Helsinki University of Technology Systems Analysis Laboratory RICHER – A Method for Exploiting Incomplete Ordinal Information in Value Trees Antti Punkka and Ahti Salo Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100, HUT, Finland

Helsinki University of Technology Systems Analysis Laboratory 2 n n m alternatives: X={x 1,…,x m }, n attributes: A={a 1,…,a n } n n Additive value function or Value tree analysis non-normalized form normalized form

Helsinki University of Technology Systems Analysis Laboratory 3 Preference elicitation n Complete information –Point estimates, e.g. w 1 =0.5 –E.g., SMART (Edwards 1977) n Incomplete information: preference programming methods –Weight ratio and weight intervals » –Intervals for normalized scores » –PAIRS (Salo and Hämäläinen 1992), PRIME (Salo and Hämäläinen 2001), Arbel’s approach (1989) n Ordinal information –Rank attributes in terms of relative importance »point estimates through, e.g., rank sum weights (Stillwell et al. 1981) »incomplete ordinal information (RICH; Salo and Punkka 2004)

Helsinki University of Technology Systems Analysis Laboratory 4 Incomplete preference information n Complete information hard to acquire –Relative importance of attributes –Alternatives’ properties –Incomplete information » n Overall value intervals for alternatives –Smallest and largest possible value from LP where S w is the feasible region for the attribute weights

Helsinki University of Technology Systems Analysis Laboratory 5 Pairwise dominance relation n Alternative x k dominates x j in the sense of pairwise dominance –Two attributes, n Several alternatives may remain non- dominated –Additional preference statements to make the feasible region smaller –Decision rules assist the DM in selection of the most preferred one V w1w w2w x 1 dominates x 2 and positive with some feasible scores and weights

Helsinki University of Technology Systems Analysis Laboratory 6 Incomplete ordinal preference information n Complete ordinal information is a complete rank-ordering of attributes or alternatives –Rankings are exactly known for each alternative –Leads to a convex set of feasible scores and weights, when interpreted as incomplete preference information n The RICH (Rank Inclusion in Criteria Hierarchies) method –Incomplete ordinal statements about relative importance of attributes –”Cost is the most important attribute” –”Environmental factors is among the three most important attributes” –Several rank-orderings can be compatible with the preference statements »e.g.: either attribute a 1 or a 2 is the most important of the three attributes n a 3 is either the second or the least important one »may lead to non-convex feasible region of the attribute weights

Helsinki University of Technology Systems Analysis Laboratory 7 Non-convex feasible region in RICH n ”Either a 1 or a 2 is the most important of the three attributes” n Calculation by dividing into compatible rank-orderings –Extreme points readily computed –Lower bounds for weights w i  b  0 n Full support provided by RICH Decisions ©, –Applications »evalution of risk management tools (Ojanen et al. 2004) »support for setting priorities for a research programme in wood material science (Salo and Liesiö 2004)

Helsinki University of Technology Systems Analysis Laboratory 8 The RICHER (RICH with Extended Rankings) method n Extends incomplete ordinal information to alternatives –”Alternatives x 1, x 2 and x 3 are the three most preferred with regard to environmental factors” –”Alternative x 1 is not among the three most preferred ones” –”Considering alternatives x 1, x 2 and x 3, the least preferred with regard to cost is x 1 ” n Statements about attribute weights incorporated as well n Comparison to the RICH method –Suitable also for statements about alternatives –Computationally much more efficient –Includes all features of RICH –Allows evaluation within subsets –Applicable in conjunction with other preference programming methods

Helsinki University of Technology Systems Analysis Laboratory 9 Modeling of incomplete ordinal information (1/4) n Rank-ordering –function r –Bijection from (sub)set of alternatives X’  X (or (sub)set of attributes) to set of rankings –E.g., r=(r(x 1 ), r(x 2 ), r(x 3 ))=(1,3,2) –The smaller the ranking, the better the alternative »e.g., ”r(x 4 )=1  the ranking of x 4 is 1, i.e. it is the most preferred” –Several rank-orderings may be compatible with the preference information –Incomplete ordinal statements about alternatives can be expressed with regard to different sets of attributes A’ »single attribute, (sub)set of attributes or holistic statements considering all attributes »e.g., one can subject statements to cost and environmental factors together

