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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 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 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 Bridge failure led to condition assessment/NBI methods “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 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 lifecycle safety and reliability for infrastructure projects Reliability index b use to measure safety 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.) 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 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 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" Xn+1, Xn+2, Xn+3, .... 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 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(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 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= [ 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 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: (Y=1) = ebX/ (1+ ebX) Where bX is our usual ‘regression’ type model Pr Example: sewer pipes