1. EMGT 5412 Operations Management Science Decision Analysis Dincer Konur Engineering Management and Systems Engineering 1

2. Outline Probability Review Introduction and Terminology Decision Making under Uncertainty –Maximax and Maximin –Laplace and Hurwicz Decision Making under Risk –Maximum Likelihood –Baye’s Decision Rule –Decision Trees –New Information Utility Theory Chapter 9 2

3. Outline Probability Review Introduction and Terminology Decision Making under Uncertainty –Maximax and Maximin –Laplace and Hurwicz Decision Making under Risk –Maximum Likelihood –Baye’s Decision Rule –Decision Trees –New Information Utility Theory Chapter 9 3

4. Discrete Random Variables Suppose we have an experiment whose outcome depends on chance –The outcome of the experiment is called random variable –The sample space of the experiment is the set of all possible outcomes. –A Random Variable, X, is said to be discrete if it can take on at most a countable (finite) number of values That is, the sample space is finite –A subset of the sample space is called event 4

5. Discrete Random Variables Rolling a standard six-sided die –6 possible outcomes Sample space, S={1,2,3,4,5,6} –X is the outcome we get X is a random variable Probability of having 4 after we roll is 1/6 event 5

6. Discrete Random Variables Rolling a standard six-sided die –What is the probability of having an even number? –6 possible outcomes after we roll –Our event E={2,4,6}  P(E)=1/6+1/6+1/6=1/2 6

7. Discrete Random Variables P(X=x) = “Probability that X takes on the value x” –0 ≤ P(X=x) ≤ 1  P(X=x) =1 Probability of an event is the sum of the probabilities of the outcomes in this event Probability distribution function (pdf) = f(X) Cumulative distribution function (cdf) = F(X) 7

8. Discrete Random Variables E[X] = “expected value of the X” –E[X] =  x P(X=x) V[X] = “variance of the X” –V[X] =  x 2 P(X=x) – E[X] 2 –V[X]=  P(X=x)(x – E[X]) 2 SD[X] = “standard deviation of X” –SD[X] = √V[X] 8

9. Discrete Random Variables Rolling a die 9

10. Discrete Random Variables Three coins are tossed. Let X be the number of heads obtained. Construct a probability distribution for X and find its mean and standard deviation. XP(X) 01 / 8 13 / 8 2 31 / 8 E[X]= 0 * (1/8) + 1 * (3/8) + 2 * (3/8) + 3 * (1/8) = 1.5 V[X]= (0 - 1.5) 2 * (1/8) + (1 - 1.5) 2 * (3/8) + (2 - 1.5) 2 * (3/8) + (3 - 1.5) 2 * (3/8)= 1.11 10

11. Expectation Recall that –E[X] =  x P(X=x) We have the following properties –E[X+c]=E[X]+c where c is a constant –E[aX]=aE[X] where a is a constant –E[aX+bY]=aE[X]+bE[Y] where Y is also a random variable 11

12. Variance Recall that –V[X] =  x 2 P(X=x) – E[X] 2 We have the following properties –V[X+c]=V[X] where c is a constant –V[aX]=a 2 V[X] where a is a constant –V[aX+bY]= a 2 V[X]+ b 2 V[Y] if X and Y are independent variables 12

13. Expectation and Variance 13

14. Conditional Probability The conditional probability of an event A is the probability that the event will occur given the knowledge that an event B has already occurred –P(A|B), notation for the probability of A given B –What is the probability of having a 4 after you roll a die and you know that you have an even number? –P(X=4| even number)=1/3 14

15. Conditional Probability 15

16. Discrete Distributions Bernoulli (yes-no) distribution: –Two outcomes: whether an event occurs or not Binomial Distribution: –Describes the number of times an event occurs in a fixed number of trials (e.g., number of heads in 10 flips of a coin) –For each trial, only two outcomes possible –Trials are independent –Probability remains the same for each trial 16

17. Discrete Distributions Geometric Distribution: –Describes the number of trials until an event occurs (e.g., number of times to roll a die until you get 6) –Same probability for each trial –Continue until succeed, unlimited trials Negative Binomial Distribution: –Describes the number of trials until an event occurs n times –Similar to geometric (when n=1, you have geometric distribution) 17

18. Discrete Distributions Poisson Distribution: –Describes the number of times an event occurs during a given period of time or space –Occurrences are independent –Any number of events is possible Discrete Uniform Distribution: –Each outcome is equally likely –Rolling a dice, each outcome is ½ probability Read Chapter 13.7 18

