session 6b
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Session 6b. Overview. Decision Analysis Uncertain Future Events Perfect Information Partial Information The Return of Rev. Thomas Bayes. - PowerPoint PPT PresentationTRANSCRIPT
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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Decision Models -- Prof. Juran
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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
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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
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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
FALSE Expected Value0.0 99.0
45.2%-102.5
FALSE Expected Value0.0 -4.1
38.7%47.5
16.1%147.5
0.3 Expected Value0.0 97.5
45.2%97.5
TRUE Expected Value0.0 97.5
38.7%97.5
16.1%97.5
Expected Value100.0TRUE Expected Value
0.0 100.020.0%-100.0
FALSE Expected Value0.0 70.0
30.0%50.0
50.0%150.0
1.0 Expected Value0.0 100.0
20.0%100.0
TRUE Expected Value0.0 100.0
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:
12 IdEV 132312221121 IsPvIsPvIsPv 652.05.97261.05.97087.05.97 50.97
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In the event of an unfavorable O ’Donnell report, the expected value of producing the pilot is:
21 IdEV 231322122111 IsPvIsPvIsPv 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:
22 IdEV 232322222121 IsPvIsPvIsPv 161.05.97387.05.97452.05.97 5.97
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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
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
FALSE Expected Value0.0 99.0
45.2%-102.5
FALSE Expected Value0.0 -4.1
38.7%47.5
16.1%147.5
0.3 Expected Value0.0 97.5
45.2%97.5
TRUE Expected Value0.0 97.5
38.7%97.5
16.1%97.5
Expected Value100.0TRUE Expected Value
0.0 100.020.0%-100.0
FALSE Expected Value0.0 70.0
30.0%50.0
50.0%150.0
1.0 Expected Value0.0 100.0
20.0%100.0
TRUE Expected Value0.0 100.0
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