Decision and Risk Analysis
Quantitative Value Function
• We have built a framework for analysis – the
Qualitative Value Function
– Could be a simple additive model
– Could be a hierarchy
• Now we need to put the numbers in
– To show relative importance or preference for the
attributes
– The value or utility of the alternatives
Decision and Risk Analysis
Evaluating Alternatives - Value Models
• Objectives related to alternatives by Attributes
• Attributes are measures of achievement of objectives
– Quantitative
– Reflect consequences
• Can be one or many (both objectives and attributes)
– One is simple and unusual
– Multiple attributes are more common
– Multiple attributes more difficult to handle
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Decision and Risk Analysis
Types of Measures (Attributes)
• Natural: In general use and have a common
interpretation to everyone.
– Ex. Profit in $, number of fatalities, Speed in Miles
per hour
• Constructed: Integrates multiple numerical and/ or
verbal descriptions into a single description of the state
of a fundamental objective. Applies to a specific
decision context.
• Proxy: Normally means objectives that have a derived,
or implied relationship to a fundamental objective.
– Ex. GNP, FER
– Could be natural measure
• Example: Vehicle weight in pounds as a proxy for
comfort........Heavier cars tend to ride better.
Decision and Risk Analysis
Important Considerations for Measurement
• Objectives (and their associated measures) must be:
– Mutually exclusive (not overlapping or double
counting)
– Collectively exhaustive (covering all the things that
are important)
– Independent
• This is the first very important principle that we must
always adhere to
• Very often violated!
• Care should be taken to make measures
understandable to enhance communication with the
stakeholders
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Decision and Risk Analysis
Value Models
• Notation:
Aj, j = 1, . . ., J is the set of alternatives
Oi, i = 1, . . ., I is the set of objectives
Xik, k = 1, . . ., K is the set of measures or attributes
– Note: attribute Xik is associated with
objective Oi
xik is the outcome of attribute Xik
– Example: O1 is the objective “Maximize Profit”
X11 is “the profit in $ millions” with x11 being $168 million
– x is the vector of expected consequences of alternative
Decision and Risk Analysis
Value Model
• a.k.a. “Objective Function”
• Denoted V(x)
• Assigns a number to the consequences, x, of an
alternative
• Used to determine preference among alternatives
• Types of value functions
– Ordinal value function - ranking only
– Measurable value function - strength of preference
– Utility function - incorporates uncertainty
• Form depends on relationships between attributes
• Convention: v(xA) > v(xB) iff we prefer alt A to alt B
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Decision and Risk Analysis
The quantitative value model is used to evaluate the
alternatives.
n
V ( x)=∑ wi vi ( x i )
i =1
This equation is an additive value model (the most
common quantitative model)
– n evaluation measures
– n evaluation measure scores, xi
– n value functions, vi(xi)
– n weights wi
Alternative evaluation is traceable to objectives, evaluation
measures, value functions, weights, and scores.
Decision and Risk Analysis
Relations Between Attributes
• Attributes will be added together to get value of
alternative
• Attributes must not overlap
• Must avoid double counting
• Attributes must be independent of each other in some
sense
• There is more than one type of independence
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Decision and Risk Analysis
Structures of Value Functions
Additive Value Function
V ( x1 , x 2 , … , x n ) =
n
∑ w v (x )
i =1
i i
i
where wi is a positive scaling constant
v and vi are value functions scaled from 0 to 1
All combinations of attributes are
Preferentially Independent!
Decision and Risk Analysis
Preferential Independence
DEFINITION:
Preferential Independence - The pair of attributes {X1,
X2} is preferentially independent of the other attributes
X3....XN if the preference order for consequences
involving only changes in X1 and X2 does not depend on
the levels at which X3....XN are fixed.
• Required for Additive Value Function
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Decision and Risk Analysis
More
Value Functions
Exponential
v~-e-cx
1
1
Concave Functions
Utility
v~e-cx
Diminishing
Returns to
Scale
Utility
0
0
0
1
NEW MEASURE (new units)
Selected Point --
Level:
Utility:
0
1
NEW MEASURE (new units)
Selected Point -- Level:
Utility:
Decision and Risk Analysis
More
Value Functions
Exponential
1
1
v~ecx
Increasing
Returns to
Scale
Utility
v~-ecx
ility
Convex Functions
0
0
0
NEW MEASURE (new units)
Selected Point --
Level:
Utility:
1
0
1
NEW MEASURE (new units)
ted Point --
Level:
Utility:
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Decision and Risk Analysis
Scale Issues
0
10
20
Maximum Speed in MPH
30
40
50
60
70
80
Actual
Acceptable
Available
Theoretically Feasible
Possible End Points
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