INFO 631
Prof. Glenn Booker
Week 9 – Chapters 24-26
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Decisions Under Risk
Ch. 24
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Decisions Under Risk
Outline
• Introducing decisions under risk
• Different techniques
– Expected value decision making
– Expectation variance
– Monte Carlo analysis
– Decision trees
– Expected value of perfect information
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Decisions Under Risk
• When you know the probabilities of the
different outcomes and will incorporate
them
– Expected value decision making
– Expectation variance
– Monte Carlo analysis
– Decision trees
– Expected value of perfect information
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Expected Value Decision Making
• The value of an alternative with multiple
outcomes can be thought of as the
average of the random individual
outcomes that would occur if that
alternative were repeated a large number
of times
– Can use PW(i), FW(i), or AE(i)
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Expected Value of a Single Alternative
• Denali project at Mountain Systems
PW(MARR)
Probability
Least
favorable Fair
-$1234
$5678
0.20
0.65
Most
favorable
$9012
0.15
• Imagine 1000 parallel universes where the Denali project
could be run at the same time
– Should expect most favorable outcome would happen in 15% or
150 of those universes
– Fair outcome would happen in 650
– Least favorable outcome would happen in 200
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Expected Value of a Single Alternative
• Total PW(i) income generated
200 * -$1234 =
-246,800
650 * $5678 = $3,690,700
150 * $9012 = $1,351,800
$4,795,700
• Average PW(i) income in each universe
$4,795,700 / 1000 = $4795.70
• Notice
(0.20 * -$1234) + (0.65 * $5678) + (0.15 * $9012) = $4795.70
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Expected Value of a Single Alternative
• General formula
n
Expect edValue   Value i  P r obabilityi 
i 1
• Can be used to help decide between
multiple alternatives
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Expected Value of Multiple Alternatives
• Same probability
• Several projects at Mountain Systems
Alternative
Denali
Shasta
Washington
Least
favorable Fair
20%
65%
-$1234
$5678
-2101
6601
-3724
4104
Most
favorable
15%
$9012
9282
9804
• Expected values
Denali
Shasta
Washington
(0.20 * -$1234) + (0.65 * $5678) + (0.15 * $9012) = $4795.70
(0.20 * -$1201) + (0.65 * $6601) + (0.15 * $9282) = $5262.75
(0.20 * -$3724) + (0.65 * $4104) + (0.15 * $9804) = $3393.40
– Choose Shasta, it has the highest expected value
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Expectation Variance
• What if probabilities were different for each alternative?
Lassen
Outcome
Least favorable
Nominal
Most favorable
Probability
45%
10%
45%
Moana Loa
AE(i)
-$3494
728
4811
Outcome
Least favorable
Low nominal
High nominal
Most favorable
Expected value = $665
Probability
AE(i)
10%
-$200
20%
108
30%
378
40%
877
Expected value = $466
• Comparing projects
– Lassen has higher expected value but win big-lose big
– Moana Loa has lower expected value but more probability of
profit
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Monte Carlo Analysis
• Randomly generate combinations of input values and
look at distribution of outcomes
– Named after gambling resort in Monaco
• Use [a variant of] Zymurgenics project (different data)
Initial investment
Operating & maintenance
Development staff cost / month
Development project duration
Income / month
Least favorable
Fair
estimate
estimate
$500,000
$400,000
$1500
$1000
$49,000
$35,000
15 months
10 months
$24,000
$40,000
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Most favorable
estimate
$360,000
$800
$24,500
7 months
$56,000
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Monte Carlo Analysis
• Simulation run results
Income range
-$75,000 to -$50,001
-$50,000 to -$25,001
-$25,000 to -$1
$0 to $24,999
$25,000 to $49,999
$50,000 to $74,999
$75,000 to $99,999
$100,000 to $124,999
$125,000 to $149,999
$150,000 to $174,999
$175,000 to $199,999
$200,000 to $224,999
$225,000 to $249,999
$250,000 to $274,999
Number of occurrences
3
32
76
258
655
921
1044
865
586
329
159
53
17
5
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Monte Carlo Analysis
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Decision Trees
• Maps out possible results when there are sequences of
decisions and future random events
– Useful when decisions can be made in stages
