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

Multiattribute value tree analysis 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

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

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?

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

Generalized SMART and SWING

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

Classification of ratio methods

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

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

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

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

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

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

Feasible region S

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

Feasible region S'

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

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

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

Implies additional constraints Feasible region S:

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

WINPRE software Weighting methods Interactive graphical user interface 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

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

Consequences table

Imprecise rating of the alternatives

Interval SMART/SWING weighting

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

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

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