Decision Support Models 2012/2013

Transcription

1 Risk Analysis Decision Support Models 2012/2013 Bibliography: Goodwin, P. and Wright, G. (2003) Decision Analysis for Management Judgment, John Wiley and Sons (chapter 7) Clemen, R.T. and Reilly, T. (2003). Making Hard Decisions With Decision Tools Suite Update 2004 Duxbury (chapter 11) 2 Models and Techniques in Decision Analysis Uncertainty Problem dominated by Complexity Revising opinion Bayesian Nets Evaluating options Choice Decision trees Influence diagrams Multicriteria Analysis (MACBETH, EQUITY)) Components decomposition Risk Analysis Resource allocation and negotiation 1

2 3 In real world situations, many factors may be subject to some uncertainty decision tree with many uncertain events: How to deal with this much uncertainty? 4 Influence diagrams are useful to structure situations with many sources of uncertainty but if there are multiple interrelated uncertain quantities represented by multiple continuous distributions, influence diagrams might not be useful 2

3 5 When the payoff of a decision depends upon a large number of uncertain factors, estimating a probability distribution for the possible values of this payoff can be a difficult task One solution: To divide the probability assessment task into smaller parts (a process sometimes referred to as credence decomposition ) E.g. we might ask the decision-maker to estimate individual probabilities for the size of the market, the market share, the launch costs, After having elicited these distributions, we need to determine the combined effect E.g. combined effect of several variables on the probability for the return on investment This might lead to an infinite number of values for circumstances that affect the return on investment in this case we cannot use a single probability tree!!! 6 One way to handle this Monte Carlo Simulation... Simulation technique based on computer simulation which allows for representing the uncertainty surrounding the possible payoffs for the different alternatives Think about the entire decision situation as an uncertain event Results from putting altogether all probability distributions for all uncertain quantities The more likely combination of circumstances will be generated most often while very unlikely combinations will be rarely generated 3

4 7 One way to handle this Monte Carlo Simulation... After running the simulation many times, we have an approximation of the probability distribution for the payoffs from the different alternatives The more simulations we carry out, the more accurate the approximation The results (both risk profiles and the average outcomes) can be used to make an appropriate decision Advantages vs. Disadvantages Flexibility and ease of use Rampant independence assumptions and a tendency to solve problems with brute force 8 Illustrating how the Monte Carlo Simulation Technique operates! Cash Inflows ($) Probability (%) Cash Outflows ($) Probability (%) 50,000 30% 50,000 45% 60,000 40% 70,000 55% 70,000 30% Sum 100% Sum 100% Use random numbers (generated similarly as if they were generated using a roulette wheel) in practice, pseudo-random numbers generated by a computer. Random numbers: Cash Inflows ($) Probability (%) Random numbers 50,000 30% ,000 40% ,000 30% Cash Outflows ($) Probability (%) Random numbers 50,000 45% ,000 55%

5 9 Illustrating how the Monte Carlo Simulation Technique operates I We should now perform the simulation run. Each simulation will generate two random numbers (one to determine the cash inflow, and another to determine the cash outflow) Ten simulations of monthly cash-flows Random no. Cash Inflow Random no. Cash outflow Net cash flow 46 60, ,000-10, , ,000 10, , ,000-20, , ,000 10, , ,000-20, , ,000-20, , ,000 20, , ,000-20, , , , , Illustrating how the Monte Carlo Simulation Technique operates II Estimating probabilities from the simulation results Net cash flow Number of simulations resulting in this net cash flow Probability estimate based on simulation Calculated probability -20, /10= , , ,

6 11 Illustrating how the Monte Carlo Simulation Technique operates II And visualising the effect of the number of simulations on the reliability of the probability estimates Net cash flow Probability estimates based on: 50 simulations Probability estimates based on: 1000 simulations Probability estimates based on: 5000 simulations Calculated probability -20, , , , Which is the minimum number of simulations? Applying simulation to a decision problem: The Elite Pottery Company The Elite Pottery Company is planning to market a special product to commemorate a major sporting event which is due to take place in a few month s time. A large number of possible products have been considered, but the list has now been winnowed down to two alternatives: a commemorative plate and a figurine. In order to make a decision between the two alternatives, the company s managing director needs to estimate the profit which would be earned by each product (the decision should be made solely on the basis of profit). There is some uncertainty about the costs of manufacturing the products and the levels of sales, although it is thought that all sales will be made in the very short period which coincides with the sporting event. 12 6

7 13 Simulation steps to be applied 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 step 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 7

8 15 Step 1: Identify the factors Profit Sales Revenue Costs Sales Price Fixed Costs Variable Costs 16 Step 2: Formulate a model Profit = (Price Variable Cost) Sales Fixed Costs 8

9 17 Step 3: Preliminary Sensitivity Analysis 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 iii. Repeat ii. but with the first factor at its highest possible value 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 Factor Most likely value Lowest possible value Highest possible value Variable costs Sales 22,000 units 10,000 units 30,000 units Fixed costs 175, , , Preliminary Sensitivity Analysis Graph shows the effect on profit if each factor changes from its lowest to its highest possible values 9

10 19 Step 4: Assess Probability Distributions Define probability distributions for variable costs, sales and fixed costs 20 Step 5: Perform the Simulation Simulation used to obtain the probability distribution for the profit of each plate, using a computer; use of three random numbers (variable costs, sales, fixed costs) Results for 500 simulations Profit No. of simulations Probability -200,000 to under -100, /500= ,000 to under to under 100, ,000 to under 200, ,000 to under 300, ,000 to under 400, Mean profit: 51,800 10

