Engineering Risk Benefit Analysis
1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82,
ESD.72, ESD.721
DA 1. The Multistage Decision Model
George E. Apostolakis
Massachusetts Institute of Technology
Spring 2007
DA 1. The Multistage Decision Model
1
Why decision analysis?
A structured way for ranking decision options by:
1. Enumerating the immediate and later choices
available to the DM
2. Characterizing the relevant uncertainties
3. Quantifying the relative desirability of outcomes
4. Providing rules for ranking the decision options,
thus helping the DM to select the “best” one.
DA 1. The Multistage Decision Model
2
Value of formal analysis
• Provides a systematic way to process large amounts
of information
• Decision making process is explicit and enhances
communication
• Provides formal rules for quantifying preferences
DA 1. The Multistage Decision Model
3
Limitations of DA
• The theory is for an individual decision maker.
This reduces considerably its applicability in
practice. (But, great normative tool.) In most cases
there is no satisfactory way to combine the utility
function of several people
• As with all formal analysis, the results are no better
than the quality of the model and its supporting
assessments
• The required inputs may not be easily obtainable
DA 1. The Multistage Decision Model
4
Manufacturing Example
• Decision: To continue producing old product (O)
or convert to a new product (N).
The payoffs depend on the market conditions:
s: strong market for the new product
m: mild market for the new product
w: weak market for the new product
DA 1. The Multistage Decision Model
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Manufacturing Example Payoffs
•
Earnings (payoffs):
L1: $150,000, old product,
P[L1/O] = 1.0
L2: $300,000, new product and the market is
strong,
P[s] = P[L2/N] = 0.3
L3: $100,000, new product and the market is mild,
P[m] = P[L3/N] = 0.5
L4: -$100,000, new product and the market is weak,
P[w] = P[L4/N] = 0.2
DA 1. The Multistage Decision Model
6
Building the decision tree
Decision
Options
Payoffs
L2 $300K
N
Payoff
depends on
market
L3 $100K
L4 -$100K
O
L1 $150K
DA 1. The Multistage Decision Model
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Decision Tree
Decision
Options
States of Nature
Payoffs
L2 $300K
P[s]= 0.3
P[m]= 0.5
L3 $100K
N
P[w]=0.2
O
L4 -$100K
P[s ∪ m ∪ w ] = 1
L1 $150K
DA 1. The Multistage Decision Model
8
Non-Probabilistic Decision Rules
• Maximin Rule:
N:
Choose option with the largest smallest
payoff (risk averse DM).
-$100
O:
$150
Choose O
• Maximax Rule:
N:
Choose option with the largest payoff
(risk taker).
$300
O:
$150
Choose N
DA 1. The Multistage Decision Model
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Probabilistic Decision Rule
• The maximin and maximax rules are incomplete
because they ignore uncertainties.
• We include probabilities by taking expected values
of the payoffs (slide 31, RPRA 2).
• Decision Rule: Maximize the expected monetary
value (EMV) of the earnings (payoffs).
• In the decision tree, work from right to left and
compute expectations.
DA 1. The Multistage Decision Model
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Calculation of the EMV
EMV[N] = 0.3x300 + 0.5x100 + 0.2x(-100) = $120K
EMV[O] = 1.0x150 = $150K
Option O has the largest EMV, therefore it should
be chosen.
DA 1. The Multistage Decision Model
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Calculation of the EMV (cont’d)
L2 $300
s, $90
m, $50
L3 $100
EMV[N]=$120
w, -$20
L4 -$100
EMV[O]=$150
$150
L1 $150
Best Decision: O
DA 1. The Multistage Decision Model
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A New Decision
• The DM considers the possibility of commissioning
a survey to be able to better judge the future
market.
• The survey costs $20,000.
• There are now two decisions (multistage model):
¾ The initial decision of whether to buy the survey
¾ The terminal decision of whether to market the new
product.
DA 1. The Multistage Decision Model
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The survey results can be:
Strong
P(s/L2) = 0.8
P(s/L3) = 0.2
P(s/L4) = 0.0
Mild
P(m/L2) = 0.2
P(m/L3) = 0.6
P(m/L4) = 0.3
Weak
P(w/L2) = 0.0
1.0
P(w/L3) = 0.2
1.0
P(w/L4) = 0.7
1.0
DA 1. The Multistage Decision Model
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The new decision tree
*Entries in earnings column
do not yet take account
of the cost of the survey.
Purchase
C
survey
Survey
response
"strong"
Survey
response
"mild"
Make old
product
Product earns*
C
300,000
D
Make new C
product
Make old
product
150,000
C
D
100,000
-100,000
150,000
300,000
Make new
C
product
100,000
-100,000
D
Survey
response
"weak"
Do not
purchase
survey
C
Make old
product
C
D
No new
information D
is obtained
150,000
300,000
Make new
C
product
Make old
product
100,000
-100,000
C
150,000
300,000
Make new
C
product
100,000
-100,000
DA 1. The Multistage Decision Model
Figure by MIT OCW.
