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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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,

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**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

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**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

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**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

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**Rough Set approach to preference modeling**

Inconsistency w.r.t. dominance principle (semantic correlation)

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**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

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**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

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**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 XU 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)

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**CRSA – formal definitions**

Approximation space U = finite set of objects (universe) C = set of condition attributes D = set of decision attributes CD= XC= – condition attribute space XD= – decision attribute space

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**CRSA – formal definitions**

Indiscernibility relation in the approximation space x is indiscernible with y by PC in XP iff xq=yq for all qP x is indiscernible with y by RD in XR iff xq=yq for all qR 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 (PC and RD) 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

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**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

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**CRSA – illustration of formal definitions**

Objects in condition attribute space attribute 1 (Investment) attribute 2 (Sales) 40 20

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**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

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**CRSA – illustration of formal definitions**

Granules of knowlegde are bounded sets IP(x) a1 40 20 20 40 a2

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**CRSA – illustration of formal definitions**

Lower approximation of class High a1 40 20 20 40 a2

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**CRSA – illustration of formal definitions**

Upper approximation of class High a1 40 20 The classes and the unions of classes 20 40 a2

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**Lower approximation of class Medium**

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

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**Upper approximation of class Medium**

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

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**CRSA – illustration of formal definitions**

Boundary set of classes High and Medium a1 40 20 a2

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**CRSA – illustration of formal definitions**

Lower = Upper approximation of class Low a1 40 20 20 40 a2

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**CRSA – formal definitions**

Basic properies of rough approximations Accuracy measures Accuracy and quality of approximation of XU by attributes PC Quality of approximation of classification Cl={Clt, t=1,...m} by attributes PC Rough membership of xU to XU, given PC 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.

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**CRSA – formal definitions**

Cl-reduct of PC, 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:

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**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

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**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

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**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{CD} (complete preorder) xq q yq : “xq is at least as good as yq on criterion q” xDPy : x dominates y with respect to PC in condition space XP if xq q yq for all criteria qP is a partial preorder Analogically, we define xDRy in decision space XR, RD XP is condition criteria space w.r.t. P

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**Dominance-based Rough Set Approach (DRSA)**

For simplicity : D={d} Id makes a partition of U into decision classes Cl={Clt, t=1,...,m} [xClr, yCls, r>s] xy (xy and not yx) 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

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**Dominance-based Rough Set Approach (DRSA)**

Granular computing with dominance cones Granules of knowledge are open sets in condition space XP (PC) (x)= {yU: yDPx} : P-dominating set (x) = {yU: 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

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**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.

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**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

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**DRSA – illustration of formal definitions**

Granular computing with dominance cones c1 40 20 c2 x The classes and the unions of classes

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**DRSA – illustration of formal definitions**

Granular computing with dominance cones c1 40 20 c2 The classes and the unions of classes

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**DRSA – illustration of formal definitions**

Lower approximation of upward union of class High c1 40 20 The classes and the unions of classes c2

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**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

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**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

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**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

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**DRSA – illustration of formal definitions**

Lower approximation of downward union of class Medium c1 40 20 c2 The classes and the unions of classes

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**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

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**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

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**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

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**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 PC Cl-reducts and Cl-core of PC

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**DRSA – induction of decision rules from rough approximations**

certain D-decision rules, supported by objects without ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possible D-decision rules, supported by objects with or without any ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possibly

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**DRSA – induction of decision rules from rough approximations**

certain D-decision rules, supported by objects without ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possible D-decision rules, supported by objects with or without any ambiguity: if xq1q1rq1 and xq2q2rq2 and … xqpqprqp, then x possibly approximate D-decision rules, supported by objects ClsCls+1…Clt without possibility of discerning to which class: if xq1q1rq1 and... xqkqkrqk and xqk+1qk+1rqk+1 and ... xqpqprqp, then xClsCls+1…Clt.

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**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

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**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

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**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

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**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

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**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)

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**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

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**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

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**DRSA – application of decision rules**

Application of decision rules: „intersection” of rules matching object x Final assignment:

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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}

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**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

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**Rough Set approach to multiple-criteria sorting**

Importance and interaction among criteria Quality of approximation of sorting P(Cl) (PC) is a fuzzy measure with the property of Choquet capacity ((Cl)=0, C(Cl)=r and R(Cl)P(Cl)r for any RPC) 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}

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**Rough Set approach to multiple-criteria sorting**

Quality of approximation of sorting students C(Cl)= [8-card({S1,S2})]/8 = 0.75

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**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,

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**DRSA – preference modeling by decision rules**

Representation axiom (cancellation property): for every dimension i=1,…,n, for every evaluation xi,yiXi and a-i,b-iX-i, and for every pair of decision classes Clr,Cls{Cl1,…,Clm}: {xia-iClr and yib-iCls} {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

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**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 aA e.g. aCl2, dClp1 U z1 zp-2 zp-1 z2 Cl1 Cl2 Clp-1 Clp U[g1(a),…,gn(a)] U[g1(d),…,gn(d)]

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**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 aA 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

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**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 aClt+1 e.g. aCl1, dClp1 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

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**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. aCl2>, because profile of a dominates partial profiles of r2 and r3 e.g. for Cl2> g1 g2 g3 gn r1 r3 r2 a

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**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

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**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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**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 BARAR 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

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**DRSA for multiple-criteria choice and ranking**

Dominance for pairs of actions (x,y),(w,z)UU For cardinal criterion qC : (x,y)Dq(w,z) if and where hqkq For subset PC of criteria: P-dominance relation on pairs of actions: (x,y)DP(w,z) if (x,y)Dq(w,z) for all qP i.e., if x is preferred to y at least as much as w is preferred to z for all qP Dq is reflexive, transitive, but not necessarily complete (partial preorder) is a partial preorder on AA x q (x,y)Dq(w,z) z y w For ordinal criterion q C:

107
**DRSA for multiple-criteria choice and ranking**

Let BUU be a set of pairs of reference objects in a given PCT Granules of knowledge positive dominance cone negative dominance cone

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**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: PC

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**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]) :

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**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

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**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

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**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

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**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)AA 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

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**DRSA for multiple-criteria choice and ranking – final recommendation**

Exploitation of the preference structure by the Net Flow Score procedure for each action xA: 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 xA

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**DRSA for multiple-criteria choice and ranking – example**

Decision table with reference objects (warehouses) ROE = profit

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**DRSA for multicriteria choice and ranking – example**

Pairwise comparison table (PCT)

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**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

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**DRSA for multicriteria choice and ranking – example**

D-decision rules

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**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)

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**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

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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

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**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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