1. Helsinki University of Technology Systems Analysis Laboratory Ahti Salo and Antti Punkka Systems Analysis Laboratory Helsinki University of Technology http://www.sal.hut.fi/ RICH - Rank Inclusion in Criteria Hierarchies

2. Helsinki University of Technology Systems Analysis Laboratory Subcontractor Schedule (a 1 ) Overall cost (a 3 ) Quality of work (a 2 ) Multi-attribute weighting References (a 4 ) Possibility of changes (a 5 ) Large firm (x 1 ) Small entrepreneur (x 2 ) Medium-sized firm (x 3 )

3. Helsinki University of Technology Systems Analysis Laboratory Weighting methods n Tradeoff method –has a sound theoretical foundation –requires continuous measurement scales –may be difficult in practice n Ratio-based methods –popular even though the theoretical foundation –SMART (Edwards 1977) –AHP (Saaty 1980) n Ordinal judgements –ask the DM to rank the attributes in terms of importance –derive a representative weight vector from this ranking »e.g., SMARTER (Edwards and Barron 1994), rank sum weights

4. Helsinki University of Technology Systems Analysis Laboratory Incomplete information n Complete information may be hard to acquire –alternatives and their impacts? –relative importance of attributes? n Examples –assessment of environmental impacts –cost of information acquisition –inability to consult all stakeholders –fluctuating preferences n What can be concluded on the basis of available information? –parametric uncertainties covered –structural uncertainties excluded

5. Helsinki University of Technology Systems Analysis Laboratory Analysis of ordinal preference statements n Earlier approaches to the analysis of ordinal information –ask the DM to rank the attributes in terms of importance –derive a representative weight vector from the ranking »e.g., SMARTER (Edwards and Barron 1994), rank sum weights n Incomplete ordinal preference information –the DM(s) may be unable to rank the attributes »statements on contentious issues may be difficult »”which is more important - economy or environmental impacts” –equal weights sometimes used as an approximation n Incomplete ordinal information in RICH –associate a set of possible rankings with a given set of attributes –these statements define possibly non-convex feasible regions

6. Helsinki University of Technology Systems Analysis Laboratory Notation n I is a set of attributes, J a set of rank numbers –r a ranking is a mapping from attributes to –r(a i ) is the rank of attribute i n Compatible rank orders n Feasible region for a given rank order r n Feasible region for rank orders compatible with the sets I and J

7. Helsinki University of Technology Systems Analysis Laboratory Preference elicitation - example 2 –The most important attribute is either a 1 or a 2 –This leads to attribute set I={a 1,a 2 } and rank set J={1} – –Compatible rank orders are (a 1,a 2,a 3 ), (a 1,a 3,a 2 ), (a 2,a 1,a 3 ), (a 2,a 3,a 1 ) – –Feasible region not convex

8. Helsinki University of Technology Systems Analysis Laboratory Preference elicitation - example 1 –Attributes a 1 and a 2 are the two most important attributes –This leads to attribute set I={a 1,a 2 } and rank set J={1,2} – –Compatible rank orders are (a 1,a 2,a 3 ) and (a 2,a 1,a 3 ) – –S p (I)=S(I,{1,…,p})

9. Helsinki University of Technology Systems Analysis Laboratory Feasible regions n Feasible region associated with certain I and J is equal to that of complement of I and complement of J n If there are more ranks in J than attributes in I, the feasible region gets smaller when attributes are added to I n If there are less ranks in J than attributes in I, the feasible region gets larger when attributes are added to I

10. Helsinki University of Technology Systems Analysis Laboratory Feasible regions n When there are less ranks in J than attributes in I, the feasible region gets smaller, when ranks are added to J n When there are more ranks in J than attributes in I, the feasible region gets bigger, when ranks are added to I

11. Helsinki University of Technology Systems Analysis Laboratory Measure for the feasible region n Measure of completeness –compares the number of compatible rank orders to the total number of rank orders

12. Helsinki University of Technology Systems Analysis Laboratory Decision criteria n Pairwise dominance n Maximax –alternative with greatest maximum value n Maximin –alternative with greatest minimum value n Minimax regret –alternative with smallest possible difference to greatest maximum value n Central values –alternative with greatest sum of maximum and minimum value

13. Helsinki University of Technology Systems Analysis Laboratory Application of decision criteria n What concluded when dominance results do not hold? –extrapolate from the available preference information –develop recommendations through alternative decision criteria –analogues include "expected value" etc. n Possible loss of value –what may be lost by terminating the analysis early? –indicates sensitivities in parameter values –can be mapped back to single-attribute scores –helps in assessing the value of additional information

14. Helsinki University of Technology Systems Analysis Laboratory Computational convergence n Questions –how effective are this kind of statements? –which decision rules are best? n Randomly generated problems –n=5,7,10 attributes; m=5,10,15 alternatives –3 different preference statements »A. DM knows the most important attribute »B. DM knows two most important attributes »C. DM knows a set of 3 attributes, which contains 2 most important –statements were compared to equal weights and complete rank orders –efficiency was studied using central values (appeared to be best) –5000 problem instances –values computed in extreme points

15. Helsinki University of Technology Systems Analysis Laboratory Percentage of correct choices

16. Helsinki University of Technology Systems Analysis Laboratory Expected loss of value

17. Helsinki University of Technology Systems Analysis Laboratory Results Ê Statements improve performance in relation to equal weights Ë Rank order is better than the studied statements Ì Statement B gives the best results –the feasible region is smallest

18. Helsinki University of Technology Systems Analysis Laboratory Conclusion n PRIME characteristics –acknowledgement of uncertainties –maintenance of consistencies –alternative elicitation processes –intermediate guidance through decision rules n PRIME Decisions –full-fledged computer implementation –interactive decision support –downloadable at http://www.hut.fi/Units/SAL/Downloadables/

19. Helsinki University of Technology Systems Analysis Laboratory References Salo, A. and R.P. Hämäläinen, "Preference Programming through Approximate Ratio Comparisons," European Journal of Operational Research 82 (1995) 458-475. Salo, A. ja R.P. Hämäläinen, "Preference Assessment by Imprecise Ratio Statements," Operations Research 40/6 (1992) 1053-1061. Salo, A., "Interactive Decision Aiding for Group Decision Support," European Journal of Operational Research 84 (1995) 134-149. R.P. Hämäläinen and M. Pöyhönen, "On-line group decision support by preference programming in traffic planning," Group Decision and Negotiation 5 (1996) 485-500. Gustafsson, J., Salo, A. and Gustafsson, T., “Prime Decisions: An Interactive Tool for Value Tree Analysis,” In Köksalan, M. and Zionts, S (Eds.), Multiple Criteria Decision Making in the New Millenium, Lecture Notes in Economics and Mathematical Systems, vol. 507, pp. 165-176, Springer-Verlag, Berlin. Mustajoki, J., Hämäläinen, R. P. and Salo, A., “Decision support by interval SMART/SWING - methods to incorporate uncertainty into multiattribute analysis,” Systems Analysis Laboratory, Helsinki University of Technology, Manuscript. Salo, A. and Hämäläinen, R. P., “Preference Ratios in Multiattribute Evaluation (PRIME) - Elicitation and Decision Procedures under Incomplete Information,” IEEE Transactions on Systems, Man, and Cybernetics (to appear).