Transcript Making Even Swaps Even Easier

Using Intervals for Global
Sensitivity and Worst Case
Analyses in Multiattribute
Value Trees
Mats Lindstedt
Raimo P. Hämäläinen
Jyri Mustajoki
Systems Analysis Laboratory
Helsinki University of Technology
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Outline
• Multiattribute value tree analysis (MAVT)
• Framework for interval sensitivity analysis
• Use of Preference Programming for
interval sensitivity analysis in MAVT
• Preference Programming framework
• Practical issues related to the analysis
• An example on nuclear emergency
management
• Conclusions
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Multiattribute Value Tree Analysis (MAVT)
• Analysis of problems with m alternatives
and n attributes
• Overall value of alternative x:
n
v ( x )   w i v i ( xi )
i 1
wi is the weight of attribute i, and wi = 1
vi(xi) is the rating (or score) of alternative x with
respect to attribute i
• Attributes can be structured hierarchically
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Value tree
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Sensitivity analyses in MAVT
• One-way sensitivity analysis
• Imprecision in a single parameter at a time
• Simulation approach
• Imprecision in multiple parameters
simultaneously
• Distributions over parameters needed
• Need of conceptually simple multiparameter analysis
 Interval sensitivity analysis
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Interval sensitivity analysis
General framework (Rios Insua and French, 1991):
• Variation allowed in several model
parameters simultaneously
• Constraints on the parameters to set the
range of allowed variation
• Changes in dominance relations studied to
see how sensitive the model is to variation
• Worst case analysis
• All the possible parameter combinations
within the given constraints allowed
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Preference Programming
• A family of methods to include imprecision
in MAVT with constraints on model
parameters
• Provides tools to apply interval sensitivity
analysis in hierarchical multi-attribute value
trees
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The PAIRS method
(Salo and Hämäläinen, 1992)
• A Preference Programming method
• Imprecise statements with intervals on
• Attribute weight ratios
• On any level of the value tree
• E.g. 1  w1 / w2  5
 Feasible region for the weights, S
• The ratings of the alternatives
• E.g. 0.6  v1(x1)  0.8
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The PAIRS method
 Intervals for the overall values
• Lower bound for the overall value of x:
n
v ( x )  min  w i v i ( xi )
wS
i 1
• Linear programming (LP) problem
• Upper bound correspondingly
• Overall value interval for x: [v(x), v(x)]
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Dominance
• Alternative x dominates alternative y if x
has higher overall value than y on each
allowed combination of weights and
ratings, i.e. if
n
min  w i [v i ( xi )  v i ( y i )]  0
wS
i 1
• Can also exist on overlapping overall
value intervals
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Possible loss of value
• Indicates how much the DM can at most
lose in the overall value when choosing
alternative x*:
max (v ( x )  v ( x )),w  S, x  X \ { x }
*
where X is the set of all alternatives
• To support analysis between nondominated alternatives
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*
Computational efficiency
• In PAIRS, LP problems are separately
solved on each branch of the value tree
• LP problems need to be solved only on the
those branches in which the changes are
made, and upwards thereof
• Usually only a few attributes on each branch
of the value tree (seldom over 10)
 Overall value intervals and dominance
relations can be quickly updated
 Makes interactive analysis possible
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WINPRE Software (Hämäläinen and Helenius, 1997)
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Different ways to assign intervals
• Worst case analysis
• Intervals to cover all the possible values
• It may happen that only few or no
alternatives become dominated
• What-if analysis
• What would be the overall intervals and
dominances, if these intervals were applied
• Interactive software needed
• E.g. to study how the dominance relations
change when varying the intervals
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Different ways to assign intervals
• Error ratios on all the weights ratios
• Each weight ratio is allowed to be at maximum
e.g. 2 times as much as the initial ratio
• Quick way to set intervals
• Confidence intervals
• E.g. 95% confidence intervals
• Interpretation of the overall intervals difficult
