© 1999 Prentice-Hall, Inc. Chap. 5 - 1

Statistics for Managers

Using Microsoft Excel/SPSS

Chapter 5

Decision Making

© 1999 Prentice-Hall, Inc. Chap. 5 - 2

Chapter Topics

•The Payoff Table and Decision Trees

•Opportunity Loss

•Criteria for Decision Making

•Expected Monetary Value

•Expected Profit Under Certainty

•Return to Risk Ratio

•Decision Making with Sample Information

•Utility

© 1999 Prentice-Hall, Inc. Chap. 5 - 3

Features of

Decision Making

•List Alternative Courses of Action

(Possible Events or Outcomes)

•Determine ‘Payoffs’

(Associate a Payoff with Each Event or Outcome)

•Adopt Decision Criteria

(Evaluate Criteria for Selecting the Best Course of

Action)

© 1999 Prentice-Hall, Inc. Chap. 5 - 4

List Possible

Actions or Events

Payoff Table Decision Tree

Two Methods

of Listing

© 1999 Prentice-Hall, Inc. Chap. 5 - 5

Event (Ei)

Cool Weather (E1) x11 =$50 x12 = $100

Warm Weather (E2) x21 = 200 x22 = 125

Payoff Table

Consider a food vendor determining

whether to sell soft drinks or hot dogs.

Course of Action (Aj)

Sell Soft Drinks (A1) Sell Hot Dogs (A2)

xij = payoff (profit) for event i and action j

© 1999 Prentice-Hall, Inc. Chap. 5 - 6

Decision Tree:Example

Food Vendor Profit Tree Diagram

x11 = $50

x21 = 200

x22 =125

x12 = 100

© 1999 Prentice-Hall, Inc. Chap. 5 - 7

Opportunity Loss: Example

Highest possible profit for an event Ei

- Actual profit obtained for an action Aj

Opportunity Loss (lij )

Event: Cool Weather

Action: Soft Drinks Profit: $50

Alternative Action: Hot Dogs Profit: $100

Opportunity Loss = $100 - $50 = $50

© 1999 Prentice-Hall, Inc. Chap. 5 - 8

Opportunity Loss: Table

Event Optimal Profit of Sell Soft Drinks Sell Hot Dogs

Action Optimal

Action

Cool Hot 100 100 - 50 = 50 100 - 100 = 0

Weather Dogs

Warm Soft 200 200 - 200 = 0 200 - 125 = 75

Weather Drinks

Alternative Course of Action

© 1999 Prentice-Hall, Inc. Chap. 5 - 9

Decision Criteria

Expected Monetary Value (EMV)

• The expected profit for taking an action Aj

Expected Opportunity Loss (EOL)

• The expected loss for not taking action Aj

Expected Value of Perfect Information (EVPI)

• The expected opportunity loss from the best

decision

© 1999 Prentice-Hall, Inc. Chap. 5 - 10

Decision Criteria -- EMV

Expected Monetary Value (EMV)

Sum (monetary payoffs of events) (probabilities of the events)

Xij Pi EMVj = N

EMVj = expected monetary value of action j

xi,j = payoff for action j and event i

Pi = probability of event i occurring

i = 1

© 1999 Prentice-Hall, Inc. Chap. 5 - 11

Decision Criteria -- EMV Table

Example: Food Vendor

Pi Event Soft xijPi Hot xijPi

Drinks Dogs

.50 Cool $50 $50 .5 = $25 $100 $100.50 = $50

.50 Warm $200 $200 .5 = 100 $125 $25.50 = 62.50

EMV Soft Drink = $125 EMV Hot Dog = $112.50

Better alternative

© 1999 Prentice-Hall, Inc. Chap. 5 - 12

Decision Criteria -- EOL

Expected Opportunity Loss (EOL)

Sum (opportunity losses of events) (probabilities of events)

EOLj = lij Pi

EOLj = expected monetary value of action j

li,j = payoff for action j and event i

Pi = probability of event i occurring

i =1

N

© 1999 Prentice-Hall, Inc. Chap. 5 - 13

Decision Criteria -- EOL Table

Example: Food Vendor Pi Event Op Loss lijPi OP Loss lijPi

Soft Drinks Hot Dogs

.50 Cool $50 $50.50 = $25 $0 $0.50 = $0

.50 Warm 0 $0 .50 = $0 $75 $75 .50 = $37.50

EOL Soft Drinks = $25 EOL Hot Dogs = $37.50

Better Choice

© 1999 Prentice-Hall, Inc. Chap. 5 - 14

Decision Criteria -- EVPI

Expected Value of Perfect Information (EVPI)

• The expected opportunity loss from the best decision

• Represents the maximum amount you are willing

to pay to obtain perfect information

Expected Profit Under Certainty

- Expected Monetary Value of the Best Alternative

EVPI (should be a positive number)

© 1999 Prentice-Hall, Inc. Chap. 5 - 15

EVPI Computation

Expected Profit Under Certainty

= .50($100) + .50($200)

= $150

Expected Monetary Value of the Best Alternative

= $125

EPVI = $25

The maximum you would be willing to

spend to obtain perfect information.