• Basic Elements
– Decision nodes – points in time where a decision maker makes a
decision (square)
– Chance nodes – points in time where the outcome is outside the
control of the decision maker (circles)
– Node sequencing
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Sample Decision Tree
P eriod 2
6 Years
P eriod 1
2 Years
$2848/y r
$1437/y r
$1200/y r
$800/y r
$835/y r
Do A
-$6000
$3615/y r
Do C
$851/y r
-$5490
-$4000
Do B
$1526/y r
$1100/y r Do D
$1000/y r
$1037/y r
$1214/y r
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Decision Tree Analysis, Part 1
1. Add the financial consequences for each
arc (PW(i), FW(i), or AE(i))
– Properly adjust for time periods as required
2. Sum financial consequences from the
root node to all leaf nodes
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Sample Decision Tree
$8917
$4500
$2000
$4917
$500
$1335
$2615
Do A
-$6000
$11,315
Do C
$2665
-$4300
-$2050
$4850
-$3800
-$4000
Do B
$4778
$1835
$2613
Do D
$1668
$3245
$3800
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$1080
$1467
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Decision Tree Analysis, Part 2
3. Write probabilities for each arc out of each
chance node
– Probabilities out of a chance node must = 1.0
4. Roll back values from leaf nodes to root
– If node is chance node, calculate expected
value at that node based on values on all nodes
to its right
– If node is decision node, select the maximum
profit (or minimum cost) from nodes to its right
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Sample Decision Tree
$4917
3/5
$3150
2/5
$500
5/8
$1200
3/8
Do A
-$2050
$4850
3/5
$1800
$1390
Do C
2/5
-$3800
$2000
Do B
$2613
5/8
Do D
3/5
$1800
3/8
$2000
2/5
$1080
$1467
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Expected Value of Perfect Information
• Value at root node is expected value of decision tree based on
current information
– Current information is known to be imperfect
• Reasonable follow-on question
“Would there be any value in taking actions that would reduce
the probability of ending up in an undesirable future state?”
– Research, experimentation, prototyping, …
– Might even be able to eliminate one or more paths through the tree
because you may discover them to be impossible
• Analyzed decision tree provides information that will help answer
that question
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Expected Value of Perfect Information
• If we had a crystal ball and knew outcomes for chance nodes, we
could find which path would be best
– Finding best path can be repeated for all possible combinations of
random variables
• Probabilities for random variables are known
– Can calculate probability for each combination of outcomes
• For each combination of outcomes, multiply its best value by
probability of that combination
• Sum the results of (value * probability) for all combinations of
outcomes
– Sum is expected value given perfect information
– Difference between sum and expected value given current information
is expected value of perfect information
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Expected Value of Perfect Information
n
EVP I   Best Valuei  P r obi   EVofDecisionTree
i 1
• EVPI is upper limit on how much to spend
to gain further knowledge
– Probably impossible to actually get perfect
information, organization should plan on
spending less
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Key Points
•
Value of an alternative with multiple outcomes is the average of the random individual
outcomes that would occur if that alternative were repeated a large number of times
(expected value)
–
•
With expectation variance, differing probabilities could influence the decision
–
•
Repeated many times and statistical distribution of outcomes is analyzed
Decision trees map out possible results when there are sequences of decisions
together with a set of future random events that have known probabilities
–
•
Alternative with lower expected value might be a better choice if it also has a much lower
probability of a negative outcome
Monte Carlo analysis generates random combinations of the input variables and
calculates results under those conditions
–
•
The alternative with the highest expected value is best
Useful with many possible future states and decisions can be made in stages
The Expected value of perfect information provides answer to, “Would there be any
value in taking actions that would reduce the probability of ending up in an
undesirable future state?”
