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RULE-BASED MULTICRITERIA DECISION SUPPORT USING ROUGH SET APPROACH

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Presentation on theme: "RULE-BASED MULTICRITERIA DECISION SUPPORT USING ROUGH SET APPROACH"— Presentation transcript:

1 RULE-BASED MULTICRITERIA DECISION SUPPORT USING ROUGH SET APPROACH
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 Examples of multiple-criteria ranking Extensions of DRSA dealing with preference-ordered data Conclusions DRSA is for multicriteria classification

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

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

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

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

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

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

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

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

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

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

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

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

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

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,

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 the boundary of 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 examples and the preference model is a set of decision 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 by induction 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  inconsistencies in the set of decision examples, due to: uncertainty of information – hesitation, unstable preferences, incompleteness of the family 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 Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad S2 S3 S4 S5 S6 S7 S8

22 Rough Set approach to preference modeling
Handling these 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 in knowledge 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, the concept of granular approximation has 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. attributes create granules 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, called lower and upper approximations The lower approximation of X consists of all the granules included in X The upper approximation of X consists of all the granules having non-empty intersection with X A rough set is defined by means of these two approximations, which coincide in the case of an ordinary set. LA - (whose elements, therefore, certainly belong to X) UA - (whose elements, therefore, possibly belong to X) Bn - (whose elements cannot be characterized with certainty as belonging or not to X, using the available information)

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 y by PC in XP iff xq=yq for all qP x is indiscernible with y by RD in XR iff 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) XP is the condition attribute space, XD is the decision attribute space

26 CRSA – illustration of formal definitions
Example Objects = firms Investments Sales Effectiveness 40 17,8 High 35 30 32.5 39 31 27.5 17.5 24 22.5 20 30.8 19 Medium 27 25 21 9.5 18 12.5 10.5 25.5 9.75 17 5 Low 11 2 10 9 13

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

28 Indiscernibility sets
CRSA – illustration of formal definitions Indiscernibility sets a1 40 20 Discretization should be meaningful for user’s perception 20 40 a2 Quantitative attributes are discretized according to perception of the user

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

30 CRSA – illustration of formal definitions
Lower approximation of class High a1 40 20 20 40 a2

31 CRSA – illustration of formal definitions
Upper approximation of class High a1 40 20 The classes and the unions of classes 20 40 a2

32 Lower approximation of class Medium
CRSA – illustration of formal definitions Lower approximation of class Medium a1 40 20 20 40 a2

33 Upper approximation of class Medium
CRSA – illustration of formal definitions Upper approximation of class Medium a1 40 20 20 40 a2

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

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

36 CRSA – formal definitions
Basic properies of rough approximations Accuracy measures Accuracy and quality of approximation of XU by attributes PC Quality of approximation of classification Cl={Clt, t=1,...m} by attributes PC Rough membership of xU to XU, given PC Rough membership may be interpreted analogously to conditional probability and may be understood as the degree of certainty (credibility) to which x belongs to X.

37 CRSA – formal definitions
Cl-reduct of PC, denoted by REDCl(P), is a minimal subset P' of P which keeps the quality of classification Cl unchanged, 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 by lower 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 semantically correlated q – weak preference relation (outranking) on U w.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 XP if xq q yq for all criteria qP is a partial preorder Analogically, we define xDRy in decision space XR, RD XP is condition criteria space w.r.t. P

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 sets in condition space XP (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 representing granules , by granules

43 DRSA – illustration of formal definitions
Example   Investments  Sales  Effectiveness  40 17,8 High 35 30 32.5 39 31 27.5 17.5 24 22.5 20 30.8 19 Medium 27 25 21 9.5 18 12.5 10.5 25.5 9.75 17 5 Low 11 2 10 9 13 Example: when we have three classes low, medium and high than we consider following unions: - at least high (consists of objects from class high) - at least medium (consists of objects from class high and medium) - at most medium (consists of objects from class low and medium) - at most low (consists of objects from class low) There is no sense to consider following unions: at least low and at most high – this is the whole set of data. For each class (object) (without the highest and lowest) it is possible to divide the set of data. For the upward union of this class and the downward union of the worst class, and vice versa.

