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 Slide 2 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 Slide 3 3
Inconsistencies in data Rough Set Theory Zdzisaw Pawlak (1926 2006)
badmediumbad S8 badmediumbad S7 good S6 good mediumgoodS5
goodmedium S4 medium S3 mediumbadmedium S2 bad mediumgoodS1 Overall
classLiteraturePhysicsMathematicsStudent Slide 4 4 Inconsistencies
in data Rough Set Theory Objects with the same description are
indiscernible and create blocks badmediumbad S8 badmediumbad S7
good S6 good mediumgoodS5 goodmedium S4 medium S3 mediumbadmedium
S2 bad mediumgoodS1 Overall classLiterature (L)Physics
(Ph)Mathematics (M)Student Slide 5 5 Inconsistencies in data Rough
Set Theory Objects with the same description are indiscernible and
create blocks badmediumbad S8 badmediumbad S7 good S6 good
mediumgoodS5 goodmedium S4 medium S3 mediumbadmedium S2 bad
mediumgoodS1 Overall classLiterature (L)Physics (Ph)Mathematics
(M)Student Slide 6 6 Inconsistencies in data Rough Set Theory
Objects with the same description are indiscernible and create
blocks badmediumbad S8 badmediumbad S7 good S6 good mediumgoodS5
goodmedium S4 medium S3 mediumbadmedium S2 bad mediumgoodS1 Overall
classLiterature (L)Physics (Ph)Mathematics (M)Student Slide 7 7
Inconsistencies in data Rough Set Theory Objects with the same
description are indiscernible and create blocks badmediumbad S8
badmediumbad S7 good S6 good mediumgoodS5 goodmedium S4 medium S3
mediumbadmedium S2 bad mediumgoodS1 Overall classLiterature
(L)Physics (Ph)Mathematics (M)Student Slide 8 8 Inconsistencies in
data Rough Set Theory Objects with the same description are
indiscernible and create blocks badmediumbad S8 badmediumbad S7
good S6 good mediumgoodS5 goodmedium S4 medium S3 mediumbadmedium
S2 bad mediumgoodS1 Overall classLiterature (L)Physics
(Ph)Mathematics (M)Student Slide 9 9 Inconsistencies in data Rough
Set Theory Objects with the same description are indiscernible and
create granules badmediumbad S8 badmediumbad S7 good S6 good
mediumgoodS5 goodmedium S4 medium S3 mediumbadmedium S2 bad
mediumgoodS1 Overall classLiterature (L)Physics (Ph)Mathematics
(M)Student Slide 10 10 Inconsistencies in data Rough Set Theory
Another information assigns objects to some classes (sets,
concepts) badmediumbad S8 badmediumbad S7 good S6 good mediumgoodS5
goodmedium S4 medium S3 mediumbadmedium S2 bad mediumgoodS1 Overall
classLiterature (L)Physics (Ph)Mathematics (M)Student Slide 11 11
Inconsistencies in data Rough Set Theory Another information
assigns objects to some classes (sets, concepts) badmediumbad S8
badmediumbad S7 good S6 good mediumgoodS5 goodmedium S4 medium S3
mediumbadmedium S2 bad mediumgoodS1 Overall classLiterature
(L)Physics (Ph)Mathematics (M)Student Slide 12 12 Inconsistencies
in data Rough Set Theory Another information assigns objects to
some classes (sets, concepts) badmediumbad S8 badmediumbad S7 good
S6 good mediumgoodS5 goodmedium S4 medium S3 mediumbadmedium S2 bad
mediumgoodS1 Overall classLiterature (L)Physics (Ph)Mathematics
(M)Student Slide 13 13 Inconsistencies in data Rough Set Theory The
granules of indiscernible objects are used to approximate classes
badmediumbad S8 badmediumbad S7 good S6 good mediumgoodS5
goodmedium S4 medium S3 mediumbadmedium S2 bad mediumgoodS1 Overall
classLiterature (L)Physics (Ph)Mathematics (M)Student Slide 14 14
Inconsistencies in data Rough Set Theory Lower approximation of
class good badmediumbad S8 badmediumbad S7 good S6 good
mediumgoodS5 goodmedium S4 medium S3 mediumbadmedium S2 bad
mediumgoodS1 Overall classLiterature (L)Physics (Ph)Mathematics
(M)Student Lower Approximation Slide 15 15 Inconsistencies in data
Rough Set Theory Lower and upper approximation of class good
badmediumbad S8 badmediumbad S7 good S6 good mediumgoodS5
