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

2. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 2 S ystems Analysis Laboratory Helsinki University of Technology Multiattribute value tree analysis Value tree: Value of an alternative x: w i is the weight of attribute i v i (x i ) is the component value of an alternative x with respect to attribute i

3. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 3 S ystems Analysis Laboratory Helsinki University of Technology Ratio methods in weight elicitation SWING 100 points to the attribute for which the swing from the lowest level to the highest is most preferred Fewer points to attributes for which the swings are less important Weights by normalizing the sum to one SMART 10 points to the least important attribute Otherwise similar

4. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 4 S ystems Analysis Laboratory Helsinki University of Technology Questions of interest Role of the reference attribute What if this is not the most or the least important as in SMART/SWING? How to incorporate preferential uncertainty? Uncertainties can be modeled as intervals of ratios instead of pointwise estimates Are there behavioral or procedural benefits?

5. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 5 S ystems Analysis Laboratory Helsinki University of Technology Generalized SMART and SWING Extensions: 1. The reference attribute can be any of the attributes 2. The DM may reply with intervals instead of exact point estimates 3. The reference attribute, too, can be assigned an interval  A family of Interval SMART/SWING methods Mustajoki, Hämäläinen and Salo, 2001

6. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 6 S ystems Analysis Laboratory Helsinki University of Technology Generalized SMART and SWING

7. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 7 S ystems Analysis Laboratory Helsinki University of Technology 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, 2001 ) Robust Portfolio Modeling (Liesiö, Mild and Salo, 2007,2008)

8. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 8 S ystems Analysis Laboratory Helsinki University of Technology Classification of ratio methods

9. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 9 S ystems Analysis Laboratory Helsinki University of Technology 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 10 S ystems Analysis Laboratory Helsinki University of Technology 1. Relaxing the reference attribute Any attribute can be selected as the reference attribute Weight ratios calculated from ratios of point assignments  Technically no difference to SMART and SWING Possibility of behavioral biases How to guide the DM? Experimental research needed

11. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 11 S ystems Analysis Laboratory Helsinki University of Technology 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 12 S ystems Analysis Laboratory Helsinki University of Technology Interval judgments about ratio estimates Max/min ratios of points constrain 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 13 S ystems Analysis Laboratory Helsinki University of Technology 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 14 S ystems Analysis Laboratory Helsinki University of Technology 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 15 S ystems Analysis Laboratory Helsinki University of Technology Feasible region S

16. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 16 S ystems Analysis Laboratory Helsinki University of Technology 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 17 S ystems Analysis Laboratory Helsinki University of Technology Feasible region S'

18. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 18 S ystems Analysis Laboratory Helsinki University of Technology Which attribute to select as the reference attribute? An attribute against which one can readily compare the other ones Possibly directly measurable (e.g. money) Elimination of remaining uncertainties through narrower intervals leads to more conclusive results

19. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 19 S ystems Analysis Laboratory Helsinki University of Technology 3. Using an interval on the reference attribute Interpretations of intervals Preferences of multiple stakeholders Ambiguous interpretations of the attribute Degree of confidence about one’s preferences Feasible region from the max/min ratios

20. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 20 S ystems Analysis Laboratory Helsinki University of Technology Interval reference A: 50-100 points B: 50-100 points C: 100-150 points

21. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 21 S ystems Analysis Laboratory Helsinki University of Technology Implies additional constraints Feasible region S:

22. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 22 S ystems Analysis Laboratory Helsinki University of Technology Using an interval on the reference attribute Are DMs able to compare against intervals? 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 23 S ystems Analysis Laboratory Helsinki University of Technology 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 24 S ystems Analysis Laboratory Helsinki University of Technology Vincent Sahid's job selection example (Hammond, Keeney and Raiffa, 1999)

25. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 25 S ystems Analysis Laboratory Helsinki University of Technology Consequences table

26. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 26 S ystems Analysis Laboratory Helsinki University of Technology Imprecise rating of the alternatives

27. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 27 S ystems Analysis Laboratory Helsinki University of Technology Interval SMART/SWING weighting

28. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 28 S ystems Analysis Laboratory Helsinki University of Technology 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 29 S ystems Analysis Laboratory Helsinki University of TechnologyConclusions 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. Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 30 S ystems Analysis Laboratory Helsinki University of TechnologyReferences 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