IndE 311 Stochastic Models and Decision Analysis

1. IndE 311Stochastic Models and Decision Analysis UW Industrial Engineering Instructor: Prof. Zelda Zabinsky

2. Operations Research “The Science of Better”

3. Operations Research Modeling Toolset 311 Queueing Theory 310 Markov Chains PERT/ CPM Network Programming Simulation Decision Analysis Stochastic Programming Inventory Theory Linear Programming Markov Decision Processes Nonlinear Programming Dynamic Programming Forecasting Integer Programming Game Theory 312

4. Decision analysis Decision making without experimentation Decision making with experimentation Decision trees Utility theory Markov chains Modeling Chapman-Kolmogorov equations Classification of states Long-run properties First passage times Absorbing states Queueing theory Basic structure and modeling Exponential distribution Birth-and-death processes Models based on birth-and-death Models with non-exponential distributions Applications of queueing theory Waiting cost functions Decision models IndE 311

5. Decision Analysis Chapter 15

6. Decision Analysis • Decision making without experimentation • Decision making criteria • Decision making with experimentation • Expected value of experimentation • Decision trees • Utility theory

7. Decision Making without Experimentation

8. Goferbroke Example • Goferbroke Company owns a tract of land that may contain oil • Consulting geologist: “1 chance in 4 of oil” • Offer for purchase from another company: $90k • Can also hold the land and drill for oil with cost $100k • If oil, expected revenue $800k, if not, nothing

9. Notation and Terminology • Actions: {a1, a2, …} • The set of actions the decision maker must choose from • Example: • States of nature: {1, 2, ...} • Possible outcomes of the uncertain event. • Example:

10. Notation and Terminology • Payoff/Loss Function: L(ai, k) • The payoff/loss incurred by taking action ai when state k occurs. • Example: • Prior distribution: • Distribution representing the relative likelihood of the possible states of nature. • Prior probabilities: P( = k) • Probabilities (provided by prior distribution) for various states of nature. • Example:

11. Decision Making Criteria Can “optimize” the decision with respect to several criteria • Maximin payoff • Minimax regret • Maximum likelihood • Bayes’ decision rule (expected value)

12. Maximin Payoff Criterion • For each action, find minimum payoff over all states of nature • Then choose the action with the maximum of these minimum payoffs

13. Minimax Regret Criterion • For each action, find maximum regret over all states of nature • Then choose the action with the minimum of these maximum regrets

14. Maximum Likelihood Criterion • Identify the most likely state of nature • Then choose the action with the maximum payoff under that state of nature

15. Bayes’ Decision Rule(Expected Value Criterion) • For each action, find expectation of payoff over all states of nature • Then choose the action with the maximum of these expected payoffs

16. Sensitivity Analysis with Bayes’ Decision Rule • What is the minimum probability of oil such that we choose to drill the land under Bayes’ decision rule?

17. Decision Making with Experimentation

18. Goferbroke Example (cont’d) • Option available to conduct a detailed seismic survey to obtain a better estimate of oil probability • Costs $30k • Possible findings: • Unfavorable seismic soundings (USS), oil is fairly unlikely • Favorable seismic soundings (FSS), oil is fairly likely

19. Posterior Probabilities • Do experiments to get better information and improve estimates for the probabilities of states of nature. These improved estimates are called posterior probabilities. • Experimental Outcomes: {x1, x2, …} Example: • Cost of experiment:  Example: • Posterior Distribution: P( = k| X = xj)

20. Goferbroke Example (cont’d) • Based on past experience: If there is oil, then • the probability that seismic survey findings is USS = 0.4 = P(USS | oil) • the probability that seismic survey findings is FSS = 0.6 = P(FSS | oil) If there is no oil, then • the probability that seismic survey findings is USS = 0.8 = P(USS | dry) • the probability that seismic survey findings is FSS = 0.2 = P(FSS | dry)

21. Bayes’ Theorem • Calculate posterior probabilities using Bayes’ theorem: Given P(X = xj |  = k), find P( = k | X = xj)

22. Goferbroke Example (cont’d) • We have P(USS | oil) = 0.4 P(FSS | oil) = 0.6 P(oil) = 0.25 P(USS | dry) = 0.8 P(FSS | dry) = 0.2 P(dry) = 0.75 • P(oil | USS) = • P(oil | FSS) = • P(dry | USS) = • P(dry | FSS) =

23. Goferbroke Example (cont’d) Optimal policies • If finding is USS: • If finding is FSS:

24. The Value of Experimentation • Do we need to perform the experiment? As evidenced by the experimental data, the experimental outcome is not always “correct”. We sometimes have imperfect information. • 2 ways to access value of information • Expected value of perfect information (EVPI) What is the value of having a crystal ball that can identify true state of nature? • Expected value of experimentation (EVE) Is the experiment worth the cost?

25. Expected Value of Perfect Information • Suppose we know the true state of nature. Then we will pick the optimal action given this true state of nature. • E[PI] = expected payoff with perfect information =

26. Expected Value of Perfect Information • Expected Value of Perfect Information: EVPI = E[PI] – E[OI] where E[OI] is expected value with original information (i.e. without experimentation) • EVPI for the Goferbroke problem =

27. Expected Value of Experimentation • We are interested in the value of the experiment. If the value is greater than the cost, then it is worthwhile to do the experiment. • Expected Value of Experimentation: EVE = E[EI] – E[OI] where E[EI] is expected value with experimental information.

