1. 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

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

3. Decision Analysis3 Contractor – submit a bid (Example 2.1) Bid Ratio, R =

4. Decision Analysis4 To determine optimal bid and optimal R Establish the objective function

5. Decision Analysis5 Case 2: Include Idling Cost

6. Decision Analysis6 Cofferdam for construction of Bridge Pier (2 yrs) (Example 2.2) h?

7. Decision Analysis7 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?

8. Decision Analysis8 Expected damage cost, C E (loss of flood)

9. Decision Analysis9 Total Cost  = 8.05 ft

10. Decision Analysis10 Cost as Functions of cofferdam elevation above normal water level 8.05

11. Decision Analysis11 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?

12. Decision Analysis12 Seepage under the Embankment (Example 2.4) Embankment Cooling Lake Pump System (100/120 gal/min ) Q = 95 or 120 gal/min? Bentonite Seal

13. Decision Analysis13 Decision tree for seepage problem Pump System B (120) Seal Q 1 (0.9) Pump System A (100) Q 2 (0.1) Add Pump System C Q 1 (0.9) Q 2 (0.1) 95 120 95 120

14. Decision Analysis14 Decision Tree Model Decision Node Chance Node AlternativesUncertainties Consequences

15. Decision Analysis15 Click to enlarge Example 2.17

16. Decision Analysis16 Click to enlarge Example 2.17

17. Decision Analysis17 Decision Criteria 1.Pessimistic  Minimize max loss  Install 2.Optimistic  Maximize max gain  Not Install

18. Decision Analysis18 3. Maximum EMV (Expected Monetary Value) E(I) = 0.1x(-2000)+0.9x(-2000) = -2000 E(II) = 0.1x(-10000)+0.9x(0) =-1000

19. Decision Analysis19 Ex. 2.9 Decision tree for construction project

20. Decision Analysis20

21. Decision Analysis21 Spillway Decisions Alternatives Capital Cost Annual OMR Cost No Change 0 0 Lengthening spillway 1.04M 0 Plus lowering crest, installing 1.30M 0 flashboard Plus considerable crest lowering, 3.90M 0 installing radial gates 50years service; Discount rate 6%

22. Decision Analysis22 Spillway Decisions Summary of Annual Costs (in Dollars) Total Annual Cost =Capital Cost x crf (i,n) +Annual DMR Cost +Expected Risk Cost (annual)

23. Decision Analysis23 Discount factors Given A to find P: Given P to find A: Where i = int. rate per period n = no. of periods

24. Decision Analysis24 E2.11 Spillway Design

25. Decision Analysis25 E2.11 Spillway Design Risk Cost

26. Decision Analysis26 Ex. 2.6 Prior Analysis A (small) B (large) EH 0.7 EL 0.3 EH 0.7 EL 0.3 0 -100 -50 -20 E(A) = 0.7 x 0 + 0.3 x (-100) = -30 E(B) = 0.7 x (-50) + 0.3 x (-20) = -41 Hence, A is the optimal alternative.

27. Decision Analysis27 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.

28. Decision Analysis28 Suppose the test indicate HR TestHR A (small) B (large) EH 0.95 EL 0.05 EH 0.95 EL 0.05 -10 -110 -60 -30

29. Decision Analysis29 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) = -58.5 > 30 good news

30. Decision Analysis30 Suppose the test indicate MR TestMR A (small) B (large) EH 0.637 EL 0.363 EH 0.637 EL 0.363 -10 -110 -60 -30 E(A|MR) = -46.3 E(B|MR) = -49.1 < -30

31. Decision Analysis31 Suppose the test indicate LR E(A|LR) = -95.7 E(B|LR) = -34.3 < -30 Only if the test showed HR,  saved money; otherwise, more money with test

32. Decision Analysis32 Should test be performed?  Preposterior analysis E(Test) = 0.59x(-15)+ 0.165 x(-46.3)+0.245 x(-34.3) = -24.86 Better than -30 (without test)

33. Decision Analysis33 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.

34. Decision Analysis34 Procedure for Preposterior Analysis B C Subtree B Subtree C

35. Decision Analysis35 Value of Information ( VI ) VI = E(T) – E( ) EMV of test alternative excluding test cost EMV of optimal alternative without Test VI = (-24.86 + 10) – (- 30) = 15.14 (max. paid for that specific Test)

36. Decision Analysis36 Suppose someone comes up with a better test, say cost 25,000, but doesn’t know that exact reliability, should the test be performed?

37. Decision Analysis37 VPI = E(PT) - E( ) P(EH 0 ) = P(EH 0 |EH) P(EH) + P(EH 0 |EL) P(EL) = 1 x 0.7 + 0 x 0.3 = 0.7 E(PT) = 0.7 x 0 + 0.3 x (-20) = -6 VPI = -6 – (-30) = 24 Max. that should be paid for any information

38. Decision Analysis38 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

39. Decision Analysis39 A B EH p EL 1-p EH p EL 1-p 0 -100 -50 -20 E(A) = p x 0 + (1-p) x (-100) E(B) = p x (-50) + (1-p) x (-20)

40. Decision Analysis40 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

41. Decision Analysis41 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

42. Decision Analysis42 Levee Elevation Decision Annual max. Flood Level: median 10, c.o.v. 20% H=10’ H=14’ H=16’ E(C)=10.594 2.731 2.641 2x10.594 pwf (20yrs, 7%)

43. Decision Analysis43 Value of Perfect Information E(C PI ) = 0.5x2.699 +0.5x2.482 = 2.59 VPI = 2.614–2.59 = $ 0.024 M Max. Amount to be paid for verifying type of distribution of annual flood level

44. Decision Analysis44 Consider a Game E(A) = 0.5 x 0 + 0.5 x 10¢ = 5 ¢ E(B) = 1.0 x 5 ¢ = 5 ¢ A B 0.5 1.0 0 10 ¢ 5 ¢ 0 $1 $0.5 0 $100 $ 50 0 $100M $ 50M EMV criteria may not be applicable We need something else!

45. Decision Analysis45 EUV criteria Expected Utility Value Definition: EUV is the true value to a decision maker with which he/she can make a proper decision based on the relative utility value.

46. Decision Analysis46 Utility function of monetary value Risk Indifferent dollars u(d) Risk aversive

47. Decision Analysis47 Maximum Expected Utility Criterion (EUV) If all consequences expressed in monetary terms:

48. Decision Analysis48 Example E( )= 0.1 u (-2000)+0.9 u (-2000) = = -1.49 E( )= 0.1 u (-10000)+0.9 u (0) =0.1x( ) +0.9x( ) = -1.64 -2000 -10000 0 0 dollars u(d)