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CONTEMPORARY MANAGEMENT SCIENCE WITH SPREADSHEETS ANDERSON SWEENEY WILLIAMS. SLIDES PREPARED BY JOHN LOUCKS. © 1999 South-Western College Publishing. Chapter 9 Decision Analysis. Structuring the Decision Problem Decision Making Without Probabilities

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CONTEMPORARY MANAGEMENT SCIENCE WITH SPREADSHEETS ANDERSON SWEENEY WILLIAMS

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Contemporary management science with spreadsheets anderson sweeney williams

CONTEMPORARY

MANAGEMENT SCIENCE

WITH SPREADSHEETS

ANDERSON SWEENEY WILLIAMS

SLIDES PREPARED BY

JOHN LOUCKS

© 1999 South-Western College Publishing


Chapter 9 decision analysis

Chapter 9Decision Analysis

  • Structuring the Decision Problem

  • Decision Making Without Probabilities

  • Decision Making with Probabilities

  • Expected Value of Perfect Information

  • Decision Analysis with Sample Information

  • Developing a Decision Strategy

  • Expected Value of Sample Information


Structuring the decision problem

Structuring the Decision Problem

  • A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.

  • The decision alternatives are the different possible strategies the decision maker can employ.

  • The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

  • For each decision alternative and state of nature, there is an outcome.

  • These outcomes are often represented in a matrix called a payoff table.


Decision trees

Decision Trees

  • A decision tree is a chronological representation of the decision problem.

  • Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives.

  • The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.

  • At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.


Decision making without probabilities

Decision Making Without Probabilities

  • If the decision maker does not know with certainty which state of nature will occur, then he is said to be doing decision making under uncertainty.

  • Three commonly used criteria for decision making under uncertainty when probability information regarding the likelihood of the states of nature is unavailable are:

  • the optimistic approach

    • the conservative approach

    • the minimax regret approach.


Optimistic approach

Optimistic Approach

  • The optimistic approach would be used by an optimistic decision maker.

  • The decision with the largest possible payoff is chosen.

  • If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.


Conservative approach

Conservative Approach

  • The conservative approach would be used by a conservative decision maker.

  • For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.)

  • If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)


Minimax regret approach

Minimax Regret Approach

  • The minimax regret approach requires the construction of a regret table or an opportunity loss table.

  • This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature.

  • Then, using this regret table, the maximum regret for each possible decision is listed.

  • The decision chosen is the one corresponding to the minimum of the maximum regrets.


Example

Example

Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits:

States of Nature

s1s2s3

d1 4 4 -2

Decisionsd2 0 3 -1

d3 1 5 -3


Example1

Example

  • Optimistic Approach

    An optimistic decision maker would use the optimistic approach. All we really need to do is to choose the decision that has the largest single value in the payoff table. This largest value is 5, and hence the optimal decision is d3.

    Maximum

    DecisionPayoff

    d1 4

    d2 3

    choose d3d3 5 maximum


Example2

Example

  • Formula Spreadsheet forOptimistic Approach


Example3

Example

  • Spreadsheet for Optimistic Approach


Example4

Example

  • Conservative Approach

    A conservative decision maker would use the conservative approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.

    Minimum

    DecisionPayoff

    d1 -2

    choose d2d2 -1 maximum

    d3 -3


Example5

Example

  • Formula Spreadsheet for Conservative Approach


Example6

Example

  • Spreadsheet for Conservative Approach


Example7

Example

  • Minimax Regret Approach

    For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 4, 0, and 1 from 4; in the second column, subtract 4, 3, and 5 from 5; etc. The resulting regret table is:

    s1s2s3

    d1 0 1 1

    d2 4 2 0

    d3 3 0 2


Example8

Example

  • Minimax Regret Approach (continued)

    For each decision list the maximum regret. Choose the decision with the minimum of these values.

