Decision tree examples operations management

Operations Management Help

Table of Contents Chapter 9 (Decision Analysis)

Decision Analysis Examples 9.2–9.3

A Case Study: The Goferbroke Company Problem (Section 9.1) 9.4–9.8

Decision Criteria (Section 9.2) 9.9–9.13

Decision Trees (Section 9.3) 9.14–9.19

Sensitivity Analysis with Decision Trees (Section 9.4) 9.20–9.24

Checking Whether to Obtain More Information (Section 9.5) 9.25–9.26

Using New Information to Update the Probabilities (Section 9.6) 9.27–9.34

Decision Tree to Analyze a Sequence of Decisions (Section 9.7) 9.35–9.38

Sensitivity Analysis with a Sequence of Decisions (Section 9.8) 9.39–9.41

Using Utilities to Better Reflect the Values of Payoffs (Section 9.9) 9.42–9.54

© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.

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?

9-2

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?

9-3

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?

9-4

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
9-5

5

Table 9.1 Prospective profits for the Goferbroke Company.

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.

9-6

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
9-7

7

Table 9.2 Prior probabilities for the first Goferbroke Co. problem.

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
9-8

8

Table 9.3 Payoff table (profit in $thousands) for the first Goferbroke Co. problem.

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
9-9

9

Table 9.4 Application of the maximax criterion to the first Goferbroke Co. problem.

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
9-10

10

Table 9.5 Application of the maximin criterion to the first Goferbroke Co. problem.

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
9-11

11

Table 9.6 Application of the maximum likelihood criterion to the first Goferbroke Co. problem.

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.

9-12

12

Figure 9.1 This spreadsheet shows the application of Bayes’ decision rule to the first Goferbroke Co. problem, where a comparison of the expected payoffs in cells F5:F6 indicates that the Drill alternative should be chosen because it has the largest expected payoff.

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.

9-13

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.

9-14

Decision Tree for Goferbroke

9-15

15

Figure 9.2 The decision tree for the first Goferbroke Co. problem as presented in Table 9.3.

Using RSPE to Construct Decision Trees

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

Choose Node>Add Node from the Decision Tree menu on the RSPE ribbon.

Specify the type of node (Decision or Event).

Label the branches and specify the value for each branch.

9-16

16

Figure 9.3 The Decision Tree dialog box used to specify that the initial node of the first Goferbroke problem is a decision node with two branches, Drill and Sell, with values (partial payoffs) of -100 and 90, respectively.

Using RSPE to Construct Decision Trees

9-17

17

Figure 9.4 The initial partial decision tree created by RSPE by selecting Add Node from the Decision Tree > Node menu on the RSPE ribbon and specifying a decision node with two branches named Drill and Sell, with partial payoffs of -100 and 90, respectively.

Using RSPE to Construct Decision Trees

Select the node at the end of the Drill branch and choose Decision Tree > Change Node

Choose Event node type and enter the name and partial payoffs for each branch.

9-18

18

Figure 9.5 The Decision Tree dialog box used to specify that the second node of the first Goferbroke problem is an event node with two branches, Oil and Dry, with values (partial payoffs) of 800 and 0, and with probabilities of 0.25 and 0.75, respectively.

RSPE 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.

9-19

19

Figure 9.5 The decision tree constructed and solved by RSPE for the first Goferbroke Co. problem as presented in Table 9.3, where the 1 in cell B9 indicates that the top branch (the Drill alternative) should be chosen.

Consolidate the Data and Results

9-20

20

Figure 9.7 In preparation for performing sensitivity analysis on the first Goferbroke Co. problem, the data and results have been consolidated on the spreadsheet below the decision tree.

Sensitivity Analysis: Prior Probability of Oil = 0.15

9-21

21

Figure 9.8 Performing sensitivity analysis for the first Goferbroke Co. problem by trying a value of 0.15 for the prior probability of oil.

Sensitivity Analysis: Prior Probability of Oil = 0.35

9-22

22

Figure 9.8 Performing sensitivity analysis for the first Goferbroke Co. problem by trying a value of 0.35 for the prior probability of oil.

Using Data Tables to Do Sensitivity Analysis

9-23

23

Figure 9.9 Expansion of the spreadsheet in Figure 9.7 to prepare for generating a data table, where the choice of E22 for the column input cell in the Table dialog box indicates that this is the data cell that is being changed in the first column of the data table.

Data Table Results The Effect of Changing the Prior Probability of Oil

9-24

24

Figure 9.10 After the preparation in Figure 9.9, clicking OK generates this data table that shows the optimal action and expected payoff for various trial values of the prior probability of oil.

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 then EVPI = EP (with perfect information) – EP (without perfect information).

9-25

Expected Payoff with Perfect Information

9-26

26

Figure 9.11 By starting with an event node involving the states of nature, RSPE uses this decision tree to obtain the expected payoff with perfect information for the first Goferbroke Co. Problem.

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.

9-27

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
9-28

28

Table 9.7 Probabilities of the possible findings from the seismic survey, given the state of nature, for the Goferbroke Co. problem.

