1. 1 Chapter 7 Applying Simulation to Decision Problems

2. 2 Introduction When the payoff of a decision depends upon a large number of factors, estimating a probability distribution for the possible values of this payoff can be a difficult task.

3. 3 The decision analysis approach to this problem is to help the decision maker by initially dividing the probability assessment task into smaller parts. Need to determine their combined effect in order to obtain a probability distribution for the return on the investment.

4. 4 There will be a large or infinite number of combinations of circumstances which could affect the return on the investment. It is clearly impractical to use an approach such as a probability tree to calculate the probability of each of these combinations of circumstances occurring.

5. 5 One answer to our problem is to use a versatile and easily understood technique called Monte Carlo simulation. This involves the use of a computer to generate a large number of possible combinations of circumstances which might occur if a particular course of action is chosen.

6. 6 Monte Carlo simulation: A simple example Cash inflows Probability outflows Probability ($) (%) ($) (%) 50 000 30 50 000 45 60 000 40 70 000 55 70 000 30 100 100

7. 7 Assigning random numbers to cash inflows Cash inflow Probability Random ($) (%) numbers 50 000 30 00 – 29 60 000 40 30 – 69 70 000 30 70 – 99

8. 8 Assigning random numbers to cash outflows Cash outflowProbabilityRandom ($) (%) numbers 50 000 4500 – 44 70 000 5545 – 99

9. 9 Ten simulations of monthly cash flows (for illustration only)

10. 10 Estimating probabilities from the simulation results

11. 11 The effect of the number of simulations on the reliability of the probability estimates

12. 12 How many simulations are needed to give an acceptable level of reliability? There is virtually no change in the estimates produced by the simulation.

13. 13 Applying simulation to a decision problem The application of simulation to a problem like this involves the following stages: (1) Identify the factors that will affect the payoffs of each course of action. (2) Formulate a model to show how the factors are related. (3) Carry out a preliminary sensitivity analysis to establish the factors for which probability distributions should be assessed. (4) Assess probability distributions for the factors which were identified in stage 3. (5) Perform the simulation. (6) Apply sensitivity analysis to the results of the simulation. (7) Compare the simulation results for the alternative courses of action and use these to identify the preferred course of action.

14. 14 Stage 1: Identify the factors The first stage in our analysis of Elite's problem is to identify the factors which it is thought will affect the profit of the two products. Figure 7.1 shows a tree for this problem. Subsequent analysis will be simplified if the factors can be identified in such a way that their probability distributions can be considered to be independent.

15. 15 Factors affecting profit earned by the plate

16. 16 Stage 2: Formulate a model Formulate a mathematical model to show how they affect the variable of interest. For Elite's problem : Profit = (Price - Variable cost) x Sales - Fixed costs A balance has to be struck between the need to keep the model simple and understandable and the need to provide a reasonable and plausible representation of the real problem.

17. 17 Stage 3: Preliminary sensitivity analysis Sensitivity analysis can be helpful at this stage in screening out those factors which do not require a probability distribution.

18. 18 Estimates of values for Elite pottery

19. 19 (i) Identify the lowest, highest and most likely values that each factor can assume. (ii) Calculate the profit which would be achieved if the first factor was at its lowest value and the remaining factors were at their most likely values.

20. 20 If variable costs are at their lowest possible value of $8 and the remaining factors are at their most likely value we have: Profit = ($25 - $8) x 22 000 - $175 000 = $199 000 (iii) Repeat (ii), but with the first factor at its highest possible value. Thus we have: Profit = ($25 - $18) x 22 000 - $175 000 = -$21000

21. 21 (iv) Repeat stages (ii) and (iii) by varying, in turn, each of the other factors between their lowest and highest possible values while the remaining factors remain at their most likely values.

22. 22 Tornado diagram for Elite pottery

23. 23 Stage 4: Assess probability distributions Figure 7.3 shows the probability distributions which were obtained for variable costs, sales and fixed costs.

24. 24 Probability distributions

25. 25 Stage 5: Perform the simulation Simulation can now be used to obtain a probability distribution for the profit which the plate will earn. The simulation involves the generation of three random numbers. For example, Profit = ($25 - $13.2) x 26500 - $125000= $187700

26. 26 Simulation results

27. 27 Probability distribution of profit earned by commemorative plate

28. 28 Stage 7: Compare alternative course of action In order to help choose between the products, the decision maker clearly needs to compare the two profit probability distributions. This comparison can be made in a number of ways.

