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Jyri Mustajoki Raimo P. Hämäläinen Ahti Salo Systems Analysis Laboratory

Decision support by interval SMART/SWING Methods to incorporate uncertainty into multiattribute analysis. Jyri Mustajoki Raimo P. Hämäläinen Ahti Salo Systems Analysis Laboratory Helsinki University of Technology www.sal.hut.fi. Multiattribute value tree analysis. Value tree:

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Jyri Mustajoki Raimo P. Hämäläinen Ahti Salo Systems Analysis Laboratory

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  1. Decision support by interval SMART/SWINGMethods to incorporate uncertainty into multiattribute analysis Jyri Mustajoki Raimo P. Hämäläinen Ahti Salo Systems Analysis Laboratory Helsinki University of Technology www.sal.hut.fi

  2. Multiattribute value tree analysis • Value tree: • Value of an alternative x: wi is the weight of attribute i vi(xi) is the component value of an alternative x with respect to attribute i

  3. Ratio methods in weight elicitation SWING • 100 points to the most important attribute range change from lowest level to the highest level • Fewer points to other attributes reflecting their relative importance • Weights by normalizing the sum to one SMART • 10 points to the least important attribute • otherwise similar

  4. Questions of interest • Role of the reference attribute • What if other than worst/best = SMART/SWING? • How to incorporate preferential uncertainty? • Uncertain replies modelled as intervals of ratios instead of pointwise estimates • Are there behavioral or procedural benefits?

  5. Generalized SMART and SWING Allow: 1. the reference attribute to be any attribute 2. the DM to reply with intervals instead of exact point estimates 3. also the reference attribute to have an interval  A family of Interval SMART/SWING methods • Mustajoki, Hämäläinen and Salo, 2001

  6. Generalized SMART and SWING

  7. Some interval methods • Preference Programming (Interval AHP) • Arbel, 1989; Salo and Hämäläinen 1995 • PAIRS (Preference Assessment by Imprecise Ratio Statements) • Salo and Hämäläinen, 1992 • PRIME (Preference Ratios In Multiattribute Evaluation) • Salo and Hämäläinen, 1999

  8. Classification of ratio methods

  9. Interval SMART/SWING = Simple PAIRS • PAIRS • Constraints on any weight ratios  Feasible region S • Interval SMART/SWING • Constraints from the ratios of the points

  10. 1. Relaxing the reference attribute • Reference attribute allowed to be any attribute • Compare to direct rating • Weight ratios calculated as ratios of the given points  Technically no difference to SMART and SWING • Possibility of behavioral biases • How to guide the DM? • Experimental research needed

  11. 2. Interval judgments about ratio estimates • Interval SMART/SWING • The reference attribute given any (exact) number of points • Points to non-reference attributes given as intervals

  12. Interval judgments about ratio estimates • Max/min ratios of points constraint the feasible region of weights • Can be calculated with PAIRS • Pairwise dominance • A dominates B pairwisely, if the value of A is greater than the value of B for every feasible weight combination

  13. Choice of the reference attribute • Only the weight ratio constraints including the reference attribute are given  Feasible region depends on the choice of the reference attribute • Example • Three attributes: A, B, C 1) A as reference attribute 2) B as reference attribute

  14. Example: A as reference • A given 100 points • Point intervals given to the other attributes: • 50-200 points to attribute B • 100-300 points to attribute C • Weight ratio between B and C not yet given by the DM

  15. Feasible region S

  16. Example: B as reference • A given 50-200 points • Ratio between A and B as before • The DM gives a pointwise ratio between B and C = 200 points for C • Less uncertainty in results  smaller feasible region

  17. Feasible region S'

  18. Which attribute to choose as a reference attribute? • Attribute agaist which one can give the most precise comparisons • Easily measurable attribute, e.g. money • The aim is to eliminate the remaining uncertainty as much as possible

  19. 3. Using an interval on the reference attribute • Meaning of the intervals • Uncertainty related to the measurement scale of the attribute • not to the ratio between the attributes (as when using an pointwise reference attribute) • Ambiguity of the attribute itself • Feasible region from the max/min ratios • Every constraint is bounding the feasible region

  20. Interval reference A: 50-100 points B: 50-100 points C: 100-150 points

  21. Implies additional constraints • Feasible region S:

  22. Using an interval on the reference attribute • Are the DMs able to compare against intevals? • Two helpful procedures: 1. First give points with pointwise reference attribute and then extend these to intervals 2. Use of external anchoring attribute, e.g. money

  23. WINPRE software • Weighting methods • Preference programming • PAIRS • Interval SMART/SWING • Interactive graphical user interface • Instantaneous identification of dominance  Interval sensitivity analysis • Available free for academic use: www.decisionarium.hut.fi

  24. Vincent Sahid's job selection example (Hammond, Keeney and Raiffa, 1999)

  25. Consequences table

  26. Imprecise rating of the alternatives

  27. Interval SMART/SWING weighting

  28. Value intervals • Jobs C and E dominated  Can be eliminated • Process continues by narrowing the ratio intervals of attribute weights • Easier as Jobs C and E are eliminated

  29. Conclusions • Interval SMART/SWING • An easy method to model uncertainty by intervals • Linear programming algorithms involved • Computational support needed • WINPRE software available for free • How do the DMs use the intervals? • Procedural and behavioral aspects should be addressed

  30. References Arbel, A., 1989. Approximate articulation of preference and priority derivation, European Journal of Operational Research 43, 317-326. Hammond, J.S., Keeney, R.L., Raiffa, H., 1999. Smart Choices. A Practical Guide to Making Better Decisions, Harvard Business School Press, Boston, MA. Mustajoki, J., Hämäläinen, R.P., Salo, A., 2005. Decision support by interval SMART/SWING – Incorporating imprecision in the SMART and SWING methods, Decision Sciences, 36(2), 317-339. 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 Trans. on SMC 31 (6), 533-545. Downloadable publications at www.sal.hut.fi/Publications

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