44 DRSA – illustration of formal definitions
Objects in condition criteria space criterion 1 (Investment) criterion 2 (Sales) 40 20 The classes and the unions of classes

45 DRSA – illustration of formal definitions
Granular computing with dominance cones c1 40 20 c2 x The classes and the unions of classes

46 DRSA – illustration of formal definitions
Granular computing with dominance cones c1 40 20 c2 The classes and the unions of classes

47 DRSA – illustration of formal definitions
Lower approximation of upward union of class High c1 40 20 The classes and the unions of classes c2

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

49 DRSA – illustration of formal definitions
Lower = Upper approximation of upward union of class Medium c1 40 20 c2 The classes and the unions of classes

50 DRSA – illustration of formal definitions
Lower = upper approximation of downward union of class Low c1 40 20 c2 The classes and the unions of classes

51 DRSA – illustration of formal definitions
Lower approximation of downward union of class Medium c1 40 20 c2 The classes and the unions of classes

52 DRSA – illustration of formal definitions
Upper approximation and the boundary of downward union of class Medium c1 40 20 c2 The classes and the unions of classes

53 Dominance-based Rough Set Approach vs. Classical RSA
Comparison of CRSA and DRSA Classes: a1 a2 40 20 c1 c2 40 20 40 20 The classes and the unions of classes

54 Variable-Consistency DRSA (VC-DRSA)
Variable-Consistency DRSA for consistency level l[0,1] e.g. l = 0.8 x The classes and the unions of classes

55 DRSA – formal definitions
Basic properies of rough approximations , for t=2,…,m Identity of boundaries , for t=2,…,m Quality of approximation of sorting Cl={Clt, t=1,...,m} by criteria PC Cl-reducts and Cl-core of PC

56 DRSA – induction of decision rules from rough approximations
certain D-decision rules, supported by objects  without ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possible D-decision rules, supported by objects  with or without any ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possibly 

57 DRSA – induction of decision rules from rough approximations
certain D-decision rules, supported by objects  without ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possible D-decision rules, supported by objects  with or without any ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possibly  approximate D-decision rules, supported by objects  ClsCls+1…Clt without possibility of discerning to which class: if  xq1q1rq1 and... xqkqkrqk and xqk+1qk+1rqk+1 and ... xqpqprqp, then xClsCls+1…Clt.

58 DRSA – decision rules Certain D-decision rules for the class High
40 -base object 20 The classes and the unions of classes If f(x,c1)35, then x belongs to class „High” 20 40 c2

59 DRSA – decision rules Possible D-decision rules for the class High
40 -base object 20 The classes and the unions of classes If f(x,c1)22.5 & f(x,c2)20, then x possibly belongs to class „High” 20 40 c2

60 Approximate D-decision rules for the class Medium or High
DRSA – decision rules Approximate D-decision rules for the class Medium or High c1 40 - base objects 20 The classes and the unions of classes If f(x,c1)[22.5, 27] & f(x,c2)[20, 25], then x belongs to class „Medium” or „High” 20 40 c2

61 Measures characterizing decision rules in system S=U, C, D
Support of a rule  : Strength of a rule  : Certainty (or confidence) factor of a rule  (Łukasiewicz, 1913): Coverage factor of a rule  : Relation between certainty, coverage and Bayes theorem: Certainty factor was first used by Łukasiewicz to estimate the probability of implications

62 Measures characterizing decision rules in system S=U, C, D
For a given decision table S, the probability (frequency) is calculated as: With respect to data from decision table S, the concepts of a priori & a posteriori probability are no more meaningful: Bayesian confirmation measure adapted to decision rules  : „ is verified more often, when  is verified, rather than when  is not verified” Monotonicity: c(,) = F[suppS(,), suppS(,), suppS(,), suppS(,)] is a function non-decreasing with respect to suppS(,) & suppS(,) and non-increasing with respect to suppS(,) & suppS(,) S.Greco, Z.Pawlak, R.Słowiński: Can Bayesian confirmation measures be useful for rough set decision rules? Engineering Appls. of Artificial Intelligence (2004)

63 Certain D-decision rules for at least Medium
DRSA – decision rules Certain D-decision rules for at least Medium c1 40 20 The classes and the unions of classes 20 40 c2

