Expert answer:Multiattribute Utility Theory
Expert answer:I need help in summarizing the multiattribute utility theory paper in the attachment and the graph. Also to come uup with a new recommended approach.
figures.pdf
mem.pdf
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Utility
1.0
0.5
Attribute level
0
Figure 1 – “Univariate utility-curve”
1
U2
GUF = αU1 + (1-α)U2
GUF = U1 + 0 . U2
Efficient
frontier
of decision
maker 1
Efficient
frontier
of decision
maker 2
U1
Figure 2 – Group-utililty-function characterization
U2
GUF = βU1 + (1-β)U2
Efficient
frontier
of decision
maker 2
Efficient
frontier of
decision
maker 1
U1
Figure 3 – Group-utility-function characterization with proxy
decision-maker efficient-frontier
political states
overall space
capability
National economy
Mission cuticality
Environment
space/ground ratio
Mission impact
Maturity of mission
Economic
commitment
Economic impact
Launch priority
Mission
performance status
Contribution to
mission
Level of Technology
Expected remaining
lifetime
Satellite
Utililty
cost/domestic
commitment
Satellite Status
Figure 4 – Model Criteria
Figure 5 – Strategic equivalence
1.0
U
A=
U
M=
cost/domestic
commitment
0.8
U
0 .7
U
0.7
A=
0.6
0.6
M=
0.6
U
U
A=
M=
0.4
0.5
U
U
M=
A=
0.
4
M=
0.4
U
0.2
0.5
.3
=0
UA
0.3
0
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0.4
Satellite Status
0.6
0.8
1.0
Legend
UA : Additive form
UM : Multiplicative form
A Multi-ATTRIBUTE-UTILITY-THEORY MODEL
THAT MINIMIZES INTERVIEW-DATA REQUIREMENTS:
A CONSOLIDATION OF SPACE LAUNCH DECISIONS
Raymond W. Staats
First Space Launch Squadron
Cape Carnaveral Air Force Station, Florida
Yupo Chan1
Air Force Institute of Technology
1
Correspondent can be reached at Department of Operational Sciences (ENS), 2950 P Street,
Wright-Patterson AFB, OH 45433.
ABSTRACT
We use multi-attribute utility theory (MAUT) to define a
mathematical representation of a decision makers utility.
By
eliminating the use of a lottery questions, the survey is simpler
to administer.
The calibrated utility function is then used in a
multi criteria optimization model for scheduling purposes.
study
is
conducted
in
which
two
different
decision
A case
makers’
preferences are combined to characterize a group utility function.
Again, a simple procedure is proposed to arrive at a group utility
function.
Keywords:
Univariate utility curves, independence properties,
ratio versus interval scales, group utility function.
1
I.
Introduction
One of the limitations in applying multi-attribute utility
theory to actual decision problems lies in a survey process.
survey
required
to
capture
the
decision
maker’s
The
preference
structure is exceptionally complex and contains questions and
methodologies which are very difficult for the interviewee to
understand.
In particular, the use of lottery questions which is
a cornerstone of MAUT is quite cumbersome. Typically, the decision
maker is given to outcomes, A and B, such that the probability that
A occurs is p and the probability that B occurs is 1-p.
decision
maker
is
then
asked
to
specify
C
such
that
The
he
is
indifferent between obtaining C with certainty and the outcome of
the lottery.
Problems with this method quickly becomes apparent
when working with a decision maker who is not familiar with MAUT.
A great deal of time must be spent by the interviewer in an attempt
to make the decision maker feel comfortable with the questions
being asked. Oftentimes, the interviewer must reveal the axioms of
probability and thoroughly introduce a lottery concept to the
decision maker.
As a result, the survey takes a great deal of time
to complete and too often the decision maker never completely
understands the question he is answering.
decision
maker
developed.
loses
confidence
in
the
As a result, the
model
that
is
being
A simpler method is needed to make the survey simpler
and shorter, and hence make MAUT a better accepted technique.
2
In many decision making situations, a hierarchy is required to
refine each criterion into subcriteria.
In the case study to
follow, an example can be found where the status of a satellite can
be refined into its mission performance status, its contribution to
the mission, its level of technology, and its expected remaining
lifetime.
