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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 1

Multiattribute value tree analysis

• Value tree: • Value of an alternative

x

:

n v

(

x

) 

w i v i

(

x i

)

i

  1

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 2

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 3

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?

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 4

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 5

Generalized SMART and SWING

Reference attribute

Least important Most important Any

Reference

10 (or any number) 100 (or any number) Any number of points

Elicitation

Point estimates Point estimates Point estimates Least important Most important Any Any 10 (or any number) 100 (or any number) Any number of points Any interval Intervals of points Intervals of points Intervals of points Intervals of points

Name

SMART SWING SMART/SWING with a free reference attribute Interval SMART Interval SWING Interval SMART/SWING Interval SMART/SWING with inteval reference attribute

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

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)

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 7

Classification of ratio methods

Minimum number of judgments More than minimum number of judgments Exact point estimates

SMART, SWING AHP, Regression analysis

Interval estimates

Interval SMART/SWING PAIRS, Preference programming

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

Interval SMART/SWING = Simple PAIRS

• PAIRS • Constraints on any  weight ratios Feasible region S • Interval

w A = w B

SMART/SWING • Constraints from the

w B = 3 w A

ratios of the points

w B w B = 3 w C S w A w A = 2 w C w C = 3 w B w C = 4 w A w C S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 9

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 10

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 11

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 13

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 14

Feasible region S

1 2 1 3  

w w B w A A w C

  1 2  1 6 

w w C B

 2

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 15

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  region smaller feasible

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

Feasible region S'

1 2 

w B w A w C

 2

w B

 2  1 

w C w A

 4

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

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 18

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 19

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

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 23

Vincent Sahid's job selection example

(Hammond, Keeney and Raiffa, 1999)

S ystems Analysis Laboratory Helsinki University of Technology Mustajoki, Hämäläinen and Salo Decision support by interval SMART/SWING / 24

Consequences table

Monthly salary Flexibility of work schedule Business skills development Job A

$2,000 Moderate Computer

Job B

$2,400 Low Manage people, computer

Job C

$1,800 High Operations, computer

Vacation (annual days) Benefits Enjoyment

14 Health, dental, retirement Great 12 Health, dental Good 10 Health Good

Job D

$1,900 Moderate

Job E

$2,200 None Organization 15 Time management, multiple tasking 12 Health, retirement Great Health, dental Boring

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

Imprecise rating of the alternatives

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

Interval SMART/SWING weighting

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

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

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

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

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

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

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