Institute for Advanced Management Systems Research
Department of Information Technologies
Faculty of Technology, Åbo Akademi University
Basics of Soft Computing - Tutorial
Robert Fullér
Directory
• Table of Contents
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c 2013
April 11, 2013
rfuller@abo.fi
Table of Contents
1. Probability, measure theory, and fuzzy sets
2. Fuzzy sets
3. Fuzzy numbers
4. Material implication
5. Fuzzy implications
6. The theory of approximate reasoning
7. Simplified fuzzy reasoning schemes
Table of Contents (cont.)
3
8. Tsukamoto’s and Sugeno’s fuzzy reasoning scheme
9. Bellman and Zadeh’s principle to fuzzy decision making
10. The continuous case
11. Aggregation operators
12. OWA Operators
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1. Probability, measure theory, and fuzzy sets
Question 1. If there a salami sandwich in the refrigerator?
The answer is 0.5.
If probability, then there either is or isn’t, with probability one half:
FIFTY/FIFTY; 50 percent YES and 50 percent NO.
If measure, then there is half a salami sandwich there. Somebody has
already eaten the other half!
If fuzzy, then there is something there, but it isn’t really a salami sandwich.
Perhaps it is some other kind of sandwich, or salami without the bread
or bread without the salami.
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2. Fuzzy sets
Fuzzy sets were introduced by Zadeh in 1965 to represent/manipulate
data and information possessing nonstatistical uncertainties.
nonstatistical uncertainties
It was specifically designed to mathematically represent uncertainty
and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems.
Fuzzy logic provides an inference morphology that enables approximate human reasoning capabilities to be applied to knowledge-based
systems. The theory of fuzzy logic provides a mathematical strength
to capture the uncertainties associated with human cognitive processes,
such as thinking and reasoning.
approximate reasoning
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Section 2: Fuzzy sets
6
Some of the essential characteristics of fuzzy logic relate to the following
(Zadeh, 1992):
• In fuzzy logic, exact reasoning is viewed as a limiting
case of approximate reasoning.
• In fuzzy logic, everything is a matter of degree.
• In fuzzy logic, knowledge is interpreted a collection of
elastic or, equivalently, fuzzy constraint on a collection
of variables.
• Inference is viewed as a process of propagation of elastic constraints.
• Any logical system can be fuzzified.
There are two main characteristics of fuzzy systems that give them better performance for specific applications.
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Section 2: Fuzzy sets
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• Fuzzy systems are suitable for uncertain or approximate reasoning, especially for the system with a mathematical model that is
difficult to derive.
• Fuzzy logic allows decision making with estimated values under
incomplete or uncertain information.
In classical set theory, a subset A of a set X can be defined by its characteristic function χA as a mapping from the elements of X to the elements of the set {0, 1},
χA : X → {0, 1}.
The value zero is used to represent non-membership, and the value one
is used to represent membership. The truth or falsity of the statement
”x is in A”
is determined by χA (x).
The statement is true if χA (x) = 1, and the statement is false if it is 0.
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Figure 1: Characteristic function of A = [a, b].
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χA (t) =
(
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0
if a ≤ t ≤ b
otherwise
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Section 2: Fuzzy sets
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Definition 2.1. (Zadeh, 1965) Let X be a nonempty set. A fuzzy set A in X
is characterized by its membership function
µA : X → [0, 1]
and µA (x) is interpreted as the degree of membership of element x in fuzzy
set A for each x ∈ X. Frequently we will write simply A(x) instead.
The value zero is used to represent complete non-membership, the value
one is used to represent complete membership, and values in between
are used to represent intermediate degrees of membership.
degree of satisfaction to a property
The degree to which the statement ”x is A” is true is determined by
A(x).
degree of truth
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Section 3: Fuzzy numbers
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3. Fuzzy numbers
A fuzzy set A of the real line R is defined by its membership function
(denoted also by A) A : R → [0, 1]. If x ∈ R then A(x) is interpreted as
the degree of membership of x in A.
Example 3.1. Assume someone wants to buy a cheap car. Cheap car can be
represented as a fuzzy set on a universe of prices, and depends on his purse.
For instance, from the Figure ’cheap car’ is roughly interpreted as follows:
• Below 3000$ cars are considered as cheap, and prices make no real
difference to buyer’s eyes.
• Between 3000$ and 4500$, a variation in the price induces a weak
preference in favor of the cheapest car.
• Between 4500$ and 6000$, a small variation in the price induces a
clear preference in favor of the cheapest car.
• Beyond 6000$ the costs are too high (out of consideration).
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Figure 2: Membership function for ”cheap car”.
Example 3.2. The membership function of the fuzzy set of real numbers
”close to 1”, is can be defined as
A(t) = exp(−β(t − 1)2 )
where β is a positive real number.
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Section 3: Fuzzy numbers
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Figure 3: A membership function for ”x is close to 1”.
Definition 3.1. A fuzzy set A is called triangular fuzzy number with peak
(or center) a, left width α > 0 and right width β > 0 if its membership
function has the following form