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.

9-29

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.

9-30

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.

9-31

Probability Tree Diagram

9-32

32

Figure 9.12 Probability tree diagram for the Goferbroke Co. problem showing all the probabilities leading to the calculation of each posterior probability of the state of nature given the finding of the seismic survey.

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
9-33

33

Table 9.8 Posterior probabilities of the states of nature, given the finding from the seismic survey, for the Goferbroke Co. problem.

Template for Posterior Probabilities

9-34

34

Figure 9.13 The Posterior Probabilities template in your MS Courseware enables efficient calculation of posterior probabilities, as illustrated here for the Goferbroke Co. problem.

Decision Tree for the Full Goferbroke Co. Problem

9-35

35

Figure 9.14 The decision tree for the full Goferbroke Co. problem (before including any numbers) when first deciding whether to conduct a seismic survey.

Decision Tree with Probabilities and Payoffs

9-36

36

Figure 9.15 The decision tree in Figure 9.14 after adding both the probabilities of random events and the payoffs.

The Final Decision Tree

9-37

37

Figure 9.16 The final decision tree that records the analysis for the full Goferbroke Co. problem when using monetary payoffs.

RSPE for the Full Goferbroke Co. Problem

9-38

38

Figure 9.17 The decision tree constructed and solved by RSPE for the full Goferbroke Co. problem that also considers whether to do a seismic survey.

Organizing the Spreadsheet for Sensitivity Analysis

9-39

39

Figure 9.18 In preparation for performing sensitivity analysis on the full Goferbroke Co. problem, the data and results have been consolidated on the spreadsheet to the right of the decision tree.

Data Table: Optimal Policy vs. Prior Probability of Oil

9-40

40

Figure 9.19 The data table that shows the optimal policy and expected payoff for various trial values of the prior probability of oil.

Optimal Policy

Let p = Prior probability of oil

If p ≤ 0.168, then sell the land (no seismic survey).

If 0.169 ≤ p ≤ 0.308, then do the survey; drill if favorable, sell if not.

If p ≥ 0.309, then drill for oil (no seismic survey).

9-41

Using Utilities to Better Reflect the Values of Payoffs

Thus far, when applying Bayes’ decision rule, we have assumed that the expected payoff in monetary terms is the appropriate measure.

In many situations, this is inappropriate.

Suppose an individual is offered the following choice:

Accept a 50-50 chance of winning $100,000.

Receive $40,000 with certainty.

Many would pick $40,000, even though the expected payoff on the 50-50 chance of winning $100,000 is $50,000. This is because of risk aversion.

A utility function for money is a way of transforming monetary values to an appropriate scale that reflects a decision maker’s preferences (e.g., aversion to risk).

9-42

A Typical Utility Function for Money

9-43

43

Figure 9.20 A typical utility function for money, where U(M) is the utility of obtaining an amount of money M.

Shape of Utility Functions

9-44

44

Figure 9.21 The shape of the utility function for (a) risk-averse, (b) risk-seeking, and (c) risk-neutral individuals.

Utility Functions

When a utility function for money is incorporated into a decision analysis approach, it must be constructed to fit the current preferences and values of the decision maker.

Fundamental Property: Under the assumptions of utility theory, the decision maker’s utility function for money has the property that the decision maker is indifferent between two alternatives if the two alternatives have the same expected utility.

When the decision maker’s utility function for money is used, Bayes’ decision rule replaces monetary payoffs by the corresponding utilities.

The optimal decision (or series of decisions) is the one that maximizes the expected utility.

9-45

Illustration of Fundamental Property

By the fundamental property, a decision maker with the utility function below-right will be indifferent between each of the three pairs of alternatives below-left.

25% chance of $100,000

$10,000 for sure

Both have E(Utility) = 0.25.

50% chance of $100,000

$30,000 for sure

Both have E(Utility) = 0.5.

75% chance of $100,000

$60,000 for sure

Both have E(Utility) = 0.75.

9-46

46

The Equivalent Lottery Method

Determine the largest potential payoff, M=Maximum. Assign U(Maximum) = 1.

Determine the smallest potential payoff, M=Minimum. Assign U(Minimum) = 0.

To determine the utility of another potential payoff M, consider the two aleternatives: A1: Obtain a payoff of Maximum with probability p. Obtain a payoff of Minimum with probability 1–p. A2: Definitely obtain a payoff of M. Question to the decision maker: What value of p makes you indifferent? Then, U(M) = p.

9-47

Generating the Utility Function for Max Flyer

The possible monetary payoffs in the Goferbroke Co. problem are –130, –100, 0, 60, 90, 670, and 700 (all in $thousands).

Set U(Maximum) = U(700) = 1.

Set U(Minimum) = U(–130) = 0.

To find U(M), use the equivalent lottery method.

For example, for M=90, consider the two alternatives: A1: Obtain a payoff of 700 with probability p Obtain a payoff of –130 with probability 1–p. A2: Definitely obtain a payoff of 90

If Max chooses a point of indifference of p = 1/3, then U(90) = 1/3.

