P(BA). If
  A and B are independent then P(A?B)P(A)P(B)
• Bayes Formula P(AB) P(A?B)/P(B)
• Random variables and probability distributions
• A random variable is a variable that takes on
  numerical values to represent the outcome of an
  experiment
• The probability distribution that assigns a
  probability to each value/ an interval values of
  a discrete/continuous random variable
• Probability density function (PDF) f(x) and
  cumulative distribution function (CDF) F(x)

4
Probability and Distributions

• Discrete Distributions
• p(x) PXx 0 p(x) 1
• F(x) P(X x)
• Mean
• Variance
• Binomial distribution
• E(X)np, Var(X)np(1-p), x number of successes
  in n independent trials
• Poisson distribution
• E(X) Var(X) ?, x number of occurrences in a
  specified time
• Continuous distributions
• 0 F(x) 1
• F(x) P(X x)
• f(x) dF(x)/dx

• P a X b
• E(X) µ
• Var(X) s2

5
Probability and Distributions

• Some popular continuous distributions
• Exponential f(t) ?e-?t, ?gt0 F(t) 1 -
  e-?t
• Weibull distribution
• 
  
• Where ? scale parameter
  ? shape parameter, ? 1
  exponential G is gamma function

f(t)
t
6
Probability and Distributions

• Gamma where ? scale parameter, ß
  shape parameter
• Erlang when ß is a positive integer, G(ß)
  (ß-1)! And
• Normal

t
7
Probability and Distributions

• Lognormal
• General distribution
• Where ?,? are the scale parameters ,ß and s are
  the shape parameters
• For m 1 weibull
• m1,s2 Reyleigh
• m1,s1 exponential
• m0 and ß 1 Extreme value
• s 1 and ß 1 Makeham
• s 0.5 and ß 1 Bathtub

8
Identifying Failure and Repair Times Distributions

• Identifying distributions
• Empirical methods (nonparametric) derive
  directly distribution (no distribution fitting)
  READING 12.2 (E286)
• Parametric methods fitting a theoretical
  distribution data collection, identifying
  candidate distributions, estimating parameters,
  Goodness-of-fit test
• Collecting the data
• Observation of failure (or repair) times t1, t2,
  , tn where ti represents the time of failure of
  the ith unit? simple sample
• Failure data operational vs. test generated
  failures, Complete vs censored data, grouped vs
  ungrouped data, large samples vs small samples,
  etc.
• Censoring data incomplete ? remove/test stoplt
  failure
• Singly censored data same test time, test is
  concluded before all units have failed?left(some
  tilttc)/right censored data (some tigttct or
  tr)
• Multiply censored data test/operating times
  differ among censor units

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Identifying Failure and Repair Times Distributions

• Identifying candidate distributions
• Construct a histogram of the failure or repair
  times number of classes of failure/repair times
• k int13.3log10n n sample size
• Compute descriptive statistics
• MTTFsampleMedianTTFsampl? normal or weilbull ?
  3-4
• MTTFsamplegtMedianTTFsampl? exponential,
  lognormal, weilbull
• Exponential sample mean sample standard
  deviation
• Analyze the empirical failure rate ?(t)
• ?(t) const. ? exponential
• ?(t) decrease ? weibull
• ?(t) increase ? weibull, normal, lognormal
• Use prior knowledge of failure process
• Use properties of the theoretical distribution
• Construct a probability plot plot the points
  (ti,F(ti)), i1,2..n, on appropriate graph
  paper, a proper fir to the distribution would
  graph as an approximate straight line use the
  rank adjustment method with F(ti)(i-0.3)/(n0.4)
  ? initial estimate parameters, small sample
  size, censored data ? least squares fit is
  recommended over a manual plot on the
  probability paper

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Identifying Failure and Repair Times Distributions

• Estimating parameters
• The join of n iid distributions is
  fx1,,xn(x1,,xn)f(x1)f(x2)f(xn), xia sample
  of size n this is the likelihood function
  probability of obtaining the observed sample
  containing the unknown parameter
• Maximum Likelihood Estimators (MLE) with complete
  data Max L(?1,,?k)?f(xi?1,,?k) ?j unknown
  parameters ? logarithm ? set partial derivative
  ?j 0
• MLE for Censored data where ? unknown
  parameter, F set of indices for the failure
  times, C set of the indices for the censored
  times, ti censored time for multiply censored
  data, for singly censored data, ti t (type I)
  or tr (Type II)