    DecisionMaximum Regret

    choose d1 d1 1 minimum

    d2 4

    d3 3


Example9

Example

  • Formula Spreadsheet for Minimax Regret Approach


Example10

Example

  • Spreadsheet for Minimax Regret Approach


Decision making with probabilities

Decision Making with Probabilities

  • Expected Value Approach

    • If probabilistic information regarding he states of nature is available, one may use the expected value (EV) approach.

    • Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring.

    • The decision yielding the best expected return is chosen.


Expected value of a decision alternative

Expected Value of a Decision Alternative

  • The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative.

  • The expected value (EV) of decision alternative di is defined as:

    where: N = the number of states of nature

    P(sj) = the probability of state of nature sj

    Vij = the payoff corresponding to decision alternative di and state of nature sj


Example burger prince

Example: Burger Prince

Burger Prince Restaurant is contemplating opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. The payoff table for the three models is as follows:

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

d1 = Model A $10,000 $15,000 $14,000

d2 = Model B $ 8,000 $18,000 $12,000

d3 = Model C $ 6,000 $16,000 $21,000


Example burger prince1

Example: Burger Prince

  • Expected Value Approach

    Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d1, d2, d3 represent the decision alternatives of models A, B, C, and s1, s2, s3 represent the states of nature of 80, 100, and 120.


Example burger prince2

Example: Burger Prince

  • Decision Tree

Payoffs

.4

s1

10,000

s2

.2

2

15,000

s3

.4

d1

14,000

s1

.4

8,000

d2

s2

.2

1

3

18,000

s3

d3

.4

12,000

s1

.4

6,000

4

s2

.2

16,000

s3

.4

21,000


Contemporary management science with spreadsheets anderson sweeney williams

Example: Burger Prince

  • Expected Value For Each Decision

  • Choose the model with largest EMV -- Model C.

EV = .4(10,000) + .2(15,000) + .4(14,000)

= $12,600

2

d1

Model A

EV = .4(8,000) + .2(18,000) + .4(12,000)

= $11,600

d2

Model B

3

1

d3

Model C

EV = .4(6,000) + .2(16,000) + .4(21,000)

= $14,000

4


Example burger prince3

Example: Burger Prince

  • Formula Spreadsheet for Expected Value Approach


Example burger prince4

Example: Burger Prince

  • Spreadsheet for Expected Value Approach


Expected value of perfect information

Expected Value of Perfect Information

  • Frequently information is available which can improve the probability estimates for the states of nature.

  • The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur.

  • The EVPI provides an upper bound on the expected value of any sample or survey information.


Expected value of perfect information1

Expected Value of Perfect Information

  • EVPI Calculation

    • Step 1:

      Determine the optimal return corresponding to each state of nature.

    • Step 2:

      Compute the expected value of these optimal returns.

    • Step 3:

      Subtract the EV of the optimal decision from the amount determined in step (2).


Example burger prince5

Example: Burger Prince

  • Expected Value of Perfect Information

    Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.

    EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000


Example burger prince6

Example: Burger Prince

  • Spreadsheet for Expected Value of Perfect Information


Decision analysis with sample information

Decision Analysis With Sample Information

  • Knowledge of sample or survey information can be used to revise the probability estimates for the states of nature.

  • Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities.

  • With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem.

  • The outcomes of this analysis are called posterior probabilities.


Posterior probabilities

Posterior Probabilities

  • Posterior Probabilities Calculation

    • Step 1:

      For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator.

    • Step 2:

      Sum these joint probabilities over all states -- this gives the marginal probability for the indicator.

    • Step 3:

      For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.


Expected value of sample information

Expected Value of Sample Information

  • The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information.


Expected value of sample information1

Expected Value of Sample Information

  • EVSI Calculation

    • Step 1:

      Determine the optimal decision and its expected return for the possible outcomes of the sample using the posterior probabilities for the states of nature.

      Step 2:

      Compute the expected value of these optimal returns.

    • Step 3:

      Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).


Efficiency of sample information

Efficiency of Sample Information

  • Efficiency of sample information is the ratio of EVSI to EVPI.

  • As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.