Helsinki University of Technology Systems Analysis Laboratory 10 Modeling of incomplete ordinal information (2/4) n Non-normalized form of value function n Set of feasible values V includes score vectors v=(v(x 1 ),..., v(x m )) –v(x k ) denotes the value of x k with regard to some set of attributes »e.g., if A’={a 2 }, then v(x k )=v 2 (x 2 k ) »e.g., if A’=A, then v(x k )=V(x k ) –Restricted by preference statements n Feasible region associated with a rank-ordering is convex

Helsinki University of Technology Systems Analysis Laboratory 11 Modeling of incomplete ordinal information (3/4) n Elicitation of the preference statements is carried out through an alternative set I  X’  X and a ranking set J  {1,...,m’}, where m’=|X’| –If |I|  |J|, the rankings in J are attained by alternatives in I –If |I|<|J|, the alternatives in I have their rankings in J –Sets subjected to X’ and A’ denoted by I(A’,X’) and J(A’,X’) n Examples –x 1 and x 2 are among the three most preferred ones with regard to cost attribute a 1. Now A’={a 1 } and X’=X, I({a 1 }; X)={x 1,x 2 }, J({a 1 }; X)={1,2,3} –Holistically (A’=A) the two least preferred are among x 4,x 5, x 8,x 9 : I(A; X)= {x 4,x 5, x 8,x 9 }, J(A; X)={m-1,m} –Holistically the most preferred of the set X’={x 1, x 2, x 7 } is x 1 : I(A; X’)={x 1 }, J(A; X’)={1}

Helsinki University of Technology Systems Analysis Laboratory 12 Modeling of incomplete ordinal information (4/4) n Sets I and J lead to compatible rank-orderings R(I,J) n The feasible region associated with many compatible rank- orderings is usually non-convex n Statements can be given with regard to different attribute sets –Several rank-orderings may be compatible with each of these sets

Helsinki University of Technology Systems Analysis Laboratory 13 Mixed integer linear programming model (1/5) n Overcoming the non-convexity –Continuous ”milestone variable” z k distinguishes between the values of alternatives with rankings k and k+1 –If x j ’s ranking is at most k, its value is at least z k and we let y k (x j )=1, else 0 –There are exactly k alternatives whose ranking is at most k »e.g., the three rankings 1, 2 and 3 are at most 3

Helsinki University of Technology Systems Analysis Laboratory 14 Mixed integer linear programming model (2/5) n Formally n For the sake of interpretational and computational matters, we set

Helsinki University of Technology Systems Analysis Laboratory 15 Mixed integer linear programming model (3/5) n Adding preference statements into the model n Assumption |J|≤|I| –For all rankings j  J, the respective alternative belongs to I n Because of the uniqueness of the rankings, there is exactly one alternative in I, for which y j-1 (x i )=0, and y j (x i )=1. For other alternatives, y j-1 (x i ) and y j (x i ) get same values n E.g., I=(x 1, x 2, x 5 ), J={2,4}, exactly one of the alternatives has the ranking 2  it is the only one with different values for y 2 (x i ) and y 1 (x i )

Helsinki University of Technology Systems Analysis Laboratory 16 Mixed integer linear programming model (4/5) n Some milestone and binary variables and the respective constraints are redundant –Given a statement that alternatives x 1 and x 2 are the two most preferred, for example variables z 1, z 3 and y 1 (x j ), y 3 (x j ) are not needed »actually only z 2 and y 2 (x j ) are needed n If set J is ”sequential”, i.e., it constitutes of consecutive positive integers, the number of variables and constraints can be substantially decreased –For example sets {3,4,5} and {1} are sequential, set {1,3} not n For the compatible rank-orderings associated with sets I and J, |I|  |J|, it holds