© 1999 Prentice-Hall, Inc. Chap. 5 - 16

Taking Account of Variability: FoodVendor

s2 for Soft Drink

= (50 -125)2 .5 + (200 -125)2 .5 = 5625

s for Soft Drink = 75

CVfor Soft Drinks = (75/125) 100% = 60%

s2 for Hot Dogs = 156.25 s for Hot dogs = 12.5

CVfor Hot dogs = 11.11%

© 1999 Prentice-Hall, Inc. Chap. 5 - 17

Return to Risk Ratio

Expresses the relationship between the return

(payoff) and the risk (standard deviation).

RRR = Return to Risk Ratio = j

jEMV

s

RRRSoft Drinks = 125/75 = 1.67 RRRHot Dogs = 9

You might wish to choose Hot Dogs. Although Soft

Drinks have the higher Expected Monetary Value, Hot

Dogs have a much larger return to risk ratio and a

much smaller CV.

© 1999 Prentice-Hall, Inc. Chap. 5 - 18

Decision Making with

Sample Information

•Permits Revising Old

Probabilities Based on New

Information

New

Information

Revised

Probability

Prior

Probability

© 1999 Prentice-Hall, Inc. Chap. 5 - 19

Additional Information: Weather forecast is COOL. When the weather is cool, the forecaster was correct 80% of the time.

When it has been warm, the forecaster was correct 70% of the time.

Revised Probabilities

Example: Food Vendor

Prior

Probability

F1 = Cool forecast

F2 = Warm forecast

E1 = Cool Weather = 0.50

E2 = Warm Weather = 0.50

P(F1 | E1) = 0.80 P(F1 | E2) = 0.30

© 1999 Prentice-Hall, Inc. Chap. 5 - 20

Revising Probabilities

Example:Food Vendor

P(E1 | F1) = P(cool) P( cool forecast | cool)

P(cool forecast)

= = .73 (.50) (.80)

(.80)(.50) + (.30)(.50)

Revised Probability

P(F1 | E1) = 0.80 P(F1 | E2) = 0.30 E1 = 0.50 E2 = 0.50

P(E2 | F1)

= P(warm) P(cool forecast | warm)

P(cool forecast) = .27

© 1999 Prentice-Hall, Inc. Chap. 5 - 21

Revised EMV Table

Example: Food Vendor

Pi Event Soft xijPi Hot xijPi

Drinks Dogs

.73 Cool $50 $36.50 $100 $73

.27 Warm $200 54 125 33.73

EMV Soft Drink = $90.50 EMV Hot Dog = $106.75

Better alternative

© 1999 Prentice-Hall, Inc. Chap. 5 - 22

Revised EOL Table

Example: Food Vendor Pi Event Op Loss lijPi OP Loss lijPi

Soft Drink Hot Dogs

.73 Cool $50 $36.50 $0 0

.27 Warm 0 $0 75 20.25

EOL Soft Drinks = 36.50 EOL Hot Dogs = $20.25

Better Choice

© 1999 Prentice-Hall, Inc. Chap. 5 - 23

Revised EVPI Computation

Expected Profit Under Certainty

= .73($100) + .27($200)

= $127

Expected Monetary Value of the Best Alternative

= $106.75

EPVI = $20.25

The maximum you would be willing to

spend to obtain perfect information.

© 1999 Prentice-Hall, Inc. Chap. 5 - 24

Taking Account of Variability: Revised Computation

s2 for Soft Drinks

= (50 -90.5)2 .73 + (200 -90.5)2 .27 = 4434.75

s for Soft Drinks = 66.59

CVfor Soft Drinks = (66.59/90.5) 100% = 73.6%

s2 for Hot Dogs = 123.1875

s for Hot dogs = 11.10

CVfor Hot dogs = (11.10/106.75) 100% = 10.4%

© 1999 Prentice-Hall, Inc. Chap. 5 - 25

Revised Return to Risk Ratio

Expresses the relationship between the return

(payoff) and the risk (standard deviation).

RRR = Return to Risk Ratio = j

jEMV

s

RRRSoft Drinks = 90.50/66.59 = 1.36 RRRHot Dogs = 9.62

You might wish to choose Hot Dogs. Hot Dogs have a

much larger return to risk ratio.

© 1999 Prentice-Hall, Inc. Chap. 5 - 26

Utility

Utility is the idea that each incremental $1 of

profit does not have the same value to every

individual:

• A risk averse person, once reaching a goal, assigns less value to each incremental $1.

• A risk seeker assigns more value to each incremental $1.

• A risk neutral person assigns the same value to each $1.

© 1999 Prentice-Hall, Inc. Chap. 5 - 27

Chapter Summary

•Described The Payoff Table and Decision Trees

•Opportunity Loss

•Provided Criteria for Decision Making

•Expected Monetary Value

•Expected Profit Under Certainty

•Return to Risk Ratio

•Discussed Decision Making with Sample Information

•Addressed the Concept of Utility