Decision making with incomplete information Some new developments. Rudolf Vetschera University of Vienna. Tamkang University May 15, 2017

Transcription

1 Decision making with incomplete information Some new developments Rudolf Vetschera University of Vienna Tamkang University May 15, 2017

2 Agenda Problem description Overview of methods Single parameter approaches Relation based approaches Volume based (probabilistic) approaches A new concept: rankings from probabilistic statements Conclusions 2/51

3 Problem description Parameters Data Model Decision 3/51

4 Problem description Parameters Data Model Decision 4/51

5 Problem description Parameters Data Model Decision How to deal with parameters and input data which are not known with certainty? 5/51

6 Sensitivity analysis The usual approach: Sensitivity analysis Model uncertainty of inputs Parameters and consider its impacts on outputs Data Model Decision 6/51

7 Decision making with incomplete information Accept uncertain inputs... Parameters Data Model Decision account for uncertainty in the model and obtain a decision that takes into account uncertainty 7/51

8 Example area of application Additive multi-attribute utility with unknown weights: u(x )= k w k u k (x k ) Note: most concepts can also be applied to other uncertain parameters (e.g. partial value functions) other preference models (e.g. outranking models) other domains (e.g. risk, group decisions,...) 8/51

9 Forms of incomplete information Intervals: weight is between... w k w k w k Rankings: attribute k is more important than attribute m w m w k Ratios: attribute k is at least twice as important as m 2 w m w k Comparison of alternatives: A i is better than A j k w k u k (a ik ) k w k u k (a jk ) In general: Linear constraints on w k 9/51

10 Admissible parameters w 1 No information: all parameter vectors fulfilling scaling conditions are admissible W = set of parameters which fulfill all constraints More information additional constraints W becomes smaller k w k 1 w 2 10/51

11 Example: Constraint from pairwise comparison of alternatives A j A i A i A j k w k u k (a ik ) = k w k u k (a jk ) 11/51

12 Decisions with incomplete information Single parameter: Identify one "best" parameter vector Approaches Relation based: Establish relations that hold for all possible parameters Volume-based: Relative size of regions in parameter space Srinivasan/Shocker 1973 UTA: Jacquet-Lagreze/Siskos 1982 Representative value functions: Greco et al Kmietowicz/Pearman 1984 Kirkwood/Sarin 1985 Park et al. 1996,1997, 2001 ROR: Greco et al 2008 Domain criterion: Starr 1962 Charnetski/Soland 1978 VIP: Climaco/Dias 2000 SMAA: Lahdelma et al 1998, /51

13 Single parameter approach w 1 "Representative" parameter vector (center of W) w 2 13/51

14 Single parameter approach w 1 Idea: be away as far as possible from all boundaries (=constraints) maximize the smallest slack Slack w 2 14/51

15 Single parameter approach: model Example: constraints from pairwise comparison of alternatives max min z ij s. t. w k (a ik a jk ) z ij =0 i, j : A i A j k z >0 15/51

16 Single parameter approach: model Getting rid of the min operator max z s. t. z z ij k z >0 i, j: A i A j w k (a ik a jk ) z ij =0 i, j : A i A j 16/51

17 Single parameter approach: model And we actually need only one slack max z s.t. w k (a ik a jk ) z 0 i, j: A i A j k z >0 17/51

18 What happens if constraints are incompatible? Allow slacks to become negative A i A j Negative slack = violation of constraints A k A l 18/51

19 Single parameter approach: model Allow for negative slack max z s.t. w k (a ik a jk ) z 0 i, j: A i A j k z 0 Optimal z is positive: Slack in all constraints, model is compatible with preferences Optimal z is negative: At least one constraint is violated, model not compatible with preferences 19/51

20 Relation based approach Consider preference between two alternatives A i and A j Can this preference hold, given the information about parameters? Possible preference relation Will this preference surely hold, given the information about parameters? Necessary preference relation 20/51

21 Possible preference w 1 A j A i A i A j w 2 21/51

22 Necessary preference w 1 A j A i A i A j w 2 22/51

23 Testing for necessary preference max u(a j w) u( A i w ) s. t. w W if optimal objective value is negative even in the best case for A j, it is worse than A i A i is necessarily preferred to A j 23/51

24 Relation based approaches Necessary preference is subset of possible preference Necessary preference is typically incomplete Possible preference is often inconclusive (holds in both directions) Relation "in between" could be useful 24/51

25 Another case of possible preference w 1 Both A j A i and A i A j are possible, but (assuming that parameters are uniformly distributed) A i A j is much more likely A j A i A i A j w 2 25/51

26 Volume-based approach: SMAA Stochastic Multiattribute Acceptability Analysis Assume that parameters are uniformly distributed Volume of subset of parameter space as probability Use to estimate probabilities of certain facts to hold an alternative is best an alternative has a certain rank in the ranking of all alternatives an alternative is ranked better than another alternative Usually done by simulation Lahdelma, /51

27 Sampling in constrained sets: "Hit and Run" method w 1 1) Start from interior point 2) Chose random direction 3) Chose random fraction of distance to boundary Tervonen et al., 2013 w 2 27/51

28 Results from volumebased methods (SMAA) Rank acceptability index: Probability r ik that alternative A i obtains rank k Pairwise winning index: Probability p ij that alternative A i is preferred to A j How to transform into a ranking of alternatives? 28/51