goodmedium S4 medium S3 mediumbadmedium S2 bad mediumgoodS1 Overall
classLiterature (L)Physics (Ph)Mathematics (M)Student Lower
Approximation Upper Approximation Slide 16 16 CRSA decision rules
induced from rough approximations Certain decision rule supported
by objects from lower approximation of Cl t (discriminant rule)
Possible decision rule supported by objects from upper
approximation of Cl t (partly discriminant rule) R.Sowiski,
D.Vanderpooten: A generalized definition of rough approximations
based on similarity. IEEE Transactions on Data and Knowledge
Engineering, 12 (2000) no. 2, 331-336 Slide 17 17 CRSA decision
rules induced from rough approximations Certain decision rule
supported by objects from lower approximation of Cl t (discriminant
rule) Possible decision rule supported by objects from upper
approximation of Cl t (partly discriminant rule) Approximate
decision rule supported by objects from the boundary of Cl t where
Cl t,Cl s,...,Cl u are classes to which belong inconsistent objects
supporting this rule Slide 18 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 Slide 19 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 Slide 20 20
Rough Set approach to preference modeling Inconsistency w.r.t.
dominance principle (semantic correlation) Slide 21 21 Rough Set
approach to preference modeling Example of inconsistencies in
preference information: Examples of classification of S1 and S2 are
inconsistent StudentMathematics (M)Physics (Ph)Literature
(L)Overall class S1goodmediumbad S2medium badmedium S3medium S4good
mediumgood S5goodmediumgood S6good S7bad S8bad mediumbad Slide 22
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., Sowiski, R.: Rough sets theory for
multicriteria decision analysis. European J. of Operational
Research, 129 (2001) no.1, 1-47 Slide 23 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 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 Slide 24 24 CRSA formal
definitions Approximation space U = finite set of objects
(universe) C = set of condition attributes D = set of decision
attributes CD= X C = condition attribute space X D = decision
attribute space Slide 25 25 CRSA formal definitions
Indiscernibility relation in the approximation space x is
indiscernible with y by PC in X P iff x q =y q for all qP x is
indiscernible with y by RD in X R iff x q =y q for all qR I P (x),
I R (x) equivalence classes including x I D makes a partition of U
into decision classes Cl={Cl t, t=1,...,m} Granules of knowledge
are bounded sets: I P (x) in X P and I R (x) in X R (PC and RD)
Classification patterns to be discovered are functions representing
granules I R (x) by granules I P (x) Slide 26 26 CRSA illustration
of formal definitions Example Objects = firms
InvestmentsSalesEffectiveness 4017,8High 3530High 32.539High
3135High 27.517.5High 242417.5High 22.520High 30.819Medium
2725Medium 219.5Medium 1812.5Medium 10.525.5Medium 9.751717Medium
17.55Low 112Low 109Low 513Low Slide 27 27 CRSA illustration of
formal definitions Objects in condition attribute space attribute 1
(Investment) attribute 2 (Sales) 0 40 20 Slide 28 28 CRSA
illustration of formal definitions a1a1 a2a2 0 40 20
Indiscernibility sets Quantitative attributes are discretized
according to perception of the user Slide 29 29 a1a1 0 40 20 CRSA
illustration of formal definitions Granules of knowlegde are
bounded sets I P (x) a2a2 Slide 30 30 a1a1 0 40 20 CRSA
illustration of formal definitions Lower approximation of class
High a2a2 Slide 31 31 a1a1 0 40 20 Upper approximation of class
High a2a2 CRSA illustration of formal definitions Slide 32 32 CRSA
illustration of formal definitions a1a1 a2a2 0 40 20 Lower
approximation of class Medium Slide 33 33 CRSA illustration of
formal definitions a1a1 a2a2 0 40 20 Upper approximation of class
Medium Slide 34 34 Boundary set of classes High and Medium CRSA
