Decision Analysis Carlos A. Santos Silva June 5 th, 2009
What is decision analysis? Often, there is more than one possible solution: Decision depends on the criteria Decision often must be made in uncertain environments: Making decisions with or without experimentation Decision analysis: decision making in face of great uncertainty; rational decision when the outcomes are uncertain due to randomness in nature It is different from Game Theory, where decision is done assuming competitive environments Decision Making Without experimentation With experimentation
Example Company owns a tract of land that may contain oil. Contracted geologist reports that chance of oil is 1 in 4. Another oil company offers 90.000 for the land. Cost of drilling is 100.000. If oil is found, expected revenue is 800.000 (expected profit is 700.000 ). Alternative Status of land Oil Payoff Dry Drill for oil 700.000 100.000 Sell the land 90.000 90.000 Chance of status 1 in 4 3 in 4
Decision making without experimentation Decision maker needs to choose one of possible decision alternatives. There are several states of nature (due to random factors). Each combination of a decision alternative and state of nature results in a known payoff, which is one entry of a payoff table. Payoff table is used to find an optimal alternative for the decision making according to a criterion. Probabilities for states of nature provided by prior distribution are prior probabilities.
Maximin payoff criterion Maximin payoff criterion: For each possible decision alternative, find the minimum payoff over all states. Next, find the maximum of these minimum payoffs. Choose the alternative whose minimum payoff gives this maximum. Choose selling!!! Best guarantee of payoff: pessimistic viewpoint. Alternative State of nature Oil Dry Minimum 1. Drill for oil 700 100 100 2. Sell the land 90 90 90 Maximin value Prior probability 0.25 0.75
Maximum likelihood criterion Maximum likelihood criterion: Identify most likely state of nature. For this state of nature, find the decision alternative with the maximum payoff. Choose this decision alternative Choose selling!!! Most likely state: ignores important information and low-probability big payoff. Alternative State of nature Oil Dry 1. Drill for oil 700 100 2. Sell the land 90 90 Maximum in this column Prior probability 0.25 0.75 Most likely
Bayes decision rule Bayes decision rule: Using prior probabilities, calculate the expected value of payoff for each decision alternative. Choose the decision alternative with the maximum expected payoff. For the prototype example: E[Payoff (drill)] = 0.25(700) + 0.75( 100) = 100. E[Payoff (sell)] = 0.25(90) + 0.75(90) = 90. Choose drilling!!! Incorporates all available information (payoffs and prior probabilities). What if probabilities are wrong?
Sensitivity analysis with Bayes rules Prior probabilities can be questionable. True probabilities of having oil are 0.15 to 0.35 (so, probabilities for dry land are from 0.65 to 0.85). p = prior probability of oil. Example: expected payoff from drilling for any p: E[Payoff (drill)] = 700p 100(1 p) = 800p 100. In figure, the crossover point is where the decision changes from one alternative to another: E[Payoff (drill)] = E[Payoff (sell)] 800p 100 = 90 p = 0.2375
Decision Making with Experimentation Improved estimates are called posterior probabilities. Example: a detailed seismic survey costs 30.000. USS: unfavorable seismic soundings: oil is fairly unlikely. FSS: favorable seismic soundings: oil is fairly likely. Based on part experience, the following probabilities are given: P(USS State=Oil) = 0.4; P(FSS State=Oil) = 1 0.4 = 0.6. P(USS State=Dry) = 0.8; P(FSS State=Dry) = 1 0.8 = 0.2. From probability theory, the Bayes theorem can be obtained: P(State = state i Finding = finding j) = = n k= 1 P(Finding = finding j State = state i) P(State = state i) P(Finding = finding j State = state k ) P(State = state k )
Probability Tree Diagram
Optimal Policy Using Bayes decision rule, the optimal policy of optimizing payoff is given by: Finding from seismic survey Optimal alternative Expected payoff excluding cost of survey Expected payoff including cost of survey USS Sell the land 90 60 FSS Drill for oil 300 270 Is it worth spending 30 to conduct the experimentation?
Value of Experimentation Before performing an experimentation, determine its potential value. Two methods: 1. Expected value of perfect information it is assumed that all uncertainty is removed. Provides an upper bound on potential value of experiment. 2. Expected value of information is the actual improvement in expected payoff.
