Session 6b. Decision Models -- Prof. Juran2 Overview Decision Analysis Uncertain Future Events...

Session 6b

Decision Models -- Prof. Juran

2

OverviewDecision Analysis• Uncertain Future Events• Perfect Information• Partial Information

– The Return of Rev. Thomas Bayes


3

F o r each p o ssib le str ateg y , w e can calcu late th e exp ected r ev en u e an d th e stan d ar d d ev iatio n o f r ev en u e.

5

6789

1 01 11 21 31 41 51 61 71 8

A B C D E F G H I J K

P u r c h a s e d 1 2 3 4 5 6 E x p e c te d R e v e n u e S td D e v o f R e v e n u e V a r ia n c e1 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 2 ,5 0 0 $ 0 02 $ 1 ,5 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 5 ,0 0 0 $ 4 ,8 2 5 $ 1 8 4 3 3 9 9 3 .7 53 $ 5 0 0 $ 4 ,0 0 0 $ 7 ,5 0 0 $ 7 ,5 0 0 $ 7 ,5 0 0 $ 7 ,5 0 0 $ 6 ,6 2 5 $ 6 2 5 3 9 0 4 6 8 .84 ( $ 5 0 0 ) $ 3 ,0 0 0 $ 6 ,5 0 0 $ 1 0 ,0 0 0 $ 1 0 ,0 0 0 $ 1 0 ,0 0 0 $ 7 ,5 5 0 $ 1 ,1 9 7 1 4 3 2 0 2 55 ( $ 1 ,5 0 0 ) $ 2 ,0 0 0 $ 5 ,5 0 0 $ 9 ,0 0 0 $ 1 2 ,5 0 0 $ 1 2 ,5 0 0 $ 7 ,4 2 5 $ 1 ,4 6 7 2 1 5 3 2 4 46 ( $ 2 ,5 0 0 ) $ 1 ,0 0 0 $ 4 ,5 0 0 $ 8 ,0 0 0 $ 1 1 ,5 0 0 $ 1 5 ,0 0 0 $ 6 ,7 7 5 $ 1 ,6 1 3 2 6 0 2 8 1 9

P r o b a b il i t y 0 .0 5 0 .1 5 0 .2 5 0 .3 0 .1 5 0 .1

P a y o f f T a b leC o n s u m e r D e m a n d

= S U M P R O D U C T ( B 1 3 :G 1 3 ,$ B $ 1 4 :$ G $ 1 4 )

= ( ( $ B $ 1 4 * ( B 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ C $ 1 4 * ( C 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ D $ 1 4 * ( D 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ E $ 1 4 * ( E 1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ F$ 1 4 * ( F1 3 - H 1 3 ) ) ^ 2 ) + ( ( $ G $ 1 4 * ( G 1 3 - H 1 3 ) ) ^ 2 )

= S Q R T ( J 1 3 )

T h e v ar ian ce fo r m u la lo o k s u g ly , bu t i t w o r k s.


4

Risk Profile

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$0 $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800

Std. Deviation of Revenue

Ex

pe

cte

d R

ev

en

ue

Buy 100 Pairs

Buy 200 Pairs

Buy 300 Pairs

Buy 400 Pairs Buy 500 Pairs

Buy 600 Pairs


5

First, click the “tree” button to start a new tree. The tree starts in the cell you click on after you click the tree button. Here, we clicked in cell I1.

12

I J1

0tree #1


6


7


8

We need to distinguish between two types of nodes.

Some events are random, and are represented by “chance” or “probability” nodes (red circles in PrecisionTree).

Other events are not random, but are management decisions. These are represented by “decision” nodes (green squares).

The first event in this problem is that management must decide how many hundreds of pairs of shoes to order. We are considering six possible order quantities here, so we select a decision node with six branches.