illustration of formal definitions a1a1 0 40 20 a2a2 Slide 35 35
a1a1 0 40 20 CRSA illustration of formal definitions Lower = Upper
approximation of class Low a2a2 Slide 36 36 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={Cl t, t=1,...m} by attributes
PC Rough membership of xU to XU, given PC Slide 37 37 CRSA formal
definitions Cl-reduct of PC, denoted by RED Cl (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: Slide 38 38 CRSA decision rules induced from rough
approximations Certain decision rule supported by objects from
lower approximation of Cl t (discriminant rule) Possible decision
rule supported by objects from upper approximation of Cl t (partly
discriminant rule) Approximate decision rule supported by objects
from the boundary of Cl t where Cl t,Cl s,...,Cl u are classes to
which belong inconsistent objects supporting this rule Slide 39 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 Slide 40 40 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) x q q y q : x q is at least as good as y q on criterion q
xD P y : x dominates y with respect to PC in condition space X P if
x q q y q for all criteria qP is a partial preorder Analogically,
we define xD R y in decision space X R, RD Dominance-based Rough
Set Approach (DRSA) to MCDA 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 Cl t )
downward union of classes, t=1,...,n-1 (at most class Cl t ) are
positive and negative dominance cones in X D, with D reduced to
single dimension d"> 41 Dominance-based Rough Set Approach
(DRSA) For simplicity : D={d} I d makes a partition of U into
decision classes Cl={Cl t, t=1,...,m} [xCl r, yCl s, 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 Cl t ) downward union of classes, t=1,...,n-1 (at
most class Cl t ) are positive and negative dominance cones in X D,
with D reduced to single dimension d Slide 42 42 Dominance-based
Rough Set Approach (DRSA) Granular computing with dominance cones
Granules of knowledge are open sets in condition space X P (PC)
(x)= {yU: yD P x} : P-dominating set (x) = {yU: xD P y} :
P-dominated set P-dominating and P-dominated sets are positive and
negative dominance cones in X P Sorting patterns (preference model)
to be discovered are functions representing granules, by granules
Slide 43 43 DRSA illustration of formal definitions Example
Investments Sales Effectiveness 4017,8High 3530High 32.539High
3135High 27.517.5High 242417.5High 22.520High 30.819Medium
2725Medium 219.5Medium 1812.5Medium 10.525.5Medium 9.751717Medium
17.55Low 112Low 109Low 513Low Slide 44 44 criterion 1 (Investment)
criterion 2 (Sales) 0 40 20 DRSA illustration of formal definitions
Objects in condition criteria space Slide 45 45 DRSA illustration
of formal definitions Granular computing with dominance cones c1c1
0 40 20 c2c2 x Slide 46 46 DRSA illustration of formal definitions
Granular computing with dominance cones c1c1 0 40 20 c2c2 Slide 47
47 c1c1 0 40 20 DRSA illustration of formal definitions Lower
approximation of upward union of class High c2c2 Slide 48 48 DRSA
illustration of formal definitions Upper approximation and the
boundary of upward union of class High c1c1 0 40 20 c2c2 Slide 49
49 DRSA illustration of formal definitions Lower = Upper
approximation of upward union of class Medium c1c1 0 40 20 c2c2
Slide 50 50 DRSA illustration of formal definitions Lower = upper
approximation of downward union of class Low c1c1 0 40 20 c2c2
Slide 51 51 DRSA illustration of formal definitions Lower
approximation of downward union of class Medium c1c1 0 40 20 c2c2
Slide 52 52 DRSA illustration of formal definitions Upper
approximation and the boundary of downward union of class Medium
c1c1 0 40 20 c2c2 Slide 53 53 40 20 Dominance-based Rough Set
Approach vs. Classical RSA Comparison of CRSA and DRSA c1c1 c2c2 0