• Overall intervals are not true confidence intervals
• Distributions of values are needed to get these
 Simulation approach
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Origins of imprecision should be considered
• Any allowed changes within the rating
intervals assumed to be independent of
each other
• Weight ratio intervals describe imprecision
in the relative importances between the
related attribute ranges
• E.g. we know that A costs twice as much as B,
but we do not know the magnitude of the costs
 Imprecision should be related into the weight
of this attribute
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An example (Mustajoki et al. 2004)
• Countermeasures for milk production in a
case of a hypothetical nuclear accident
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Alternatives
• Combinations of different countermeasures
for weeks 2-5 and 6-12 after the accident:
--Fod
Prod
Ban
=
=
=
=
Do nothing
Provide clean fodder to cattle
Production change from milk to e.g. cheese
Ban the milk
• E.g. Fod+Fod = providing clean fodder for
both periods
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No imprecision
 Pointwise overall values
• Fod+Fod is the most preferred alternative
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Imprecision in weight assessment
• Error ratio 2 on each weight ratio
• Fod+Fod still dominates all the other
alternatives
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Imprecision in value estimation
• ±10 % of the rating interval in each sociopsychological attribute
• Fod+Fod dominates all the other alternatives
except Prod+Fod
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Imprecision both in weight assessment
and value estimation
• ---+--- is the only dominated alternative
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Results
• Imprecision in either weights or ratings
 No considerable effects on dominances
• Imprecision simultaneously in both
 Almost all the dominances disappear
• The analysis can be continued by
interactively studying with which intervals
the dominance relations change
• The DM can e.g. tighten the intervals and
study in which points some alternative
becomes dominated
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Conclusions
Interval sensitivity analysis with Preference
Programming:
• Imprecision simultaneously in all the model
parameters
• Conceptually simple
• Computationally efficient
• Flexible  different ways to assign
imprecision intervals
• WINPRE software available for interactive
analyses
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References
Hämäläinen, R.P., 2003. Decisionarium - Aiding Decisions, Negotiating and
Collecting Opinions on the Web, Journal of Multi-Criteria Decision
Analysis 12, 101-110.
Hämäläinen, R.P., 2000. Decisionarium – Global Space for Decision
Support. Systems Analysis Laboratory, Helsinki University of Technology.
(www.decisionarium.hut.fi)
Hämäläinen, R.P., Helenius, J., 1997. WINPRE - Workbench for Interactive
Preference Programming. Computer software. Systems Analysis
Laboratory, Helsinki University of Technology. (Downloadable at
www.decisionarium.hut.fi)
Lindstedt, M., Hämäläinen, R.P., Mustajoki, J. 2001. Using Intervals for
Global Sensitivity Analyses in Multiattribute Value Trees, in M. Köksalan
and S. Zionts (eds.), Lecture Notes in Economics and Mathematical
Systems 507, 177-186.
Mustajoki, J., Hämäläinen, R.P., Sinkko, K., 2004. Interactive Computer
Support in Decision Conferencing: Two Cases on Off-site Nuclear
Emergency Management. Manuscript.
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References
Proll, L.G., Salhi, A., Rios Insua, D., 2001. Improving an optimization-based
framework for sensitivity analysis in multi-criteria decision-making. Journal
of Multi-Criteria Decision Analysis 10, 1-9.
Rios Insua, D., French, S., 1991. A framework for sensitivity analysis in
discrete multi-objective decision-making. European Journal of Operational
Research 54, 176-190.
Salo, A., Hämäläinen, R.P., 1992. Preference assessment by imprecise ratio
statements. Operations Research 40(6), 1053-1061.
Salo, A., Hämäläinen, R.P., 1995. Preference programming through
approximate ratio comparisons. European Journal of Operational
Research 82, 458-475.
Salo, A., Hämäläinen, R.P., 2001. Preference Ratios in Multiattribute
Evaluation (PRIME) - Elicitation and Decision Procedures under
Incomplete Information, IEEE Transactions on Systems, Man, and
Cybernetics - Part A: Systems and Humans 31(6), 533-545.
Salo, A., Hämäläinen, R.P., 2004. Preference Programming. Manuscript.
(Downloadable at http://www.sal.hut.fi/Publications/pdf-files/msal03b.pdf)
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