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Decisions Under Uncertainty
Ch. 25
Slides adapted from Steve Tockey – Return on Software
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Decisions Under Uncertainty
Outline
• Introducing decisions under uncertainty
• Different Techniques
– Payoff matrix
– Laplace Rule
– Maximin Rule
– Maximax Rule
– Hurwicz Rule
– Minimax Regret Rule
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Decisions Under Uncertainty
• Used when impossible to assign probabilities to
outcomes
– Can also be used when you don’t want to put probabilities on
outcomes, e.g., safety-critical software system where a failure
could threaten human life
• People may not react well to an assigned probability of
fatality
• If probabilities can be assigned, Decision Making under
Risk should be used
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Payoff Matrix
• Shows all possible outcomes to consider
– One axis lists mutually exclusive alternatives
– Other axis lists different states of nature
• Each state of nature is a future outcome the decision maker
doesn’t have control over
– Cells have PW(i), FW(i), AE(i), …
Alternative
A1
A2
A3
A4
A5
State1
-4010
948
-2005
0
1005
State2
1002
1101
1516
2020
3014
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State3
2001
4021
6004
5104
2008
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Reduced Payoff Matrix
• One alternative may be “dominated” by another
– Another alternative has equal or better payoff under every state
of nature
• Reduced payoff matrix has no dominated alternatives
– Less work if dominated alternatives are removed
Alternative
A1
A2
A3
A4
A5
State1
-4010
948
-2005
0
1005
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State2
1002
1101
1516
2020
3014
State3
2001
4021
6004
5104
2008
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Laplace Rule
• Assumes each state of nature is equally
likely
– Sometimes called “principle of insufficient
reason”
• Calculate average payoff for each
alternative across all states of nature
– Same as expected value analysis for multiple
alternatives with equal probabilities
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Laplace Rule
• Example
Alternative
A2
A3
A4
A5
State1
948
-2005
0
1005
State2
1101
1516
2020
3014
State3
4021
6004
5104
2008
Average payoff
1933
1838
2374
2009
– Alternative A4 is chosen; the highest payoff
always wins!
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Maximin Rule
• Assumes worst state of nature will happen
– Most pessimistic technique
– Pick alternative that has best payoff from all
worst payoffs
• Formula


i  j Pij 


max min
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Maximin Rule
• Example
Alternative
A2
A3
A4
A5
State1
948
-2005
0
1005
State2
1101
1516
2020
3014
State3
4021
6004
5104
2008
Worst payoff
948
-2005
0
1005
– Alternative A5 is chosen
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Maximax Rule
• Assumes best state of nature will happen
– Most optimistic technique
– Pick alternative that has best payoff from all
best payoffs
• Formula


i  j Pij 


max max
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Maximax Rule
• Example
Alternative
A2
A3
A4
A5
State1
948
-2005
0
1005
State2
1101
1516
2020
3014
State3
4021
6004
5104
2008
Best payoff
4021
6004
5104
3014
– Alternative A3 is chosen
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Hurwicz Rule
• Assumes that without guidance people will
tend to focus on extremes
– Blends optimism and pessimism using a
selected ratio
• Index of optimism, a, between 0 and 1
 a = 0.2 means more pessimism than optimism
 a = 0.1 means more pessimism than a = 0.2
 a = 0.85 means lots of optimism but a small
amount of pessimism (15%) remains
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Hurwicz Rule
• Formula
 max 
min  
i a  j Pij   1  a  j Pij  



 
max
• Example
 a = 0.2
Alternative
A2
A3
A4
A5
State1
948
-2005
0
1005
State2
1101
1516
2020
3014
State3
4021
6004
5104
2008
Blended payoff
(0.2 * 4021) + (0.8 * 948) = 1563
(0.2 * 6004) + (0.8 * -2005) = -403
(0.2 * 5104) + (0.8 * 0) = 1021
(0.2 * 3014) + (0.8 * 1005) = 1407
– Alternative A2 is chosen
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Hurwicz Rule
A3
6000
6000
A4
A2
4000
4000
A5
2000
2000
0
0
0.5
-2000
-2000
.25
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Minimax Regret Rule
• Minimize regret you would have if you chose wrong
alternative under each state of nature
– If you selected A1 and state of nature happened where A1 had
the best payoff then you would have no regrets
– If you selected A1 and state of nature happened where another
alternative was better, you can quantify regret as difference
between payoff you chose and best payoff under that state of
nature
• Regret matrix
– Need to calculate
– Difference between payoff you chose and best payoff under that
state of nature
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Minimax Regret Rule –
Calculate Regret matrix
• Regret matrix
– Difference between payoff you chose and best payoff under that
state of nature
• For State 1 – A2
o 1005 – 948 = 57
• For State 1 – A3
o 1005 – (-2005) = 3010
o Etc.
