Using Intervals for Global Sensitivity and Worst Case Analyses in Multiattribute Value Trees

1. 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

2. 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

3. 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

4. Value tree

5. 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

6. 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

7. 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

8. 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

9. 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)]

10. 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

11. 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

12. 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

13. WINPRE Software (Hämäläinen and Helenius, 1997)

14. 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

15. 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

16. 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

17. An example (Mustajoki et al. 2004) • Countermeasures for milk production in a case of a hypothetical nuclear accident

18. Alternatives • Combinations of different countermeasures for weeks 2-5 and 6-12 after the accident: - - - = Do nothing Fod = Provide clean fodder to cattle Prod = Production change from milk to e.g. cheese Ban = Ban the milk • E.g. Fod+Fod = providing clean fodder for both periods

19. No imprecision  Pointwise overall values • Fod+Fod is the most preferred alternative

20. Imprecision in weight assessment • Error ratio 2 on each weight ratio • Fod+Fod still dominates all the other alternatives

21. Imprecision in value estimation • ±10 % of the rating interval in each socio-psychological attribute • Fod+Fod dominates all the other alternatives except Prod+Fod

22. Imprecision both in weight assessment and value estimation • ---+--- is the only dominated alternative

23. 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

24. 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

25. 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.

26. 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)