Seo and Sakawa (1988) refer to this method as the
“nesting of preferences.”
Where subcriteria are defined, pairwise
comparison — a cornerstone of MAUT — are done only between
subcriteria within the same group. A multivariate utility function
is formulated for each group of subcriteria under a criterion and
are called the criterion functions.
Pairwise comparisons are then
conducted for the second tier of criteria, just as they were for
the subcriteria.
Seo and Sakawa show that MAUT techniques are
equally applicable to a tiered model.
The nesting approach is
advantageous as it allows us to work with a model that has many
criteria without becoming overburdened with pairwise comparisons.
Aggregating individual utility functions into a group utility
function has always been the last frontier for MAUT.
It will be
shown that through the use of sensitivity analysis of multicriteria
optimization.
This will be demonstrated by a two decision maker
example in which we are able to state the conditions under which
one individual decision maker’s decision will be guaranteed to
prevail in a group environment.
We are able to state these
conditions without determining the efficient frontier of the second
3
decision maker, and without specifying the form of the group
utility function.
II.
Background
The use of fractile of bracketing method to determine the
shape of the univariate utility function is cumbersome in as much
as it hinges around the lottery method.
Kirkwood (1994) offers a
simplifying assumption that alleviates the problem.
extensive
empirical
research
which
concludes
that
He cites
univariate
utility functions are well approximated using an exponential form.
Hence, the decision maker needs only to indicate one point from
which the constant to the exponential function, called the risk
attitude constant, is derived.
The univariate utility curve is
then defined for the entire range of the attribute.
Verifying the independence of attributes is another important
step of MAUT.
When there is a large number of criteria, the survey
again gets very impractical to conduct.
Let us examine the case of
a decision problem with five attributes to show preferential
independence, preferences shown over a pair attributes compared
with a third attribute at a fixed value, and then the comparisons
are repeated as a level of the third attribute is varied across its
range.
With
comparisons.
five
attributes,
this
requires
thirty
sets
of
To verify mutual utility independence, twenty more
pairwise comparisons must be made.
Clearly, the survey quickly
becomes too burdensome for a decision maker to complete within a
4
reasonable amount of time.
Keeney and Raiffa (1976) come to our
rescue by stating the following theorem:
Given
attributes
equivalent:
x1,
x2,
…
xn,
the
following
are
(a) attributes x1, x2, … xn, are mutually
utility independent, and (b) x1 is utility independent
and { x1, xi is preferentially independent for i = 1, 2,
3, …, n for n $ 3}
This immediately eliminates fifteen pairwise comparisons from the
survey.
In addition, if we carefully define our criteria, we can
reasonably
thereby
make
the
assumption
eliminating
forbidding portion
thirty
more
of
preferential
sets
of
independence,
comparisons.
Now
a
of the survey has been reduced to a manageable
size.
Traditionally, in MAUT criterion weights are determined using
lottery questions.
As stated before, this methodology is often
confusing to the decision maker.
achieve consistent responses.
Simplification is necessary to
Seo and Sakawa (1988) suggest a
method to break this process down into smaller, more manageable
steps.
First, we ask the decision maker to rank the attributes in
descending order of importance, which is normally a fairly easy
task.
Next, we assess relative weights.
Using one attribute as
the base, we can examine tradeoffs between the base attribute and
the other attributes.
one ranked the highest.
A good choice of the base attribute is the
We then ask the question, “How much of the
base attribute can be given up to gain an additional unit of
another attribute?”
In this manner, we collect information on the
5
preference intensities between the attributes.
Consistency can be
checked by using a different attribute as the base and re-asking
the same questions. Finally, the weight of our base attribute must
be determined. Here we substitute the swing weight method proposed
by Clemens (1991) for the traditional lottery question.
In this
method, we start with all attributes at their worst level (the
worst possible alternative) assigning this hypothetical alternative
a utility of 0.
Next, we “swing” the base attribute to its best
possible level, and ask the decision maker to assign a utility that
describes his/her assessment of such an alternative.
The utility
thus assigned can be mathematically shown to be the weight of the
base
attribute.
Together
with
the
relative
weights
already
determined, we now can derive all the attribute weights.
The key
benefit of this methodology is that we have completely eliminated
the use of lottery questions.
