Systems Engineering - Decision Analysis in Engineering Michael Havbro Faber Aalborg University, Denmark

Transcription

1 Risk and Safety Management Aalborg University, Esbjerg, 2017 Systems Engineering - Decision Analysis in Engineering Michael Havbro Faber Aalborg University, Denmark 1/50 M. H. Faber Systems Engineering, April 2017

2 Contents of Presentation Context of Engineering Decision Making Decisions and preferences Uncertainty Probability Decision Ranking Introduction to Decision Analysis 2/50 M. H. Faber Systems Engineering, April 2017

3 Context of Engineering Decision Making What are we up against? Corrosion Fatigue 3/50 M. H. Faber Systems Engineering, April 2017 Richardson Lecture:

4 Context of Engineering Decision Making What are we up against? Tornados and strong winds 4/50 M. H. Faber Systems Engineering, April 2017 Richardson Lecture:

5 Context of Engineering Decision Making What are we up against? Earthquakes 5/50 M. H. Faber Systems Engineering, April 2017 Richardson Lecture:

6 Context of Engineering Decision Making What are we up against? Earth slide Rock fall 6/50 M. H. Faber Systems Engineering, April 2017 Richardson Lecture:

7 Context of Engineering Decision Making What are we up against? Fires Explosions 7/50 M. H. Faber Systems Engineering, April 2017 Richardson Lecture:

8 Context of Engineering Decision Making What are we up against? Over load Design error 8/50 M. H. Faber Systems Engineering, April 2017 Richardson Lecture:

9 Context of Engineering Decision Making What are we up against? Bombs Airplane impacts 9/50 M. H. Faber Systems Engineering, April 2017 Richardson Lecture:

10 Context of Engineering Decision Making What are we up against? Deepwater Horizon April 20, fatalities 17 injured Oil spill > 5 million barrels Health effects? Eco. imp. > 10 billion $US BP response 14 billion $US lost jobs 10/50 M. H. Faber Systems Engineering, April 2017

11 Context of Engineering Decision Making What are we up against? Hurricane Katrina August 23, 2005 > 1800 fatalities Eco. imp. > 80 billion $US 11/50 M. H. Faber Systems Engineering, April 2017

12 Context of Engineering Decision Making What are we up against? Fukushima Nuclear Event March 11, 2011 No fatalities..? Eco. imp. > 75 billion $US 12/50 M. H. Faber Systems Engineering, April 2017

13 Context of Engineering Decision Making What are we up against? SARS, 2003 Fatalities: < 800 Eco. imp. 2% GDP 200 billion $US 13/50 M. H. Faber Systems Engineering, April 2017

14 Context of Engineering Decision Making What are we up against? Food borne diseases - USA Affects 76 million per year Hospitalizations: per year Fatalities: 5000 pr year 14/50 M. H. Faber Systems Engineering, April 2017

15 Decisions and Preferences Attributes of decision outcomes Decisions aim to achieve an objective The degree of achievement is measured by attributes - natural attributes (measurable, e.g. costs and loss of lives) - constructed attributes (a function of natural attributes e.g. GDP) - proxy attributes (indicators which measure the perceived degree of fulfilment of an objective) 15/50 M. H. Faber Systems Engineering, April 2017

16 Decisions and Preferences Preferences among attributes - utility The attributes associated with a decision outcome may be translated into a degree of achievement of the objective by means of a utility function different attributes are brought together on one or several scales multi attribute decision making implies a weighing of different attributes 16/50 M. H. Faber Systems Engineering, April 2017

17 Decisions and Preferences Constraints on decision making In principle any society may define what they consider to be acceptable decisions Typically decisions are constrained e.g. in terms of maximum acceptable risks to - persons - qualities of the environment 17/50 M. H. Faber Systems Engineering, April 2017

18 Uncertainty Different types of uncertainties influence decision making Inherent natural variability aleatory uncertainty - result of throwing dices - variations in material properties - variations of wind loads - variations in rain fall Model uncertainty epistemic uncertainty - lack of knowledge (future developments) - inadequate/imprecise models (simplistic physical modelling) Statistical uncertainties epistemic uncertainty - sparse information/small number of data 18/50 M. H. Faber Systems Engineering, April 2017

19 Probability 19/50 M. H. Faber Systems Engineering, April 2017

20 Probability 20/50 M. H. Faber Systems Engineering, April 2017

21 Probability 21/50 M. H. Faber Systems Engineering, April 2017

22 Probability Formulate hypothesis about the world Utilize existing knowledge Combine with data Learn how to develop knowledge! 22/50 M. H. Faber Systems Engineering, April 2017

23 Probability Conditional probabilities are of special interest as they provide the basis for utilizing new information in decision making. The conditional probability of an event E 1 given that event E 2 has occured is written as: P( E E ) 1 2 P( E E ) PE ( ) 2 Not defined if PE ( ) 0 The event E 1 is said to be probabilistically independent of the event E 2 if: P( E E ) P( E ) /50 M. H. Faber Systems Engineering, April 2017

