1. 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

2. 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

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
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 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. 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:
points to attribute B
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 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: points
B: points
C: points

21. Implies additional constraints
Feasible region S:

22. 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. 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:

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., 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.
Mustajoki, J., Hämäläinen, R.P., Salo, A., Decision support by interval SMART/SWING – Incorporating imprecision in the SMART and SWING methods, Decision Sciences, 36(2),
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., Preference ratios in multiattribute evaluation (PRIME) - elicitation and decision procedures under incomplete information. IEEE Trans. on SMC 31 (6),
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