19. Continuous Random Variables A random variable is called continuous if it can assume all possible values in the possible range of the random variable. –The interarrival times –The age of a bulp –The weight of a fish caught –The heat in a day 19

20. Continuous Random Variables 20

21. Continuous Distributions Uniform Distribution Triangular Distribution Normal Distribution Exponential Distribution Gamma Distribution Erlang Distribution Lognormal Distribution Read Chapter 13.7 21

22. Continuous Distributions Uniform distribution example 22

23. Outline Probability Review Introduction and Terminology Decision Making under Uncertainty –Maximax and Maximin –Laplace and Hurwicz Decision Making under Risk –Maximum Likelihood –Baye’s Decision Rule –Decision Trees –New Information Utility Theory Chapter 9 23

24. Decision Analysis: Introduction Managers often need to make decisions under uncertainty –Introducing a new product –A financial firm’s investment decisions –Agricultural firm’s mix of crop planning –Oil company’s drilling decisions –Storage Wars!!!! –Me preparing exams!!! Which questions to ask? 24

25. Decision Analysis: Example The Goferbroke Company develops oil wells in unproven territory –A consulting geologist has reported that there is a one- in-four chance of oil on a particular tract of land. –Drilling for oil on this tract would require an investment of about $100,000. If the tract contains oil, it is estimated that the net revenue generated would be approximately $800,000. –Another oil company has offered to purchase the tract of land for $90,000. Question: Should Goferbroke drill for oil or sell the tract? 25

26. Decision Analysis: Example Prospective profits Profit Status of LandOilDry Alternative Drill for oil$700,000–$100,000 Sell the land90,000 Chance of status1 in 43 in 4 26

27. Decision Analysis: Terminology The decision maker is the individual or group responsible for making the decision. The alternatives are the options for the decision to be made. The outcome is affected by random factors outside the control of the decision maker. These random factors determine the situation that will be found when the decision is executed. Each of these possible situations is referred to as a possible state of nature. The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities. Each combination of a decision alternative and a state of nature results in some outcome. The payoff is a quantitative measure of the value to the decision maker of the outcome. It is often the monetary value. 27

28. Decision Analysis: Prior Probability The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities. State of NaturePrior Probability The tract of land contains oil0.25 The tract of land is dry (no oil)0.75 28

29. Decision Analysis: Payoff Table Each decision alternative can have different values (payoffs) under different states of nature State of Nature AlternativeOilDry Drill for oil$700K$–100K Sell the land$90K Prior probability0.250.75 29

30. Outline Probability Review Introduction and Terminology Decision Making under Uncertainty –Maximax and Maximin –Laplace and Hurwicz Decision Making under Risk –Maximum Likelihood –Baye’s Decision Rule –Decision Trees –New Information Utility Theory Chapter 9 30

31. Decision Analysis: Decisions? When we have certainty, we can use –Linear programming, integer programming, –Binary programming, nonlinear programming What if we do not have certainty? –Uncertainty vs. Risk Uncertainty: Probabilities are not known Risk (Stochastic): Probabilities are known –We can still use mathematical programming for both cases!! 31

32. Maximax Criterion The maximax criterion is the decision criterion for the eternal optimist. It focuses only on the best that can happen. Plan for the best worst case Procedure: –Identify the maximum payoff from any state of nature for each alternative. –Find the maximum of these maximum payoffs and choose this alternative. 32

33. Maximax Criterion Drill for oil State of Nature AlternativeOilDryMaximum in Row Drill for oil700–100 700  Maximax Sell the land90 33

34. Maximax Criterion Another example… Future stateF1F1 F2F2 F3F3 Alternative(Payoff) A1A1 100 400 A2A2 -200150600 A3A3 0200500 A4A4 100300200 A1A2A3A4A1A2A3A4 Best Best Case 400 (j=3) 600 (j=3) 500 (j=3) 300 (j=2) i j Risky Payoff 34

35. Maximin Criterion The maximin criterion is the decision criterion for the total pessimist. It focuses only on the worst that can happen. Plan for the best worst case Procedure: –Identify the minimum payoff from any state of nature for each alternative. –Find the maximum of these minimum payoffs and choose this alternative. 35

36. Maximin Criterion Sell the land Note that the focus is on payoffs… –Do not confuse with cost vs. profits State of Nature AlternativeOilDryMinimum in Row Drill for oil700–100 Sell the land90 90  Maximin 36

