Decision Analysis: Theory and Methodology: Decision analysis has a major role to play in helping decision makers to gain a greater understanding of the problems they face, and provides tools for quantitatively analyzing decision with uncertainty and/or multiple conflicting objectives. Decision Analysis Includes: 1. 2. 3. 4. 5. 6. 7.
How to model a decision problem? How to model the uncertainty with probability? probability? How to model the preference preference with utility? Decision making with uncertainty. Decision making with risk. Bayesian Analysis. Multiple criteria decision making
1.1 Decision Problem & Decision Making We make decisions all the time. For example: Shall I bring the umbrella today? – The decision depends on something which I do not know, namely whether it will rain or not. a. Alternatives/Options/Action Alternatives/Options/Actions: s: Do I take my umbrella or not? b. States/Uncertainty: States/Uncertainty: Whether or not rain? c. Outcomes/Payoff: Outcome1: If I take and it is sunny, but it’s inconvenient.
Outcome2: If I take and it rains, it is good. Outcome3: If I don’t take and it is sunny, that is best. Outcome4: If I don’t take and it rains, I ruin my suit.
Alternatives
S ta tates
Outcomes Out1
Out2
Out3
Out4
1.2 Conditions for Decision Making:A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs. Condition 1: Decision Making under Certainty Condition 2: Decision Making under Uncertainty Condition 3: Decision Making under Risk Decision Making under Certainty: The decision maker knows all possible alternatives, and can exactly say at what probability each occurs. Decision maker knows with certainty the consequences of every alternative or decision choice The decision maker chooses the action that will result in the most desirable outcome. A company has decided that it faces three alternatives: It can manufacture/assemble the keyboard itself. It can buy the keyboards from a domestic manufacturer. It can buy the keyboards from a manufacturer in the Far East. The objective of the decision: profit. In this case there is a single, known state of nature. Although this case appears simpler than those of non-certainty, the problem of calculating the payoff for each alternative action, or at least of identifying an action that would result in an outcome which was satisfactory, may not be trivial. Methods of Operational Research, such as Linear Programming and Dynamic Programming, may be needed. Options
States of Nature
Objectives/Goals
Manufacture itself
Payoff ? Sale level is low for sure
Buy Domestic
Payoff ??
Buy Abroad
Payoff ???
Break -Even Analysis Total Revenue ) 0 0 0 $ ( s t s o C / s e u n e v e R
80
Break-even point Profit area
70
Break-Even Point
60
Variable Costs
50 40
Loss Area
30
BE
Fixed Costs
20 10 10
20
30
40 50 Output (000)
60
70
TFC P-VC
Decision Making under Uncertainty: There are several outcomes for each action, depending on the state of nature. The outcome corresponding to the chosen action are uncertain, but the probabilities associated with each outcome is known. The decision maker does not know the probabilities of the various outcomes There are several criteria for making decisions under uncertainty. (1) Maximax (optimistic) (2) Maximin (pessimistic) (3) Criterion of realism (Hurwicz) (4) Equally likely (Laplace) (5) Minimax regret (Savage) Example: Thompson Lumber Company Step 1 – Define the problem Expand by manufacturing and marketing a new product, backyard storage sheds Step 2 – List alternatives Construct a large new plant A small plant No plant at all Step 3 – Identify possible outcomes The market could be favorable or unfavorable Step 4 – List the payoffs Identify conditional values for the profits for large, small, and no plants for the two possible market conditions Step 5 – Select the decision model Depends on the environment and amount of risk and uncertainty Step 6 – Apply the model to the data Solution and analysis used to help the decision making Table 1 Decision Matrix (Payoff table) STATE OF NATURE
ALTERNATIVE
Construct a large plant Construct a small plant Do nothing
FAVORABLE MARKET ($)
UNFAVORABLE MARKET ($)
200,000
–180,000
100,000
–20,000
0
0
(1) Maximax (optimistic) Used to find the alternative that maximizes the maximum payoff.Locate the maximum payoff for each alternative.Select the alternative with the maximum number
(2) Maximin (pessimistic) Used to find the alte rnative that maximizes the minimum payoff.Locate the minimum payoff for each alternative.Select the alternative with the maximum number
(3) Criterion of realism (Hurwicz)
(4) Equally likely (Laplace) Considers all the payoffs for each alternative.Find the average payoff for each alternative.Select the alternative with the highest average
(5) Minimax regret (Savage) Based on opportunity loss or regret, the difference between the optimal profit and actual payoff for a decision.Create an opportunity loss table by determining the opportunity loss for not choosing the best alternative.Opportunity loss is calculated by subtracting each payoff in the column from the best payoff in the column.Find the maximum opportunity loss for each alternative and pick the alternative with the minimum number Opportunity Loss Tables
STATE OF NATURE
STATE OF NATURE ALTERNATIVE
FAVORABLE UNFAVORABLE MARKET ($) MARKET ($)
Construct a large plant
0
180,000
Construct a small plant
100,000
20,000
Do nothing
200,000
0
ALTERNATIVE Construct a large plant
FAVORABL UNFAVORAB MAXIMUM E MARKET LE MARKET IN A ROW ($) ($) ($) 0
180,000
180,000
Construct a small plant
100,000
20,000
100,000
Do nothing
200,000
0
200,000
Decision Making under Risk Risk refers to the situation in which the outcome of each action is not certain, but where the probabilities of the different states of nature (and hence of the alternative outcomes) can be determined. Decision making when there are several possible states of nature and we know the probabilities associated with each possible state. The decision maker knows the probabilities of the various outcomes Most popular method is to choose the alternative with the highest expected monetary value (EMV) Probability: Objective / Subjective Decision criterion: EMV / EU EMV (alternative i
) = (Payoff of first state of nature) x (probability of first state of nature) + (payoff of second state of nature) x (probability of second state of nature) + … + (payoff of last state of nature) x (probability of last state of nature)
(large plant) = (0.50)($200,000) + (0.50)( –$180,00 = $10,000 EMV (small plant) = (0.50)($100,000) + (0.50)( –$20,000 = $40,000 EMV (do nothing) = (0.50)($0) + (0.50)($0) = $0 EMV
Each market has a probability of 0.50 Which alternative would give the highest EMV? The calculations are:
EMV for
Thompson Lumber
STATE OF NATURE ALTERNATIVE
Each market has a probability of 0.50 Which alternative would give the highest EMV?
