Helsinki University of Technology Systems Analysis Laboratory Incomplete Ordinal Information in Value Tree Analysis Antti Punkka and Ahti Salo Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100, TKK, Finland

Helsinki University of Technology Systems Analysis Laboratory 2 n n m alternatives X={x 1,…,x m } n n n twig-level attributes A={a 1,…,a n } n n Additive value n n Set of possible non-normalized scores – –v i (x i 0 )=0 and v i (x i * ) = w i Value tree analysis non-normalized form normalized form

Helsinki University of Technology Systems Analysis Laboratory 3 Preference information n Complete information –Point estimates for weights and scores –Examples »SWING (von Winterfeldt and Edwards 1986) »SMART (Edwards 1977) n Incomplete information –Modeled through linear constraints on weights and scores –Provides dominance relations and value intervals for alternatives –Supports ex ante sensitivity analysis in view of feasible parameters –Examples »PAIRS (Salo and Hämäläinen 1992) »PRIME (Salo and Hämäläinen 2001)

Helsinki University of Technology Systems Analysis Laboratory 4 Incomplete weight information n Forms of incomplete information (Park and Kim 1997): 1.weak ranking w i  w j 2.strict ranking w i – w j   3.ranking with multiples w i   w j 4.interval form  ≤ w i ≤  +  5.ranking of differences w i – w j  w k – w l n A feasible region for attribute weights S w

Helsinki University of Technology Systems Analysis Laboratory 5 Preference Programming with ordinal information Incomplete ordinal information about: 1. 1.Relative importance of attributes 2. 2.Values of alternatives w.r.t. - A single twig-level attribute - Several attributes (e.g. higher-level attributes) - All attributes (holistic comparisons) Dominance relations, Decision rules and Overall value intervals MILP model Decision Other forms of incomplete information: 1. 1.Weak ranking 2. 2.Strict ranking 3. 3.Ranking with multiples 4. 4.Interval form 5. 5.Ranking of differences Results sufficiently conclusive for the DM? Additional information yes no

Helsinki University of Technology Systems Analysis Laboratory 6 Ordinal preference information n Comparative statements between objects –No information about how much ’better’ or more important an object is than another –Can be useful in evaluation w.r.t. qualitative attributes –Complete information = a rank-ordering of all attributes or alternatives n Uses in preference elicitation –Rank attributes in terms of relative importance »Obtain point estimates through, e.g., rank sum weights (Stillwell et al. 1981), rank order centroid (Barron 1992) –Rank alternatives with regard to one or several attributes –Holistic comparisons: ”alternative x 1 preferred to alternative x 2 overall”

Helsinki University of Technology Systems Analysis Laboratory 7 Incomplete ordinal preference information n A complete rank-ordering, too, may be difficult to obtain –Identification of best performing alternative with regard to some attribute »which office facility has the best public transport connections? –Comparison of attributes »which attribute is the most important one? n Rank Inclusion in Criteria Hierarchies (RICH) –Salo and Punkka (2005), European Journal of Operational Research 163/2, pp –Admits incomplete ordinal information about the importance of attributes »”the most important attribute is either cost or durability” »”environmental factors is among the three most important attributes”

Helsinki University of Technology Systems Analysis Laboratory 8 Non-convex feasible region RICH n ”Either attribute a 1 or a 2 is the most important of the three attributes” n Four rank-orderings compatible with this statement Supported by RICH Decisions ©, n Selection of risk management methods (Ojanen et al. 2005) n Participatory priority-setting for a Scandinavian research program (Salo and Liesiö 2006)

Helsinki University of Technology Systems Analysis Laboratory 9 RICHER - RICH with Extended Rankings n Admits incomplete ordinal information about alternatives –”Alternatives x 1, x 2 and x 3 are the three most preferred ones with regard to environmental factors” –”Alternative x 1 is the least preferred among x 1, x 2 and x 3 w.r.t. cost” –”Alternative x 1 is not among the three most preferred ones overall” n Ordinal statements w.r.t. different attribute sets –Twig-level attributes –Higher-level attributes A’  A –Holistic statements w.r.t. all attributes

Helsinki University of Technology Systems Analysis Laboratory 10 Modeling of incomplete ordinal information (1/3) n The smaller the ranking, the more preferred the alternative –r(x 4 )=1  the ranking of x 4 is 1  it is the most preferred n Rank-orderings r=(r 1,..., r m’ ) on alternatives X’  X –Bijections from alternatives X’  X to corresponding rankings 1,...,|X’|=m’ –Notation: r i = r(x j ), s.t. j is the i-th smallest index in X’ –Convex feasible region »A’={a i }

