Decision Analysis

Decision Analysis

And Decision Tree Software

Decision Analysis

• Managers often must make decisions in environments that are fraught with uncertainty.

• Some Examples– A manufacturer introducing a new product into the marketplace

• What will be the reaction of potential customers?

• How much should be produced?

• Should the product be test-marketed?

• How much advertising is needed?

– A financial firm investing in securities

• Which are the market sectors and individual securities with the best prospects?

• Where is the economy headed?

• How about interest rates?

• How should these factors affect the investment decisions?

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Decision Analysis

• Managers often must make decisions in environments that are fraught with uncertainty.

• Some Examples– A government contractor bidding on a new contract.

• What will be the actual costs of the project?• Which other companies might be bidding?• What are their likely bids?

– An agricultural firm selecting the mix of crops and livestock for the season.• What will be the weather conditions?• Where are prices headed?• What will costs be?

– An oil company deciding whether to drill for oil in a particular location.• How likely is there to be oil in that location?• How much?• How deep will they need to drill?• Should geologists investigate the site further before drilling?

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Steps in Decision Making

• Clearly define the problem

• List all possible decision alternatives

• Identify the possible future outcomes for each decision

• Identify the payoff ( usually profit or cost)for each combination of alternatives and outcomes

• Select one of the decision theory modeling techniques discussed. Apply the decision model and make your decision

The Goferbroke Company Problem

• The Goferbroke Company develops oil wells in unproven territory.

• A consulting geologist has reported that there is a one-in-four chance of oil on a particular tract of land.

• Drilling for oil on this tract would require an investment of about $100,000.

• If the tract contains oil, it is estimated that the net revenue generated would be approximately $800,000.

• Another oil company has offered to purchase the tract of land for $90,000.

Question: Should Goferbroke drill for oil or sell the tract?

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Prospective Profits

Profit

Status of Land Oil Dry

Alternative

Drill for oil $700,000 –$100,000

Sell the land 90,000 90,000

Chance of status 1 in 4 3 in 4

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Decision Analysis Terminology

• The decision maker is the individual or group responsible for making the decision.

• The alternatives are the options for the decision to be made.

• The outcome is affected by random factors outside the control of the decision maker. These random factors determine the situation that will be found when the decision is executed. Each of these possible situations is referred to as a possible state of nature.

• The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities.

• Each combination of a decision alternative and a state of nature results in some outcome. The payoff is a quantitative measure of the value to the decision maker of the outcome. It is often the monetary value.

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Prior Probabilities

State of Nature Prior Probability

The tract of land contains oil 0.25

The tract of land is dry (no oil) 0.75

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Payoff Table (Profit in $Thousands)

State of Nature

Alternative Oil Dry

Drill for oil 700 –100

Sell the land 90 90

Prior probability 0.25 0.75

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The Maximax Criterion

• The maximax criterion is the decision criterion for the eternal optimist.

• It focuses only on the best that can happen.

• Procedure:– Identify the maximum payoff from any state of nature for each alternative.– Find the maximum of these maximum payoffs and choose this alternative.

State of Nature

Alternative Oil Dry Maximum in Row

Drill for oil 700 –100 700 Maximax

Sell the land 90 90 90

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The Maximin Criterion

• The maximin criterion is the decision criterion for the total pessimist.

• It focuses only on the worst that can happen.

• Procedure:– Identify the minimum payoff from any state of nature for each alternative.– Find the maximum of these minimum payoffs and choose this alternative.

State of Nature

Alternative Oil Dry Minimum in Row

Drill for oil 700 –100 –100

Sell the land 90 90 90 Maximin

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The Maximum Likelihood Criterion

• The maximum likelihood criterion focuses on the most likely state of nature.

• Procedure:– Identify the state of nature with the largest prior probability– Choose the decision alternative that has the largest payoff for this state of nature.

State of Nature

Alternative Oil Dry

Drill for oil 700 –100 –100

Sell the land 90 90 90 Step 2: Maximum

Prior probability 0.25 0.75

Step 1: Maximum

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The Minimax Regret Criterion• The minimax regret criterion gets way from the focus on optimism versus pessimism

Focus is on choosing a decision that minimizes the regret that can be felt afterward if the decision does not turn out well.

• Procedure:

– Regret = maximum payoff – actual payoff where maximum payoff is the largest payoff that could have been obtained from any decision alternative for the observed state of nature

– Choose the decision alternative that has the minimum of the maximum regrets.

