S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 1 Decision analysis by interval SMART/SWING Jyri Mustajoki Raimo P. Hämäläinen Systems Analysis Laboratory Helsinki University of Technology

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 2 Multiattribute Value Tree Analysis Value tree: Value of an alternative x (additive): w i is the weight of attribute i v i (x i ) is the component value of an alternative x in respect of an attribute i

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 3 Ratio methods in weight elicitation Questions of interest - new alternative ways: Reference attribute (Are there other than worst/best = SMART/SWING?) Relationship to direct weighting? Uncertain replies modelled as intervals Uncertain reference considered as an interval Behavioral and procedural benefits and problems

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 4 Attribute weighting SWING 100 points to the most important attribute change from its lowest level to the highest level Fewer points to other attributes denoting their relative importance Weights elicited by normalizing the sum of the points to one SMART 10 points to the least important attribute

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 5 Interval decision analysis methods Intervals used to describe impreciseness 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

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 6 Generalizing SMART and SWING Relaxing the reference attribute to be any attribute Allowing the DM to reply with intervals instead of exact point estimates Allowing also the reference attribute to be an interval

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 7 Generalizing SMART and SWING

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 8 Simplified PAIRS PAIRS Constraints on any weight ratios  Feasible region S Generalized ratio methods simplified cases of PAIRS

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 9 Relaxing the reference attribute to be any attribute Generalization of SMART/SWING or direct weighting Weight ratios calculated as ratios of the given points  Technically no difference to SMART and SWING Possibility of behavioral biases Proper guidance to the DMs More research needed

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 10 Interval SMART/SWING The reference attribute given any (exact) number of points Points to non-reference attributes given as intervals

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 11 Interval SMART/SWING Max/min ratios of points constraint the feasible region of weights Values 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

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 12 An example Three attributes: A, B, C Preferences of the DM: Two cases considered: 1. A chosen as reference attribute (100 points)  Other attributes (B, C) given points 2. B chosen as reference attribute (100 points)  A given points, C given 100 points

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 13 Reference attribute A as a reference attribute

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 14 Feasible region A as a reference attribute

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 15 Reference attribute B as a reference attribute

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 16 Feasible region B as a reference attribute

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 17 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 Choice of the reference attribute? Attribute with least uncertainty Easily measurable attribute, e.g. money

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 18 Using an interval on the reference attribute Meaning of the intervals Ambiguity Constraints for the weight ratios: Every constraint is bounding the feasible region

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 19 Using an interval on the reference attribute An example

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 20 Using an interval on the reference attribute Feasible region S

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 21 Using an interval on the reference attribute Are the DMs able to compare the intevals? The final step of generalizations is to relax the weight ratio constraints to be any constraints  PAIRS method

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 22 WINPRE software Weighting methods Preference programming PAIRS Interval SMART/SWING

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 23 An example Vincent Sahid's job selection (Hammond, Keeney and Raiffa, 1999)

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 24 Value Tree

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 25 Imprecise rating of the alternatives

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 26 Interval SMART/SWING weighting

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 27 PAIRS

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 28 The results

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 29 The results Jobs C and E dominated  Eliminated from subsequent analyses Process could be continued by defining the attributes more accurately Easier as fewer alternative

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 30 Conclusions Interval SMART/SWING An easy method to model uncertainty by intervals Linear programming algorithms involved Software needed WINPRE introduced Does the DMs understand the intervals? More research needed

S ysteemianalyysin Laboratorio Teknillinen korkeakoulu Mustajoki and Hämäläinen Decision analysis by interval SMART/SWING / 31 References Arbel, A., Approximate articulation of preference and priority derivation, European Journal of Operational Research 43, Hammond, J.S., Keeney, R.L., Raiffa, H., Smart Choices. A Practical Guide to Making Better Decisions, Harvard Business School Press, Boston, MA. Salo, A., Hämäläinen, R.P., Preference assessment by imprecise ratio statements, Operations Research 40 (6) Salo, A., Hämäläinen, R.P., Preference programming through approximate ratio comparisons, European Journal of Operational Research 82, Salo, A., Hämäläinen, R.P., PRIME - Preference ratios in multiattribute evaluation. Manuscript. Downloadable at Publications/pdf-files/Prime.pdf