64 Decision rule wih a hyperplane for at least Medium
DRSA – decision rules Decision rule wih a hyperplane for at least Medium c1 40 20 The classes and the unions of classes If f(x,c1)9.75 and f(x,c2)9.5 and 1.5 f(x,c1)+1.0 f(x,c2)22.5, then x is at least Medium 20 40 c2

65 DRSA – application of decision rules
Application of decision rules: „intersection” of rules matching object x Final assignment:

66 Rough Set approach to multiple-criteria sorting
Example of preference information about students: Examples of classification of S1 and S2 are inconsistent Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad S2 S3 S4 S5 S6 S7 S8

67 Rough Set approach to multiple-criteria sorting
If we eliminate Literature, then more inconsistencies appear: Examples of classification of S1, S2, S3 and S5 are inconsistent Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad S2 S3 S4 S5 S6 S7 S8

68 Rough Set approach to multiple-criteria sorting
Elimination of Mathematics does not increase inconsistencies: Subset of criteria {Ph,L} is a reduct of {M,Ph,L} Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad S2 S3 S4 S5 S6 S7 S8

69 Rough Set approach to multiple-criteria sorting
Elimination of Physics also does not increase inconsistencies: Subset of criteria {M,L} is a reduct of {M,Ph,L} Intersection of reducts {M,L} and {Ph,L} gives the core {L} Student Mathematics (M) Physics (Ph) Literature (L) Overall class S1 good medium bad S2 S3 S4 S5 S6 S7 S8

70 Rough Set approach to multiple-criteria sorting
Let us represent the students in condition space {M,L} :   bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

71 Rough Set approach to multiple-criteria sorting
Dominance cones in condition space {M,L} :   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

72 Rough Set approach to multiple-criteria sorting
Dominance cones in condition space {M,L} :   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

73 Rough Set approach to multiple-criteria sorting
Dominance cones in condition space {M,L} :   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

74 Rough Set approach to multiple-criteria sorting
Lower approximation of at least good students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

75 Rough Set approach to multiple-criteria sorting
Upper approximation of at least good students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

76 Rough Set approach to multiple-criteria sorting
Lower approximation of at least medium students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

77 Rough Set approach to multiple-criteria sorting
Upper approximation of at least medium students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

78 Rough Set approach to multiple-criteria sorting
Boundary region of at least medium students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

79 Rough Set approach to multiple-criteria sorting
Lower approximation of at most medium students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

80 Rough Set approach to multiple-criteria sorting
Upper approximation of at most medium students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

81 Rough Set approach to multiple-criteria sorting
Lower approximation of at most bad students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

82 Rough Set approach to multiple-criteria sorting
Upper approximation of at most bad students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

83 Rough Set approach to multiple-criteria sorting
Boundary region of at most bad students:   P={M,L} bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8

84 Rough Set approach to multiple-criteria sorting
Decision rules in terms of {M,L} :   D> - certain rule bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8 If M  good & L  medium, then student  good

85 Rough Set approach to multiple-criteria sorting
Decision rules in terms of {M,L} :   D> - certain rule bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8 If M  medium & L  medium, then student  medium

86 Rough Set approach to multiple-criteria sorting
Decision rules in terms of {M,L} :   D>< - approximate rule bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8 If M  medium & L  bad, then student is bad or medium

87 Rough Set approach to multiple-criteria sorting
Decision rules in terms of {M,L} :   D< - certain rule bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8 If M  medium, then student  medium

88 Rough Set approach to multiple-criteria sorting
Decision rules in terms of {M,L} :   D< - certain rule bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8 If L  bad, then student  medium

89 Rough Set approach to multiple-criteria sorting
Decision rules in terms of {M,L} :   D< - certain rule bad medium good Mathematics Literature good  medium  bad S5,S6 S7 S2 S1 S4 S3 S8 If M  bad, then student  bad

90 Rough Set approach to multiple-criteria sorting
Set of decision rules in terms of {M, L} representing preferences: If M  good & L  medium, then student  good {S4,S5,S6} If M  medium & L  medium, then student  medium {S3,S4,S5,S6} If M  medium & L  bad, then student is bad or medium {S1,S2} If M  medium, then student  medium {S2,S3,S7,S8} If L  bad, then student  medium {S1,S2,S7} If M  bad, then student  bad {S7,S8}