37. Maximin Criterion Another example… Future stateF1F1 F2F2 F3F3 Alternative(Payoff) A1A1 100 400 A2A2 -200150600 A3A3 0200500 A4A4 100300200 A1A2A3A4A1A2A3A4 Best Worst Case 100 (j=1 or 2) -200 (j=1) 0 (j=1) 100 (j=1) Conservative i j Payoff 37

38. Laplace Criterion Assume that probability of each occurrence is identical ProbabilityN/A Future StateF1F1 F2F2 F3F3 Alternative(Payoff) A1A1 100 400 A2A2 -200150600 A3A3 0200500 A4A4 100300200 1/3 1/3 1/3 A1A2A3A4A1A2A3A4 Average Payoff 200=(100+100+400)/3 183=(-200+150+600)/3 233=(0+200+500)/3. 200=(100+300+200)/3 38

39. Hurwicz (Realism) Criterion Combination of maximin maximax What if = 0?  Maximin What if = 1?  Maximax How optimistic you are 39

40. Hurwicz (Realism) Criterion (Adapted From: Blanchard and Fabrycky, “System Engineering and Analysis, Prentice Hall, 1998) Values for the Hurwicz Rule for Four Alternatives $600 400 200 0 $600 400 200 0 -200 0 0.2 0.4 0.6 0.8 1.0 A2A3A1A4A2A3A1A4 i=1 i=3 i=2 Risky Conservative 40

41. Decision Making Under Uncertainity When you have no information about probabilities –Maximax –Minimax –Laplaca –Hurwicz “Cost-stable truck scheduling at a cross-dock facility with unknown truck arrivals” Konur and Golias (2013) You do not know when the trucks will arrive Schedule to minimize maximum costs? –Stability: range between maximum and minimum possible costs –Minimize the range while minimizing the average costs 41

42. Decision Making under Risk Now suppose that you know some information about the probabilities –You can still use maximax, minimax, laplace, hurwicz Risk has three primary components: –an event –a probability of occurrence of that event –the impact of that event You can utilize the probabilities that you know –Maximum likelihood criterion –Baye’s decision rule 42

43. Outline Probability Review Introduction and Terminology Decision Making under Uncertainty –Maximax and Maximin –Laplace and Hurwicz Decision Making under Risk –Maximum Likelihood –Baye’s Decision Rule –Decision Trees –New Information Utility Theory Chapter 9 43

44. Maximum Likelihood Criterion The maximum likelihood criterion focuses on the most likely state of nature. –Procedure: Identify the state of nature with the largest prior probability Choose the decision alternative that has the largest payoff for this state of nature. State of Nature AlternativeOilDry Drill for oil700–100 Sell the land90 90  Step 2: Maximum Prior probability0.250.75  Step 1: Maximum 44

45. Maximum Likelihood Criterion Practice… Probability0.30.20.5 Future StateF1F1 F2F2 F3F3 Alternative(Payoff) A1A1 100 400 A2A2 -200150600 A3A3 0200500 A4A4 100300200 Which alternative will you choose based on Maximum likelihood criterion? 45

46. Baye’s Decision Rule Bayes’ decision rule directly uses the prior probabilities. Procedure: –For each decision alternative, calculate the weighted average of its payoff by multiplying each payoff by the prior probability and summing these products. This is the expected payoff (EP). –Choose the decision alternative that has the largest expected payoff. 46

47. Baye’s Decision Rule State of Nature AlternativeOilDry Drill for oil$700K$–100K Sell the land$90K Prior probability0.250.75 EP=700*1/4-100*3/4=100 EP=90*1/4+100*3/4=90 47

48. Baye’s Decision Rule Another example… E[X] =  x p(x) A 1 :100(0.3) + 100(0.2) + 400(0.5) = $250 A 2 :-200(0.3) + 150(0.2) + 600(0.5) = $270 A 3 :0(0.3) + 200(0.2) + 500(0.5) = $290 A 4 :100(0.3) + 300(0.2) + 200(0.5) = $190 Must choose Alternative 3! Probability0.30.20.5 Future State F1F1 F2F2 F3F3 Alternative A1A1 100 400 A2A2 -200150600 A3A3 0200500 A4A4 100300200 Payoffs 48