9


10

123456789

1011121314

I J KTRUE 1

0 0

FALSE 0

0 0

FALSE 0

0 0

Decision

0

FALSE 0

0 0

FALSE 0

0 0

FALSE 0

0 0

Buy Shoes

branch

branch

branch

branch

branch

branch


11


12

123456789

1011121314

I J KTRUE 1

0 0

FALSE 0

0 0

FALSE 0

0 0

Decision

0

FALSE 0

0 0

FALSE 0

0 0

FALSE 0

0 0

Buy Shoes

Buy 100

Buy 200

Buy 300

Buy 400

Buy 500

Buy 600


13


14

1234567891011121314151617181920212223242526

I J K L50.0% 0.5

0 0

0.0% 0

0 0

0.0% 0

0 0

TRUE Demand = 100

0 0

0.0% 0

0 0

0.0% 0

0 0

50.0% 0.5

0 0

FALSE 0

0 0

FALSE 0

0 0

Decision

0

FALSE 0

0 0

FALSE 0

0 0

FALSE 0

0 0

Buy Shoes

Buy 100

Buy 200

Buy 300

Buy 400

Buy 500

Buy 600

branch

branch

branch

branch

branch

branch


15

123456789

1011121314

K L50.0% 0.5

0 0

0.0% 0

0 0

0.0% 0

0 0

Demand = 100

0

0.0% 0

0 0

0.0% 0

0 0

50.0% 0.5

0 0

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500


16

Each of the branches at the far right of the diagram is characterized by two elements: a probability and a payoff value. For example, the “Sell 100” branch has a probability of 0.05 and a payoff (in the case of having purchased 100 units of shoes) of $2,500 (see cell B8 in the payoff table).

These entries are tedious, so we want to use copy-and-paste as much as possible. The payoffs will vary across the different purchase quantities, but the probabilities will not.


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1234567891011121314

J K5.0%

0

15.0%

0

25.0%

0

TRUE Demand = 100

0 0

30.0%

0

15.0%

0

10.0%

0

Buy 100

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500


18


19

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374

I J K L5.0% 0.05

$0 0

15.0% 0.15

0 0

25.0% 0.25

0 0

TRUE Demand = 100

0 0

30.0% 0.3

0 0

15.0% 0.15

0 0

10.0% 0.1

0 0

5.0% 0

$0 0

15.0% 0

0 0

25.0% 0

0 0

FALSE Demand = 100

0 0

30.0% 0

0 0

15.0% 0

0 0

10.0% 0

0 0

5.0% 0

$0 0

15.0% 0

0 0

25.0% 0

0 0

FALSE Demand = 100

0 0

30.0% 0

0 0

15.0% 0

0 0

10.0% 0

0 0

Decision

0

5.0% 0

$0 0

15.0% 0

0 0

25.0% 0

0 0

FALSE Demand = 100

0 0

30.0% 0

0 0

15.0% 0

0 0

10.0% 0

0 0

5.0% 0

$0 0

15.0% 0

0 0

25.0% 0

0 0

FALSE Demand = 100

0 0

30.0% 0

0 0

15.0% 0

0 0

10.0% 0

0 0

5.0% 0

$0 0

Buy Shoes

Buy 100

Buy 200

Buy 300

Buy 400

Buy 500

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500

Sell 100


20

123456789

10111213141516171819202122232425262728

J K L5.0% 0

$2,500 2500

15.0% 0

$2,500 2500

25.0% 0

$2,500 2500

FALSE Demand = 100

0 2500

30.0% 0

$2,500 2500

15.0% 0

$2,500 2500

10.0% 0

$2,500 2500

5.0% 0

$1,500 1500

15.0% 0

$5,000 5000

25.0% 0

$5,000 5000

FALSE Demand = 100

0 4825

30.0% 0

$5,000 5000

15.0% 0

$5,000 5000

10.0% 0

$5,000 5000

Buy 100

Buy 200

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500

Sell 100

Sell 600

Sell 200

Sell 300

Sell 400

Sell 500


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PrecisionTree uses “True” and “False” to indicate which branch of a decision node is optimal. Here, the best policy is to order 400 units and have an expected profit of $7,550.


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Example 2: TV Production

Witkowski TV Productions is considering a pilot for a comedy series for a major television network. The network may reject the pilot and the series, or it may purchase the program for one or two years. Witkowski may decide to produce the pilot or transfer the rights for the series to a competitor for $100,000.


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Witkowski’s profits are summarized in the following profit ($1000s) payoff table:

If the probability estimates for the states of nature are P(Reject) = 0.20, P(1 Year) = 0.30, and P(2 Years) = 0.50, what should Witkowski do?