91 Rough Set approach to multiple-criteria sorting
Set of decision rules in terms of {M,Ph,L} representing preferences: If M  good & L  medium, then student  good {S4,S5,S6} If M  medium & L  medium, then student  medium {S3,S4,S5,S6} If M  medium & L  bad, then student is bad or medium {S1,S2} If Ph  medium & L  medium then student  medium {S1,S2,S3,S7,S8} If M  bad, then student  bad {S7,S8} The preference model involving all three criteria is more concise

92 Rough Set approach to multiple-criteria sorting
Importance and interaction among criteria Quality of approximation of sorting P(Cl) (PC) is a fuzzy measure with the property of Choquet capacity ((Cl)=0, C(Cl)=r and R(Cl)P(Cl)r for any RPC) Such measure can be used to calculate Shapley value or Benzhaf index, i.e. an average „contribution” of criterion q in all coalitions of criteria, q{1,…,m} Fuzzy measure theory permits, moreover, to calculate interaction indices (Murofushi & Soneda, Grabisch or Roubens) for pairs (or larger subsets) of criteria, i.e. an average „added value” resulting from putting together q and q’ in all coalitions of criteria, q,q’{1,…,m}

93 Rough Set approach to multiple-criteria sorting
Quality of approximation of sorting students C(Cl)= [8-card({S1,S2})]/8 = 0.75

94 DRSA – preference modeling by decision rules
A set of (D D D)-rules induced from rough approximations represents a preference model of a Decision Maker Traditional preference models: utility function (e.g. additive, multiplicative, associative, Choquet integral, Sugeno integral), binary relation (e.g. outranking relation, fuzzy relation) Decision rule model is the most general model of preferences: a general utility function, Sugeno or Choquet inegral, or outranking relation exists if and only if there exists the decision rule model Słowiński, R., Greco, S., Matarazzo, B.: “Axiomatization of utility, outranking and decision-rule preference models for multiple-criteria classification problems under partial inconsistency with the dominance principle”, Control and Cybernetics, 31 (2002) no.4,

95 DRSA – preference modeling by decision rules
Representation axiom (cancellation property): for every dimension i=1,…,n, for every evaluation xi,yiXi and a-i,b-iX-i, and for every pair of decision classes Clr,Cls{Cl1,…,Clm}: {xia-iClr and yib-iCls}  {yia-i or xib-i } The above axiom constitutes a minimal condition that makes the weak preference relation i a complete preorder This axiom does not require pre-definition of criteria scales gi, nor the dominance relation, in order to derive 3 preference models: general utility function, outranking relation, set of decision rules D or D Greco, S., Matarazzo, B., Słowiński, R.: Conjoint measurement and rough set approach for multicriteria sorting problems in presence of ordinal criteria. [In]: A.Colorni, M.Paruccini, B.Roy (eds.), A-MCD-A: Aide Multi Critère à la Décision – Multiple Criteria Decision Aiding, European Commission Report EUR EN, Joint Research Centre, Ispra, 2001, pp

96 Comparison of decision rule preference model and utility function
Value-driven methods The preference model is a utility function U and a set of thresholds zt, t=1,…,p-1, on U, separating the decision classes Clt, t=0,1,…,p A value of utility function U is calculated for each action aA e.g. aCl2, dClp1 U z1 zp-2 zp-1 z2 Cl1 Cl2 Clp-1 Clp U[g1(a),…,gn(a)] U[g1(d),…,gn(d)]

97 Comparison of decision rule preference model and outranking relation
ELECTRE TRI Decision classes Clt are caracterized by limit profiles bt, t=0,1,…,p The preference model, i.e. outranking relation S, is constructed for each couple (a, bt), for every aA and bt, t=0,1,…,p g1 g2 g3 gn b0 b1 bp-2 bp-1 bp b2 Cl1 Cl2 Clp-1 Clp a d

98 Comparison of decision rule preference model and outranking relation
ELECTRE TRI Decision classes Clt are caracterized by limit profiles bt, t=0,1,…,p Compare action a successively to each profile bt, t=p-1,…,1,0; if bt is the first profile such that aSbt, then aClt+1 e.g. aCl1, dClp1 g1 g2 g3 gn b0 b1 bp-2 bp-1 bp b2 Cl1 Cl2 Clp-1 Clp a d comparison of action a to profiles bt

