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Risk and Decision Modeling

Risk and Decision Modeling. Objectives. Explore decision making environments Examine the concepts of risk and uncertainty Review various approaches to decision modeling Examine decision modeling with imprecise information. Introduction.

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Risk and Decision Modeling

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  1. Risk and Decision Modeling

  2. Objectives Explore decision making environments Examine the concepts of risk and uncertainty Review various approaches to decision modeling Examine decision modeling with imprecise information.

  3. Introduction Decision making (DM)) is the process of selecting an action where we have different alternatives to follow. DM is very common in our lifes. From decision making in business environments, for example: Selection of human resources Selection of investments to the individual decisions of our lives, for example: Selection of an apartment Selection of a car Selection of food. 3

  4. Introduction Usually, we use judgment when making these decisions but it is important to note that implicitly, there is a mathematical (statistical) model that describes these processes. Moreover, DM also exists in a more complex way that affects our subconscious, for example: The decision of saying the appropriate words when speaking The decisions given to our body when walking or doing a sport activity. 4

  5. Decision Making Environments In general, the decision processes are affected by one of the following environments: DM under certainty DM under risk DM under uncertainty 5

  6. DM under Certainty Situations where we know what is going to happen in the future. In these situations, we should be able to solve the problem. Various methods are available, including: Linear programming. 6

  7. DM under Risk Situations where we do not know what is going to happen in the future but we have some probabilistic information to assess the problem. Note that depending on the available information we can use: Objective probabilities (based on historical or experimental data). Subjective probabilities (based on our beliefs). Methods are available, including: The expected value approach. 7

  8. DM under Uncertainty Situations where we do not know what is going to happen in the future and we do not have probabilistic information available. Methods: Optimistic approach: Max {ai}. Pessimistic approach: Min {ai}. Laplace criteria: (1 / n) × (Σai). Hurwicz criteria: α× Max {ai} + (1 –α) × Min {ai}. 8

  9. DM under Uncertainty These classical methods are particular cases of a more general model called the ordered weighted averaging (OWA) operator. OWA = where bj is the jth largest of the ai. 9

  10. DM under Uncertainty As we can see in the OWA, if: w1 = 1 and wj = 0 for j ≠ 1  Max {ai}. (Optimistic approach). wn = 1 and wj = 0 for j ≠ 1  Min {ai}. (Pessimistic approach). w1 = α, wn = (1 – α), and wj = 0 for j ≠ 1, n  α× Max {ai} + (1 –α) × Min {ai}. (Hurwicz criteria). wj = 1 / n, for all j (1 / n) × (Σai). (Laplace criteria). 10

  11. DM under Uncertainty - Example Example: Assume a decision maker wants to invest some money in a company and he considers 4 alternatives. A1: Invest in a automobile company. A2: Invest in a computer company. A3: Invest in a pharmaceutical company. A4: Invest in a food company. 11

  12. DM under Uncertainty - Example He assumes that the key factor that affects the results obtained with each investment is the economic situation of the country. He considers the following states of nature. S1: Very good economic situation. S2: Good economic situation. S3: Regular economic situation. S4: Bad economic situation. S5: Very bad economic situation. 12

  13. DM under Uncertainty - Example Payoff matrix: 13

  14. DM under Uncertainty - Example With the optimistic criteria: A1 = 80. A2 = 70. A3 = 70. A4 = 60. 14

  15. DM under Uncertainty - Example With the pessimistic criteria: A1 = 20. A2 = 30. A3 = 40. A4 = 50. 15

  16. DM under Uncertainty - Example With the Laplace criteria: A1 = (1/5) (80 + 60 + 50 + 30 + 20) = 48. A2 = (1/5) (70 + 70 + 40 + 40 + 30) = 50. A3 = (1/5) (70 + 60 + 50 + 50 + 40) = 54. A4 = (1/5) (60 + 50 + 50 + 50 + 60) = 54. 16

  17. DM under Uncertainty - Example With the Hurwicz criteria (α = 0.4): A1 = 0.4 × 80 + 0.6 × 20 = 44. A2 = 0.4 × 70 + 0.6 × 30 = 46. A3 = 0.4 × 70 + 0.6 × 40 = 52. A4 = 0.4 × 60 + 0.6 × 60 = 60. 17

  18. DM under Uncertainty - Example With the OWA operator (W = (0.1, 0.2, 0.2, 0.2, 0.3)). This means that we are a bit pessimistic because we put more weight at the end of the weighting vector W. A1 = 0.1 × 80 + 0.2 × 60 + 0.2 × 50 + 0.2 × 30 + 0.3 × 20 = 42. A2 = 0.1 × 70 + 0.2 × 70 + 0.2 × 40 + 0.2 × 40 + 0.3 × 30 = 46. A3 = 0.1 × 70 + 0.2 × 60 + 0.2 × 50 + 0.2 × 50 + 0.3 × 40 = 51. A4 = 0.1 × 60 + 0.2 × 60 + 0.2 × 50 + 0.2 × 50 + 0.3 × 50 = 53. 18

