You cannot get to where you are going if you don’t know where you are! Today’s philosophical point
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
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
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 +
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
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
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
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
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
Exponential growth of DT’s
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.
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
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
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
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?
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
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
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
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
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
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?
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.
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.