Helsinki University of Technology Systems Analysis Laboratory 17 Mixed integer linear programming model (5/5) n Partitioning of J into a minimal number of sequential sets J i –For example, J={1,2,6,7} is partitioned into J 1 ={1,2} and J 2 ={6,7} –At most 2 milestone and 2m’ binary variables needed to represent the statement associated to a sequential set J i n Representation of the feasible region S(I,J) as the intersection of the feasible regions S(I,J i ) –Constraints for all pairs I, J i are set in the same model n If contradictionary to the assumption it holds |J|>|I|, the feasible region is constructed with the help of complement sets I C =X’\I and J C ={1,...,m’}\J –Now |J C |≤|I C | –S(I,J)=S(I C,J C ) n All linear inequalities can be included in the model

Helsinki University of Technology Systems Analysis Laboratory 18 An illustrative example (1/6) n A company is about to choose the facility for a new office –Eight attributes relevant: a 1 : size of the office, a 2 : rental costs, a 3 : renovation need, a 4 : car park opportunities, a 5 : means of communication, a 6 : distance to city center, a 7 : other facilities in the neighborhood, a 8 : habitability –12 alternatives

Helsinki University of Technology Systems Analysis Laboratory 19 An illustrative example (2/6) n Attributes size, rental costs, car park opportunities and distance to city center are assessed through [0,1]-normalized scores or score intervals

Helsinki University of Technology Systems Analysis Laboratory 20 An illustrative example (3/6) n Other information is turned into incomplete ordinal statements –E.g., alternative x 2 is the only one with no renovation need (a 3 ), hence the ranking 1 –E.g., alternative x 2 is the least preferred w.r.t. habitability (a 8 )

Helsinki University of Technology Systems Analysis Laboratory 21 An illustrative example (4/6) n Information on attributes’ relative importance –Complete rank-ordering of the attributes is r(a 1,a 2,...,a 8 )=(1,2,...,8) –A weight of 0.50 is assigned to the most important attribute, size of the office –Weights are lower bounded by w i  1/3n n A holistic preference for x 4 over x 1 over x 3

Helsinki University of Technology Systems Analysis Laboratory 22 An illustrative example (5/6) n Pairwise bounds (minima of overall value differences) indicate that there are 5 non-dominated alternatives x 5, x 7, x 8, x 9 and x 10

Helsinki University of Technology Systems Analysis Laboratory 23 An illustrative example (6/6) n To discriminate between non-dominated alternatives, decision rules are applied n Each rule recommends alternative x 5 n XPress-MP was used in solving the example n Calculation of the example (pairwise bounds, overall value intervals for each 12 alternatives; 156 MILPs) took 14 seconds on a Pentium III at 800 MHz with 256 MB RAM

Helsinki University of Technology Systems Analysis Laboratory 24 Conclusion n The DM can give incomplete ordinal information about the alternatives with regard to a single attribute, a set of attributes or holistically n Statements about relative importance of attributes are allowed, as well n Based on a linear model and hence it can be used in conjunction with other preference programming methods n Computationally far more efficient than RICH, and more flexible as it contains all features of RICH n Software implementation of RICHER Decisions © ongoing n Future research directions –Modeling of classification procedures with RICHER methodology (cf. the example in this presentation) –Application of RICHER methodology to voting or other group decision processes –Application of incompelete ordinal information in Robust Portfolio Modeling (RPM)

Helsinki University of Technology Systems Analysis Laboratory 25 Related references Arbel, A., “Approximate Articulation of Preference and Priority DerivationApproximate Articulation of Preference and Priority Derivation”, European Journal of Operations Research 43 (1989) 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 (to appear). Salo, A. ja R. P. Hämäläinen, "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 Programme”, submitted manuscript. Salo, A. and Punkka, A., “Rank Inclusion in Criteria Hierarchies”, European Journal of Operations Research (to appear). 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)