session 6b

Session 6b

Decision Models -- Prof. Juran

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

– The Return of Rev. Thomas Bayes


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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.

56789

1 01 11 21 31 41 51 61 71 8

A B C D E F G H I J K

P u rc 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 i l 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

= SU 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 )

= SQ 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 rk s.


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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

Expe

cted

Rev

enue

Buy 100 Pairs

Buy 200 Pairs

Buy 300 Pairs

Buy 400 Pairs Buy 500 Pairs

Buy 600 Pairs


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6

12345

6789101112131415161718192021222324252627

A B C D E F G H I J K L M N O P$45 Cost of each pair of new tennis shoes$70 Selling price of each pair of shoes$35 Closeout sale price of each leftover pair

Purchased 1 2 3 4 5 61 $2,500 $2,500 $2,500 $2,500 $2,500 $2,5002 $1,500 $5,000 $5,000 $5,000 $5,000 $5,0003 $500 $4,000 $7,500 $7,500 $7,500 $7,5004 ($500) $3,000 $6,500 $10,000 $10,000 $10,0005 ($1,500) $2,000 $5,500 $9,000 $12,500 $12,5006 ($2,500) $1,000 $4,500 $8,000 $11,500 $15,000

Probability 0.05 0.15 0.25 0.3 0.15 0.1

Alternative 10

0 01

0Alternative 2

00 0

Payoff TableConsumer Demand


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Ctrl + Shift + T to modify a selected node


8


9

181920212223242526272829303132333435363738394041

I J K L M N O

Buy 1000

0 0

Buy 2000

0 0

Buy 3001 0

0 0 0

Buy 4000

0 0

Buy 5000

0 0


10

1718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

H I J K L M N O P Q R0.2Outcome 6

00 0

0.2Outcome 7

00 0

0.2Buy 100 Outcome 8

00 0 0 0

0.2Outcome 9

00 0

0.2Outcome 10

00 0

Buy 2001 0

0 0 0

Buy 3000

0 0

Buy 4000

0 0

Buy 5000

0 0

Type in probabilities in the cells above “outcome” labels


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Type in payoffs in the cells below “outcome” labels, and ONLY below outcome labels

117118119120121122123124125126127128129130131132133134135136137138139140

K L M N O P Q R0.05Demand 100

-1500-1500 -1500

0.2Demand 200

20002000 2000

0.3Buy 500 Demand 300

55000 6550 5500 5500

0.3Demand 400

90009000 9000

0.15Demand 500

1250012500 12500

117118119120121122123124125126127128129130131132133134135136137138139140

K L M N O P Q R0.05Demand 100

00 0

0.2Demand 200

00 0

0.3Buy 500 Demand 300

00 0 0 0

0.3Demand 400

00 0

0.15Demand 500

00 0


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TreePlan JargonOutcomes (states of nature)

Event Node (a.k.a. chance or probability node; tree splits into multiple outcomes)

Decision Node (a.k.a. choice node; tree splits into multiple decision alternatives)

Terminal Node (the end of a branch; a specific combination of decisions and events)

TreePlan-185-Guide.pdf

(pretty good documentation)

Ctrl+Shift+T (context-specific TreePlan menu)


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TreePlan IssuesLimited to 5 branches from any nodeRuns slowly; Freezes frequentlyGraphics don’t appear until you click near themCopy and Paste Subtree doesn’t always work• Copy from Event Nodes only?• Paste grayed out?• Clipboard error messagesWhen Paste Subtree does work, can take a LONG time


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Example 2: TV ProductionWitkowski 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 Value0 70

30.0%50

50.0%150

Expected Value100

20.0%100

TRUE Expected Value0 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.


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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 ti e s S ta te s P r i o r C o n d i t i 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 j

j

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 j

j

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 Value0.0 99.7

26.1%47.5

65.2%147.5

0.7 Expected Value0.0 99.7

8.7%97.5

FALSE Expected Value0.0 97.5

26.1%97.5

65.2%97.5


45.2%-102.5

FALSE Expected Value0.0 -4.1

38.7%47.5

16.1%147.5


45.2%97.5


38.7%97.5

16.1%97.5

Expected Value100.0TRUE Expected Value

0.0 100.020.0%-100.0


30.0%50.0

50.0%150.0


20.0%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


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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.


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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?


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I n th e ev en t o f a fav o r ab le O ’D o n n el l rep o r t, th e exp ected v alu e o f p r o 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 r ab 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:



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


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



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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


26.1%47.5

65.2%147.5


8.7%97.5


26.1%97.5

65.2%97.5


45.2%-102.5

FALSE Expected Value0.0 -4.1

38.7%47.5

16.1%147.5


45.2%97.5


38.7%97.5

16.1%97.5

Expected Value100.0TRUE Expected Value

0.0 100.020.0%-100.0


30.0%50.0

50.0%150.0


20.0%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


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Expected Value of Sample Information

The expected value of sample information is calculated using this formula: EV SI = EVwSI - 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: EVSI = EV wSI - EV woSI

10098.98 02.1


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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$


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Efficiency of Sample InformationThe 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


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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

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