Notice that we have combined the ratio scale with interval
scale in the above proposal, where ratio scale is used to compare
between criteria and interval scale used to score alternatives.
This combination is not without precedent.
Seo and Sakawa specify
this combination in their approach to measuring utility functions.
Marvin and Hutchinson (1994) also reported success in using this
methodology. The method proposed takes advantage of both scales to
measure the criterion weights.
A ratio scale measurement requires
an explicit (or at least an implicit) zero point.
The swing method
specifies the zero point as a base when the multivariate utility
does not increase when the criterion is varied from its low value
6
to its high value.
That is, the criterion weight is zero.
Therefore, it is valid to express one criterion weight as a ratio
to another criterion weight.
Once all weights are expressed as
ratios to one another, the swing weight experiment only needs to be
performed once to place the weights on an interval scale.
The
advantage is that ratio comparisons are easier to obtain from the
decision maker than swing weights.
Once we have examined the underlying theory required to make
our model plausible, we need to turn our attention to the decision
maker.
Since every individual has a uniquely different preference
structure, with whom we select to conduct our interviews is very
important.
Most major decisions are made by a group of decision
makers, rather than by an individual.
But MAUT provides the
framework for deriving at individuals utility function, but not a
groups. This is often identified as the limitation of the state of
the art in this field.
One approach is to aggregate an individual
utility function into a group utility function.
the form of the aggregate function be?
But what should
de Neufville (1990)
indicates that finding an appropriate form to represent a group
utility
function
is
satisfactory results.
problematic
and
usually
does
not
yield
Seo and Sakawa show that under certain
conditions an edited form of a group utility function may be
appropriate via their “representation theorem for a group utility
function” however.
They suggest two methods to determine the
weighing factors for each decision maker, the “benevolent dictator”
approach and the “collective response” approach.
In the former,
7
the weights are specified by a knowledgeable individual.
This
approach is trivial in its application but often unsatisfactory in
its
results.
interpersonal
The
latter
comparison
of
approach
requires
preferences
and
an
an
extensive
interpersonal
comparison of differences.
Another approach to model the groups choices is that of an
individual.
functional
This
form
eliminates
since
we
the
used
problem
the
MAUT
of
determining
process.
The
the
well
publicized “Arrows paradox” finds fault with this approach, in as
much as a series of individually expressed preferences is shown to
be intransitive. Keeney and Raiffa (1976) suggest that in deciding
whether to use an individual as the decision maker or a group, we
need to step back and examine the purpose of the study.
Are we
trying to describe the decision process, or prescribe what decision
should be made?
They propose that a unitary decision maker is
appropriate for the prescriptive approach — i.e., one is assessing
what solution the decision maker should propose. In this approach,
we can incorporate into the model the decision makers perceived
notions
about
what
others
might
do
(i.e.,
the
environment), as part of the uncertainties he faces.
political
When faced
with a limited resource, oftentimes we use a multi criterion
optimization model to allocate resources in accordance with the
calibrated utility functions.
Such an approach arrives at an
optimal
a
decision
allocation.
representing
particular
way
of
resource
Using vector sensitivity analysis, we can then define
some limits for weighing factors that will effect the optimum
8
decision.
Wendell (1985) outlines a “tolerance” approach that
determines
how
much
each
objective
function
simultaneously and independently vary.
function
coefficient
alternative,
we
can
coefficient
can
Suppose the objective
represents
utilities
of
a
analytically
determine
how
particular
the
resulting
variance of the original objective function coefficients from their
original values effects the optimal solution for prioritizing
alternatives.
This provides a way of incorporating a second
decision makers preferences and in an indirect way arrives at a
group utility function.
More will be said about this important
subject in sequel.
III. The Survey
Here in this section we will provide the details of the survey
procedure.
mind:
The proposed procedure was designed with four goals in
clarity, simplicity, brevity, and consistency.
is designed to be used in a face-to-face interview.
The survey
The analyst
provides initial background information, guides the decision maker
through the questions, and records responses.
As de Neufville
points out, an experienced analyst is important to this process.
The analyst should ideally conduct a few practice sessions with
trial decision makers before conducting the survey with the actual
decision maker. The decision maker must be gradually introduced to
the concepts of utility theory.
