Decision Analysis1 Ultimate objective of all engineering analysis Uncertainty always exist, hence satisfactory performance not guaranteed More conservative design reduces risk Same design SF for all? Component vs. System Risk Proper tradeoff between risk and investment

Decision Analysis2 Solution by Calculus Set up objective function where ’s are decision variables From solution to yields optimal values of decision variables

Decision Analysis3 Cofferdam for construction of Bridge Pier (2 yrs) (Example 2.2) h?

Decision Analysis4 Information Floods occur according to Poisson process with mean rate of 1.5/yr Elevation of each flood – exponential with mean 5 feet Each overtopping  loss due to pumping + delay = $25,000 Construction cost, h?

Decision Analysis5 Expected damage cost, C E (loss of flood)

Decision Analysis6 Total Cost  = 8.05 ft

Decision Analysis7 Cost as Functions of cofferdam elevation above normal water level 8.05

Decision Analysis8 Limitation of this Approach Objective function may not be continuous function of decision variables Alternatives may be discrete e.g. dam for flood control (height, location, other schemes) Consequences may be more than monetary costs Alternative may include acquiring new information before final decision Should we acquire or not?

Decision Analysis9 Decision Tree Model Decision Node Chance Node AlternativesUncertainties Consequences

Decision Analysis10 Decision Criteria 1.Pessimistic  Minimize max loss  Install 2.Optimistic  Maximize max gain  Not Install

Decision Analysis11 3. Maximum EMV (Expected Monetary Value) E(I) = 0.1x(-2000)+0.9x(-2000) = E(II) = 0.1x(-10000)+0.9x(0) =-1000

Decision Analysis12 Ex. 2.9 Decision tree for construction project

Decision Analysis13

Decision Analysis14 Spillway Decisions Alternatives Capital Cost Annual OMR Cost No Change 0 0 Lengthening spillway 1.04M 0 Plus lowering crest, installing 1.30M 2,000 flashboard Plus considerable crest lowering, 3.90M 10,000 installing radial gates 50years service; Discount rate 6%

Decision Analysis15 Spillway Decisions Summary of Annual Costs (in Dollars) Total Annual Cost =Capital Cost x crf (i,n) +Annual DMR Cost +Expected Risk Cost (annual)

Decision Analysis16 Discount factors Given A to find P: Given P to find A: Where i = int. rate per period n = no. of periods

Decision Analysis17 E2.11 Spillway Design

Decision Analysis18 E2.11 Spillway Design Risk Cost

Decision Analysis19 Ex. 2.6 Prior Analysis A (small) B (large) EH 0.7 EL 0.3 EH 0.7 EL E(A) = 0.7 x x (-100) = -30 E(B) = 0.7 x (-50) x (-20) = -41 Hence, A is the optimal alternative.

Decision Analysis20 Lab. Model test on Efficiency (Cost $10,000) will indicate: HR (high rating) MR (medium rating) LR (Low rating) HR 0.8 HR 0.1 If EH MR 0.15 If EL MR 0.2 LR 0.05 LR 0.7 e.g. If the process is actually high efficiency (EH), then the probability that the test will indicate HR is 0.8.

Decision Analysis21 Suppose the test indicate HR TestHR A (small) B (large) EH 0.95 EL 0.05 EH 0.95 EL

Decision Analysis22 Suppose the test indicate HR Similarly, P(EL|HR) = 0.05 E(A|HR) =0.95x(-10)+0.05x(-110) = -15 E(B|HR) =0.95x(-10)+0.05x(-110) = > 30 good news

Decision Analysis23 Should test be performed?  Preposterior analysis E(Test) = 0.59x(-15) x(-46.3) x(-34.3) = Better than -30 (without test)

Decision Analysis24 Procedure for Preposterior Analysis Determine updated probabilities using Bayes Theorem; Sub-tree analysis –Identify optimal alternative and maximum utility; Determine the best alternative in the next decision node (to the left); If Experimental alternative is optimal, wait for experimental outcome and select corresponding optimal alternative.

Decision Analysis25 Value of Information ( VI ) VI = E(T) – E( ) EMV of test alternative excluding test cost EMV of optimal alternative without Test VI = ( ) – (- 30) = (max. paid for that specific Test)

Decision Analysis26 Suppose someone comes up with a better test, say cost 25,000, but doesn’t know that exact reliability, should the test be performed?

Decision Analysis27 VPI = E(PT) - E( ) P(EH 0 ) = P(EH 0 |EH) P(EH) + P(EH 0 |EL) P(EL) = 1 x x 0.3 = 0.7 E(PT) = 0.7 x x (-20) = -6 VPI = -6 – (-30) = 24 Max. that should be paid for any information

Decision Analysis28 Sensitivity Analysis If the probability estimates are off by +10%, would the alternative previously chosen be still optimal? Method 1: Repeat analysis with several values of p Method 2: Determine value of probability p that decision is switched

Decision Analysis29 A B EH p EL 1-p EH p EL 1-p E(A) = p x 0 + (1-p) x (-100) E(B) = p x (-50) + (1-p) x (-20)

Decision Analysis30 Sensitivity of Decision to Probability p<0.62  E(B) >E(A) P>0.62  E(B) <E(A) E(PT) =px0+(1-p)(-20) = -20(1-p) E(T) VPI VI

Decision Analysis31 Levee Elevation Decision Annual max. Flood Level: median 10, c.o.v. 20% Cost of construction: a 1 : $ 2 million a 2 : $ 2.5 million Service Life: 20 years Average annual damage cost due to inadequate protection: $ 2 million

Decision Analysis32 Levee Elevation Decision Annual max. Flood Level: median 10, c.o.v. 20% H=10’ H=14’ H=16’ E(C)= x pwf (20yrs, 7%)

Decision Analysis33 Value of Perfect Information E(C PI ) = 0.5x x2.482 = 2.59 VPI = 2.614–2.59 = $ M Max. Amount to be paid for verifying type of distribution of annual flood level