49. Baye’s Decision Rule Features of Bayes’ Decision Rule –It accounts for all the states of nature and their probabilities. –The expected payoff can be interpreted as what the average payoff would become if the same situation were repeated many times. Therefore, on average, repeatedly applying Bayes’ decision rule to make decisions will lead to larger payoffs in the long run than any other criterion. Criticisms of Bayes’ Decision Rule –There usually is considerable uncertainty involved in assigning values to the prior probabilities. –Prior probabilities inherently are at least largely subjective in nature, whereas sound decision making should be based on objective data and procedures. –It ignores typical aversion to risk. By focusing on average outcomes, expected (monetary) payoffs ignore the effect that the amount of variability in the possible outcomes should have on decision making. 49

50. Other criteria Aspiration-Level Criterion - desired/undesired level of achievement –Suppose that you do not want to have a possibility of losing money, then you will sell the land Mean-Variance Criterion - based on “average” outcome and “variance” –You may want to have variance of an alternative to be less than a specific value 50

51. Decision Trees A decision tree can apply Bayes’ decision rule while displaying and analyzing the problem graphically. A decision tree consists of nodes and branches. –A decision node, represented by a square, indicates a decision to be made. The branches represent the possible decisions. –An event node, represented by a circle, indicates a random event. The branches represent the possible outcomes of the random event. 51

52. Decision Trees Decision tree of the Goferbroke Company E[Payoff] = 700(.25) – 100(.75) = $100 E[Payoff] = 90(1) = $90 52

53. Sensitivity Analysis At what probability of oil are you indifferent? E[Payoff of Drill] = E[Payoff of Sell] 700p -100(1-p) = $90  p =.2375 53

54. Value of More Information Might it be worthwhile to spend money for more information to obtain better estimates? –What if we knew for sure whether or not there was oil? –We can make the decision after we learn the true state A quick way to check is to pretend that it is possible to actually determine the true state of nature (“perfect information”). 54

55. Value of More Information EP (with perfect information) = Expected payoff if the decision could be made after learning the true state of nature. EP (without perfect information) = Expected payoff from applying Bayes’ decision rule with the original prior probabilities. The expected value of perfect information is then EVPI = EP (with perfect information) – EP (without perfect information). 55

56. Value of More Information For the Goferbroke Company –When we have perfect information on the state, we automatically select the best option for that state If Dry, we will sell If Oil, we will drill 56

57. Value of More Information For the Goferbroke Company –EP(without perfect information)=100 –EP(with perfect information)=242.5 Expected value of perfect information: –EVPI= EP(with perfect information)- EP(without perfect information) –EVPI=242.5-100=142.5 57

58. Value of More Information Let’s say you can pay $C to have perfect information –A seismic survey to obtain better estimates If –EPVI>C, it might be worthwhile to do the survey –EPVI<C, it is not worthwhile to do the survey 58

59. Posterior Probabilities The prior probabilities of the possible states of nature often are quite subjective in nature. –They may only be rough estimates. It is frequently possible to do additional testing or surveying (at some expense) to improve these estimates. –The improved estimates are called posterior probabilities. 59

60. Using the New Information Goferbroke can obtain improved estimates of the chance of oil by conducting a detailed seismic survey of the land, at a cost of $30,000. –Possible findings from a seismic survey: FSS: Favorable seismic soundings; oil is fairly likely. USS: Unfavorable seismic soundings; oil is quite unlikely. –P(finding | state) =Probability that the indicated finding will occur, given that the state of nature is the indicated one. 60

61. Using the New Information Prior Probabilities –P(Oil) = 0.25 –P(Dry) = 0.75 Conditional Probabilities P(finding | state) State of NatureFavorable (FSS)Unfavorable (USS) OilP(FSS | Oil) = 0.6P(USS | Oil) = 0.4 DryP(FSS | Dry) = 0.2P(USS | Dry) = 0.8 61

62. Using the New Information Each combination of a state of nature and a finding will have a joint probability determined by the following formula: –P(state and finding) = P(state) P(finding | state) P(Oil and FSS) = P(Oil) P(FSS | Oil) = (0.25)(0.6) = 0.15. P(Oil and USS) = P(Oil) P(USS | Oil) = (0.25)(0.4) = 0.1. P(Dry and FSS) = P(Dry) P(FSS | Dry) = (0.75)(0.2) = 0.15. P(Dry and USS) = P(Dry) P(USS | Dry) = (0.75)(0.8) = 0.6. 62

63. Using the New Information Given the joint probabilities of both a particular state of nature and a particular finding, the next step is to use these probabilities to find each probability of just a particular finding, without specifying the state of nature. –P(finding) = P(Oil and finding) + P(Dry and finding) P(FSS) = 0.15 + 0.15 = 0.3. P(USS) = 0.1 + 0.6 = 0.7. 63