 1 − a − t if a − α ≤ t ≤ a



α

t
−
a
A(t) =
if a ≤ t ≤ a + β
1−



β



0
otherwise
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Section 3: Fuzzy numbers
13
and we use the notation A = (a, α, β). A triangular fuzzy number with
center a may be seen as a fuzzy quantity
”x is close to a” or ”x is approximately equal to a”.
4
1. the
Fuzzyproperty
Sets and Fuzzy
x = a satisfies
”aLogic
is close to a” with degree one
[A] = ”a
[a1 (γ),
a2 (γ)].
x = a − α satisfies the property
−α
is close to a” with degree zero
γ
The support of A is the open interval (a1 (0), a2 (0)).
A is not athe
fuzzyproperty
number then
an γ ∈to
[0,a”
1] such
thatdegree
[A]γ
x = a + β Ifsatisfies
”athere
+ βexists
is close
with
zero
is not a convex subset of R.
1
a-!
a
a+"
Fig. 1.2. Triangular fuzzy number.
Figure 4: A triangular fuzzy number.
Definition 1.1.4 A fuzzy set A is called triangular fuzzy number with peak
width α > 0 and right width β > 0 if its membership
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The support
of A is (a − α, b + β).
Section 3: Fuzzy
numbers
14
1
a-!
a
b
b+"
Fig. 1.3. Trapezoidal fuzzy number.
Figure 5: Trapezoidal fuzzy number.
A trapezoidal fuzzy number may be seen as a fuzzy quantity
”x is approximately in the interval [a, b]”.
DefinitionDefinition
3.2. A 1.1.6
fuzzyAny
setfuzzy
A isnumber
called
fuzzy
A ∈trapezoidal
F can be described
as number if its mem % form
&
bership function has the following
a−t

 

if t ∈ [a − α, a]
L


α t
a
−





 1
−b] α ≤ t ≤ a
−

1
if if
t ∈a[a,


% α &
A(t)
=



t − b)

 1

≤b
if if
t ∈a[b,≤
b +t β]

R
β


A(t) =




0 t − b otherwise


1−
if a ≤ t ≤ b + β


β

where [a, b] is the peak
or
core
of A,



L : [0,01] → [0, 1], R : otherwise
[0, 1] → [0, 1]
are continuous and non-increasing shape functions with L(0) = R(0) = 1 and
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= L(1) = 0. II
We call this fuzzy
of LR-typeBack
and refer to
by
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Section 3: Fuzzy numbers
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and we use the notation A = (a, b, α, β). A trapezoidal fuzzy number may
be seen as a fuzzy quantity ”x is approximately in the interval [a, b]”.
Figure 6: Membership functions for monthly ”small salary” and ”big
salary”.
If x0 is the amount of the salary then x0 belongs to fuzzy set
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A1 = small
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Section 3: Fuzzy numbers
16
with degree of membership
A1 (x0 ) =
and to


1−
 0
x0 − 2000
4000
if 2000 ≤ x0 ≤ 6000
otherwise
A2 = big
with degree of membership


1


6000 − x0
A2 (x0 ) =
1−

4000

 0
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if x0 ≥ 6000
if 2000 ≤ x0 ≤ 6000
otherwise
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Section 4: Material implication
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4. Material implication
Let p = ’x is in A’ and q = ’y is in B’ are crisp propositions, where A and
B are crisp sets for the moment. The full interpretation of the material
implication p → q is that: the degree of truth of p → q quantifies to
what extend q is at least as true as p, i.e.
1 if τ (p) ≤ τ (q)
τ (p → q) =
0 otherwise
τ (p)
τ (q)
1
0
0
1
1
1
0
0
τ (p → q)
1
1
1
0
Table 1: Truth table for the material implication.
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Section 5: Fuzzy implications
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5. Fuzzy implications
Consider the implication statement
if pressure is high then volume is small
Figure 7: Membership function for ”big pressure”.
The membership function of the fuzzy set A, big pressure, can be interpreted as
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Section 5: Fuzzy implications
19
• 1 is in the fuzzy set big pressure with grade of membership 0
• 4 is in the fuzzy set big pressure with grade of membership 0.75
• x is in the fuzzy set big pressure with grade of membership 1, x ≥ 5