9-48

Max’s Utility Function for Money

9-49

49

Figure 9.22 Max’s utility function for money as the owner of Goferbroke Co.

Utilities for the Goferbroke Co. Problem

Monetary Payoff, M Utility, U(M)
–130 0.00
–100 0.05
60 0.30
90 0.33
670 0.97
700 1.00
9-50

50

Table 9.9 Utilities for the Goferbroke Co. problem.

Decision Tree with Utilities

9-51

51

Figure 9.23 The final decision tree constructed and solved by RSPEfor the full Goferbroke Co. problem when using Max’s utility function for money to maximize expected utility.

Exponential Utility Function

The procedure for constructing U(M) requires making many difficult decisions about probabilities.

An alternative approach assumes a certain form for the utility function and adjusts this form to fit the decision maker as closely as possible.

A popular form is the exponential utility function U(M) = 1 – e–M/R where R is the decision maker’s risk tolerance.

An easy way to estimate R is to pick the value that makes you indifferent between the following two alternatives:

A 50-50 gamble where you gain R dollars with probability 0.5 and lose R/2 dollars with probability 0.5.

Neither gain nor lose anything.

9-52

Setting up RSPE to use the Exponential Utility Function

9-53

53

Figure 9.24 The Tree tab of the RSPE Options dialog box allows you to set several options for how the decision tree is solved. Here the options are set to use the exponential utility function, to maximize profit, and to use an R value of 1,000.

Decision Tree with an Exponential Utility Function

9-54

54

Figure 9.25 The final decision tree constructed and solved by RSPE for the full Goferbroke Co. problem when using an exponential utility function.

1

2

3

4

5

6

7

8

A

B

C

D

E

F

Bayes' Decision Rule for the Goferbroke Co.

Payoff Table

Expected

Alternative

Oil

Dry

Payoff

Drill

700

-100

100

Sell

90

90

90

Prior Probability

0.25

0.75

State of Nature

Bayes
Bayes' Decision Rule for the Goferbroke Co.
Payoff Table State of Nature Expected Range Name Cells
Alternative Oil Dry Payoff DrillPayoff C5:D5
Drill 700 -100 100 ExpectedPayoff F5:F6
Sell 90 90 90 PriorProbability C8:D8
SellPayoff C6:D6
Prior Probability 0.25 0.75
Sheet1
A

B

Payoff

-100

90

700

Oil (0.25)

Dry (0.75)

Drill

Sell

1

2

3

4

5

6

7

8

9

ABCDEFG

Sell

90

9090

1

90

Decision 2

0

00

1

2

3

4

5

6

7

8

9

10

11

12

13

14

ABCDEFGHIJK

25%

Oil

700

Drill 800700

-10010075%

Dry

-100

1 0-100

100

Sell

90

9090

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

ABCDEFGHIJK

25%

Oil

700

Drill 800700

-10010075%

Dry

-100

1 0-100

100

Sell

90

9090

Data

Cost of Drilling100

Revenue if Oil800

Revenue if Sell90

Revenue if Dry0

Probability of Oil25%

ActionDrill

Expected Payoff100

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

ABCDEFGHIJK

15%

Oil

700

Drill 800700

-1002085%

Dry

-100

2 0-100

90

Sell

90

9090

Data

Cost of Drilling100

Revenue if Oil800

Revenue if Sell90

Revenue if Dry0

Probability of Oil15%

ActionSell

Expected Payoff90

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

ABCDEFGHIJK

15%

Oil

700

Drill 800700

-1002085%

Dry

-100

2 0-100

90

Sell

90

9090

Data

Cost of Drilling100

Revenue if Oil800

Revenue if Sell90

Revenue if Dry0

Probability of Oil15%

ActionSell

Expected Payoff90

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

ABCDEFGHIJKLM

15%

Oil

700

Drill 800700

-1002085%

Dry

-100

2 0-100

90

Sell

90

9090

ProbabilityExpected

Data of OilActionPayoff

Cost of Drilling100 Sell90

Revenue if Oil800 15%

Revenue if Sell9017%

Revenue if Dry019%

Probability of Oil15% 21%

23%

ActionSell25%

27%

Expected Payoff9029%

31%

33%

35%

Select these cellsbefore choosing Data Table from the What-If Analysis menu on the Data tab.

image1.png

16

17

18

19

20

21

22

23

24

25

26

27

28

29

I

J

K

Probability

Expected

of Oil

Action

Payoff

Drill

100

0.15

Sell

90

0.17

Sell

90

0.19

Sell

90

0.21

Sell

90

0.23

Sell

90

0.25

Drill

100

0.27

Drill

116

0.29

Drill

132

0.31

Drill

148

0.33

Drill

164

0.35

Drill

180

Decision Tree
0.25
Oil
700 Range Name Cell
Drill 800 700 Action E24
CostOfDrilling E18
-100 100 0.75 ExpectedPayoff E26
Dry ProbabilityOfOil E22
-100 RevenueIfDry E21
1 0 -100 RevenueIfOil E19
100 RevenueIfSell E20
Sell
90