24 Probability From P( E E ) 1 2 P( E E ) 1 2 PE ( ) 2 it follows that P( E E ) P( E ) P( E E ) and when E 1 and E 2 are statistically independent it is P( E E ) P( E ) P( E ) /50 M. H. Faber Systems Engineering, April 2017

25 Probability Consider the sample space exclusive events E 1, E 2,, E n divided up into n mutually E 1 E 2 E 3 E 4 A E 5 E 6 E 7 E 8 n i1... P A P A E P A E P A E... P( E ) 1 2 n P A E P E P A E P E P A E P E P A E n n i i 25/50 M. H. Faber Systems Engineering, April 2017

26 Probability as there is P( A E ) P( A E ) P( E ) P( E A) P( A) i i i i we have Likelihood Prior P( E A) i P( A Ei ) P( Ei ) P( A Ei ) P( Ei ) n PA ( ) P( A E ) P( E ) i1 i i Posterior Bayes Rule Reverend Thomas Bayes ( ) 26/50 M. H. Faber Systems Engineering, April 2017

27 Decision Ranking Emperor Qianlong Qing dynasty Reign : Daniel Bernoulli 1738 Expected utility hypothesis von Neumann and Morgenstern Axioms of utility theory: Ranking based on expected value of utility (VNM rational) 27/50 M. H. Faber Systems Engineering, April 2017

28 Decision Ranking Risk is a characteristic of an activity relating to all possible events n E which may follow as a result of the activity The risk contribution R Ei from the event E i is defined through the product between the event probability P Ei and the consequences of the event C Ei The risk associated with a given activity R A may then be written as n n R A E i1 R E i E i1 P E i C E i 28/50 M. H. Faber Systems Engineering, April 2017

29 Decision Optimization Prior decision analysis Information is bought by choice of prior density a X b( a, x) Decision Event Benefit Optimal decision maximizes the expected value of utility (benefit) (von Neumann & Morgenstern) * B0 max E b( a, X ) max b( a, x) f X ( x, a) dx a a 29/50 M. H. Faber Systems Engineering, April 2017

30 Decision Optimization Posterior decision analysis z By sampling information using an experiment we may update the probabilistic description of X e f ( x, a z) X L( x z) f ( x, a) X L( x z) f ( x, a) X Of course the likelihood of the sample z depends on the experiment e why we write L( x z) L( x z, e) 30/50 M. H. Faber Systems Engineering, April 2017

31 Decision Optimization Posterior decision analysis a X b( a, x) Decision Event Benefit zˆ max E b( a, X ) max b( a, x) f ( x, a ) dx a a X 31/50 M. H. Faber Systems Engineering, April 2017

32 Decision Optimization Pre-posterior decision analysis (extensive form) e Z a X b( e, a, x) Decision Event Decision Event Benefit The optimal experiment e may be found from * B1 max E max b( e, a, x) f X ( x, a ) dx Z e a Z 32/50 M. H. Faber Systems Engineering, April 2017

33 Decision Optimization Value of Information b( e, a, x) e Ẑ 1 Z a X 0 b( a, x) a X Decision Decision Event Decision Event Benefit The value of information VoI is determined from: VoI max E max b( e, a, x) f X( x, a ) dx max b( a, x) f Z X( x, a) dx e a Z a 33/50 M. H. Faber Systems Engineering, April 2017

34 Decision Optimization Games and Risk Rules (exogenous) - Nature Rules (endogenous) - Knowledge - Best practices - Rules and standards - Culture - Ethics Drivers/Challenges - Preferences - Psychology - Asymmetric information Game Rules Rules Nature Player Other players 34/50 M. H. Faber Systems Engineering, April 2017

35 Decision Analysis in Engineering Pile The decision tree Depth of rock bed 40ftor 50ft? Action alternatives Outcome Consequence Utility(consequence) depth = 40 ft none 0 40 ft Pile depth = 50 ft splice 400 depth = 40 ft cutting ft Pile depth = 50 ft none 0 35/50 M. H. Faber Systems Engineering, April 2017

36 Decision Analysis in Engineering The different types of decision analysis - Prior - Posterior - Pre-posterior Illustrated on an example : Question : What pile length should be applied? Pile Alternatives : a 0 : Choose a 40 ft pile a 1 : Choose a 50 ft pile States of nature (depth to rock bed) θ 0 : Rock bed at 40 ft θ 1 : Rock bed at 50 ft Depth of rock bed 40ft or 50 ft? 36/50 M. H. Faber Systems Engineering, April 2017