4020 a1a1 a2a2 0 40 20 Classes: Slide 54 54 Variable-Consistency
DRSA (VC-DRSA) Variable-Consistency DRSA for consistency level l
[0,1] x e.g. l = 0.8 Slide 55 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={Cl
t, t=1,...,m} by criteria PC Cl-reducts and Cl-core of PC Slide 56
56 Induction of decision rules from rough approximations certain D
-decision rules, supported by objects without ambiguity: if x q1 q1
r q1 and x q2 q2 r q2 and x qp qp r qp, then x possible D -decision
rules, supported by objects with or without any ambiguity: if x q1
q1 r q1 and x q2 q2 r q2 and x qp qp r qp, then x possibly DRSA
induction of decision rules from rough approximations Slide 57 57
DRSA induction of decision rules from rough approximations
Induction of decision rules from rough approximations certain D
-decision rules, supported by objects without ambiguity: if x q1 q1
r q1 and x q2 q2 r q2 and x qp qp r qp, then x possible D -decision
rules, supported by objects with or without any ambiguity: if x q1
q1 r q1 and x q2 q2 r q2 and x qp qp r qp, then x possibly
approximate D -decision rules, supported by objects Cl s Cl s+1 Cl
t without possibility of discerning to which class: if x q1 q1 r q1
and... x qk qk r qk and x qk+1 qk+1 r qk+1 and... x qp qp r qp,
then xCl s Cl s+1 Cl t. Slide 58 58 c1c1 0 40 20 DRSA decision
rules Certain D -decision rules for the class High -base object
c2c2 If f(x,c 1 )35, then x belongs to class High Slide 59 59 c1c1
0 40 20 DRSA decision rules Possible D -decision rules for the
class High -base object c2c2 If f(x,c 1 )22.5 & f(x,c 2 )20,
then x possibly belongs to class High Slide 60 60 c1c1 0 40 20 DRSA
decision rules Approximate D -decision rules for the class Medium
or High - base objects c2c2 If f(x,c 1 )[22.5, 27] & f(x,c 2
)[20, 25], then x belongs to class Medium or High Slide 61 61
Support of a rule : Strength of a rule : Certainty (or confidence)
factor of a rule (ukasiewicz, 1913) : Coverage factor of a rule :
Relation between certainty, coverage and Bayes theorem: Measures
characterizing decision rules in system S=U, C, D Slide 62 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[supp S (,), supp S (,), supp S (,),
supp S (,)] is a function non-decreasing with respect to supp S (,)
& supp S (,) and non-increasing with respect to supp S (,)
& supp S (,) S.Greco, Z.Pawlak, R.Sowiski: Can Bayesian
confirmation measures be useful for rough set decision rules?
Engineering Appls. of Artificial Intelligence (2004) Slide 63 63
c1c1 0 40 20 DRSA decision rules Certain D -decision rules for at
least Medium c2c2 Slide 64 64 c1c1 0 40 20 DRSA decision rules
Decision rule wih a hyperplane for at least Medium c2c2 If f(x,c 1
)9.75 and f(x,c 2 )9.5 and 1.5 f(x,c 1 )+1.0 f(x,c 2 )22.5, then x
is at least Medium Slide 65 65 DRSA application of decision rules
Application of decision rules: intersection of rules matching
object x Final assignment: Slide 66 66 Rough Set approach to
multiple-criteria sorting Example of preference information about
students: Examples of classification of S1 and S2 are inconsistent
StudentMathematics (M)Physics (Ph)Literature (L)Overall class
S1goodmediumbad S2medium badmedium S3medium S4good mediumgood
S5goodmediumgood S6good S7bad S8bad mediumbad Slide 67 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 StudentMathematics (M)Physics
(Ph)Literature (L)Overall class S1goodmediumbad S2medium badmedium
S3medium S4good mediumgood S5goodmediumgood S6good S7bad S8bad
mediumbad Slide 68 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}
StudentMathematics (M)Physics (Ph)Literature (L)Overall class
S1goodmediumbad S2medium badmedium S3medium S4good mediumgood
S5goodmediumgood S6good S7bad S8bad mediumbad Slide 69 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} StudentMathematics (M)Physics (Ph)Literature