FAVORABLE UNFAVORABL MARKET ($) E MARKET ($)
EMV
($)
Construct a large plant
200,000
–180,000
10,000
Construct a small plant
100,000
–20,000
40,000
Do nothing
0
0
Probabilities
0.50
0.50
0
Expected Value of Perfect Information (EVPI) Scientific Marketing, Inc. offers analysis that will provide certainty about market conditions (favorable) Additional information will cost $65,000 Is it worth purchasing the information?
Compute EVwPI
State of Nature Alternative
Favorable Market (p=0.5)
Unfavorable Market (p=0.5)
EMV
Construct a large plant
200,000
-180,000
10,000
Construct a small plant
100,000
-20,000
40,000
0
0
0
200,000
0
EVwPI = 100,000
Do nothing Perfect Information
The best alternative with a favorable market is to build a large plant w ith a payoff of $200,000. In an unfavorable market the choice is to do nothing with a payoff of $0
EVwPI = ($200,000)*.5 + ($0)(.5) = $100,000
Compute EVPI = EVwPI – max EMV = $100,000 - $40,000 = $60,000
The most we should pay for any information is $60,000
1. Best alternative for favorable state of nature is build a large plant with a payoff of $200,000 Best alternative for unfavorable state of nature is to do nothing with a payoff of $0 EVwPI
= ($200,000)(0.50) + ($0)(0.50) = $100,000
2. The maximum EMV without additional information is $40,000 = EVwPI – Maximum = $100,000 - $40,000 = $60,000 EVPI
EMV
Expected Opportunity Loss:
Expected opportunity loss (EOL) is the cost of not picking the best solution
First construct an opportunity loss table For each alternative, multiply the opportunity loss by the probability of that loss for each possible outcome and add these together Minimum EOL will always result in the same decision as maximum EMV Minimum EOL will always equal EVPI
Thompson Lumber: Payoff Table
The Opportunity Loss Table STATE OF NATURE ALTERNATIVE
FAVORABLE UNFAVORABL MARKET ($) E MARKET ($)
EOL
Construct a large plant
200,000 200,000
0-(-180,000)
90,000
Construct a small plant
200,000 100,000
0-(-20,000)
60,000
Do nothing
200,000 - 0
0-0
Probabilities
0.50
0.50
100,000
Expected Opportunity Loss
STATE OF NATURE FAVORABLE MARKET ($)
ALTERNATIVE
UNFAVORABLE MARKET ($)
EOL
Construct a large plant
0
180,000
90,000
Construct a small plant
100,000
20,000
60,000
Do nothing
200,000
0
100,000
0.50
0.50
Probabilities
EOL
(large plant)= (0.50)($0) + (0.50)($180,000) = $90,000
EOL
(small plant)=(0.50)($100,000) + (0.50)($20,000) = $60,000
EOL
(do nothing)= (0.50)($200,000) + (0.50)($0) = $100,000
The minimum EOL will always result in the same decision (NOT value) as the maximum EMV
The EVPI will always equal the minimum EOL EVPI = minimum EOL
(3) Sensitivity Analysis Sensitivity analysis examines how our decision might change with different input data For the Thompson Lumber example P = probability of a favorable market (1 – P) = probability of an unfavorable market
EMV(Large
Plant)
= $200,000 P – $180,000)( 1 – P ) = $200,000 P – $180,000 + $180,000 P = $380,000 P – $180,000
EMV(Small
Plant)
= $100,000 P – $20,000)(1 – P ) = $100,000 P – $20,000 + $20,000 P = $120,000 P – $20,000
EMV(Do
Nothing)
= $0 P + 0(1 – P ) = $0
Point 1: EMV(do nothing) = EMV(small plant)
Point 2: EMV(small plant) = EMV(large plant)
The end
