chapter 3 decision analysis. the six steps in decision making 1.clearly define the problem at hand...
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Chapter 3
Decision Analysis
The Six Steps in Decision Making
1. Clearly define the problem at hand2. List the possible alternatives3. Identify the possible outcomes or states
of nature4. List the payoff or profit of each
combination of alternatives and outcomes
5. Select one of the mathematical decision theory models
6. Apply the model and make your decision
Case of Maria Rojas
Maria Rojas is considering the possibility of opening a small dress shop on Fairbanks Avenue, a few blocks from the university. She has located a good mall that attracts students.
Her options are to open as mall shop, a medium-sized shop, or no shop at all. The market for a dress shop can be good, average, or bad. The probabilities is 1/3 for each market.
The net profit or loss for the medium-sized and small shops for the various market conditions are given in the following table. Building no shop at all yields no loss and no gain.
Case of Maria Rojas
Types of Decision-Making Environments
Type 1:Type 1: Decision making under certainty Decision maker knows with certaintyknows with certainty the
consequences of every alternative or decision choice
Type 2:Type 2: Decision making under uncertainty The decision maker does not knowdoes not know the
probabilities of the various outcomesType 3:Type 3: Decision making under risk
The decision maker knows the knows the probabilitiesprobabilities of the various outcomes
Decision Making Under Uncertainty
1. Maximax (optimistic)
2. Maximin (pessimistic)
3. Criterion of realism (Hurwicz)
4. Equally likely (Laplace)
5. Minimax regret
There are several criteria for making decisions under uncertainty
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)
MAXIMUM IN A ROW
($)
Small shop 75,000 25,000 -40,000 75,000
Medium-sized shop 100,000 35,000 -60,000 100,000
Do nothing 0 0 0 0
MaximaxUsed to find the alternative that maximizes the maximum payoff
Locate the maximum payoff for each alternative Select the alternative with the maximum
number
MaximaxMaximax
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)MINIMUM IN A ROW ($)
Small shop 75,000 25,000 -40,000 -40,000
Medium-sized shop 100,000 35,000 -60,000 -60,000
Do nothing 0 0 0 0
Maximin
Used to find the alternative that maximizes the minimum payoff
Locate the minimum payoff for each alternative Select the alternative with the maximum
number
MaximinMaximin
Criterion of Realism (Hurwicz)
A weighted averageweighted average compromise between optimistic and pessimistic
Select a coefficient of realism Coefficient is between 0 and 1 A value of 1 is 100% optimistic Compute the weighted averages for each
alternative Select the alternative with the highest value
Weighted average = (maximum in row) + (1 – )(minimum in row)
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)
CRITERION OF REALISM ( = 0.8)$
Small shop 75,000 25,000 -40,000 52,000
Medium-sized shop 100,000 35,000 -60,000 68,000
Do nothing 0 0 0 0
Criterion of Realism (Hurwicz)
For the small shop alternative using = 0.8(0.8)(75,000) + (1 – 0.8)(–40,000) = 52,000
For the medium-sized shop alternative using = 0.8 (0.8)(100,000) + (1 – 0.8)(–60,000) = 68,000
RealismRealism
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)
ROW AVERAGE ($)
Small shop 75,000 25,000 -40,000 20,000
Medium-sized shop 100,000 35,000 -60,000 25,000
Do nothing 0 0 0 0
Equally Likely (Laplace)
Considers all the payoffs for each alternative Find the average payoff for each alternative Select the alternative with the highest average
Equally likelyEqually likely
Minimax Regret
Based on opportunity lossopportunity loss or regretregret, 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
Minimax Regret
Table 3.7
Opportunity Loss Tables
STATE OF NATURE
GOOD MARKET ($)
AVERAGE MARKET
($)
BAD MARKET
($)
100,000 - 75,000 35,000 - 25,000 0 – (- 40,000)
100,000 - 100,000 35,000 - 35,000 0 – (- 60,000)
100,000 - 0 35,000 - 0 0
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)
Small shop 25,000 10,000 40,000
Medium-sized shop 0 0 60,000
Do nothing 100,000 35,000 0
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)
MAXIMUM IN A ROW
($)
Small shop 25,000 10,000 40,000 40,000
Medium-sized shop 0 0 60,000 60,000