Payoff Table

State of Nature

Alternative Oil Dry

Drill for oil 700 –100

Sell the land 90 90

Maximum payoff 700 90

Regret Table

State of Nature

Alternative Oil Dry

Drill for oil 700 90

-700 -(-100)

0 190

Sell the land 700 90

-90 -90

610 0

Bayes’ Decision Rule

• Bayes’ decision rule directly uses the prior probabilities.

• Procedure:– For each decision alternative, calculate the weighted average of its payoff by

multiplying each payoff by the prior probability and summing these products. This is the expected payoff (EP).

– Choose the decision alternative that has the largest expected payoff.

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A B C D E F

Bayes' Decision Rule for the Goferbroke Co.

Payoff Table ExpectedAlternative Oil Dry Payoff

Drill 700 -100 100Sell 90 90 90

Prior Probability 0.25 0.75

State of Nature

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Bayes’ Decision Rule

• Features of Bayes’ Decision Rule– It accounts for all the states of nature and their probabilities.

– The expected payoff can be interpreted as what the average payoff would become if the same situation were repeated many times. Therefore, on average, repeatedly applying Bayes’ decision rule to make decisions will lead to larger payoffs in the long run than any other criterion.

• Criticisms of Bayes’ Decision Rule– There usually is considerable uncertainty involved in assigning values to the prior

probabilities.

– Prior probabilities inherently are at least largely subjective in nature, whereas sound decision making should be based on objective data and procedures.

– It ignores typical aversion to risk. By focusing on average outcomes, expected (monetary) payoffs ignore the effect that the amount of variability in the possible outcomes should have on decision making.

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Decision Trees

• A decision tree can apply Bayes’ decision rule while displaying and analyzing the problem graphically.

• A decision tree consists of nodes and branches.– A decision node, represented by a square, indicates a decision to be made. The

branches represent the possible decisions.

– An event node, represented by a circle, indicates a random event. The branches represent the possible outcomes of the random event.

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Decision Tree for Goferbroke

A

B

Payoff

-100

90

700Oil (0.25)

Dry (0.75)

Drill

Sell

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Using TreePlan

TreePlan,, can be used to construct and analyze decision trees on a spreadsheet.

1. Choose Decision Tree on the Add-Ins tab (Excel 2007 or 2010) or the Tools menu (for other versions).

2. Click on New Tree, and it will draw a default tree with a single decision node and two branches, as shown below.

3. The labels in D2 and D7 (originally Decision 1 and Decision 2) can be replaced by more descriptive names (e.g., Drill and Sell).

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Using TreePlan

4. To replace a node (such as the terminal node of the drill branch in F3) by a different type of node (e.g., an event node), click on the cell containing the node, choose Decision Tree again from the Add-Ins tab (Excel 2007 or 2010) or Tools menu (other versions of Excel), and select “Change to event node”.

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Using TreePlan

5. Enter the correct probabilities in H1 and H6.

6. Enter the partial payoffs for each decision and event in D6, D14, H4, and H9.

9-20

TreePlan Results

• The numbers inside each decision node indicate which branch should be chosen (assuming the branches are numbered consecutively from top to bottom).

• The numbers to the right of each terminal node is the payoff if that node is reached.

• The number 100 in cells A10 and E6 is the expected payoff at those stages in the process.

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Consolidate the Data and Results

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A B C D E F G H I J K0.25

Oil700

Drill 800 700

-100 100 0.75Dry

-1001 0 -100

100

Sell90

90 90

DataCost of Drilling 100Revenue if Oil 800

Revenue if Sell 90Revenue if Dry 0

Probability Of Oil 0.25

Action Drill

Expected Payoff 100

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Sensitivity Analysis: Prior Probability of Oil = 0.15

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Oil700

Drill 800 700

-100 20 0.85Dry

-1002 0 -100

90

Sell90

90 90




Action Sell

Expected Payoff 90

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Sensitivity Analysis: Prior Probability of Oil = 0.35

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Oil700

Drill 800 700

-100 180 0.65Dry

-1001 0 -100

180

Sell90

90 90




Action Drill

Expected Payoff 180

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Using Data Tables to Do Sensitivity Analysis

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A B C D E F G H I J K L M0.25

Oil700

Drill 800 700

-100 100 0.75Dry

-1001 0 -100

100

Sell90

90 90

Probability ExpectedData of Oil Action Payoff

Cost of Drilling 100 Drill 100Revenue if Oil 800 0.15

Revenue if Sell 90 0.17Revenue if Dry 0 0.19

Probability Of Oil 0.25 0.210.23

Action Drill 0.250.27

Expected Payoff 100 0.290.310.330.35

Select thesecells (I18:K29), before choosing Table from the Data menu.