64. Using the New Information The posterior probabilities give the probability of a particular state of nature, given a particular finding from the seismic survey. –P(state | finding) = P(state and finding) / P(finding) P(Oil | FSS) = 0.15 / 0.3 = 0.5. P(Oil | USS) = 0.1 / 0.7 = 0.14. P(Dry | FSS) = 0.15 / 0.3 = 0.5. P(Dry | USS) = 0.6 / 0.7 = 0.86. 64

65. Calculating the Posterior Probabilities The formula for a posterior probability (Baye’s Theorem) P(state | finding)= P(state)P(finding | state) P(oil)P(finding | oil)+P(dry)P(finding | dry) P(state | finding) FindingOilDry Favorable (FSS)P(Oil | FSS) = 1/2P(Dry | FSS) = 1/2 Unfavorable (USS)P(Oil | USS) = 1/7P(Dry | USS) = 6/7 65

66. Decision Trees Decision tree with the survey option 66

67. Decision Trees And the payoffs (0.7) 67

68. Decision Making Sell Drill Do survey 68

69. Decision Making Best decision –Do the seismic survey If the result is unfavorable, sell the land If the result is favorable, drill for oil –The expected payoff is 123 69

70. Outline Probability Review Introduction and Terminology Decision Making under Uncertainty –Maximax and Maximin –Laplace and Hurwicz Decision Making under Risk –Maximum Likelihood –Baye’s Decision Rule –Decision Trees –New Information Utility Theory 70

71. Utility Theory Thus far, when applying Bayes’ decision rule, we have assumed that the expected payoff in monetary terms is the appropriate measure. –In many situations, this is inappropriate. Accept a 50-50 chance of winning $100,000. Receive $40,000 with certainty. Many would pick $40,000, even though the expected payoff on the 50-50 chance of winning $100,000 is $50,000. This is because of risk aversion. –A utility function for money is a way of transforming monetary values to an appropriate scale that reflects a decision maker’s preferences (e.g., aversion to risk). 71

72. Utility Theory The people!! –U(M): utility function for money 72

73. Utility Theory When a utility function for money is incorporated into a decision analysis approach, it must be constructed to fit the current preferences and values of the decision maker. When the decision maker’s utility function for money is used, Bayes’ decision rule replaces monetary payoffs by the corresponding utilities. The optimal decision (or series of decisions) is the one that maximizes the expected utility. 73

74. Utility Theory Fundamental Theory: –Under the assumptions of utility theory, the decision maker’s utility function for money has the property that the decision maker is indifferent between two alternatives if the two alternatives have the same expected utility. 74

75. Utility Theory Fundamental Theory: 25% chance of $100,000 = $10,000 for sure Both have E(Utility) = 0.25. 50% chance of $100,000 = $30,000 for sure Both have E(Utility) = 0.5. 75% chance of $100,000 = $60,000 for sure Both have E(Utility) = 0.75. 75

76. Utility Theory Determine the largest potential payoff, M=Maximum. –Assign U(Maximum) = 1. Determine the smallest potential payoff, M=Minimum. –Assign U(Minimum) = 0. To determine the utility of another potential payoff M, consider the two aleternatives: –A 1 :Obtain a payoff of Maximum with probability p. Obtain a payoff of Minimum with probability 1–p. –A 2 : Definitely obtain a payoff of M. Question to the decision maker: What value of p makes you indifferent? Then, U(M) = p. 76

77. Utility Theory The possible monetary payoffs in the Goferbroke Co. problem are – 130, –100, 0, 60, 90, 670, and 700 (all in $thousands). Set U(Maximum) = U(700) = 1. Set U(Minimum) = U(–130) = 0. To find U(M), use the equivalent lottery method. For example, for M=90, consider the two alternatives: A 1 :Obtain a payoff of 700 with probability p Obtain a payoff of –130 with probability 1–p. A 2 :Definitely obtain a payoff of 90 If Max chooses a point of indifference of p = 1/3, then U(90) = 1/3 77

78. Utility Theory Utility function will be 78

79. Utility Theory Risk Averse –U(t a ) = R(1-e -t/R ) Risk Neutral –U(t n ) = at + b Risk Seekers –U(t s ) = t 2 /c 79

80. Further study… Read Chapter 9 Practice problems –9.1, 9.2, 9.3, 9.4, 9.7, 9.19, 9.20, 9.27 80