  19. DM under Uncertainty - Example Note that with the OWA: Optimistic criteria: W = (1, 0, 0, 0, 0). Pessimistic criteria: W = (0, 0, 0, 0, 1). Laplace criteria: W = (0.2, 0.2, 0.2, 0.2, 0.2). Hurwicz criteria: W = (0.4, 0, 0, 0, 0.6). 19

  20. DM under Uncertainty - Example Note that in DM under Risk, we assume that we have some probabilistic information regarding the states of nature. For example, if P = (0.1, 0.1, 0.4, 0.3, 0.1), we are saying that the probability that economic situation is “regular”, is 40%, “bad” 30%, and so on. Thus: A1 = 0.1 × 80 + 0.1 × 60 + 0.4 × 50 + 0.3 × 30 + 0.1 × 20 = 45. A2 = 0.1 × 70 + 0.1 × 70 + 0.4 × 40 + 0.3 × 40 + 0.1 × 30 = 45. A3 = 0.1 × 70 + 0.1 × 60 + 0.4 × 50 + 0.3 × 50 + 0.1 × 40 = 52. A4 = 0.1 × 60 + 0.1 × 50 + 0.4 × 50 + 0.3 × 50 + 0.1 × 60 = 52. 20

  21. Other DM models under Uncertainty There are a lot of other decision making models for dealing with uncertainty, for example: Savage criteria: DM with minimization of regret. The Analytic Hierarchy Process (AHP). The TOPSIS method. 21

  22. Recent approaches – Risk - Uncertainty Some recent methods have tried to consider both risk and uncertainty in the same problem, that is, situations where we deal with probabilities and with the attitudinal character (degree of optimism) of the decision maker. Immediate probabilities. Probabilistic OWA operator. 22

  23. Other DM models More complex DM models can be used in the analysis depending on the problem we are studying. For example: Group DM. Game theory. Utility theory. Sequential DM: Decision trees. Dempster-Shafer belief structure. 23

  24. Dealing with imprecise information Sometimes, the available information is not clear and we cannot assess it with the “classical” exact numbers. Therefore, a better approach may be the use of other methods such as: Interval numbers. Fuzzy numbers. Linguistic variables Probabilistic sets. 24

  25. Interval numbers Example: Do you know the US inflation for the next year ? 1.7% ? 2% ? 3.2% ? 2.3% ? We don’t know and we cannot provide a number because our knowledge (information) is imprecise. 25

  26. Interval numbers Thus, we need to use another technique such as the use of interval numbers. In this case, we could assume that the inflation could be: [1.7, 3.2]% That is, at least 1.7% and no more than 3.2%. With this information, we know at least that the inflation is going to be in this interval. Although we don’t know the inflation for the next year, we have this general information that permits us to make further analysis. 26

  27. Interval analysis For example, if we can make an investment and expect to gain [20, 40] million dollars, then we would gain a lot of money with this investment no matter what happens. On the other hand, imagine that the expected benefits are: [–20, 80]. If we use the expected value of 30 to represent the payoffs, this outcome seems to be positive. However, as you can see the real analysis shows that this investment is not safe because we may lose a lot of money. 27

  28. Interval numbers Types: Interval (2-tuple): [a1, a2]. Ex: [4, 7]. Triplet: [a1, a2, a3], where a1 ≤ a2 ≤ a3. Ex: [2, 4, 9]. Quadruplet: [a1, a2, a3, a4], where a1 ≤ a2 ≤ a3 ≤ a4. Ex: [2, 4, 6, 9]. Etc. 28

  29. Interval numbers Basic operations with intervals (2-tuples). Let A = [a1, a2] and B = [b1, b2]. Addition (A + B): [a1 + b1, a2 + b2]. Subtraction (A – B): [a1–b2, a2–b1]. Multiplication (A × B): [min {a1b1, a1b2, a2b1, a2b2}, max {a1b1, a1b2, a2b1, a2b2}]. If R+: [a1 × b1, a2 × b2]. Division (A  B): [min {a1 b1, a1 b2, a2 b1, a2 b2}, max {a1 b1, a1 b2, a2 b1, a2 b2}]. If R+: [a1b2, a2b1]. 29

  30. Interval numbers Examples. Let A = [6, 8] and B = [2, 3]. A + B = [6 + 2, 8 + 3] = [8, 11]. A – B = [6 – 3, 8 – 2] = [3, 6]. A × B = [6 × 2, 8 × 3] = [12, 24]. A  B = [6  3, 8  2] = [2, 4]. 30

