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  1. RULE-BASED MULTICRITERIA DECISION SUPPORT USING ROUGH SETAPPROACH Roman Slowinski Laboratory of Intelligent Decision Support Systems Institute of Computing Science Poznan University of Technology  Roman Slowinski 2006

  2. Plan • Rough Set approach to preference modeling • Classical Rough Set Approach (CRSA) • Dominance-based Rough Set Approach (DRSA) for multiple-criteria sorting • Granular computing with dominance cones • Induction of decision rules from dominance-based rough approximations • Examples of multiple-criteria sorting • Decision rule preference model vs. utility function and outranking relation • DRSA for multicriteria choice and ranking • Dominance relation for pairs of objects • Induction of decision rules from dominance-based rough approximations • Examples of multiple-criteria ranking • Extensions of DRSA dealing with preference-ordered data • Conclusions

  3. Student Mathematics Physics Literature Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Zdzisław Pawlak (1926 – 2006)

  4. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Objects with the same description are indiscernible and create blocks

  5. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Objects with the same description are indiscernible and create blocks

  6. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Objects with the same description are indiscernible and create blocks

  7. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Objects with the same description are indiscernible and create blocks

  8. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Objects with the same description are indiscernible and create blocks

  9. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Objects with the same description are indiscernible and create granules

  10. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Another information assigns objects to some classes(sets, concepts)

  11. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Another information assigns objects to some classes(sets, concepts)

  12. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Another information assigns objects to some classes(sets, concepts)

  13. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • The granules of indiscernible objects are used to approximate classes

  14. Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good Lower Approximation S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Lower approximation ofclass „good”

  15. Student Mathematics (M) Physics (Ph) Literature (L) Overall class Upper Approximation S1 good medium bad bad S2 medium medium bad medium S3 medium medium medium medium S4 medium medium medium good S5 good medium good good Lower Approximation S6 good good good good S7 bad bad medium bad S8 bad bad medium bad Inconsistencies in data – Rough Set Theory • Lower and upper approximation ofclass „good”

  16. CRSA – decision rules induced from rough approximations • Certain decision rule supported by objects from lower approximation of Clt (discriminant rule) • Possible decision rule supported by objects from upper approximation of Clt (partly discriminant rule) R.Słowiński, D.Vanderpooten: A generalized definition of rough approximations based on similarity. IEEE Transactions on Data and Knowledge Engineering, 12 (2000) no. 2, 331-336

  17. CRSA – decision rules induced from rough approximations • Certain decision rule supported by objects from lower approximation of Clt (discriminant rule) • Possible decision rule supported by objects from upper approximation of Clt (partly discriminant rule) • Approximate decision rule supported by objects from theboundaryof Clt where Clt,Cls,...,Clu are classes to which belong inconsistent objects supporting this rule

  18. Rough Set approach to preference modeling • The preference information has the form of decision examplesand the preference model is a set ofdecision rules • “People make decisions by searching for rules that provide good justification of their choices” (Slovic, 1975) • Rules make evidence of a decision policy and can be used for both explanation of past decisions &recommendation of future decisions • Construction of decision rules is done by induction (kind of regression) • Induction is a paradigm of AI: machine learning, data mining, knowledge discovery • Regression aproach has also been used for other preference models: • utility (value) function, e.g. UTA methods, MACBETH • outranking relation, e.g. ELECTRE TRI Assistant

  19. Rough Set approach to preference modeling • Advantages of preference modeling byinduction from examples: • requires less cognitive effort from the agent, • agents are more confident exercising their decisions than explaining them, • it is concordant with the principle of posterior rationality(March, 1988) • Problem inconsistenciesin the set of decision examples, due to: • uncertainty of information – hesitation, unstable preferences, • incompleteness of thefamily of attributes and criteria, • granularity of information

  20. Rough Set approach to preference modeling • Inconsistency w.r.t. dominance principle (semantic correlation)

  21. Rough Set approach to preference modeling • Example of inconsistencies in preference information: • Examples of classification of S1 and S2 are inconsistent

  22. Rough Set approach to preference modeling • Handlingthese inconsistencies is of crucial importance for knowledge discovery and decision support • Rough set theory (RST), proposed by Pawlak (1982), provides an excellent framework for dealing with inconsistency inknowledge representation • The philosophy of RST is based on observation that information about objects (actions) is granular, thus their representation needs a granular approximation • In the context of multicriteria decision support, theconcept of granular approximationhas to be adapted so as to handle semantic correlation between condition attributes and decision classes Greco, S., Matarazzo, B., Słowiński, R.: Rough sets theory for multicriteria decision analysis. European J. of Operational Research,129 (2001) no.1, 1-47

  23. Classical Rough Set Approach (CRSA) • Let U be a finite universe of discourse composed of objects (actions) described by a finite set of attributes • Sets of objects indiscernible w.r.t. attributescreategranules of knowledge (elementary sets) • Any subset XU may be expressed in terms of these granules: • either precisely– as a union of the granules • or roughly– by two ordinary sets, calledlowerandupper approximations • The lower approximation of Xconsists of all the granules included in X • The upper approximation of X consists of all the granuleshavingnon-empty intersection with X

  24. CRSA – formal definitions • Approximation space U = finite set of objects (universe) C = set of condition attributes D = set of decision attributes CD= XC= – condition attribute space XD= – decision attribute space