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Data Table ResultsThe Effect of Changing the Prior Probability of Oil

1617181920212223242526272829

I J KProbability Expected

of Oil Action PayoffDrill 100

0.15 Sell 900.17 Sell 900.19 Sell 900.21 Sell 900.23 Sell 900.25 Drill 1000.27 Drill 1160.29 Drill 1320.31 Drill 1480.33 Drill 1640.35 Drill 180

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Checking Whether to Obtain More Information

• Might it be worthwhile to spend money for more information to obtain better estimates?

• A quick way to check is to pretend that it is possible to actually determine the true state of nature (“perfect information”).

• EP (with perfect information) = Expected payoff if the decision could be made after learning the true state of nature.

• EP (without perfect information) = Expected payoff from applying Bayes’ decision rule with the original prior probabilities.

• The expected value of perfect information is thenEVPI = EP (with perfect information) – EP (without perfect

information).

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Expected Payoff with Perfect Information

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B C DPayoff Table

Alternative Oil DryDrill 700 -100Sell 90 90

Maximum Payoff 700 90

Prior Probability 0.25 0.75

EP (with perfect info) 242.5

State of Nature

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Expected Payoff with Perfect Information

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A B C D E F G H I J K

Drill0.25 700

Oil 700 7001

0 700Sell

9090 90

242.5Drill

0.75 -100Dry -100 -100

20 90

Sell90

90 90

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Using New Information to Update the Probabilities

• The prior probabilities of the possible states of nature often are quite subjective in nature. They may only be rough estimates.

• It is frequently possible to do additional testing or surveying (at some expense) to improve these estimates. The improved estimates are called posterior probabilities.

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Seismic Survey for Goferbroke• Goferbroke can obtain improved estimates of the chance of oil by conducting

a detailed seismic survey of the land, at a cost of $30,000.

• Possible findings from a seismic survey:– FSS: Favorable seismic soundings; oil is fairly likely.

– USS: Unfavorable seismic soundings; oil is quite unlikely.

• P(finding | state) = Probability that the indicated finding will occur, given that the state of nature is the indicated one.

P(finding | state)

State of Nature Favorable (FSS) Unfavorable (USS)

Oil P(FSS | Oil) = 0.6 P(USS | Oil) = 0.4

Dry P(FSS | Dry) = 0.2 P(USS | Dry) = 0.8

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Calculating Joint Probabilities

• Each combination of a state of nature and a finding will have a joint probability determined by the following formula:

P(state and finding) = P(state) P(finding | state)

• P(Oil and FSS) = P(Oil) P(FSS | Oil) = (0.25)(0.6) = 0.15.

• P(Oil and USS) = P(Oil) P(USS | Oil) = (0.25)(0.4) = 0.1.

• P(Dry and FSS) = P(Dry) P(FSS | Dry) = (0.75)(0.2) = 0.15.

• P(Dry and USS) = P(Dry) P(USS | Dry) = (0.75)(0.8) = 0.6.

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Probabilities of Each Finding

• Given the joint probabilities of both a particular state of nature and a particular finding, the next step is to use these probabilities to find each probability of just a particular finding, without specifying the state of nature.

P(finding) = P(Oil and finding) + P(Dry and finding)

• P(FSS) = 0.15 + 0.15 = 0.3.

• P(USS) = 0.1 + 0.6 = 0.7.

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Calculating the Posterior Probabilities

• The posterior probabilities give the probability of a particular state of nature, given a particular finding from the seismic survey.

P(state | finding) = P(state and finding) / P(finding)

• P(Oil | FSS) = 0.15 / 0.3 = 0.5.

• P(Oil | USS) = 0.1 / 0.7 = 0.14.

• P(Dry | FSS) = 0.15 / 0.3 = 0.5.

• P(Dry | USS) = 0.6 / 0.7 = 0.86.

