Reliability Life Cycle Cost Analysis

1
Reliability / Life Cycle Cost Analysis

• H. Scott Matthews
• February 17, 2004

2
Recap of Last Lecture

• Why performance measurement is difficult
• Data availability, lack of common language for
  metrics and use
• Overview of performance metrics at the global
  scale
• Intro to reliability

3
Examples (No User Costs)

• Project B
• Construction 350k
• Prevent. Maint. _at_ Yr 8 40k
• Major Rehab _at_ Yr 15 300k
• Prevent. Maint. _at_ Yr 20 40k
• Prevent. Maint. _at_ Yr 25 60k
• Salvage_at_ 30 105k
• NPV 610k

• Project A
• Construction 500k
• Prevent. Maint. _at_ Yr 15 40k
• Major Rehab _at_ Yr 20 300k
• Salvage_at_ 30 150k
• NPV 705k

4
An Energy Example

• Could consider life cycle costs of people using
  electricity in Texas
• Assume coal-fired power plants used
• Coal comes from Wyoming
• Option 1 (current) coal mined, sent by train to
  Texas, burned there
• Option 2 coal mined, burned in Wyoming into
  electricity, sent via transmission line to Texas
• Which might be cheaper in cost? What are
  components of cost that may be relevant? Are
  there other user costs?

5
Reliability-Based Management

• From Frangopol (2001) paper
• Funds are scarce, need a better way
• Have been focused on condition-based
• Unclear which method might be cheaper
• Bridge failure led to condition assessment/NBI
  methods
• Which emphasized need for 4Rs
• Eventually money got more scarce
• Bridge Management Systems (BMS) born
• PONTIS, BRIDGIT, etc.
• Use deterioration and performance as inputs into
  economic efficiency measures

6
BMS Features

• Elements characterized by discrete condition
  states noting deterioration
• Markov model predicts probability of state
  transitions (e.g. good-bad-poor)
• Deterioration is a single step function
• Transition probabilities not time variant

7
Reliability Assessment

• Decisions are made with uncertainty
• Should be part of the decision model
• Uses consideration of states, distribution
  functions, Monte Carlo simulation to track
  life-cycle safety and reliability for
  infrastructure projects
• Reliability index b use to measure safety
• Excellent State 5, b gt 9, etc.
• No guarantee that new bridge in State 5!
• In absence of maintenance, just a linear,
  decreasing function (see Fig 1)

8
Reliability (cont.)

• Not only is maintenance effect added, but
  random/state/transitional variables are all given
  probability distribution functions, e.g.
• Initial performance, time to damage,
  deterioration rate w/o maintenance, time of first
  rehab, improvement due to maint, subsequent
  times, etc..
• Used Monte Carlo simulation, existing bridge data
  to estimate effects
• Reliability-based method could have significant
  effect on LCC (savings) Why?

9
Condition State Transitions and Deterioration
Models
10
Linear Regression (in 1 slide)

• Arguably simplest of statistical models, have
  various data and want to fit an equation to it
• Y (dependent variable)
• X vector of independent variables
• b vector of coefficients
• e error term
• Y BX e
• Use R-squared, related metrics to test model and
  show how robust it is

11
Markov Processes

• Markov chain - a stochastic process with what is
  called the Markov property
• Discrete and continuous versions
• Discrete consists of sequence X1,X2,X3,.... of
  random variables in a "state space", with Xn
  being "the state of the system at time n".
• Markov property - conditional distribution of the
  "future" Xn1, Xn2, Xn3, .... given the "past
  (X1,X2,X3,...Xn), depends on the past only
  through Xn.
• i.e. no memory of how Xn reached
• Famous example random walk

12
Markov (cont.)

• i.e., knowledge of the most recent past state of
  the system renders knowledge of less recent
  history irrelevant.
• Markov chain may be identified with its matrix of
  "transition probabilities", often called simply
  its transition matrix (T) .
• Entries in T given by pij P(Xn1 j Xn i
  )
• pij probability that system in state j
  "tomorrow" given that it is in state i "today".
• ij entry in the k th power of the matrix of
  transition probabilities is the conditional
  probability that k "days" in the future the
  system will be in state j, given that it is in
  state i "today".

13
Markov Applications

• Markov chains used to model various processes in
  queuing theory and statistics , and can also be
  used as a signal model in entropy coding
  techniques such as arithmetic coding.
• Note Markov created this theory from analyzing
  patterns in words, syllables, etc.

14
Infrastructure Application

• Used to predict/estimate transitions in states,
  e.g. for bridge conditions
• Used by Bridge Management Systems, e.g. PONTIS,
  to help see portfolio effects of assets under
  control
• Helps plan expenditures/effort/etc.
• Need empirical studies to derive parameters
• Source for next few slides Chase and Gaspar,
  Journal of Bridge Engineering, November 2000.

15
Sample Transition Matrix


T

• Thus pii suggests probability of staying in same
  state, 1- pii probability of getting worse
• Could simplify this type of model by just
  describing vector P of pii probabilities (1 -
  pii) values are easily calculated from P
• Condition distribution of bridge originally in
  state i after M transitions is CiTM

16
Superstructure Condition

• NBI instructions
• Code 9 Excellent
• Code 0 Failed/out of service
• If we assume no rehab/repair effects, then
  bridges only get worse over time
• Thus transitions (assuming they are slow) only go
  from Code i to Code i-1
• Need 10x10 matrix T
• Just an extension of the 5x5 example above

17
Empirical Results

• P 0.71, 0.95, 0.96, 0.97, 0.97, 0.97, 0.93,
  0.86, 1
• Could use this kind of probabilistic model result
  to estimate actual transitions

18
More Complex Models

• What about using more detailed bridge parameters
  to guess deficiency?
• Binary deficient or not
• What kind of random variable is this?
• What types of other variables needed?

19
Logistic Models

• Want Pr(j occurs) Pr (Yj) F(effects)
• Logistic distribution
• Pr (Y1) ebX/ (1 ebX)
• Where bX is our usual regression type model
• Example sewer pipes