  25. CRSA – formal definitions • Indiscernibility relation in the approximation space x is indiscernible with yby PCin XPiff xq=yq for all qP x is indiscernible with yby RDin XRiff xq=yq for all qR IP(x), IR(x) – equivalence classes including x ID makes a partition of U into decision classes Cl={Clt, t=1,...,m} • Granules of knowledge are bounded sets: IP(x) in XP and IR(x) in XR(PC and RD) • Classification patterns to be discovered are functions representing granules IR(x) by granules IP(x)

  26. CRSA – illustration of formal definitions • Example Objects = firms

  27. attribute 1 (Investment) 40 20 attribute 2 (Sales) 0 20 40 CRSA – illustration of formal definitions Objects in condition attribute space

  28. CRSA – illustration of formal definitions Indiscernibility sets a1 40 20 0 a2 20 40 Quantitative attributes are discretized according to perception of the user

  29. CRSA – illustration offormal definitions Granules of knowlegde are bounded sets IP(x) a1 40 20 0 a2 20 40

  30. CRSA – illustration offormal definitions Lower approximation of class High a1 40 20 0 a2 20 40

  31. CRSA – illustration offormal definitions Upper approximation of class High a1 40 20 0 a2 20 40

  32. CRSA – illustration offormal definitions Lower approximation of class Medium a1 40 20 0 a2 20 40

  33. CRSA – illustration offormal definitions Upper approximation of class Medium a1 40 20 0 a2 20 40

  34. a1 40 20 0 a2 20 40 CRSA – illustration of formal definitions Boundary set of classes High and Medium

  35. CRSA – illustration offormal definitions Lower = Upper approximation of class Low a1 40 20 0 a2 20 40

  36. CRSA – formal definitions • Basic properies of rough approximations • Accuracy measures • Accuracyandquality of approximation of XU by attributes PC • Quality of approximation of classificationCl={Clt, t=1,...m}by attributes PC • Rough membershipof xU to XU,given PC

  37. CRSA – formal definitions • Cl-reduct of PC, denoted byREDCl(P),is a minimal subset P' of P which keeps the quality of classification Clunchanged, i.e. • Cl-core is the intersection of all the Cl-reducts of P:

  38. CRSA – decision rules induced from rough approximations • Certain decision rule supported by objects from lower approximation of Clt (discriminant rule) • Possible decision rule supported by objects from upper approximation of Clt (partly discriminant rule) • Approximate decision rule supported by objects from the boundary of Clt where Clt,Cls,...,Clu are classes to which belong inconsistent objects supporting this rule

  39. CRSA – summary of useful results • Characterization of decision classes (even in case of inconsistency) in terms of chosen attributes bylower and upper approximation • Measure of the quality of approximation indicating how good the chosen set of attributes is for approximation of the classification • Reduction of knowledge contained in the table to the description by relevant attributes belonging to reducts • The core of attributes indicating indispensable attributes • Decision rules induced from lower and upper approximations of decision classes show classification patterns existing in data

  40. Dominance-based Rough Set Approach (DRSA) to MCDA • Sets of condition (C) and decision(D) criteria are semanticallycorrelated • q – weak preference relation(outranking) on Uw.r.t. criterion q{CD} (complete preorder) • xq q yq : “xq is at least as good as yq on criterion q” • xDPy:x dominates y with respect to PC in condition space XPif xq q yq for all criteria qP • is a partial preorder • Analogically,we definexDRyin decision space XR, RD

  41. Dominance-based Rough Set Approach (DRSA) • For simplicity : D={d} • Id makes a partition of U into decision classes Cl={Clt, t=1,...,m} • [xClr, yCls, r>s]  xy (xy and not yx) • In order to handle semantic correlation between condition and decision criteria: – upward union of classes, t=2,...,n(„at least” class Clt) – downward union of classes, t=1,...,n-1(„at most” class Clt) • are positive and negative dominance cones in XD, with D reduced to single dimension d

  42. Dominance-based Rough Set Approach (DRSA) • Granular computing with dominance cones • Granules of knowledge are open setsin condition spaceXP(PC) (x)= {yU: yDPx} : P-dominating set (x) = {yU: xDPy} : P-dominated set • P-dominating and P-dominated sets are positive and negative dominance cones in XP • Sorting patterns(preference model) to be discovered are functions representinggranules , by granules

  43. DRSA – illustration of formal definitions • Example  

  44. criterion 1 (Investment) 40 20 0 criterion 2 (Sales) 20 40 DRSA – illustration of formal definitions Objects in condition criteria space

  45. c1 40 x 20 0 c2 20 40 DRSA – illustration of formal definitions Granular computing with dominance cones

  46. c1 40 20 0 c2 20 40 DRSA – illustration of formal definitions Granular computing with dominance cones

  47. c1 40 20 0 20 40 DRSA – illustration of formal definitions Lower approximation of upward union of class High c2

  48. c1 40 20 0 c2 20 40 DRSA – illustration of formal definitions Upper approximation and the boundary of upward union of class High

  49. c1 40 20 0 c2 20 40 DRSA – illustration of formal definitions Lower = Upper approximation of upward union of class Medium

  50. c1 40 20 0 c2 20 40 DRSA – illustration of formal definitions Lower = upper approximation of downward union of class Low