using intervals for global sensitivity and worst case analyses in multiattribute value trees

S ystemsAnalysis LaboratoryHelsinki University of Technology

Using Intervals for Global Sensitivity and Worst Case Analyses in Multiattribute

Value Trees

Mats LindstedtRaimo P. Hämäläinen

Jyri MustajokiSystems Analysis Laboratory

Helsinki University of Technology


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


Multiattribute Value Tree Analysis (MAVT)

• Analysis of problems with m alternatives and n attributes

• Overall value of alternative x:

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

n

iiii xvwxv

1

)()(


Value tree


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 multi-parameter analysis

Interval sensitivity analysis


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


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


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


The PAIRS method

Intervals for the overall values• Lower bound for the overall value of x:

• Linear programming (LP) problem• Upper bound correspondingly

• Overall value interval for x: [v(x), v(x)]

n

iiii

Swxvwxv

1

)(min)(


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

• Can also exist on overlapping overall value intervals

0)]()([min1

ii

n

iiii

Swyvxvw


Possible loss of value

• Indicates how much the DM can at most lose in the overall value when choosing alternative x*:

where X is the set of all alternatives

• To support analysis between non-dominated alternatives

}{\,)),()((max ** xXxSwxvxv


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


WINPRE Software (Hämäläinen and Helenius, 1997)


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


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


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


An example (Mustajoki et al. 2004)

• Countermeasures for milk production in a case of a hypothetical nuclear accident


Alternatives

• Combinations of different countermeasures for weeks 2-5 and 6-12 after the accident:

- - - = Do nothingFod = Provide clean fodder to cattleProd = Production change from milk to e.g.

cheeseBan = Ban the milk

• E.g. Fod+Fod = providing clean fodder for both periods


No imprecision

Pointwise overall values• Fod+Fod is the most preferred alternative


Imprecision in weight assessment

• Error ratio 2 on each weight ratio• Fod+Fod still dominates all the other

alternatives


Imprecision in value estimation• ±10 % of the rating interval in each socio-

psychological attribute• Fod+Fod dominates all the other alternatives

except Prod+Fod


Imprecision both in weight assessment and value estimation

• ---+--- is the only dominated alternative


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


ConclusionsInterval sensitivity analysis with PreferenceProgramming:• Imprecision simultaneously in all the model

parameters• Conceptually simple• Computationally efficient• Flexible different ways to assign

imprecision intervals• WINPRE software available for interactive

analyses


ReferencesHämäläinen, 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.


ReferencesProll, 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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