  31. Interval numbers Sometimes it is not clear which interval is higher. Thus, we need to use a method for ranking interval numbers. We recommend to use the average (or weighted average) of the interval. That is: Let A = [10, 20] and B = [15, 17]. A = (10 + 20) / 2 = 15. B = (15 + 17) / 2 = 16. Thus, we assume that B > A. 31

  32. Fuzzy numbers The FNs are similar to the interval numbers but more complete because they provide more information concerning the possibility that the internal values of the interval will occur. There are a lot of types of FNs, for example: Triangular FNs. Trapezoidal FNs. Interval – valued FNs. 32

  33. Fuzzy numbers Triangular FNs: (a1, a2, a3) = (a1 + (a2–a1) × α, a3–(a3–a2) × α). α = 0  (a1, a3)  the minimum and the maximum values. α = 1  (a2, a2) = a2 the most possible value. Example: (3, 6, 8) = (3 + 3α, 8 – 2α). α = 0  (3, 8). α = 1  (6, 6) = 6. α = 0.4  (3 + 3 × 0.4, 8 – 2 × 0.4) = (4.2, 7.2). 33

  34. DM under Uncertainty and Imprecise Information Assume an investment decision making problem: 34

  35. DM under Uncertainty and Imprecise Information With the optimistic criteria: A1 = (70, 90). A2 = (70, 80). A3 = (60, 70). A4 = (55, 60). 35

  36. DM under Uncertainty and Imprecise Information With the pessimistic criteria: A1 = (10, 30). A2 = (20, 30). A3 = (35, 45). A4 = (46, 50). 36

  37. DM under Uncertainty and Imprecise Information With the Laplace criteria: A1 = (1/5) [(70, 90) + (50, 70) + (40, 70) + (30, 50) + (10, 30)] = (40, 62). A2 = (1/5) [(70, 80) + (60, 70) + (40, 50) + (30, 40) + (20, 30)] = (44, 54). A3 = (1/5) [(60, 70) + (50, 60) + (45, 50) + (42, 50) + (35, 45)] = (46.4, 55). A4 = (1/5) [(55, 60) + (50, 53) + (48, 50) + (46, 50) + (52, 60)] = (50.2, 54.6). 37

  38. DM under Uncertainty and Imprecise Information In this case, it is not clear which value is higher, so we have to calculate the average of the interval: A1 = (40 + 62) / 2 = 51. A2 = (44 + 54) / 2 = 49. A3 = (46.4 + 54) / 2 = 50.2. A4 = (50.2 + 54.6) / 2 = 52.4. 38

  39. DM under Uncertainty and Imprecise Information With the Hurwicz criteria (α = 0.4): A1 = 0.4 × (70, 90) + 0.6 × (10, 30) = (34, 54). A2 = 0.4 × (70, 80) + 0.6 × (20, 30) = (40, 50). A3 = 0.4 × (60, 70) + 0.6 × (35, 45) = (45, 55). A4 = 0.4 × (55, 60) + 0.6 × (46, 50) = (49.6, 54). 39

  40. DM under Uncertainty and Imprecise Information In this case, it is not clear which value is higher, so we have to calculate the average of the interval: A1 = (34 + 54) / 2 = 44. A2 = (40 + 50) / 2 = 45. A3 = (45 + 55) / 2 = 50. A4 = (49.6 + 54) / 2 = 51.8. 40

  41. DM under Uncertainty and Imprecise Information With the OWA operator (W = (0.1, 0.2, 0.2, 0.2, 0.3)). A1 = 0.1 × (70, 90) + 0.2 × (50, 70) + 0.2 × (40,70) + 0.2 × (30, 50) + 0.3 × (10, 30) = (34, 56). A2 = 0.1 × (70, 80) + 0.2 × (60, 70) + 0.2 × (40, 50) + 0.2 × (30, 40) + 0.3 × (20, 30) = (39, 49). A3 = 0.1 × (60, 70) + 0.2 × (50, 60) + 0.2 × (45, 50) + 0.2 × (42, 50) + 0.3 × (35, 45) = (43.9, 52.5). A4 = 0.1 × (55, 60) + 0.2 × (52, 60) + 0.2 × (50, 53) + 0.2 × (48, 50) + 0.3 × (46, 50) = (49.3, 53.6). 41

  42. DM under Uncertainty and Imprecise Information In this case, it is not clear which value is higher, so we have to calculate the average of the interval: A1 = (34 + 56) / 2 = 45. A2 = (39 + 49) / 2 = 44. A3 = (43.9 + 52.5) / 2 = 48.2. A4 = (49.3 + 53.6) / 2 = 51.45. 42

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