Transcript lec11-04-reliab

Reliability / Life Cycle Cost
Analysis
H. Scott Matthews
February 17, 2004
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
Examples (No User Costs)
 Project A:
 Construction
$500k
 Prevent. Maint. @
Yr 15
$40k
 Major Rehab @ Yr
20
$300k
 Salvage@ 30 $150k
 NPV
$705k
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Project B:
 Construction $350k
 Prevent. Maint. @ Yr 8
$40k
 Major Rehab @ Yr 15
$300k
 Prevent. Maint. @ Yr 20
$40k
 Prevent. Maint. @ Yr 25
$60k
 Salvage@ 30 $105k
 NPV
$610k
An Energy Example
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Could consider life cycle costs of people using
electricity in Texas
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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’?
Reliability-Based Management
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From Frangopol (2001) paper
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Bridge failure led to condition assessment/NBI
methods
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“Funds are scarce, need a better way”
Have been focused on “condition-based”
Unclear which method might be cheaper
Which emphasized need for 4R’s
Eventually money got more scarce
Bridge Management Systems (BMS) born
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PONTIS, BRIDGIT, etc.
Use deterioration and performance as inputs into economic
efficiency measures
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
Reliability Assessment
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Decisions are made with uncertainty
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Should be part of the decision model
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Uses consideration of states, distribution
functions, Monte Carlo simulation to track lifecycle safety and reliability for infrastructure
projects
 Reliability index b use to measure safety
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Excellent: State 5, b >= 9, etc.
No guarantee that new bridge in State 5!
In absence of maintenance, just a linear,
decreasing function (see Fig 1)
Reliability (cont.)
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Not only is maintenance effect added, but
random/state/transitional variables are all
given probability distribution functions, e.g.
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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?
Condition State Transitions
and Deterioration Models
Linear Regression (in 1 slide)
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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
Markov Processes
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Markov chain - a stochastic process with what is
called the Markov property
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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" Xn+1, Xn+2, Xn+3, .... given the "past”
(X1,X2,X3,...Xn), depends on the past only through Xn.
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i.e. ‘no memory’ of how Xn reached
Famous example: random walk
Markov (cont.)
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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) .
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Entries in T given by pij =P(Xn+1 = 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".
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.
Infrastructure Application
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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
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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.
Sample Transition Matrix
T=
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[
p11
0
0
0
0
1-p11
p22
0
0
0
0
1-p22
p33
0
0
0
0
1-p33
p44
0
0
0
0
1-p44
p55
]
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
Superstructure Condition
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NBI instructions:
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Code 9 = Excellent
Code 0 = Failed/out of service
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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
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Just an extension of the 5x5 example above
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
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?
Logistic Models
 Want
Pr(j occurs) = Pr (Y=j) = F(effects)
 Logistic distribution:
(Y=1) = ebX/ (1+ ebX)
 Where bX is our usual ‘regression’ type
model
 Pr
 Example:
sewer pipes