The document discusses key concepts in defining and structuring decision problems. It defines the three components of a problem statement as the current state, desired state, and central objective. Decision trees and influence diagrams are presented as tools to structure choices and uncertainties. Deterministic, stochastic, and simulation models are described based on their mathematical focus. Probability is discussed in terms of frequentist, subjective, and logical interpretations, and methods for forecasting and decomposing complex probabilities are outlined. Calibration and sensitivity analysis are introduced as ways to evaluate probability estimates and assumptions.

1. Modelling Decision Processes Reading: Chapter 4 Problem definition Decision models available You cannot get to where you are going if you don’t know where you are! Today’s philosophical point

2. Defining the Problem and Its Structure Remember to define the problem, not the symptom A fully formed problem statement contains three key components: The current state of affairs The desired state of affairs A statement of the central objective(s) that distinguish the two The content of problem statement requires analysis

3. Problem Definition Errors Failing to identify and define the problem fully may result in a great solution that does not solve the right problem A common error: premature focus on the set of solutions rather than the problem itself The decision maker may be left with a solution looking for a problem to solve E.g. The enthusiastic researcher may focus on a technique but have no application

4. E.G. Problem Statement “Should convicted murderers be subject to a death penalty” is a solution looking for a problem “Why should convicted murderers be put to death?” – explores the reasons “ Many convicted murderers are being released back into society and murder again. A determination must be made regarding how to prevent convicted murderers from killing again ” Current Desired objective +

5. Problem Scope The problem may be worth solving but the scope is beyond the available resources or time constraints or cognitive limitations In such cases, the scope must be reduced to a focus that allows a solution One method to limit the scope is to identify its breadth by asking questions about people involved, cost and magnitude

6. Problem Structure Design of problem structure is similar to design of many other entities (e.g. car, building) What is the final appearance? What are the elemental details? What are the relationships between those elements? Regardless of context, a problem structure can be described in terms of choices, uncertainties and objectives

7. Problem Structure (cont.) Choices: there are always at least two alternatives (one is “do nothing”) Uncertainties: situations beyond the direct control of the decision maker; their individual probability of occurrence is only estimable within a certain range Objectives: methods of establishing the criteria used to measure the value of the outcome The closer the outcome matches the criteria the more desirable “ to increase profits” is vague “ to increase them by 10% over last year” is specific and better

8. Structuring Tools Influence diagram: a simple method of graphing the components of a decision and linking them to show the relationships between them Decision Objective Uncertainty Decision Uncertainty Decision tree: another diagram that models choices and uncertainties and can be extended to include multiple, sequential decisions

9. Decision Tree Example You have £500 You have a decision on what to do with it .. Win £5000 Lose wager Make large profit Lose most of stake Lose/gain nothing 10:1 bet on a horse Invest in stock Do nothing Horse wins Horse loses Significant rise Stocks fall Minor profit/loss Steady change http:// www.psychwww.com/mtsite/dectree.html

10. Exponential growth of DT’s

11. Common Decision Structures These are frequently used in managerial activities Basic Risky Decision: decision maker takes a choice in the face of uncertainty. Success is a function of the choice and outcome. Certainty: a multiple-objective decision with little risk. Success is a function of the trade-off between objectives. Sequential: several risky decisions over time. Earlier outcomes may affect later choices.

12. Decision Models Decision models can be classified in a number of ways: Is time a factor? Models that do not include time are “static” versus “dynamic” What is the technique’s mathematical focus? Some abstract model types are deterministic, stochastic, simulation and domain specific

13. Model Classification Examples Deterministic: linear programming, production planning large number of elements, many relationships same input values give same outputs Stochastic: queuing theory, linear regression analysis uncertainty in variables Simulation: production modeling, transportation analysis Real experiments are not feasible or too expensive (e.g. flight, flow of oil) Domain-specific: supply-demand, medicine DSS must be aware of incorporating these

14. Conceptual Models A formal mathematical approach is not always appropriate Conceptual models are formulated under the notion that even though all problems are unique, no problem is completely new Decision makers can recall and combine a variety of past experiences to create an accurate model of the current situation Criticism for being too subjective/biased But the choice of abstract model is also dependent on DM’s experience Howard’s clarity test states that every component of the decision structure should not be open to interpretation

15. Types of Probability Used to quantify the uncertainties Three requirements of probability: All probabilities are in the range 0 to 1 The probabilities of all outcomes of an event must add up to the probability of their union The total probability of a complete set of outcomes must equal 1 But how accurate are our estimates?

16. Lets look at a Decision Tree again 0.5 Lets keep the maths easy by using p=0.5 for all uncertainties 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 P=0.25 P=0.25 P=0.5 P=0.25 P=0.25 P=0.25 P=0.25 A A1 B1 B2 B ALL uncertainties for events = 1

17. How Are Probabilities Generated? Long-run frequency: with enough “history”, you can estimate an event’s probability by its relative frequency Subjective: probability represents an individual’s “degree of belief” that an event will occur Logic: a probability may be derivable, but its accuracy may not be acceptable E.g. In life or death situations

18. Forecasting Probabilities? Direct probability forecasting — an expert is simply asked to estimate the chance that an outcome will occur Odds forecasting — a series of bets are proposed to determine how strongly the bettor feels an event will occur Comparison forecasting — similar to odds forecasting except that one game has known probabilities

19. Calculating Odds in a D/Tree Bet on team A Bet against team A Team A wins Team A loses Team A loses Team A wins £Xx -£X £Y -£Y P(Team A wins)=Y/(X+Y) see text for detail, pg 125/6

20. Decomposing Complex Probabilities Probabilities for complex events may be more easily generated by using conditional probabilities within subsets of the events For example, it may be easier to forecast sales of a weather-related product by forecasting sales under good weather, then bad weather and then considering the probability of bad weather

21. Conditional and Total Probabilities Severe winter P(B)=0.7 Moderate winter 1 - P(B)=0.3 Sales > 1000 P(A|B) Sales >= 1000 1 - P(A|B) Sales > 1000 P(A|B`) Sales <= 1000 1 - P(A|B`) 0.8 0.2 0.5 0.5 P(A)=P(A|B) * P(B) = P(A|B`) * P(B) = 0.71

22. Calibration A decision maker is said to be well calibrated if his probability forecasts are correct at about the same rate as his confidence in them (9 out of 10 times his 90% confidence intervals should be correct). Probabilities are a kind of average Calibration requires years of experience and feedback to develop. most of us are NOT well calibrated Better to use confidence intervals Most of us are too optimistic and our intervals are too tight. Give your 90% confidence interval for: What was the population of Scotland in 2001? How many stripes are there on the american flag? How wide did you make your intervals?

23. Sensitivity Analysis A method for testing the degree to which a set of assumptions affects the results from a model. – “What-If” analysis. If a small change in the value of a variable yields a measurable change in output, that variable is said to be highly sensitive. Variables that are not sensitive may be treated as fixed, reducing the model’s complexity.

24. Value Analysis We always need to be concerned that enough reliable information is available to make a successful decision. We can determine how much we are willing to pay for better info by computing its expected value. This involves a comparison of the expected return with the info to the expected return without the info.

25. Key Points Problem definition Current and desired states Decision Tree representations Types of decision models Deterministic etc Types of probability Frequentist, subjective, logical Forecasting probabilities Odds, complex Calibration and sensitivity