(L)Overall class S1goodmediumbad S2medium badmedium S3medium S4good
mediumgood S5goodmediumgood S6good S7bad S8bad mediumbad Slide 70
70 Rough Set approach to multiple-criteria sorting Let us represent
the students in condition space {M,L} : bad medium good medium
Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3 S8
Slide 71 71 Rough Set approach to multiple-criteria sorting
Dominance cones in condition space {M,L} : P={M,L} bad medium good
medium Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3
S8 Slide 72 72 Dominance cones in condition space {M,L} : P={M,L}
bad medium good medium Mathematics Literature good medium bad S5,S6
S7 S2 S1 S4 S3 S8 Rough Set approach to multiple-criteria sorting
Slide 73 73 Dominance cones in condition space {M,L} : P={M,L}
Rough Set approach to multiple-criteria sorting bad medium good
medium Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3
S8 Slide 74 74 Lower approximation of at least good students:
P={M,L} Rough Set approach to multiple-criteria sorting bad medium
good medium Mathematics Literature good medium bad S5,S6 S7 S2 S1
S4 S3 S8 Slide 75 75 Upper approximation of at least good students:
P={M,L} Rough Set approach to multiple-criteria sorting bad medium
good medium Mathematics Literature good medium bad S5,S6 S7 S2 S1
S4 S3 S8 Slide 76 76 Lower approximation of at least medium
students: P={M,L} Rough Set approach to multiple-criteria sorting
bad medium good medium Mathematics Literature good medium bad S5,S6
S7 S2 S1 S4 S3 S8 Slide 77 77 Upper approximation of at least
medium students: P={M,L} Rough Set approach to multiple-criteria
sorting bad medium good medium Mathematics Literature good medium
bad S5,S6 S7 S2 S1 S4 S3 S8 Slide 78 78 Boundary region of at least
medium students: P={M,L} Rough Set approach to multiple-criteria
sorting bad medium good medium Mathematics Literature good medium
bad S5,S6 S7 S2 S1 S4 S3 S8 Slide 79 79 Lower approximation of at
most medium students: P={M,L} Rough Set approach to
multiple-criteria sorting bad medium good medium Mathematics
Literature good medium bad S5,S6 S7 S2 S1 S4 S3 S8 Slide 80 80
Upper approximation of at most medium students: P={M,L} Rough Set
approach to multiple-criteria sorting bad medium good medium
Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3 S8
Slide 81 81 Lower approximation of at most bad students: P={M,L}
Rough Set approach to multiple-criteria sorting bad medium good
medium Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3
S8 Slide 82 82 Upper approximation of at most bad students: P={M,L}
Rough Set approach to multiple-criteria sorting bad medium good
medium Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3
S8 Slide 83 83 Boundary region of at most bad students: P={M,L}
Rough Set approach to multiple-criteria sorting bad medium good
medium Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3
S8 Slide 84 84 Rough Set approach to multiple-criteria sorting
Decision rules in terms of {M,L} : D > - certain rule bad medium
good medium Mathematics Literature good medium bad S5,S6 S7 S2 S1
S4 S3 S8 If M good & L medium, then student good Slide 85 85
Decision rules in terms of {M,L} : D > - certain rule Rough Set
approach to multiple-criteria sorting bad medium good medium
Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3 S8 If M
medium & L medium, then student medium Slide 86 86 Decision
rules in terms of {M,L} : D >< - approximate rule Rough Set
approach to multiple-criteria sorting bad medium good medium
Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3 S8 If M
medium & L bad, then student is bad or medium Slide 87 87
Decision rules in terms of {M,L} : D < - certain rule Rough Set
approach to multiple-criteria sorting bad medium good medium
Mathematics Literature good medium bad S5,S6 S7 S2 S1 S4 S3 S8 If M
medium, then student medium Slide 88 88 Decision rules in terms of