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Identifying Failure and Repair Times Distributions

• Goodness-of-fit tests
• Hypothesis tests
• H0 the failure times came from the specified
  dist.
• H1 the failure times did not come from the
  specified dist.
• General tests Chi-Square goodness-of-fit test
• Where k number of classes Oi observed number
  of failures in the ith class Ei npi5
  expected number of failures in the ith class n
  sample size pi F(ai)-F(ai-1) probability of
  a failure occurring in the ith class if H0 is
  true. ith class ai-1,ai) where a00 ak t,
  tr, 8
• Specific test Bartletts test for exponential
  distribution (E399)
• Specific test Manns test for Weibull
  distribution (E400)
• Specific test Kolmogorov-Smirnov test for normal
  and lognormal distributions (E402)

12
Experimental Design

• Experimental design collect and analyze data ?
  critical factors significantly affect failures or
  repair times
• Factorial Design collect data at all
  combinations of the levels of the factors ?
  simultaneous evaluation of the factors ? number
  of experiments mk, k factors each at m levels
• Two-factor factorial experiment model
• Yijk µai ßj (aß)ij eijk where
  µ overall mean effect ai the main
  effect of factor A at level i
• ßj the main effect for factor B at level
  j
  (aß)ij the interaction effect
  with factor A at level i and factor B at level
  j eijk random error of the kth
  replication with factor A at level i and factor B
  at level j Yijk the value of the
  response variable at the kth replication with
  factor A at level i and factor B at level j

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Experimental Design

• The statistical hypotheses of interest
• H0 ai 0 for all i
• H0 ßj 0 for all j
• H0 (aß)ij 0 for all i,j
• H1 ai ? 0 for at least one i
• H1 ßj ? 0 for at least one j
• H1 (aß)ij ? for at least one i,j
• Test hypotheses ANOVA

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Experimental Design

• Where
• a the number of levels of factor A
• b the number of level of factor B
• n the number of replications

15
Derivative Integral Calculations

• Derivative and indefinite integral calculations
• Definite integrals

16
Laplace Transform and Differential Equations

• Laplace transforms
• The first order differential equations
• Basic conversion

17
Markov Model
?

• Transition Model consider state i
• Pi(t ?t) Pi(t) (1 outgoing transition rate
  ?t) incoming transition ?t
• ratePj(t)
• Transition rate ?t prob. That the system is at
  result state in time ?t
• Steady state model dPi(t)/dt 0 when t ??? then
• ?j(rate into state i from state j)Pj
  (rate out of state i) Pi

18
Basic Economics

• Present value of single amount
• Present value of multiple equal-payments

19
Examples

• Example 1 the following 35 failure times (in
  operating hours) were obtained from field data
  over a 6-month period. Identify the failure times
  distribution
• 1476 300 98 221 157 182
  499 552 1563 36
• 246 442 20 796 31 47 438 400 279 247
• 210 284 553 767 1297 214 248 597 2025
  185
• 467 401 210 289 1204
• Example 2an aircraft manufacturer is concerned
  with the large number of failures of the use of
  the auxiliary power unit (APU) a particular model
  of its aircraft. The APU is a gas turbine engine
  mounted internally in the lower rear of the
  fuselage. It provides the aircraft with a source
  of power, independent of the main engines, for
  ground operations, main engine starting, and
  in-flight emergencies. Its reliability is
  measured by the number of unscheduled removals
  from the air craft. The manufacturer is
  interested in establishing whether there are
  significant differences in the removal rate that
  depend on carrier type (factor A) and fleet size
  (factor B). Carrier type was defined to be either
  domestic or foreign, and fleet size was
  categorized as small, medium, and large. The
  companys maintenance data collection system
  provided the following information over a
  three-year period. Each years worth of data
  constitutes a single replication. The response
  variable is the number of removals per 100 flying
  hours
• Factor A (type) Factor B (fleet size)
• Small (1-10) Medium (11-22) Large (over 22)
• Domestic 0.82/1.267/0.9 0.8/0/56/0.7867 0.74/0.7
  4/0.76
• Foreign 0.7865/0.57/0.74 0.545/0.41/0.63 0.63/0
  .54/0.58