29 Model for rank acceptability index Assignment problem N alt k =1 N alt i=1 x ik =1 i x ik =1 k x ik {0,1} each alternative is assigned to one rank to each rank, one alternative is assigned (omit to allow indifference) x ik : Alternative A i is assigned to rank k 29/51

30 Objective functions Average probability of assignments max i, k : x ik =1 r ik N alt = i=1 N alt k=1 r ik x ik Joint probability max i, k : x ik =1 r ik N = alt i =1 N alt k =1 log(r ik ) x ik Minimum probability of assignment max z z r ik +(1 x ik ) i, k 30/51

31 Models for pairwise winning indices Construct complete order relation: Complete and asymmetric y ij + y ji =1 i j Irreflexive y ii =0 i With indifference y ij + y ji 1 i j z ij = y ij + y ji 1 Transitive y ij y ik + y kj 1.5 k i, j y ij : Alternative A i is preferred to (at least as good as) A j z ij : Indifference between A i and A j 31/51

32 Linking models Rank from assignment model R i = k k x ik Rank from relation model R i =1+ j y ji = 32/51

33 Computational study Generate problem Marginal utilities "True" weights Generate information levels (pairwise comparisons) Perform SMAA Solve models and benchmarks Rank correlation to "true" ranking N All levels Y 33/51

34 Computational study Problem dimensions: 3, 5, 7 attributes, 6, 9, 12, 15 alternatives 2 methods for generating comparisons (by alternative# and by "true" ranking ) available information: Vol(W)/Vol(Unit simplex) 500 randomly generated problems for each problem dimension and information method 34/51

35 Effects of problem dimensions Correlation to true ranking Attributes.Alternatives 35/51

36 Effect of information Correlation to true ranking Quantile of volume 36/51

37 Differences between indices Correlation with true ranking PWI RAI Quantile of volume 37/51

38 Differences between objective functions 0.9 Correlation with true ranking MM Prod Sum Quantile of volume 38/51

39 Regression model M1 M2 M3 M4 (Intercept) *** *** *** *** NAlt=9 *** *** *** *** Nalt=12 *** *** *** *** Nalt=15 *** *** *** *** Ncrit=5 *** *** *** *** Ncrit=7 *** *** *** *** Vol *** *** *** Rank based *** *** Indifference ** *** Obj. Prod. *** *** Obj. Sum *** *** Interaction Vol with... Rank based *** Indifference Obj. Prod. *** Obj. Sum *** R /51

40 Interpretation of interaction terms Coefficient Rank based Indifference Obj. Prod. Obj. Sum Vol 40/51

41 Information in one experiment 41/51

42 Information anomalies True parameter vector Additional constraint Initial admissible set Revised estimate Initial estimate 42/51

43 Occurrence of information anomalies 25% 20% Anomalies 15% 10% 5% 0% Quantile of Vol 43/51

44 Differences between indices 25% 20% Anomalies 15% 10% PWI RAI 5% 0% Quantile of Vol 44/51

45 Differences between objective functions 25% 20% 15% 10% MM Prod Sum 5% 0% /51

46 Regression results m1 m2 m3 m4 (Intercept) *** *** *** *** Nalt=9 *** *** *** *** Nalt=12 *** *** *** *** Nalt=15 *** *** *** *** Nkrit=5 *** *** *** *** Nkrit=7 *** *** *** *** Vol *** *** *** Vol 2 *** *** *** Rank based *** ** Indifference *** *** Obj. Prod. *** *** Obj. Sum *** *** Interaction Vol.. Rank based *** Indifference *** Obj. Prod. ** Obj. Sum * AIC Logistic regression on occurrence of anomaly 46/51

47 Interpretation of interactions 0.20 Coefficient Rank based Indifference Obj. Prod. Obj. Sum Vol 47/51

48 Summary of methods Single parameter: Identify one "best" parameter vector Low effort Loss of information about uncertainty Well-defined ranking Approaches Relation based: Establish relations that hold for all possible parameters Medium effort No clear ranking Incomplete relation Most robust decision Volume-based: Relative size of regions in parameter space High effort (simulation) Rich information No clear ranking Complete ranking via models presented 48/51

49 Conclusions Uncertainty of parameters is important for realistic decision models Different approaches available: Single parameter vector Relation-based Volume-based Represent a scale between richness of information and effort New approaches to generate rankings from probabilistic information 49/51

50 References Single parameter Srinivasan, V. and A. D. Shocker (1973). "Estimating the Weights for Multiple Attributes in a Composite Criterion Using Pairwise Judgements." Psychometrika Jacquet-Lagreze, E. and J. Siskos (1982). "Assessing a set of additive utility functions for multicriteria decision-making, the UTA method." European Journal of Operational Research Kadzinski, M., S. Greco, et al. (2012). "Selection of a representative value function in robust multiple criteria ranking and choice." European Journal of Operational Research 217( ). Relations Greco, S., V. Mousseau, et al. (2008). "Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions." European Journal of Operational Research 191(2): Probabilistic Lahdelma, R., J. Hokkanen, et al. (1998). "SMAA - Stochastic multiobjective acceptability analysis." European Journal of Operational Research 106(1): Kadziński, M. and T. Tervonen (2013). "Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements." European Journal of Operational Research 228(1): Relations from probabilistic Vetschera, R. (2017). Deriving rankings from incomplete preference information: A comparison of different approaches. European Journal of Operational Research 258 (1:) /51

51 Thank you for your attention!