States of Nature s1 = Reject s2 = 1 Year s3 = 2 Years

Produce Pilot d1 -100 50 150 Sell to Competitor d2 100 100 100


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20.0%

-100

FALSE Expected Value

0 70

30.0%

50

50.0%

150

Expected Value

100

20.0%

100

TRUE Expected Value

0 100

30.0%

100

50.0%

100

Witkowski

Produce Pilot

Sell to Competitor

Reject

1 Year

2 Years

Reject

1 Year

2 Years


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Value of Perfect Information

State of Nature Probabilities

Net Payout if Pilot is Produced

Net Payout if Sold to Competitor Optimal Decision

Reject 0.2 -100 100 Sell to Competitor 1 Year 0.3 50 100 Sell to Competitor 2 Years 0.5 150 100 Produce Pilot

We calculate the expected value with perfect information by summing up the probability-weighted best payoffs for each state of nature. For this example:

EVwPI 150*50.0100*30.0100*20.0

753020

125


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Perfect information (if it were available) would be worth up to 125 - 100 = 25 thousand dollars to Witkowski.

This is referred to as expected value of perfect information.


27

For a consulting fee of $2,500, the O’Donnell agency will review the plans for the comedy series and indicate the overall chance of a favorable network reaction.

O’Donnell Results I1 = Favorable I2 = Unfavorable

Reject 30.011 sIP 70.012 sIP

1 year 60.021 sIP 40.022 sIP

2 years 90.031 sIP 10.032 sIP


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U s i n g B a y e s ’ L a w , w e c a n u s e t h e s e c o n d i t i o n a l p r o b a b i l i t i e s t o c a l c u l a t e p o s t e r i o r p r o b a b i l i t i e s ( p r o b a b i l i t i e s f o r e a c h s t a t e o f n a t u r e g i v en e a c h p o s s i b l e o u t c o m e o f t h e O ’ D o n n e l l r e p o r t ) :

P r o b a b i l i t y c a l c u l a t i o n s : P r o b a b i l i t i e s

S ta te s P r i o r C o n d i ti o n a l J o i n t P o s te r i o r

js jsP jsIP 1 jjj sIPsPIsP 11 1

11 IP

IsPIsP

jj

R e j e c t 0 .2 0 0 .3 0 0 .0 6 0 .0 8 7 1 Y e a r 0 .3 0 0 .6 0 0 .1 8 0 .2 6 1 2 - Y e a r 0 .5 0 0 .9 0 0 .4 5 0 .6 5 2

T o ta l 69.01 IP

T h e t o t a l p r o b a b i l i t y o f a f a v o r a b l e O ’ D o n n e l l r e p o r t i s 6 9 % .


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Probabilities States Prior Conditional Joint Posterior

js jsP jsIP 2 jjj sIPsPIsP 22 2

22 IP

IsPIsP

jj

Reject 0.20 0.70 0.14 0.452 1 Year 0.30 0.40 0.12 0.387 2-Year 0.50 0.10 0.05 0.161

Total 31.02 IP

The total probability of an unfavorable O ’Donnell report is 31%.


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Now we can calculate an expected value for each decision alternative for each possible outcome of the O’Donnell project, and we can calculate an overall expected value.

A revised payoff table:

States of Nature

s1 = Reject s2 = 1 Year s3 = 2 Years

d1 = Produce Pilot -100 50 150 No Report d2 = Sell to Competitor 100 100 100

d1 = Produce Pilot -102.5 47.5 147.5 I1 = Favorable Report

d2 = Sell to Competitor 97.5 97.5 97.5

d1 = Produce Pilot -102.5 47.5 147.5

Get Report

I2 = Unfavorable Report d2 = Sell to Competitor 97.5 97.5 97.5


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8.7%

-102.5

TRUE Expected Value

0.0 99.7

26.1%

47.5

65.2%

147.5

0.7 Expected Value

0.0 99.7

8.7%

97.5


0.0 97.5

26.1%

97.5

65.2%

97.5


0.0 99.0

45.2%

-102.5


0.0 -4.1

38.7%

47.5

16.1%

147.5

0.3 Expected Value

0.0 97.5

45.2%

97.5

TRUE Expected Value

0.0 97.5

38.7%

97.5

16.1%

97.5

Expected Value

100.0

TRUE Expected Value

0.0 100.0

20.0%

-100.0


0.0 70.0

30.0%

50.0

50.0%

150.0

1.0 Expected Value

0.0 100.0

20.0%

100.0

TRUE Expected Value

0.0 100.0

30.0%

100.0

50.0%

100.0

Witkowski

Get O'Donnell Report

Do Not Get O'Donnell Report

Favorable

Unfavorable

Produce Pilot

Sell to Competitor

Reject

1 Year

2 Years

Reject

1 Year

2 Years

Produce Pilot

Sell to Competitor

Reject

1 Year

2 Years

Reject

1 Year

2 Years

Produce Pilot

Sell to Competitor

Reject

1 Year

2 Years

Reject

1 Year

2 Years


32

What should Witkowski’s strategy be? What is the expected value of this strategy?