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Probability Tree Diagram

0.25(0.6) = 0.15

Oil and FSS Oil, given FSS

0.25(0.4) = 0.1Oil and USS

0.75(0.2) = 0.15

0.75(0.8) = 0.6Dry and USS

Dry and FSS

Dry, given USS

Dry, given FSS

Oil, given USS

= 0.50.15 0.3

0.1 0.7

= 0.14

0.15 0.3

= 0.5

0.6 0.7

= 0.86

PriorProbabilities

P(state)

Conditional Probabilities

P(finding | state)

Joint Probabilities

P(state and finding)

Posterior Probabilities

P(state | finding)

Unconditional probabilities: P(FSS) = 0.15 + 0.15 = 0.3

P(finding) P(USS) = 0.1 + 0.6 = 0.7

0.6 FSS, given Oil

0.4 USS, given Oil

0.2 FSS, given Dry

0.8 USS, given Dry

0.25 Oil

0.75 Dry

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Posterior Probabilities

P(state | finding)

Finding Oil Dry

Favorable (FSS) P(Oil | FSS) = 1/2 P(Dry | FSS) = 1/2

Unfavorable (USS) P(Oil | USS) = 1/7 P(Dry | USS) = 6/7

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Template for Posterior Probabilities

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B C D E F G HData:

State of PriorNature Probability FSS USS

Oil 0.25 0.6 0.4Dry 0.75 0.2 0.8

PosteriorProbabilities:

Finding P(Finding) Oil DryFSS 0.3 0.5 0.5USS 0.7 0.1429 0.8571

P(State | Finding)State of Nature

P(Finding | State)Finding

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Decision Tree for the Full Goferbroke Co. Problem

a

b

c

d

e

f

g

h

Do seismic survey

No seismic survey

Unfavorable

Favorable

Drill

Sell

Drill

Sell

Oil

Dry

Oil

Dry

Oil

Dry

Sell

Drill

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Decision Tree with Probabilities and Payoffs

a

b

c

d

e

f

g

h

Payoff

670

-130

60

670

-130

60

700

-100

90

Do seismic survey

No seismic survey

Unfavorable

Favorable

Drill-100

90Sell

Drill-100

90Sell

Drill-100

90Sell

Oil (0.143)800

0Dry(0.857)

Oil (0.5)800

0

Dry (0.5)

Oil (0.25)800

Dry (0.75)

0

(0.3)-30

0

0

0

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The Final Decision Tree

a

b

c

d

e

f

g

h

Payoff

670

-130

60

670

-130

60

700

-100

90

100

270

60

123

123

-15.7

270

100

Do seismic survey

No seismic survey

-30

0

Unfavorable

0

0

Favorable (0.3)

Drill-100

90Sell

Drill -100

90Sell

Drill-100

90Sell

Oil (0.143)800

0Dry (0.857)

Oil (0.5)800

0Dry (0.5)

Oil (0.25)

800

0

Dry (0.75)

9-40

TreePlan for the Full Goferbroke Co. Problem12345678910111213141516171819202122232425262728293031323334353637383940414243444546

A B C D E F G H I J K L M N O P Q R S

Decision Tree for Goferbroke Co. Problem (With Survey)

0.143Oil

670Drill 800 670

-100 -15.714 0.8570.7 Dry

Unfavorable -1302 0 -130

0 60

Sell60

90 60Do Survey

0.5-30 123 Oil

670Drill 800 670

-100 270 0.50.3 Dry

Favorable -1301 0 -130

0 270

1 Sell123 60

90 60

0.25Oil

700Drill 800 700

-100 100 0.75Dry

No Survey -1001 0 -100

0 100

Sell90

90 90

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Organizing the Spreadsheet for Sensitivity Analysis

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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA0.143 Data

Oil Cost of Survey 30670 Cost of Drilling 100

Drill 800 670 Revenue if Oil 800Revenue if Sell 90

-100 -15.714 0.857 Revenue if Dry 00.7 Dry Prior Probability Of Oil 0.25

Unfavorable -130 P(FSS|Oil) 0.62 0 -130 P(USS|Dry) 0.8

0 60

Sell Action60 Do Survey? Yes

90 60Do Survey If No If Yes

0.5-30 123 Oil Drill Drill If Favorable

670 Sell If UnfavorableDrill 800 670

-100 270 0.50.3 Dry Expected Payoff

Favorable -130 ($thousands)1 0 -130 123

0 270

1 Sell123 60 Data:

90 60 State of PriorNature Probability FSS USS

0.25 Oil 0.25 0.6 0.4Oil Dry 0.75 0.2 0.8

700Drill 800 700

-100 100 0.75Dry Posterior

No Survey -100 Probabilities:1 0 -100 Finding P(Finding) Oil Dry

0 100 FSS 0.3 0.5 0.5USS 0.7 0.143 0.857

Sell90

90 90

P(State | Finding)

Finding

State of Nature

P(Finding | State)

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Documents

Decision Analysis