Expected value of perfect information State of nature Alternative Oil Dry 1. Drill for oil 700 100 2. Sell the land 90 90 Maximum payoff 700 90 Prior probability 0.25 0.75 Expected payoff with perfect information = 0.25(700) + 0.75(90) = 242.5 Expected value of perfect information (EVPI) is: EVPI = expected payoff with perfect information expected payoff without experimentation Example: EVPI = 242.5 100 = 142.5. As 142.5 exceeds 30, the seismic survey should be done
Expected value of information Requires expected payoff with experimentation: Expected payoff with experimentation= Example: j P(Finding = finding j) E[payoff Finding = finding j] see probability tree diagram, where: P(USS) = 0.7, P(FSS) = 0.3. Expected payoff (excluding cost of survey) was obtained in optimal policy: E(Payoff Finding = USS) = 90, E(Payoff Finding = FSS) = 300. So, expected payoff with experimentation is Expected payoff with experimentation = 0.7(90) + 0.3(300) = 153. Expected value of experimentation (EVE) is: EVE = expected payoff with experimentation expected payoff without experimentation (EVE = 153 100 = 53) As 53 exceeds 30, the seismic survey should be done
Decision tree with analysis (see class 7)
Optimal policy for prototype example The decision tree results in the following decisions: Do the seismic survey. If the result is unfavorable, sell the land. If the result is favorable, drill for oil. The expected payoff (including the cost of the seismic survey) is 123 (123 000 ). Same result as obtained with experimentation For any decision tree, the backward induction procedure always will lead to the optimal policy
NOW WHAT?
So far we learned Operations Research arrive at optimal or near optimal solutions to complex problems using Mathematical modeling, Statistics, Algorithms Linear and Nonlinear Programming Dynamic Programming Metaheuristics Multiobjective Optimization Decision Analysis Prescribing the recommended action using Statistics, Decision Trees
Is this enough? Modeling optimization problems (cost functions and constraints) Sometimes it is difficult to have a mathematic description of what we want We need new tools to help us describe the reality Modeling systems Sometimes it is difficult to describe a system using mathematical expressions It is necessary to build a model only using data Soft Computing Computer techniques that tolerant of imprecision, uncertainty, partial truth, and approximation Fuzzy Systems Neural Networks
Soft Computing (slides from Prof. João Sousa) Carlos A. Santos Silva June 5 th, 2009
FUZZY SYSTEMS
Precision vs. Relevancy A 1500 kg mass is approaching your your head head at 45,3m/s. at 45.3 m/sec. OUT!! LOOK OUT!
Probability vs. Possibility Event u: Hans ate X eggs for breakfast. Probability distribution: P X (u) Possibility distribution: π X (u) u 1 2 3 4 5 6 7 8 P X (u) 0.1 0.8 0.1 0 0 0 0 0 π X (u) 1 1 1 1 0.8 0.6 0.4 0.2
Introduction Imprecision or vagueness in natural language does not imply a loss of accuracy or meaningfulness! Proposed in 1965 by Lotfi Zadeh Fuzzy Sets, Information Control, 8, pp. 338-353. How to describe very complex systems? Allow some degree of uncertainty in their description! How to deal mathematically with uncertainty? Using probabilistic theory (stochastic). Using the theory of fuzzy sets (nonstochastic). Applications Modeling, control, optimization and decision making
Classical set (binary logic) Example: set of old people A = {age age 70} 1 A 0.5 0 50 60 70 80 90 100 age [years]
Logic propositions Nick is old... true or false? Nick s age: age Nick = 70, µ A (70) = 1 (true) age Nick = 69.9, µ A (69.9) = 0 (false) age Nick = 90, µ A (90) = 1 (true) 1 0.5 A 0 50 60 70 80 90 100 age [years]
Fuzzy set Graded membership, element belongs to a set to a certain degree. The world is not only black and white, yes or no,..the world is not binary 1 A Membership membership grade de 0.5 0 50 60 70 80 90 100 age [years]
Fuzzy proposition Nick is old... degree of truth age Nick = 70, µ A (70) = 0.5 age Nick = 69.9, µ A (69.9) = 0.49 age Nick = 90, µ A (90) = 1 1 A Membership grade membership grade 0.5 0 50 60 70 80 90 100 age [years]
Types of membership functions (a) Triangular MF (b) Trapezoidal MF 1 1 0.5 0.5 0 0 20 40 60 80 100 1 0.5 (c) Gaussian MF 0 0 20 40 60 80 100 0 0 20 40 60 80 100 1 0.5 (d) Generalized Bell MF 0 0 20 40 60 80 100
Linguistic values describe variables Membership membershi ip grade 1 young middle-age old young middle age old infan t 0 20 40 60 80 100 age [years]
Fuzzy if-then rules Fuzzy propositions x is A, y is B Linguistic (Mamdani) fuzzy if-then rule If x is A then y is B (or A B) Antecedent or premise: x is A Consequent or conclusion: y is B
Fuzzy Inference System Data base Knowledge database Rule base If Xis then Yis input fuzzification inference defuzzification output
Example How white is the hair depending on the age?
