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 B:  Construction $350k  Prevent. Yr 8 $40k  Major Yr 15 $300k  Prevent. Yr 20 $40k  Prevent. Yr 25 $60k  30 $105k  NPV $610k  Project A:  Construction $500k  Prevent. Yr 15 $40k  Major Yr 20 $300k  30 $150k  NPV $705k

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’?

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 4R’s  Eventually money got more scarce  Bridge Management Systems (BMS) born  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  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  use to measure safety  Excellent: State 5,  >= 9, etc.  No guarantee that new bridge in State 5!  In absence of maintenance, just a linear, decreasing function (see Fig 1)

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?

Condition State Transitions and Deterioration Models

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   : vector of coefficients   : error term  Y =  X +   Use R-squared, related metrics to test model and show how ‘robust’ it is

Markov Processes  Markov chain - a stochastic process with what is called the Markov property  Discrete and continuous versions  Discrete: consists of sequence X 1,X 2,X 3,.... of random variables in a "state space", with X n being "the state of the system at time n".  Markov property - conditional distribution of the "future" X n+1, X n+2, X n+3,.... given the "past” (X 1,X 2,X 3,...X n ), depends on the past only through X n.  i.e. ‘no memory’ of how X n reached  Famous example: random walk

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 p ij =P(X n+1 = j | X n = i )  p ij : 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  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.

Sample Transition Matrix T = [ ]  Thus p ii suggests probability of staying in same state, 1- p ii probability of getting worse  Could ‘simplify’ this type of model by just describing vector P of p ii probabilities (1 - p ii ) values are easily calculated from P  Condition distribution of bridge originally in state i after M transitions is C i T M

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

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:  Pr (Y=1) = e  X / (1+ e  X )  Where  X is our usual ‘regression’ type model  Example: sewer pipes