o NOTE: use numbers from original matrix
Alternative
A2
A3
A4
A5
State1
57
3010
1005
0
State2
2003
1498
994
0
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State3
1983
0
900
3966
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Minimax Regret Rule
• Choose alternative with smallest maximum regret
Alternative
A2
A3
A4
A5
State1
57
3010
1005
0
State2
2003
1498
994
0
State3
1983
0
900
3966
Maximum regret
2003
3010
1005
3996
– Alternative A4 is chosen
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Summary of Uncertainty Rules
Decision rule
Laplace
Maximin
Maximax
Hurwicz (a=0.2)
Minimax regret
Alternative selected
A4
A5
A3
A2
A4
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Optimism or pessimism
Neither
Pessimism
Optimism
Blend
Pessimism
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Key Points
•
•
•
Uncertainty techniques used when impossible, or impractical, to assign
probabilities to outcomes
Payoff matrix shows all possible outcomes to consider
Laplace rule assumes each state of nature is equally likely
– Essentially expected value with equal probabilities
•
Maximin rule is most pessimistic
– Pick alternative with best payoff from all worst payoffs
•
Maximax rule is most optimistic
– Pick alternative with best payoff from all best payoffs
•
Hurwicz Rule assumes that without guidance people will tend to focus on
the extremes
– Blend optimism and pessimism using selected ratio
•
Minimax Regret rule minimizes regret you would have if you chose the
wrong alternative under each state of nature
– Choose alternative with smallest maximum regret
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Multiple Attribute Decisions
Ch. 26
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Multiple Attribute Decisions
Outline
•
•
•
•
•
•
•
Introducing multiple attribute decisions
Case study: Fly-by-Night Air
Different kinds of “value”
Choosing attributes
Measurement scales
Non-compensatory techniques
Compensatory techniques
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Introducing
Multiple Attribute Decisions
• Previous chapters explained how to make decisions
using a single criterion, money
– Alternative with best PW(i), AE(i), incremental IRR, incremental
benefit-cost ratio, etc. is selected
• Aside from technical feasibility, money is almost always
the most important decision criterion
– But not the only one
– Often, other criteria (“attributes”) must be considered and can’t
be cast in terms of money
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Case Study: Fly-by-Night (FBN) Airlines
• 10-year old regional airline with above average growth
• Moving into nationwide market as no-frills carrier
• As part of strategic planning, IT department charged with
examining airline reservations systems
– 10 year planning horizon, effective income tax rate=37%,
after-tax MARR=15%
• Research has identified five technically-viable
alternatives
–
–
–
–
–
–
Keep existing software
Buy Jupiter commercial system
Buy Sword commercial system
Buy Guppy commercial system
Develop new software in-house
Develop new software offshore
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Different Kinds of “Value”
• Decision process is all about maximizing value
– Choose from available alternatives the one that
maximizes value
• When value is expressed as money, decision
process may be complex but is straightforward
– Money isn’t the only kind of value
– Money is really only a way to quantify value
• Two kinds of value
– Use-value - the ability to get things done, the
properties of the object that cause it to perform
– Esteem value - the properties that make it desirable
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Choosing Attributes
• Decisions should be based on appropriate attributes
– Each attribute should capture a unique dimension of decision
– Set of attributes should cover important aspects of decision
– Differences in attribute values should be meaningful in
distinguishing among alternatives
– Each attribute should distinguish at least two alternatives
• Selection of attributes may be subjective
– Too many attributes is unwieldy
– Too few attributes gives poor differentiation
– Potential for better decisions needs to be balanced with extra
effort of more attributes
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FBN Air: Decision Attributes
• Total cost of ownership
• In-service availability
• Liffey performance index
– From Liffey Consultancy, Ltd in Dublin,
Ireland
• Alignment with existing business
processes
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Measurement Scales
• Each alternative will be evaluated on each attribute
• Many ways to measure things
– In fact, different “classes” of measurements
– Within a class, some manipulations make sense and others don’t
• So it’s important for you to know what the different classes of
measurements are, how to recognize them, and what can
and can’t be done with them.