Clemens very neatly laid out a set
of “axioms of expected utility” that is useful in accomplishing
this.
Furthermore, the decision maker must be reminded that there
9
are no “right or wrong” answers.
Remember the goal of the survey
is to capture the decision makers preference structure.
The survey is designed to take no more than a definable amount
of time to complete.
(In our case study to follow, this is limited
to no more than two hours.)
An exhausted decision maker is
unlikely to give reliable or consistent responses.
An experienced
analyst or interviewer can survey completion time at the second and
subsequent interviews, as both the analyst and the decision maker
become more familiar with the process.
achieved by using a written survey.
Finally, consistency is
As Clemens astutely points
out, how questions are posed can greatly influence the answers
given.
By using a written survey, we are assured that individual
decision makers interviewed are given identical survey instruments.
The first section of the survey maps out the subcriterion
utility functions, where a subcriterion is defined as a lower tier
than a regular criterion.
Subcriteria often arise in a complex
problem where each criterion needs to be broken down further into
components.
The decision maker is given some information and
answers a question concerning each subcriterion.
subcriterion is defined.
First, the
Each subcriterion is scaled from zero to
one, and discrete levels of the subcriterion between these points
are defined.
Next, the decision maker is told that the lowest
level of the attribute is assigned a minimum utility of zero, or
the highest level of the attribute is assigned a maximum utility of
one.
That is
10
(1)
The decision maker is then asked to define an attribute level such
that he/she feels has a utility of 0.5.
(2)
These three points allow the univariate utility function to be
drawn,
using
the
method
proposed
by
Kirkwood
as
discussed
previously. An example of such a univariate utility curve is shown
in Figure 1.
The mathematical representation of this univariate
utility function takes the form:
(3)
where X is between 0 and 1 (including the end points) and r is the
risk attitude constant (r
0).
r is determined from the 0.5
utility point specified by the decision maker.
The risk attitude
constant is available using a “0.5 utility point versus RAC” lookup
table that was created and documented in Kirkwood (1994).
If the
decision maker indicates that the utility midpoint (0.5) occurs at
the attribute midpoint, then the univariate utility function is
linear, and r = 0.
In this case, the mathematical expression of
the utility function is simply
(4)
11
Session two of the survey verifies independence properties.
recommended that preferential independence be assumed.
It is
As Keeney
points out, preferential independence is a reasonable assumption
for most multi-attribute decision models and cases were it does not
hold are fairly rare.
Intuitively, most analysts select criteria
that do not replicate one another and in so doing preferential
independence is usually achieved.
A single subcriterion is then
chosen as a basis for comparison.
We ask a series of questions to
determine whether this subcriterion is utility independent of each
of the other twelve subcriterion.
The decision maker is asked
whether the utility midpoint chosen in section one of the survey
for the base subcriterion is effected by changing the level of any
of the other subcriteria.
If it is not, then the base of criterion
is utility independent of the other subcriterion. Here we make use
of Keeney’s weaker conditions for utility independence as described
previously.
Hence, mutual utility independence is verified.
The next survey section determines multivarieate criterion
utility functions.
respective criteria.
Here the subcriteria are grouped in their
The decision maker is asked the rank, in
descending order of importance, the subcriterion in each group.
Once the subcriteria are rank ordered, the decision maker is asked
to indicate the relative importance.
The next step is perhaps the most difficult section of the
survey.
The subcriteria are again grouped into their respective
criteria, and then for each group the following definitions are
given.
12
(5)
Given these definitions, the decision maker is now asked to assign
a
utility
subcriterion
value
is
to
set
a
at
satellite
its
where
maximum
the
level,
highest
while
the
ranked
other
subcriteria are set at their respective minimum levels.
(6)
Seo and Sakawa proved that the utility value given by the decision
maker for this type of formulation is the weighing factor of the
maximized subcriterion, x1.
The process is repeated for each group
of subcriterion.
The next two sections of the survey form creates the overall
utility function.
Here the decision maker ranks the criteria in
descending order, then assess their relative weights.
The next
section has proven to be a little bit tricky for these reasons.
Maximizing a criterion takes place when all of its respective
subcriteria are maximized.
Similarly, a criterion is minimized
when its subcriteria are minimized.
Hence, the decision maker is
attempting to assign a utility to a sit …
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