Example burger prince7

Example: Burger Prince

  • Sample Information

    Burger Prince must decide whether or not to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable" or "unfavorable". The conditional probabilities are:

    P(favorable | 80 customers per hour) = .2

    P(favorable | 100 customers per hour) = .5

    P(favorable | 120 customers per hour) = .9

    Should Burger Prince have the survey performed by Stanton Marketing?


Example burger prince8

Example: Burger Prince

  • Posterior Probabilities

    Favorable Survey Results

    StatePriorConditionalJointPosterior

    80 .4 .2 .08 .148

    100 .2 .5 .10 .185

    120 .4 .9 .36.667

    Total .54 1.000

    P(favorable) = .54


Example burger prince9

Example: Burger Prince

  • Posterior Probabilities

    Unfavorable Survey Results

    StatePriorConditionalJointPosterior

    80 .4 .8 .32 .696

    100 .2 .5 .10 .217

    120 .4 .1 .04.087

    Total .46 1.000

    P(unfavorable) = .46


Example burger prince10

Example: Burger Prince

  • Formula Spreadsheet for Posterior Probabilities


Example burger prince11

Example: Burger Prince

  • Spreadsheet for Posterior Probabilities


Example burger prince12

Example: Burger Prince

  • Decision Tree (top half)

s1 (.148)

$10,000

s2 (.185)

4

$15,000

d1

s3 (.667)

$14,000

s1 (.148)

$8,000

d2

s2 (.185)

5

2

$18,000

s3 (.667)

d3

$12,000

I1

(.54)

s1 (.148)

$6,000

s2 (.185)

6

$16,000

s3 (.667)

1

$21,000


Example burger prince13

Example: Burger Prince

  • Decision Tree (bottom half)

s1 (.696)

1

$10,000

s2 (.217)

7

$15,000

I2

(.46)

s3 (.087)

d1

$14,000

s1 (.696)

$8,000

d2

s2 (.217)

8

3

$18,000

s3 (.087)

$12,000

d3

s1 (.696)

$6,000

s2 (.217)

9

$16,000

s3 (.087)

$21,000


Example burger prince14

Example: Burger Prince

  • EMV = .148(10,000) + .185(15,000)

    • + .667(14,000) = $13,593

4

d1

  • $17,855

d2

  • EMV = .148 (8,000) + .185(18,000)

    • + .667(12,000) = $12,518

5

2

d3

I1

(.54)

  • EMV = .148(6,000) + .185(16,000)

    • +.667(21,000) = $17,855

6

1

  • EMV = .696(10,000) + .217(15,000)

    • +.087(14,000)= $11,433

7

d1

I2

(.46)

d2

  • EMV = .696(8,000) + .217(18,000)

    • + .087(12,000) = $10,554

8

3

d3

  • $11,433

  • EMV = .696(6,000) + .217(16,000)

    • +.087(21,000) = $9,475

9


Example burger prince15

Example: Burger Prince

  • Decision Strategy Assuming the Survey is Undertaken:

    • If the outcome of the survey is favorable, choose Model C.

    • If it is unfavorable, choose Model A.


Example burger prince16

Example: Burger Prince

  • Question:

    Should the survey be undertaken?

  • Answer:

    If the Expected Value with Sample Information (EVwSI) is greater, after deducting expenses, than the Expected Value without Sample Information (EVwoSI), the survey is recommended.


Example burger prince17

Example: Burger Prince

  • Expected Value with Sample Information (EVwSI)

    EVwSI = .54($17,855) + .46($11,433) = $14,900.88

  • Expected Value of Sample Information (EVSI)

    EVSI = EVwSI - EVwoSI

    assuming maximization

    EVSI= $14,900.88 - $14,000 = $900.88


Example burger prince18

Example: Burger Prince

  • Conclusion

    EVSI = $900.88

    Since the EVSI is less than the cost of the survey ($1000), the survey should not be purchased.


Example burger prince19

Example: Burger Prince

  • Efficiency of Sample Information

    The efficiency of the survey:

    EVSI/EVPI = ($900.88)/($2000) = .4504


The end of chapter 9

The End of Chapter 9


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