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Measurement Scales
Scale type
Description
Example
Nominal
Two things are assigned the
same symbol if they have the
same value
Ordinal
The order of the symbols reflects
an order defined on the attribute
Letter grades in school
(A, B, C, ...)
Interval
Differences between the numbers
reflect differences in the attribute
Temperature in degrees
Fahrenheit or Celsius,
Calendar date
=, <>,
<, >, <=, =>
=, <>,
<, >, <=, =>,
+, -
Ratio
Differences and ratios between the
numbers reflect differences and
ratios of the attribute
Length in centimeters,
Duration in seconds,
Temperature in Kelvin
=, <>,
<, >, <=, =>,
+, -, *, /
House style (Colonial,
Contemporary, Ranch,
Craftsman, Bungalow, …)
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Operations
=, <>
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FBN Air: Evaluation and Attribute Scales
Alternative
Existing
Jupiter
Sword
Guppy
New in-house
New off-shore
Cost
PW(i)
-$1.8M
-$15.4M
-$21.6M
-$16.7M
-$30.3M
-$17.5M
Availability
Months
3
6
5
8
14
18
Attribute
Cost
Availability
Liffey index
Alignment
Liffey index
[65..135]
99
115
128
105
105
105
Alignment
[Ex, Vg,Ok,Pr, Vpr]
Excellent
Poor
Ok
Very poor
Excellent
Very good
Scale
Ratio
Ratio
Interval
Ordinal
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Dimensionality of Decision Techniques
• Two families of decision techniques
– Differ in how attributes used
• Non-compensatory, or fully dimensioned, techniques
– Each attribute treated as separate entity
– No tradeoffs among attributes
• Compensatory, or single-dimensioned, techniques
– Collapse attributes onto single figure of merit
– Lower score in one attribute can be compensated by—or traded
off against—higher score in others
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Non-compensatory Decision Techniques
• Three will be described
– Dominance
– Satisficing
– Lexicography
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Dominance
• Compare each pair of alternatives on attribute-by-attribute basis
– Look for one alternative to be at least as good in every attribute and
better in one or more
• When found, no problem deciding
– One alternative is clearly superior to the other, inferior can be discarded
• May not lead to selecting one single alternative
– Good for filtering alternatives and reducing work using other techniques
• In FBN Air, Jupiter dominates Guppy
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Satisficing
• Sometimes called “method of feasible
ranges”
– Establish acceptable ranges of attribute
values
– Alternatives with any attributes outside
acceptable range are discarded
• May not lead to selecting one single
alternative
– Good for filtering alternatives and reducing
work using other techniques
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Satisficing
• Can lead to selecting one alternative when used with an
iterative propose-then-evaluate process
Repeat
Propose a new solution
Evaluate that solution against the decision attributes
Until the solution is within the acceptable range for all decision attributes
Note: Stops when 1st acceptable solution is proposed
• Iterative version is appropriate when satisfactory
performance, rather than optimal performance, is good
enough
– If optimal performance needed, always identify several
alternatives that meet satisficing criteria then do further decision
analysis with one of other techniques
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Lexicography
• Two previous techniques assume attributes have equal importance
– If one attribute is far more important than others, final choice could be
made on that one attribute alone
• If alternatives have identical values for most-important attribute, use
next-most-important attribute to break tie
– If still tied, compare next most important attribute, …
– Continue until a single alternative chosen or all alternatives evaluated
• FBN Air
– Alignment might be #1, eliminates all but Existing and In-house
– Cost might be #2, eliminates in-house
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Compensatory Decision Techniques
• Attribute values converted into common
“figure of merit”
– Units for common scale are usually arbitrary
– If common scale is at least interval scale then
scores can be compared meaningfully
• Two will be presented
– Nondimensional scaling
– Additive Weighting
– Analytical Hierarchy Process (see text)
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Non-Dimensional Scaling
•
Convert attribute values into common scale so they can be added together
to make composite score for each alternative
– Alternative with best composite score is selected
– All attributes are defined to have equal importance
•
Common scale needs same range for all attributes
– Must also follow same trend on desirability; most-preferred value needs to
always be biggest or always be smallest common scale value
•
Formula for converting attributes, as long as interval or ratio-scaled, into the
common scale
Rating  Range *
WorstValue  ValueToMak eDimension less
WorstValue  BestValue
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FBN Air: Scaled Attributes
Alternative
Existing
Jupiter
Sword
Guppy
New in-house
New off-shore
Cost
[0..50]
50.0
26.1
15.3
23.9
0.0
22.5
Availability
[0..50]
50.0
40.0
43.3
33.3
13.3
0.0
Liffey index
[0..50]
0.0
27.6
50.0
10.3
10.3
10.3
Total
100.0
93.7
108.6
67.5
23.6
32.8
Note: Let’s entirely arbitrarily chose the common scale to be 0..50. In FBN’s
case,
• lower cost is better so lowest cost alternative highest common rating
• higher Liffey Index (LI) is better so the highest LI alternative highest common
rating.