{M,L} : D < - certain rule Rough Set approach to
multiple-criteria sorting bad medium good medium Mathematics
Literature good medium bad S5,S6 S7 S2 S1 S4 S3 S8 If L bad, then
student medium Slide 89 89 Decision rules in terms of {M,L} : D
< - certain rule Rough Set approach to multiple-criteria sorting
bad medium good medium Mathematics Literature good medium bad S5,S6
S7 S2 S1 S4 S3 S8 If M bad, then student bad Slide 90 90 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} Rough Set approach to
multiple-criteria sorting Slide 91 91 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
Rough Set approach to multiple-criteria sorting Slide 92 92 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} Slide 93 93 Rough Set approach to
multiple-criteria sorting Quality of approximation of sorting
students C (Cl)= [8-card({S1,S2})]/8 = 0.75 Slide 94 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 Sowiski, 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, 1005-1035 Slide 95 95 Representation
axiom (cancellation property): for every dimension i=1,,n, for
every evaluation x i,y i X i and a -i,b -i X -i, and for every pair
of decision classes Cl r,Cl s {Cl 1,,Cl m }: {x i a -i Cl r and y i
b -i Cl s } {y i a -i or x i b -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 g i, 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., Sowiski, 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 Critre la Dcision
Multiple Criteria Decision Aiding, European Commission Report EUR
19808 EN, Joint Research Centre, Ispra, 2001, pp. 117-144 DRSA
preference modeling by decision rules Slide 96 96 Value-driven
methods The preference model is a utility function U and a set of
thresholds z t, t=1,,p-1, on U, separating the decision classes Cl
t, t=0,1,,p A value of utility function U is calculated for each
action aA e.g. aCl 2, dCl p1 Comparison of decision rule preference
model and utility function U z1z1 z p-2 z p-1 z2z2 Cl 1 Cl 2 Cl p-1
Cl p U[g 1 (a),,g n (a)]U[g 1 (d),,g n (d)] Slide 97 97 ELECTRE TRI
Decision classes Cl t are caracterized by limit profiles b t,
t=0,1,,p The preference model, i.e. outranking relation S, is
constructed for each couple (a, b t ), for every aA and b t,
t=0,1,,p Comparison of decision rule preference model and
outranking relation g1g1 g2g2 g3g3 gngn b0b0 b1b1 b p-2 b p-1 bpbp
b2b2 Cl 1 Cl 2 Cl p-1 Cl p a d Slide 98 98 ELECTRE TRI Decision
classes Cl t are caracterized by limit profiles b t, t=0,1,,p
Compare action a successively to each profile b t, t=p-1,,1,0; if b
t is the first profile such that aSb t, then aCl t+1 e.g. aCl 1,
dCl p1 Comparison of decision rule preference model and outranking
relation g1g1 g2g2 g3g3 gngn b0b0 b1b1 b p-2 b p-1 bpbp b2b2 Cl 1
Cl 2 Cl p-1 Cl p a d comparison of action a to profiles b t Slide
99 99 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.
aCl 2 >, because profile of a dominates partial profiles of r 2
and r 3 Comparison of decision rule preference model and outranking
relation a e.g. for Cl 2 > g1g1 g2g2 g3g3 gngn r1r1 r3r3 r2r2
Slide 100 100 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 experts decisions and support diagnosis of new buses Slide
101 101 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 Slide 102 102 Slide 103 103 Slide 104 104 Slide
105 105 Input (preference) information: Rerefence actions Criteria
Rank in order q1q1 q2q2...qmqm xf(x, q 1 )f(x, q 2 )...f(x, q m
)1st yf(y, q 1 )f(y, q 2 )...f(y, q m )2nd uf(u, q 1 )f(u, q 2
)...f(u, q m )3rd... zf(z, q 1 )f(z, q 2 )...f(z, q m )k-th Pairs
of ref. actions Difference on criteria Preference relation q1q1
q2q2...qmqm (x,y)(x,y) 1(x,y)1(x,y)2(x,y)2(x,y) m(x,y)m(x,y) xSy
(y,x)(y,x) 1(y,x)1(y,x)2(y,x)2(y,x)... m(y,x)m(y,x) yS c x
(y,u)(y,u) 1(y,u)1(y,u)2(y,u)2(y,u)... m(y,u)m(y,u) ySu...