99 Comparison of decision rule preference model and outranking relation
Rule-based classification The preference model is a set of decision rules for unions , t=2,…,p A decision rule compares an action profile to a partial profile using a dominance relation e.g. aCl2>, because profile of a dominates partial profiles of r2 and r3 e.g. for Cl2> g1 g2 g3 gn r1 r3 r2 a

100 An impact of DRSA on Knowledge Discovery from data bases
Example of diagnostic data 76 buses (objects) 8 symptoms (attributes) Decision = technical state: 3 – good state (in use) 2 – minor repair 1 – major repair (out of use) Discover patterns = find relationships between symptoms and the technical state Patterns explain expert’s decisions and support diagnosis of new buses

101 DRSA – example of technical diagnostics
76 vehicles (objects) 8 attributes with preference scale decision = technical state: 3 – good state (in use) 2 – minor repair 1 – major repair (out of use) all criteria are semantically correlated with the decision inconsistent objects: 11, 12, 39

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105 DRSA for multiple-criteria choice and ranking
Input (preference) information: Rerefence actions Criteria Rank in order q1 q2 ... qm x f(x, q1) f(x, q2) f(x, qm) 1st y f(y, q1) f(y, q2) f(y, qm) 2nd u f(u, q1) f(u, q2) f(u, qm) 3rd z f(z, q1) f(z, q2) f(z, qm) k-th AR BARAR Pairs of ref. actions Difference on criteria Preference relation q1 q2 ... qm (x,y) 1(x,y) 2(x,y) m(x,y) xSy (y,x) 1(y,x) 2(y,x) m(y,x) yScx (y,u) 1(y,u) 2(y,u) m(y,u) ySu (v,z) 1(v,z) 2(v,z) m(v,z) vScz Pairwise Comparison Table (PCT) S – outranking Sc – non-outranking i – difference on qi

106 DRSA for multiple-criteria choice and ranking
Dominance for pairs of actions (x,y),(w,z)UU For cardinal criterion qC : (x,y)Dq(w,z) if and where hqkq For subset PC of criteria: P-dominance relation on pairs of actions: (x,y)DP(w,z) if (x,y)Dq(w,z) for all qP i.e., if x is preferred to y at least as much as w is preferred to z for all qP Dq is reflexive, transitive, but not necessarily complete (partial preorder) is a partial preorder on AA x q (x,y)Dq(w,z) z y w For ordinal criterion q C:

107 DRSA for multiple-criteria choice and ranking
Let BUU be a set of pairs of reference objects in a given PCT Granules of knowledge positive dominance cone negative dominance cone

108 DRSA for multiple-criteria choice and ranking – formal definitions
P-lower and P-upper approximations of outranking relations S: P-lower and P-upper approximations of non-outranking relation Sc: P-boundaries of S and Sc: PC

109 DRSA for multiple-criteria choice and ranking – formal definitions
Basic properties: Quality of approximation of S and Sc: (S,Sc)-reduct and (S,Sc)-core Variable-consistency rough approximations of S and Sc (l(0,1]) :

110 DRSA for multiple-criteria choice and ranking – decision rules
Decision rules (for criteria with cardinal scales) D-decision rules if x y and x y and ... x y, then xSy D-decision rules if x y and x y and ... x y, then xScy D-decision rules if x y and x y and ... x y and x y and ... x y, then xSy or xScy

111 Example of application of DRSA
Acquiring reference objects Preference information on reference objects: making a ranking pairwise comparison of the objects (x S y) or (x Sc y) Building the Pairwise Comparison Table (PCT) Inducing rules from rough approximations of relations S and Sc u q1 = q2 = … qm = 3.0 q1 = q2 = … qm = 5.7 q1 = q2 = … qm = 7.1 q1 = q2 = … qm = 9.0 Pairs of objects Difference of evaluations on each criterion Preference relation  q1  q2  qn (x,y) 1(x,y) = -1.1 2(x,y) = 1.0 ... n(x,y) = 2.7 x Sc y (y,z) 1(y,z) = -2.4 2(y,z) = 4.0 n(y,z) = -4.1 y S z (y,u) 1(y,u) = 1.8 2(y,u) = 6.0 n(y,u) = -6.0 y S u (u,z) 1(u,z) = -4.2 2(u,z) = -2.0 n(u,z) = 1.9 u Sc z y x z