Do nothing 100,000 35,000 0 100,000
Minimax Regret
Table 3.8
MinimaxMinimax
Decision Making Under Risk
Decision making when there are several possible states of nature and we know the probabilities associated with each possible state
Most popular method is to choose the alternative with the highest expected monetary value (expected monetary value (EMVEMV))
EMV(alternative i) = (payoff of 1st state of nature) x (prob. of 1st state of nature)
+ (payoff of 2nd state of nature) x (prob. of 2nd state of nature)
+ …
+ (payoff of last state of nature) x (prob. of last state of nature)
EMV for Maria Rojas
Each market has a probability of 1/3 Which alternative would give the highest EMV? The calculations are
EMV (small shop) = (1/3)($75,000) + (1/3)($25,000) + (1/3)($-40,000) = $20,000
EMV (medium shop) = (1/3)($100,000) + (1/3)($35,000) + (1/3)($-60,000) = $25,000
EMV (do nothing) = (1/3)($0) + (1/3)($0) + (1/3)($0) = $0
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)
ROW AVERAGE ($)
Small shop 75,000 25,000 -40,000 20,000
Medium-sized shop 100,000 35,000 -60,000 25,000
Do nothing 0 0 0 0
Probability 1/3 1/3 1/3
EMV for Maria Rojas
Largest Largest EMVEMVEMV (small shop) = (1/3)($75,000) + (1/3)(25,000) + (1/3)(-40,000)
= $20,000EMV (medium shop) = (1/3)($100,000) + (1/3)(35,000) + (1/3)($-60,000)
= $25,000EMV (do nothing) = (1/3)($0) + (1/3)($0) + (1/3)($0) = $0
Expected Value of Perfect Information (EVPI)
EVwPI (Expected Value with Perfect Information) is the long run average return if we have perfect information before a decision is made
EVwPI = (best payoff for 1st SoN)x P1st SoN
+ (best payoff for 2nd SoN)x P2nd SoN
+ … + (best payoff for nth SoN)x Pnth
SoN
EVPI (Expected Value of Perfect Information) places an upper bound on what you should pay for additional information
EVPI = EVwPI – Maximum EMV
Expected Value of Perfect Information (EVPI)
Scientific Marketing, Inc. offers analysis that will provide certainty about market conditions (favorable)
Additional information will cost $25,000 Is it worth purchasing the information?
STATE OF NATURE
ALTERNATIVEGOOD
MARKET ($)
AVERAGE MARKET
($)
BAD MARKET
($)
ROW AVERAGE ($)
Small shop 75,000 25,000 -40,000 20,000
Medium-sized shop 100,000 35,000 -60,000 25,000
Do nothing 0 0 0 0
Probability 1/3 1/3 1/3
Expected Value of Perfect Information (EVPI)
1. Best alternative for good state of nature is opening a medium shop with a payoff of $100,000Best alternative for average state of nature is opening a medium shop with a payoff of $35,000
Best alternative for bad state of nature is to do nothing with a payoff of $0
EVwPI = ($100,000)(1/3) + ($35,000)(1/3) + ($0)(1/3) = $45,000
2. The maximum EMV without additional information is $25,000
EVPI = EVwPI – Maximum EMV = $45,000 - $25,000 = $20,000
1. Best alternative for good state of nature is opening a medium shop with a payoff of $100,000Best alternative for average state of nature is opening a medium shop with a payoff of $35,000
Best alternative for bad state of nature is to do nothing with a payoff of $0
EVwPI = ($100,000)(1/3) + ($35,000)(1/3) + ($0)(1/3) = $45,000
2. The maximum EMV without additional information is $25,000
EVPI = EVwPI – Maximum EMV = $45,000 - $25,000 = $20,000
Expected Value of Perfect Information (EVPI)
So the maximum Maria should pay for the additional information is $20,000
Expected Opportunity Loss
Expected opportunity lossExpected opportunity loss (EOL) is the cost of not picking the best solution
First construct an opportunity loss table For each alternative, multiply the opportunity
loss by the probability of that loss for each possible outcome and add these together
Minimum EOL will always result in the same decision as maximum EMV
Minimum EOL will always equal EVPI
STATE OF NATURE
ALTERNATIVE
GOOD MARKET
($)
AVERAGE MARKET
($)
BAD MARKET
($)
MAXIMUM IN A ROW
($)
Small shop 25,000 10,000 40,000 25,000
Medium-sized shop 0 0 60,000 20,000
Do nothing 100,000 35,000 0 45,000
Probability 1/3 1/3 1/3
Expected Opportunity Loss
Minimum Minimum EOLEOL
Opportunity loss table
EOL (small shop) = (1/3)($25,000) + (1/3)($10,000) + (1/3)($40,000) = $25,000
EOL (medium shop) = (1/3)($0) + (1/3)($0) + (1/3)($60,000) = $20,000
EOL (do nothing) = (1/3)($100,000) + (1/3)($35,000) + (1/3)($0) = $45,000
Homework 03
Prob. 3.16, 3.18, 3.19, 3.22, 3.26, 3.27