(v,z)(v,z)1(v,z)1(v,z)2(v,z)2(v,z) m(v,z)m(v,z)vS c z S outranking
S c non-outranking i difference on q i DRSA for multiple-criteria
choice and ranking Pairwise Comparison Table (PCT) ARAR BARARBARAR
Slide 106 106 DRSA for multiple-criteria choice and ranking
Dominance for pairs of actions (x,y),(w,z)UU For cardinal criterion
qC : (x,y)D q (w,z) if and where h q k q For subset PC of criteria:
P-dominance relation on pairs of actions: (x,y)D P (w,z) if (x,y)D
q (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 D q is reflexive, transitive, but
not necessarily complete (partial preorder) is a partial preorder
on AA x q q (x,y)D q (w,z) z y w For ordinal criterion q C: Slide
107 107 Let BUU be a set of pairs of reference objects in a given
PCT Granules of knowledge positive dominance cone negative
dominance cone DRSA for multiple-criteria choice and ranking Slide
108 108 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 S c : P-boundaries of S and S c : PC Slide 109 109 DRSA
for multiple-criteria choice and ranking formal definitions Basic
properties: Quality of approximation of S and S c : (S,S c )-reduct
and (S,S c )-core Variable-consistency rough approximations of S
and S c ( l (0,1]) : Slide 110 110 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 xS c y D
-decision rules if x y and x y and... x y and x y and... x y, then
xSy or xS c y DRSA for multiple-criteria choice and ranking
decision rules Slide 111 111 q 1 = 2.3 q 2 = 18.0 q m = 3.0 q 1 =
1.2 q 2 = 19.0 q m = 5.7 q 1 = 4.7 q 2 = 14.0 q m = 7.1 q 1 = 0.5 q
2 = 12.0 q m = 9.0 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 S
c y) Building the Pairwise Comparison Table (PCT) Inducing rules
from rough approximations of relations S and S c x y u z Pairs of
objects Difference of evaluations on each criterion Preference
relation q 1 q 2 q n (x,y) 1 (x,y) = -1.1 2 (x,y) = 1.0... n (x,y)
= 2.7 x Sc yx Sc y (y,z) 1 (y,z) = -2.4 2 (y,z) = 4.0... n (y,z) =
-4.1 y S zy 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 S c
z Slide 112 112 Induction of decision rules from rough
approximations of S and S c if q 1 (a,b) -2.4 & q 2 (a,b) 4.0
then aSb if q 1 (a,b) -1.1 & q 2 (a,b) 1.0 then aS c b if q 1
(a,b) 2.4 & q 2 (a,b) -4.0 then aS c b if q 1 (a,b) 4.1 & q
2 (a,b) 1.9 then aSb q 1 q 2 x S c y z S c y u S c y u S c z y S x
y S z y S u t S c q u S z Slide 113 113 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)AA can match the decision
rules in one of four ways: xSy and not xS c y, that is true
outranking (xS T y), xS c y and not xSy, that is false outranking
(xS F y), xSy and xS c y, that is contradictory outranking (xS K
y), not xSy and not xS c y, that is unknown outranking (xS U y). x
y x xx yy y xSTyxSTy xSFyxSFy xSKyxSKy xSUyxSUy S S ScSc ScSc The
4-valued outranking underlines the presence and the absence of
positive and negative reasons of outranking Slide 114 114 DRSA for
multicriteria choice and ranking Net Flow Score x v u y z S S ScSc
ScSc weakness of xstrength of x NFS(x) = strength(x) weakness(x)
(,) (+,) (+,+) (,+) S positive argument S c negative argument Slide
115 115 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)} xAxA Slide 116 116 DRSA for multiple-criteria
choice and ranking example Decision table with reference objects
(warehouses) Slide 117 117 DRSA for multicriteria choice and
ranking example Pairwise comparison table (PCT) Slide 118 118 DRSA
for multicriteria choice and ranking example Quality of
approximation of S and Sc by criteria from set C is 0.44 RED PCT =
CORE PCT = {A 1,A 2,A 3 } D -decision rules and D -decision
rules
