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Chapters 19 – 20: Dealing With Uncertainties

Chapters 19 – 20: Dealing With Uncertainties. TPS Example: Uncertain outcome Decision rules for complete uncertainty Utility functions Expected value of perfect information Statistical decision theory The value-of-information procedure. TPS Decision Problem 7. OS Development Options:

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Chapters 19 – 20: Dealing With Uncertainties

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  1. Chapters 19 – 20: Dealing With Uncertainties • TPS Example: Uncertain outcome • Decision rules for complete uncertainty • Utility functions • Expected value of perfect information • Statistical decision theory • The value-of-information procedure

  2. TPS Decision Problem 7 OS Development Options: • Option B-Conservative (BC) • Sure to work (cost=$600K) • Performance = 4000 tr/sec • Option B-Bold (BB) • If successful, Cost=$600K Perf.= 4667 tr/sec • If not, Cost=$100K Perf.= 4000 tr/sec Which option should we choose?

  3. Operating System Development Options

  4. Payoff Matrix for OS Options BB, BC

  5. Decision Rules for Complete Uncertainty • Maximin Rule. Determine minimum payoff for each option. Choose option maximizing the minimum payoff. • Maximax Rule. Determine max-payoff for each option. Choose option maximizing max-payoff. • Laplace Rule. Assume all states of nature equally likely. Choose option with maximum expected value. BB = 0.5(250) + 0.5(-50) = 100 BC = 0.5(250) + 0.5(50) = 50

  6. Difficulty with Laplace Rule U1 : performance failure U2 : reliability failure

  7. Breakeven Analysis • Treat uncertainty as a parameter P= Prob (Nature is favorable) • Compute expected values as f(P) EV (BB) = P(250) + (1-P)(-50) = -50 + 300P EV (BC) = P(50) + (1-P)(50) = 50 • Determine breakeven point P such that EV (BB) = EV (BC): -50 + 300P = 50, P=0.333 If we feel that actually P>0.333, Choose BB P<0.333, Choose BC

  8. Utility Functions Is a Guaranteed $50K The same as A 1/3 chance for $250K and A 2/3 chance for -$50K?

  9. Utility Function for Two Groups of Managers Utility 20 10 -3 -2 -1 1 2 3 4 5 Payoff ($M) -10 Group 1 -20 Group 2

  10. Possible TPS Utility Function 20 (250,20) (170,15) 10 (50,5) (100,10) 100 200 300 -100 -10 (-50,-20) -20 (-80,-25)

  11. Utility Functions in S/W Engineering • Managers prefer loss-aversion • EV-approach unrealistic with losses • Managers’ U.F.’S linear for positive payoffs • Can use EV-approach then • People’s U.F.’S aren’t • Identical • Easy to predict • Constant

  12. TPS Decision Problem 7 • Available decision rules inadequate • Need better information • Info. has economic value

  13. Expected Value of Perfect Information (EVPI) Build a Prototype for $10K • If prototype succeeds, choose BB Payoff: $250K – 10K = $240K • If prototype fails, choose BC Payoff: $50K – 10K = $40K If equally likely, Ev = 0.5 ($240K) + 0.5 ($40K) = $140K Could invest up to $50K and do better than before • thus, EVPI = $50K

  14. However, Prototype Will Give Imperfect Information That is, P(IB|SF) = 0.0 Investigation (Prototype) says choose bold state of nature: Bold will fail P(IB|SS) = 1.0 Investigation (prototype) says choose bold state of nature: bold will succeed

  15. Suppose we assess the prototype’s imperfections as P(IB|SF) = 0.20, P(IB|SS) = 0.90 And Suppose the states of nature are equally likely P(SF) = 0.50 P(SS) = 0.50 We would like to compute the expected value of using the prototype EV(IB,IC) = P(IB) (Payoff if use bold) +P(IC) (Payoff if use conservative) = P(IB) [ P(SS|IB) ($250K) + P(SE|IB) (-$50K) ] + P(IC) ($50K) But these aren’t the probabilities we know

  16. How to get the probabilities we need P(IB) = P(IB|SS) P(SS) + P(IB|SF) P(SF) P(IC) = 1 – P(IB) P(SS|IB) = P(IB|SS) P(SS) (Bayes’ formula) P(IB) P(SF|IB) = 1 – P (SS|IB) P(SS|IB) = Prob (we will choose Bold in a state of nature where it will succeed) Prob (we will choose Bold)

  17. 8 NET EV, $K 4 0 10 20 30 Net Expected Value of Prototype PROTO COST, $K

  18. Conditions for Successful Prototyping (or Other Info-Buying) • There exist alternatives whose payoffs vary greatly depending on some states of nature. • The critical states of nature have an appreciable probability of occurring. • The prototype has a high probability of accurately identifying the critical states of nature. • The required cost and schedule of the prototype does not overly curtail its net value. • There exist significant side benefits derived from building the prototype.

  19. Pitfalls Associated by Success Conditions • Always build a prototype or simulation • May not satisfy conditions 3,4 • Always build the software twice • May not satisfy conditions 1,2 • Build the software purely top-down • May not satisfy conditions 1,2 • Prove every piece of code correct • May not satisfy conditions 1,2,4 • Nominal-case testing is sufficient • May need off-nominal testing to satisfy conditions 1,2,3

  20. Statistical Decision Theory: Other S/W Engineering Applications How much should we invest in: • User information gathering • Make-or-buy information • Simulation • Testing • Program verification How much should our customers invest in: • MIS, query systems, traffic models, CAD, automatic test equipment, …

  21. The Software Engineering Field Exists Because Processed Information Has Value

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