• Best = Sword
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Non-Dimensional Scaling and Ordinal
Attributes
• When decision includes ordinal scaled attributes, you will need to:
– Ignore ordinal-scaled attributes
– Refine ordinal-scaled attributes to use interval or ratio scales and
include them in nondimensional scaling
– Do nondimensional scaling for all interval- and ratio-scaled attributes
then finish using a non-compensatory technique
Alternative
Existing
Jupiter
Sword
Guppy
New in-house
New off-shore
Total
100.0
93.7
108.6
67.5
23.6
32.8
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Alignment
Excellent
Poor
Ok
Very poor
Excellent
Very good
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Additive Weighting
• Identical to non-dimensional scaling except attributes
have different “weights” or degrees of influence on the
decision
– An attribute that’s more important will have more influence on
outcome
– Most popular
• Step 1: select common scale and convert all interval and
ratio-scaled attribute values into that scale
– Just like non-dimensional scaling
• Step 2: assign weights based on relative importance
– Many different approaches to this
– Recommended approach is
• Each attribute given “points” corresponding to importance
• Weight for each attribute is its points divided by sum of points
across all attributes
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FBN Air: Weighting the Attributes
• Suppose FBN Air gives point values as
shown for ratio and interval-scaled
attributes
Attribute
Cost
Availability
Liffey index
Points
50
10
25
Weight
50 / ( 50 + 10 + 25 ) = 0.588
10 / ( 50 + 10 + 25 ) = 0.118
25 / ( 50 + 10 + 25 ) = 0.294
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Additive Weighting
• Step 3: calculate each alternative’s total weighted score
– Example Existing = (0.588*50)+(0.118*50)+(0.294*0) = 35.3
Alternative
Existing
Jupiter
Sword
Guppy
New in-house
New off-shore
Cost
(0.588)
50.0
26.1
15.3
23.9
0.0
22.5
Availability
(0.118)
50.0
40.0
43.3
33.3
13.3
0.0
Liffey index
(0.294)
0.0
27.6
50.0
10.3
10.3
10.3
Total
35.3
28.2
28.8
21.0
4.6
16.3
• Same as non-dimensional scaling, decision is made on
total score if there are no relevant ordinal-scaled
attributes
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Key Points
•
•
•
Aside from technical feasibility, money is almost always the most important
decision criterion but it’s not always the only one
Use values can usually be quantified in terms of money
Esteem values can't be quantified in terms of money
– Decisions involving more than one attribute are almost inevitable
•
•
Choose decision attributes to cover all relevant use values and esteem
values
Several different classes of measurement
– Nominal, Ordinal, Interval, and Ratio
– Within each class, some comparisons will make sense and others won’t
•
Non-compensatory techniques treat each attribute as a separate entity
– Dominance, Satisficing, Lexicography
•
Compensatory techniques allow better performance on one attribute to
compensate for poorer performance in another
– Nondimensional Scaling, Additive Weighting
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