15
New inputs
• The earnings must be reduced by the survey cost of
$20K: L1 = $130K, L2 = $280K, L3 = $80K,
L4 = -$120K
• The probabilities of the states of nature (the
probabilities of earnings) must also be updated to
reflect the survey findings.
DA 1. The Multistage Decision Model
16
Bayes’ Theorem (Slide 16, RPRA 2)
• The mutually exclusive and exhaustive states are:
L2, L3, and L4
• Evidence: “Survey result is strong”
P( L j / s ) =
P (s / L j )P ( L j )
4
∑ P(s / L j )P( L j )
2
j = 2, 3,4
DA 1. The Multistage Decision Model
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Calculations for “survey result is s”
Payoff Prior Likelihood
Prob.
L2
L3
L4
0.3
0.5
0.2
1.0
P(s/ L2)=0.8
P(s/ L3)=0.2
P(s/ L4)=0.0
Product
0.24
0.10
0.00
0.34
DA 1. The Multistage Decision Model
Posterior
Probability
P(L2/s)=0.706
P(L3/s)=0.294
P(L4/s)=0.000
1.000
18
Results for m and w
P(L2/m) = 0.143
P(L2/w) = 0.000
P(L3/m) = 0.714
P(L3/w) = 0.417
P(L4/m) = 0.143
1.000
P(L4/w) = 0.583
1.000
DA 1. The Multistage Decision Model
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The updated decision tree
s
O
C
1
L1
N
L2
C
.706
.294
0
O
C
1
L1
N
C
.143
.714
.143
L2
L3
O
C
1
L1
N
C
0
.417
.583
L2
L3
O
C
1
L1
150,000
L2
C
.3
.5
.2
300,000
100,000
-100,000
D
.34
S
C
.42
m
D
.24
D
w
S
C
1
D
D
N
130,000
280,000
L3
80,000
L4 -120,000
130,000
280,000
80,000
L4 -120,000
130,000
280,000
80,000
L4 -120,000
L3
L4
Figure by MIT OCW.
DA 1. The Multistage Decision Model
20
Optimal terminal decisions
1. Solve “backwards in time.”
2. Determine the best solution at every terminal
node, conditional on the DM’s being there.
3. Find the EMV for each terminal node and the
decision option that maximizes the EMV.
4. An arrow indicates the best decision for each
terminal node.
DA 1. The Multistage Decision Model
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Decision tree solution
130,000
221,300
s
D
O
C
221,300
N
C
.34
130,000
C
. 42
m
D
O
. 24
C
80,000
N
C
130,000
130,000
D
w
D
O
C
-36,600
N
C
150,000
150,000
S
L1
130,000
.706
.294
0
L2
L3
L4
280,000
80,000
-120,000
1
L1
130,000
.143
.714
.143
L2
L3
1
L1
0
.417
.583
L2
L3
1
L1
.3
.5
.2
L2
130,000
161,008
S
1
C
1
O
150,000
D
C
120,000
N
C
280,000
80,000
L4 -120,000
130,000
280,000
80,000
L4 -120,000
150,000
300,000
L3 100,000
L4 -100,000
Figure by MIT OCW.
DA 1. The Multistage Decision Model
22
Best decision
1. Assume that the DM makes the best decision at
each terminal node, if it is reached.
2. Find the EMV for the initial node.
3. Buy survey:
EMV = $161,008
Do not buy survey:
EMV = $150,000
4. Best initial decision:
Buy survey
DA 1. The Multistage Decision Model
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Best terminal decision
If survey result is “strong” ⇒ Market new product
(EMV = $221,200)
If survey result is “mild”
⇒ Market old product
(EMV = $130,000)
If survey result is “weak” ⇒ Market old product
(EMV = $130,000)
DA 1. The Multistage Decision Model
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General form of a decision tree
G1
P(H1 |C1 )
H1
C1
P(C 1 )
D1
D2
P(C 2 )
P(C n )
C2
G2
Gk
Cn
D3
Dm
GENERAL FORM OF A DECISION TREE
DA 1. The Multistage Decision Model
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The multistage decision model
1.
Each stage consists of a decision node followed by a
chance node for each of the decision options available in
this stage.
2.
The DM must select one of the initial acts Aj.
3.
The Aj may be viewed as “learning experiments”
providing, at specified costs, opportunities for obtaining
partial or complete information about present
uncertainties.
4.
Following the probabilistic results a1, a2, …, of the initial
decision, the DM must select the next decision B1, B2, …
DA 1. The Multistage Decision Model
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The multistage decision model (cont’d)
5.
Finally, a terminal chance node, b1, b2,…, occurs.
6.
As a result of the initial decision Aw, its chance outcome
ax, the terminal decision By, and its chance outcome bz, the
total consequence C(Aw, ax, By, bz) is obtained.
7.
The solution proceeds sequentially backwards in time, i.e.,
we first identify the best decision at each terminal node,
then the best decision one stage earlier assuming that,
whatever chance event results from this stage, it will be
followed by the best of the available terminal decisions.
DA 1. The Multistage Decision Model
27