11 21 Step 5: Perform the Simulation Probability distribution for profit earned by the commemorative plate 22 Step 6: Sensitivity Analysis on the Results of the Simulation Monte Carlo simulation is a form of sensitivity analysis, in particular useful when there are doubts with regard to the probability distributions Changes in the structure of the model on results might be tested 11

12 23 Step 7: Comparing Alternative Courses of Action 1. Plotting the two distributions 2. Determining the option with the highest expected utility 3. Applying stochastic dominance 1. First degree stochastic dominance (=deterministic) 2. Second degree stochastic dominance (=stochastic) 4. The mean-standard deviation approach 24 Step 7.1: Plotting the Two Distributions Mean? Standard deviation? Which is more uncertain? 12

13 25 Step 7.2: Determining the Option with the Highest Expected Utility Because the two products offer different levels of risk, utility theory can be used to identify the option that the decision maker would use More on utility later 26 Step 7.3: Applying Stochastic Dominance (first degree) Short cut method Underlying assumption: when money is the attribute under consideration, the main assumption is simply that higher monetary values have a higher utility Product Q has always the highest expected utility; for any level of profit, Q offers the smallest probability of falling below that profit 13

14 27 Step 7.3: Applying Stochastic Dominance (second degree) When the CDFs for the options intersect each other at least once, it may be possible to identify the preferred option if, in addition to the weak assumption made for first degree stochastic dominance, we make the assumption that the decision maker is risk averse for the range of values under consideration. For the range between 0 and 15, Product R is the dominant product, while in the range 15 to 25, S dominates; since area X is larger than area Y, Product R stochastically Dominates Product S Step 7.3: The simpler case of Deterministic Dominance Stochastic but deterministic 28 14

15 29 Step 7.4: The Mean-Standard Deviation Approach Applying the same principles as in portfolio theory, a risk averse decision maker has to choose between a large number of possible investment portfolios 5 alternative products Objectives to maximize return and minimize risk or uncertainty Product A dominates B; C dominates B; D dominates E A, D and C belong to the efficient frontier, and only these products would survive the screening process In order to this approach to be valid, the probability distributions should be close to the Normal distribution shape. 30 Laboratory handout for performing Monte Carlo simulations 15

16 31 In summary, simulation useful to Model uncertainty Develop risk profiles for decision alternatives Be used as a subsidiary modelling tool to construct a probability model for a particular part of a problem But problem of modelling interdependencies and complexity... The value of some variables might depend on the value of others alternative methods: Conditional sampling (elicitation of a series of probability distributions for one of the variables dependent variable with each distribution being elicited on the assumption that a particular value of the other variable independent variable has occurred) Use of a correlation coefficient showing the degree of association 32 Other types of simulation models Discrete event stochastic simulation models 16

17 33 Simul8 software package simul8.com/ Conceptual model 34 PCC Primary Care Centre DH Distrital Hospital CH Central Hospital Source: Farinha, R., Oliveira, M.D., Sá, A.B. (2008) Networks of primary and secondary care services: How to organize services so as to promote efficiency and quality in access while reducing costs? Quality in Primary Care, 16,

18 35 Study area Setubal Health Sub-Region 36 General view of the implemented model Modelling a Primary Care Centre Modelling a Hospital 18

19 37 Scenarios tested Scenario 1 Testing the capacity of the system to cope with population ageing and higher use of services increase in demand of 10%. Scenario 2 Simulating a change in the focus from secondary to primary care, changing the mix of general practice doctors / specialists. Scenario 3 Restructuring primary care, closing emergency services and focusing on ambulatory consultations. 38 Model outputs I 19

20 39 Model outputs II Mónica Oliveira, 2012/2013 Mónica Oliveira, MAD 2012/ Fit distributions to data - Selecting the appropriate probability distributions - Using other features 20

21 41 Fitting distributions to data p pdf 5 0,2 6 0,3 7 0,2 8 0,1 9 0, ,05 42 Choosing which probability distributions? Uniform distribution Decision-maker indicates: Min value Max value No clue about intermediate values Ex: Load between 10 and 12 MW Triangular distribution Decision-maker indicates: Min value Max value Most likely value Ex: Load between 10 and 12 MW, most likely 10,5 21

22 43 Choosing which probability distributions? Normal distribution Sum of many variables Ex: Total Cost = CostA? +CostB? +CostC? +CostD? Lognormal distribution Multiplication of many variables Ex: Net Income = Gross Income*(1-taxA?)*(1- taxb?)*(1-taxc?) 44 Some features when risk modelling Projecting today s known values into the future Will the flood occur or the competitor enter the market? These values will be affected by what happens someplace else 22

23 45 Projecting today s known values into the future An increasingly uncertain future or increasing variability Time has an impact on estimates they become less and less certain the further out in time projections extend variable.xls 46 Will the flood occur or the competitor enter the market? Modelling uncertain chance events Change events might impact on a model results... And future decisions also... The competitor will enter the market or not? If he does... Need to use discrete functions Decomposing quantities in two parts: Probability of a competitor entering Variability on sales if the competitor enters discrete.xls 23

24 47 These values will be affected by what happens someplace else Very often we do not know how precise are the argument values for a distribution Use dependency relationships using variable arguments and correlations Variable arguments: Uncertainty in parameters and in probability distributions Correlations in sampling: Relating sample values dep.xls 24