112 Induction of decision rules from rough approximations of S and Sc
 q1  q2 x Sc y z Sc y u Sc y u Sc z y S x y S z y S u t Sc q u S z if q1(a,b) ≥ -2.4 & q2(a,b) ≥ 4.0 then aSb if q1(a,b) ≤ -1.1 & q2(a,b) ≤ 1.0 then aScb if q1(a,b) ≤ 2.4 & q2(a,b) ≤ -4.0 then aScb if q1(a,b) ≥ 4.1 & q2(a,b) ≥ 1.9 then aSb

113 Application of decision rules for decision support
Application of decision rules (preference model) on the whole set A induces a specific preference structure on A Any pair of objects (x,y)AA can match the decision rules in one of four ways: xSy and not xScy, that is true outranking (xSTy), xScy and not xSy, that is false outranking (xSFy), xSy and xScy, that is contradictory outranking (xSKy), not xSy and not xScy, that is unknown outranking (xSUy). x y xSTy xSFy xSKy xSUy S Sc The 4-valued outranking underlines the presence and the absence of positive and negative reasons of outranking

114 DRSA for multicriteria choice and ranking – Net Flow Score
x v u y z S Sc weakness of x strength of x NFS(x) = strength(x) – weakness(x) (–,–) (+,–) (+,+) (–,+) S – positive argument Sc – negative argument (+,-): positive argument (ySx) in disfavor of x (-,+): negative argument (yScx) in favor of x

115 DRSA for multiple-criteria choice and ranking – final recommendation
Exploitation of the preference structure by the Net Flow Score procedure for each action xA: NFS(x) = strength(x) – weakness(x) Final recommendation: ranking: complete preorder determined by NSF(x) in A best choice: action(s) x*A such that NSF(x*)= max {NSF(x)} (+,-): positive argument (ySx) in disfavor of x (-,+): negative argument (yScx) in favor of x xA

116 DRSA for multiple-criteria choice and ranking – example
Decision table with reference objects (warehouses) ROE = profit

117 DRSA for multicriteria choice and ranking – example
Pairwise comparison table (PCT)

118 DRSA for multicriteria choice and ranking – example
Quality of approximation of S and Sc by criteria from set C is 0.44 REDPCT = COREPCT = {A1,A2,A3} D-decision rules and D-decision rules If warehouse x is at least weakly preferred to warehouse y w.r.t. capacity of sales staff and perceived quality of goods, then warehouse x is at least as profitable as y

119 DRSA for multicriteria choice and ranking – example
D-decision rules

120 DRSA for multicriteria choice and ranking – example
Ranking of warehouses for sale by decision rules and the NFS procedure Final ranking: (2',6')  (8')  (9')  (1')  (4')  (5')  (3')  (7',10') Best choice: select warehouse 2' and 6' having maximum score (11)

121 Other extensions of DRSA
DRSA for choice and ranking with graded preference relations DRSA for choice and ranking with Lorenz dominance relation DRSA for decision with multiple decision makers DRSA with missing values of attributes and criteria DRSA for hierarchical decision making Fuzzy set extensions of DRSA DRSA for decision under risk Discovering association rules in preference-ordered data sets Building a strategy of efficient intervention based on decision rules

122 Conclusions Preference information is given in terms of decision examples on reference actions Preference model induced from rough approximations of unions of decision classes (or preference relations S and Sc) is expressed in a natural and comprehensible language of "if..., then..." decision rules Decision rules do not convert ordinal information into numeric one and can represent inconsistent preferences Heterogeneous information (attributes, criteria) and scales of preference (ordinal, cardinal) can be processed within DRSA DRSA supplies useful elements of knowledge about decision situation: certain and doubtful knowledge distinguished by lower and upper approximations relevance of particular attributes or criteria and information about their interaction reducts of attributes or criteria conveying important knowledge contained in data the core of indispensable attributes and criteria decision rules can be used for both explanation of past decisions and decision support

123 Free software available
ROSE ROugh Set data Explorer 4eMka Dominance-based Rough Set Approach to Multicriteria Classification JAMM A New Decision Support Tool for Analysis and Solving of Multicriteria Classification Problems


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