The best thing to do is to forget about O’Donnell and sell the rights for $100,000.


33

What is the expected value of the O’Donnell agency’s sample information? Is the information worth the $2,500 fee? What is the efficiency of O’Donnell’s sample information?


34

I n th e ev en t o f a fav o rab le O ’D o n n el l rep o r t, th e exp ected v alu e o f p ro d u cin g th e p i lo t is:

11 IdEV 131312121111 IsPvIsPvIsPv

652.05.147261.05.47087.05.102

65.99

I n th e ev en t o f a fav o rab le O ’D o n n el l rep o r t, th e exp ected v alu e o f sel l in g to th e com p eti to r is:


652.05.97261.05.97087.05.97

50.97


35

In the event of an unfavorable O ’Donnell report, the expected value of producing the pilot is:


161.05.147387.05.47452.05.102

20.4

In the event of an unfavorable O ’Donnell report, the expected value of selling to the competitor is:


161.05.97387.05.97452.05.97

5.97


36

Decision Alternative Expected Value

Produce Pilot 99.65 OptimalFavorable O’Donnell Report

Sell to Competitor 97.50

Produce Pilot -4.20 Unfavorable O’Donnell Report

Sell to Competitor 97.50 Optimal

The overall expected value with sample information (EVwSI) is:

( ) ( ) ( ) ( )2211 IPIEVIPIEV + ( ) ( ) 98.9831.0*5.9769.0*65.99 =+=

(Note that we are assuming here that we will always adopt the optimal strategy in light of whatever information O’Donnell provides.)


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8.7%

-102.5

TRUE Expected Value

0.0 99.7

26.1%

47.5

65.2%

147.5

0.7 Expected Value

0.0 99.7

8.7%

97.5


0.0 97.5

26.1%

97.5

65.2%

97.5


0.0 99.0

45.2%

-102.5


0.0 -4.1

38.7%

47.5

16.1%

147.5

0.3 Expected Value

0.0 97.5

45.2%

97.5

TRUE Expected Value

0.0 97.5

38.7%

97.5

16.1%

97.5

Expected Value

100.0

TRUE Expected Value

0.0 100.0

20.0%

-100.0


0.0 70.0

30.0%

50.0

50.0%

150.0

1.0 Expected Value

0.0 100.0

20.0%

100.0

TRUE Expected Value

0.0 100.0

30.0%

100.0

50.0%

100.0

Witkowski

Get O'Donnell Report

Do Not Get O'Donnell Report

Favorable

Unfavorable

Produce Pilot

Sell to Competitor

Reject

1 Year

2 Years

Reject

1 Year

2 Years

Produce Pilot

Sell to Competitor

Reject

1 Year

2 Years

Reject

1 Year

2 Years

Produce Pilot

Sell to Competitor

Reject

1 Year

2 Years

Reject

1 Year

2 Years


38

Expected Value of Sample Information

The expected value of sample information is calculated using this formula:

EV SI = EV wSI - EV woSI

w here

EV SI = expected value of sample information

EV wSI = expected value w ith sample information about the states of nature

EV woSI = expected value w ithout sample information about the states of nature

In our example, the expected value of sample information is:

EV SI = EV wSI - EV woSI

10098.98

02.1


39

What is the expected value of the O’Donnell agency’s sample information? Is the information worth the $2,500 fee?

If we pay O’Donnell the $2,500 fee, our overall expected value drops by $1,020. This implies that the O’Donnell report is worth

We would be willing to pay up to (but no more than) $1,480 for the O’Donnell report.

(This is one way to address the question, “How much should Witkowski be prepared to pay for the research study?”)

020,1$500,2$ 480,1$


40

Efficiency of Sample Information

The efficiency of sample information is calculated using this formula:

In other words, the market research project gives us information with less than 6% of the utility of having perfect information.

E EVPIEVSI

000,25$480,1$

0592.0


41

Conclusions

• Don’t buy the O’Donnell report• Sell the script to the competitor• Earn $100,000


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SummaryDecision Analysis• Uncertain Future Events• Perfect Information• Partial Information

– The Return of Rev. Thomas Bayes

Session 6b. Decision Models -- Prof. Juran2 Overview Decision Analysis Uncertain Future Events...

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