29. 29 Probability distribution for profit on the figurine

30. 30 The decision maker can compare the probabilities of each product making a loss or the probabilities that each product would reach a target level of profit.

31. 31 A comparison -- plotting the two probability distributions

32. 32 The distribution for the figurine has a greater spread. It implies that there is more uncertainty about the profit which will actually be achieved. The spread of a distribution is often used as a measure of the risk. A distribution's spread can be measured by calculating its standard deviation.

33. 33 Determining the option with the highest expected utility It is difficult to construct the utility function, since each option has a very large number of possible outcomes. One way around this problem is to find a mathematical function which will approximate the decision maker's utility function.

34. 34 Stochastic dominance Sometimes the alternative with the highest expected utility can be identified by a short cut method which is based on a concept known as stochastic dominance. This exists where the expected utility of one option is greater than that of another for an entire class of utility functions. All we need to establish is that the utility function has some basic characteristics.

35. 35 First- and second-degree stochastic dominance are the two most useful forms which the CDFs can reveal.

36. 36 First-degree stochastic dominance This concept requires some very unrestrictive assumptions about the nature of the decision maker's utility function. When money is the attribute under consideration, the main assumption is simply that higher monetary values have a higher utility.

37. 37 Cumulative probability distributions

38. 38 First-degree stochastic dominance

39. 39 Because Q's CDF is always to the right of P's, we can say that Q exhibits first-degree stochastic dominance over P.

40. 40 Second-degree stochastic dominance When the CDFs for the options intersect each other at least once it may still be possible to identify the preferred option if, in addition to the weak assumptions we made for first-degree stochastic dominance, we can also assume that the decision maker is risk averse (i.e. his utility function is concave) for the range of values under consideration.

41. 41 To determine which is the dominant product overall we need to compare both the lengths of the ranges for which the products are dominant and the extent to which they are dominant within these ranges (i.e. the extent to which one curve falls below the other).

42. 42 Two more cumulative distributions

43. 43 Second-degree stochastic dominance

44. 44 The mean-standard deviation approach When a decision problem involves a large number of alternative courses of action it is helpful if inferior options can be screened out at an early stage. In these situations the mean- standard deviation approach can be useful. Maximize the expected mean and minimize the standard deviation (risk-averse)

45. 45 The mean-standard deviation screening method

46. 46 The non-dominated products. A, C and D, are therefore said to lie on the efficient frontier, and only these products would survive the screening process and be considered further. The choice between A, C and D will depend on how risk averse the decision maker is.

47. 47 Assumptions for the mean-standard deviation screening process First, the probability distributions for profit should be fairly close to the normal distribution shape. Second, the decision maker should have a utility function which not only indicates risk aversion but which also has (at least approximately) a quadratic form. U(x) = c + bx + ax 2

48. 48 A normal distribution for profit

49. 49 Examples of quadratic utility functions

50. 50 Applying simulation to investment decisions The net present value (NPV) method This implies that money which will be earned in the future should be discounted so that its value can be compared with sums of money which are being held now. The process involved is referred to as ‘ discounting to present value ’.

51. 51 An example: Annual cash flows for two machines (initial cost: $30000)

52. 52 Calculating the NPV for the Alpha machine

53. 53 Calculating the NPV for the Beta machine

54. 54 According to the NPV criterion, the Alpha machine offers the most attractive investment opportunity. In general, there will be uncertainty about the size of the future cash flows and about the lifetime of the project. Expressing the cash flow estimates as single figures creates an illusion of accuracy, but it also means that we have no idea as to how reliable the resulting NPV is.

55. 55 Clearly, the approach would be more realistic if we could incorporate our uncertainty about the cash flows into the analysis. We could assess the chances of the project producing a negative NPV or the probability that the NPV from one project will exceed that of a competing project.

56. 56 Using simulation Factors would affect the return on the investment: (i) The price of the machine ($30000); (ii) The revenue resulting from the machine's output in years 1 to 4; (iii) Maintenance costs in years 1 to 4; (iv) The scrap value of the machine at the end of year 4.

57. 57 Probability distributions for Alpha machine

58. 58 NPV probability distributions for the two machines