Premises
Conclusions
Rules
If you are 33. You have 47,4% of white hair
CONTROL
Example: Liquid level in a tank If level is low then increase valve opening If level is OK then maintain valve opening If level is high then decrease valve opening h R João Miguel da Costa Sousa / Alexandra Moutinho 39
Fired fuzzy rules
Aggregation and defuzzifucation
CLUSTERING
Example of linguistic model Ifincomeis Lowthentaxis Low Ifincomeis Highthentaxis High
Fuzzy c-means example 1
Fuzzy c-means example 2 1 1 MF 0.5 MF 0.5 0 1 0.5 Y 0 0 X 0.5 1 0 1 0.5 Y 0 0 X 0.5 1 1 MF 0.5 0 1 0.5 0.5 1 Y 0 0 X
DECISION ANALYSIS
Example A person is driving a car on a cold winter day down a road. Suddenly, a dog jumps in front of the car. The driver can decide between two actions: 1. he can break hard applying full power to the brakes, or 2. he can brake soft knowing that the car cannot come to a stop before a collision with the animal. What should the driver do? Θ ={slippery road, not slippery road} A={brake soft, brake hard}
Decision tree brake soft hit dog slightly D 1 brake hard slipandhittree D 2 roadisnotslippery brake soft hit dog slightly D 3 brake hard do not hit anything D 4 Θ states alternatives A solution set κ consequences Ξ D preference ordering
OPTIMIZATION
Fuzzy goal Goal: Product concentration should be about 80%. 1 membership gr rade 0.5 About80% 0 70 75 80 85 90 95
Fuzzy constraint Constraint: Product concentration should be not substantially higher than 75%. 1 membership grad de 0.5 Not substantially higherthan75% 0 70 75 80 85 90 95
Optimal fuzzy decision Fuzzy decision F should satisfy decision goals G as well as decision constraints membership grade x m FuzzyDecisionµ D Symmetric model Maximizing decision using min: a* = arg max µ ( a) µ ( a) a A G C
NEURAL NETWORKS
Neural networks Motivation: Humans are able to process complex tasks efficiently (perception, pattern recognition, reasoning, etc.). Ability to learn from examples. Adaptability and fault tolerance. Engineering applications: Nonlinear approximation and classification. Learning (adaptation) from data: black-box modeling. Very-Large-Scale Integration implementation.
Biological neuron Soma: body of the neuron. Dendrites: receptors (inputs) of the neuron. Axon: output of neuron; connected to dendrites of other neurons via synapses. Synapses: transfer of information between neurons (electrochemical signals).
Neural networks Biological neural networks Neuron switching time: 0.001 s Number of neurons: 10 billion Connections per neuron (synapses): 10,000 Face recognition time: 0.1 s Artificial neural networks Weighted connections amongst units Highly parallel, distributed process Emphasis on tuning weights automatically Use when Biological neural network Soma Dendrite Artificial neural network Neuron Input Input and output are high-dimensional, mathematical form of system is unknown and interpretability of identified model is unimportant Applications Axon Synapse Pattern recognition, Classification, Prediction, Modeling Output Weight
Artificial Network x 1 4 x 2 5 3 6 8 x 8 7 9 x 9 Layer 1 Layer 2 Layer 3 (Input) (Hidden) (Output)
Artificial neuron x 1 x 2... w 2 Neuron y x n w n x i : i-th input of the neuron w i : synaptic strengh (weight) for x i y = σ (Σw i x i ): output signal
Types of neurons McCulloch and Pits (1943) Threshold θ: n y= sign wixi θ i= 1 Other types of activation functions (net= Σw i x i ) 1 1 0 1, if 0 step net y = 0, if net< 0 y linear = y + sigmoid 1 net 1 e = net
FORECAST
Example