Helsinki University of Technology Systems Analysis Laboratory 11 Modeling of incomplete ordinal information (2/3) n Specified as a set of alternatives I  X ’  X and corresponding rankings J  {1,...,m’} –X’ = subset of alternatives under comparison and m’ = |X’| its cardinality n If |I|<|J|, alternatives in I have their rankings in J –x 4 and x 5 belong to the three most preferred alternatives –I = {x 4, x 5 }, J = {1,2,3} n If |I|  |J|, rankings in J are attained by alternatives in I –The least preferred alternative in X={x 1,...,x 10 } is among x 1, x 2, x 3, x 4 –I = {x 1, x 2, x 3, x 4 }, J = {10}

Helsinki University of Technology Systems Analysis Laboratory 12 Modeling of incomplete ordinal information (3/3) n Sets I and J lead to compatible rank-orderings R(I,J) for each combination of X’, A’ n Feasible region associated with compatible rank-orderings n Sets S(I,J) have several useful properties, for example –S(I,J) = S(I C,J C ), where I C is the complement of I in X’ –Set inclusions: I 2  I 1, |I i |  |J| => S(I 2,J)  S(I 1,J)

Helsinki University of Technology Systems Analysis Laboratory 13 Linear inequality formulation for S(I,J) (1/3) n Values of alternatives with rankings k and k+1 are separated by milestone variable z k –If the ranking of x j is ”worse” than k, its value is at most z k –Binary variable y k (x j )=1 iff the value of x j is at least z k –Milestone, binary and value variables subjected to A’ and X’

Helsinki University of Technology Systems Analysis Laboratory 14 n n There are exactly k alternatives whose ranking is k or better n n If the ranking of x j is better than k-1, it is also better than k Linear inequality formulation for S(I,J) (2/3) decreasing value

Helsinki University of Technology Systems Analysis Laboratory 15 Linear inequality formulation for S(I,J) (3/3) n Feasible region S(I,J) characterized by linear constraints on binary variables n By using milestone and binary variables for each set pair (A’, X’) used in elicitation, all constraints are in the same linear model n Characteristics of incomplete ordinal information used to enhance computational properties –E.g., only the relevant milestone and binary variables are introduced »given a statement that alternatives x 1 and x 2 are the two most preferred, only z 2 is needed

Helsinki University of Technology Systems Analysis Laboratory 16 Pairwise dominance n Value intervals may overlap, but n Example with two attributes –Interval statement on weights –Point estimates for scores –x1 dominates x2 –x3 is also non-dominated n Non-dominated alternatives –Calculation through LP V w1w w2w x 1 dominates x 2 and strictly positive with some feasible scores and weights Alternative x k dominates x j Value intervals

Helsinki University of Technology Systems Analysis Laboratory 17 Decision rules n Maximize max overall value (’maximax’) => x 1 n Maximize min overall value (’maximin’) => x 3 n Maximize avg of max and min values (’central values’) => x 1 n Minimize greatest possible loss relative to another alternative (’minimax regret’) => x 1 V w1w w2w maximax maximin central values minimax regret

Helsinki University of Technology Systems Analysis Laboratory 18 RICHER n Key features –Extends preference elicitation techniques by admitting incomplete ordinal information about attributes and alternatives –Converts preference statements into a linear inequality formulation »can thus be combined with any other Preference Programming methods –Offers recommendations through pairwise dominance and decision rules n Decision support tools –Experiments suggest that MILP model is reasonably efficient –Software implementation of RICHER Decisions© ongoing n Future research directions –Sorting / classification procedures in score elicitation –Analyses of voting behavior (e.g., acceptance voting) Submitted manuscript downloadable at

Helsinki University of Technology Systems Analysis Laboratory 19 References Barron, F. H., “Selecting a Best Multiattribute Alternative with Partial Information about Attribute Weights”, Acta Psychologica 80 (1992) Edwards, W., “How to Use Multiattribute Utility Measurement for Social Decision Making”, IEEE Transactions on Systems, Man, and Cybernetics 7 (1977) Ojanen, O., Makkonen, S. and Salo, A., “A Multi-Criteria Framework for the Selection of Risk Analysis Methods at Energy Utilities”, International Journal of Risk Assessment and Management 5 (2005) Park, K. S. and Kim, S. H., “Tools dor Interactive Decision Making with Incompletely Identified Information”, European Journal of Operational Research 98 (1997) Salo, A. and Hämäläinen, R. P., "Preference Assessment by Imprecise Ratio Statements”, Operations Research 40 (1992) 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 31 (2001) Salo, A. and Liesiö, J., “A Case Study in Participatory Priority-Setting for a Scandinavian Research Program”, International Journal of Information Technology and Decision Making (to appear). Salo, A. and Punkka, A., “Rank Inclusion in Criteria Hierarchies”, European Journal of Operations Research 163 (2005) Stillwell, W. G., Seaver, D. A. and Edwards, W., “A Comparison of Weight Approximation Techniques in Multiattribute Utility Decision Making”, Organizational Behavior and Human Performance 28 (1981) von Winterfeldt, D., Edwards, W., ”Decision Analysis and Behavioral Research”, Cambridge University Press (1986).