28. Goferbroke Example (cont’d) • Expected Value of Experimentation: EVE = E[EI] – E[OI] EVE =

29. Decision Trees

30. Decision Tree • Tool to display decision problem and relevant computations • Nodes on a decision tree called __________. • Arcs on a decision tree called ___________. • Decision forks represented by a __________. • Chance forks represented by a ___________. • Outcome is determined by both ___________ and ____________. Outcomes noted at the end of a path. • Can also include payoff information on a decision tree branch

31. Goferbroke Example (cont’d)Decision Tree

32. Analysis Using Decision Trees • Start at the right side of tree and move left a column at a time. For each column, if chance fork, go to (2). If decision fork, go to (3). • At each chance fork, calculate its expected value. Record this value in bold next to the fork. This value is also the expected value for branch leading into that fork. • At each decision fork, compare expected value and choose alternative of branch with best value. Record choice by putting slash marks through each rejected branch. • Comments: • This is a backward induction procedure. • For any decision tree, such a procedure always leads to an optimal solution.

33. Goferbroke Example (cont’d)Decision Tree Analysis

34. Painting problem • Painting at an art gallery, you think is worth $12,000 • Dealer asks $10,000 if you buy today (Wednesday) • You can buy or wait until tomorrow, if not sold by then, can be yours for $8,000 • Tomorrow you can buy or wait until the next day: if not sold by then, can be yours for $7,000 • In any day, the probability that the painting will be sold to someone else is 50% • What is the optimal policy?

35. Drawer problem • Two drawers • One drawer contains three gold coins, • The other contains one gold and two silver. • Choose one drawer • You will be paid $500 for each gold coin and $100 for each silver coin in that drawer • Before choosing, you may pay me $200 and I will draw a randomly selected coin, and tell you whether it’s gold or silver and which drawer it comes from (e.g. “gold coin from drawer 1”) • What is the optimal decision policy? EVPI? EVE? Should you pay me $200?

36. Utility Theory

37. Validity of Monetary Value Assumption • Thus far, when applying Bayes’ decision rule, we assumed that expected monetary value is the appropriate measure • In many situations and many applications, this assumption may be inappropriate

38. Choosing between ‘Lotteries’ • Assume you were given the option to choose from two ‘lotteries’ • Lottery 150:50 chance of winning $1,000 or $0 • Lottery 2Receive $50 for certain • Which one would you pick? .5 $1,000 .5 $0 1 $50

39. Choosing between ‘lotteries’ • How about between these two? • Lottery 150:50 chance of winning $1,000 or $0 • Lottery 2Receive $400 for certain • Or these two? • Lottery 150:50 chance of winning $1,000 or $0 • Lottery 2Receive $700 for certain .5 $1,000 .5 $0 1 $400 .5 $1,000 .5 $0 1 $700

40. Utility • Think of a capital investment firm deciding whether or not to invest in a firm developing a technology that is unproven but has high potential impact • How many people buy insurance?Is this monetarily sound according to Bayes’ rule? • So... is Bayes’ rule invalidated?No- because we can use it with the utility for money when choosing between decisions • We’ll focus on utility for money, but in general it could be utility for anything (e.g. consequences of a doctor’s actions)

41. A Typical Utility Function for Money u(M) 4 3 What does this mean? 2 1 M 0 $500 $1,000 $100 $250

42. Decision Maker’s Preferences • Risk-averse • Avoid risk • Decreasing utility for money • Risk-neutral • Monetary value = Utility • Linear utility for money • Risk-seeking (or risk-prone) • Seek risk • Increasing utility for money • Combination of these u(M) M u(M) M u(M) M u(M) … M

43. Constructing Utility Functions • When utility theory is incorporated into a real decision analysis problem, a utility function must be constructed to fit the preferences and the values of the decision maker(s) involved • Fundamental property:The decision maker is indifferent between two alternative courses of action that have the same utility

44. Indifference in Utility • Consider two lotteries • The example decision maker we discussed earlier would be indifferent between the two lotteries if • p is 0.25 and X is … • p is 0.50 and X is … • p is 0.75 and X is … p $1,000 1 $X 1-p $0

45. Goferbroke Example (with Utility) • We need the utility values for the following possible monetary payoffs: 45° u(M) M

46. Constructing Utility FunctionsGoferbroke Example • u(0) is usually set to 0. So u(0)=0 • We ask the decision maker what value of p makes him/her indifferent between the following lotteries: • The decision maker’s response is p=0.2 • So… p 700 1 0 1-p -130

47. Constructing Utility FunctionsGoferbroke Example • We now ask the decision maker what value of p makes him/her indifferent between the following lotteries: • The decision maker’s response is p=0.15 • So… p 700 1 90 1-p 0

48. Constructing Utility FunctionsGoferbroke Example • We now ask the decision maker what value of p makes him/her indifferent between the following lotteries: • The decision maker’s response is p=0.1 • So… p 700 1 60 1-p 0

49. Goferbroke Example (with Utility)Decision Tree

50. Exponential Utility Functions • One of the many mathematically prescribed forms of a “closed-form” utility function • It is used for risk-averse decision makers only • Can be used in cases where it is not feasible or desirable for the decision maker to answer lottery questions for all possible outcomes • The single parameter R is the one such that the decision maker is indifferent between 0.5 R 1 (approximately) and 0 0.5 -R/2