37 Decision Analysis in Engineering Prior Analysis p=0.70 P [q 0 ] = 0.70 a 1 q 0 P [q 1 ] = p=0.30 q 1 The expected utility is calculated to be equal to 0 1 E ' u min{ u a, u a } min{ P ' q0 u q0 a0 P ' q1 u q1 a0, P ' q0 u q0 a1 P ' q1 u q1 a1 } min{ , } min{120,70} 70 Decision for a 1 (50ft Pile) a p=0.70 q 0 q 1 p=0.30 u = 0 u = 400 (Pile is spliced) u = 100 (Pile is cut) u = 0 37/50 M. H. Faber Systems Engineering, April 2017

38 Decision Analysis in Engineering a 0 a p=0.70 q 0 q 1 q 0 p=0.30 p=0.70 u = 0 u = 400 (Pile is spliced) u = 100 (Pile is cut) 70 q 1 p=0.30 u = 0 Choice of pile a 1 (50ft Pile) 38/50 M. H. Faber Systems Engineering, April 2017

39 Decision Analysis in Engineering Posterior Analysis P ''( q ) i 'q P z k qi P i P zk q P ' q j j j 39/50 M. H. Faber Systems Engineering, April 2017

40 Decision Analysis in Engineering Posterior Analysis Prior Posterior Likelihood Prior Posterior Likelihood Prior Likelihood Posterior 40/50 M. H. Faber Systems Engineering, April 2017

41 Decision Analysis in Engineering Posterior Analysis Ultrasonic tests to determine the depth to bed rock P ''( q ) i 'q P z k qi P i P zk q P ' q j j j Test result True state q 0 40 ft depth q 1 50 ft depth z 0-40 ft indicated z 1-50 ft indicated z 2-45 ft indicated Likelihoods of the different indications/test results given the various possible states of nature ultrasonic test methods Pzk q j 41/50 M. H. Faber Systems Engineering, April 2017

42 Decision Analysis in Engineering Posterior Analysis P ''( q ) It is assumed that a test gives a 45 ft indication i 'q P z k qi P i P zk q P ' q j j j q Pq z Pz q Pq = 0.21 P' ' x q Pq z Pz q Pq = 0.06 P' ' x P'' P'' z q z q = 0.78 = /50 M. H. Faber Systems Engineering, April 2017

43 Decision Analysis in Engineering Posterior Analysis Test result indicates 45ft to rock bed a 0 a p=0.78 q 0 q 1 q 0 p=0.22 p=0.78 q 1 p=0.22 u = 0 u = 400 (Pile is spliced) u = 100 (Pile is cut) u = 0 Choice of alternative a 1 (50ft Pile) 43/50 M. H. Faber Systems Engineering, April 2017

44 Decision Analysis in Engineering Posterior Analysis E '' u z2 min{ E '' u( a j ) z 2 } j q q q q min{ P '' 0 P '' 400, P '' 100 P '' 0} min{ , } a 0 a p=0.78 q 0 q 1 q 0 p=0.22 p=0.78 q 1 p=0.22 u = 0 u = 400 (Pile is spliced) u = 100 (Pile is cut) u = 0 min{88, 78} 78 Choice of alternative a 1 (50ft Pile) 44/50 M. H. Faber Systems Engineering, April 2017

45 Decision Analysis in Engineering Pre-posterior Analysis n j1, E u P ' z E '' u z P ' z min{ E '' u( a ) z } n i i i j i m i1 i1 q q q q P ' z P z P ' P z P ' i i 0 0 i 1 1 q q q q P ' z0 P z0 0 P ' 0 P z0 1 P ' q q q q P ' z1 P z1 0 P ' 0 P z1 1 P ' q q q q P ' z2 P z2 0 P ' 0 P z2 1 P ' /50 M. H. Faber Systems Engineering, April 2017

46 Decision Analysis in Engineering Pre-posterior Analysis E '' u z0 min{ E '' u( a j ) z 0 } j a 0 a 1 do nothing splicing cutting do nothing min{ P '' q z 0 P '' q z 400, P '' q z 100 P '' q z 0} min{ , } /50 M. H. Faber Systems Engineering, April 2017

47 Decision Analysis in Engineering Pre-posterior Analysis E '' u z1 min{ E '' u( a j ) z 1 } j a 0 a 1 do nothing splicing cutting do nothing min{ P '' q z 0 P '' q z 400, P '' q z 100 P '' q z 0} min{ , } /50 M. H. Faber Systems Engineering, April 2017

48 Decision Analysis in Engineering Pre-posterior Analysis The minimum expected costs based on pre-posterior decision analysis not including costs of experiments n E u P ' zi E '' u zi i1 Allowable costs for the experiment E ' u E u /50 M. H. Faber Systems Engineering, April 2017

49 Decision Analysis in Engineering Pre-posterior Analysis Allowable costs for experiments E ' u E u /50 M. H. Faber Systems Engineering, April 2017

50 Thanks for your attention 50/50 M. H. Faber Systems Engineering, April 2017