1



if u ≥ 5
5−u
A(u) =
1−
if 1 ≤ u ≤ 5

4


0
otherwise
The membership function of the fuzzy set B, small volume, can be interpreted as

1
if v ≤ 1



v−1
B(v) =
1−
if 1 ≤ v ≤ 5

4


0
otherwise
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Section 5: Fuzzy implications
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Figure 8: Membership function for ”small volume”.
• 5 is in the fuzzy set small volume with grade of membership 0
• 2 is in the fuzzy set small volume with grade of membership 0.75
• x is in the fuzzy set small volume with grade of membership 1,
x≤1
If p is a proposition of the form ’x is A’ where A is a fuzzy set, for
example, big pressure and q is a proposition of the form ’y is B’ for
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Section 5: Fuzzy implications
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example, small volume then we define the implication p → q as
A(x) → B(y)
For example,
x is big pressure → y is small volume ≡ A(x) → B(y)
Remembering the full interpretation of the material implication
1 if τ (p) ≤ τ (q)
p→q=
0 otherwise
We can use the definition
A(x) → B(y) =
1
0
if A(x) ≤ B(y)
otherwise
4 is big pressure → 1 is small volume = 0.75 → 1 = 1
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Section 6: The theory of approximate reasoning
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6. The theory of approximate reasoning
In 1979 Zadeh introduced the theory of approximate reasoning. This
theory provides a powerful framework for reasoning in the face of imprecise and uncertain information.
How to make inferences in fuzzy environment?
<1 :
if
observation
pressure is BIG then
pressure is 4
conclusion
<1 :
volume is ?
if
observation
pressure is BIG then
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volume is SMALL
pressure is 4
conclusion
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volume is SMALL
volume is 2
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Figure 9: BIG(4) = SMALL(2) = 0.75.
7. Simplified fuzzy reasoning schemes
Suppose that we have the following rule base
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Section 7: Simplified fuzzy reasoning schemes
<1 :
also
<2 :
<n :
fact:
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if
x is A1 then
y is z1
if
x is A2 then
............
x is An then
y is z2
if
y is zn
x is x0
y is z0
action:
where A1 , . . . , An are fuzzy sets.
Suppose further that our data base consists of a single fact x0 . The
problem is to derive z0 from the initial content of the data base, x0 , and
from the fuzzy rule base < = {<1 , . . . , <n }.
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Section 7: Simplified fuzzy reasoning schemes
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<1 :
if
salary is small then
loan is z1
<2 :
if
salary is big then
loan is z2
also
salary is x0
fact:
loan is z0
action:
A deterministic rule base can be formed as follows
<1 :
if
2000 ≤ s ≤ 6000 then
loan is max 1000
<3 :
if
s ≥ 6000 then
loan is max 2000
<4 :
if
s ≤ 2000 then
no loan at all
The data base contains the actual salary, and then one of the rules is
applied to obtain the maximal loan can be obtained by the applicant.
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Section 7: Simplified fuzzy reasoning schemes
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Figure 10: Discrete causal link between ”salary” and ”loan”.
In fuzzy logic everything is a matter of degree.
If x is the amount of the salary then x belongs to fuzzy set
• A1 = small with degree of membership 0 ≤ A1 (x) ≤ 1
• A2 = big with degree of membership 0 ≤ A2 (x) ≤ 1
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Figure 11: Membership functions for ”small” and ”big”.
In fuzzy rule-based systems each rule fires.
The degree of match of the input to a rule (which is the firing strength)
is the membership degree of the input in the fuzzy set characterizing
the antecedent part of the rule.
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Section 7: Simplified fuzzy reasoning schemes
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The overall system output is the weighted average of the individual rule
outputs, where the weight of a rule is its firing strength with respect to
the input.
To illustrate this principle we consider a very simple example mentioned above
<1 :
if
salary is small then
loan is z1
<2 :
if
salary is big then
loan is z2
also
fact:
salary is x0
loan is z0
action:
Then our reasoning system is the following
• input to the system is x0
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Section 7: Simplified fuzzy reasoning schemes
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Figure 12: Example of simplified fuzzy reasoning.
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30
• the firing level of the first rule is α1 = A1 (x0 )
• the firing level of the second rule is α2 = A2 (x0 )
• the overall system output is computed as the weighted average of
the individual rule outputs
α1 z1 + α2 z2
z0 =
α1 + α2
that is
A1 (x0 )z1 + A2 (x0 )z2
z0 =
A1 (x0 ) + A2 (x0 )
A1 (x0 ) =
1 − (x0 − 2000)/4000
0

 1
1 − (6000 − x0 )/4000
A2 (x0 ) =

0
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if 2000 ≤ x0 ≤ 6000
otherwise
if x0 ≥ 6000
if 2000 ≤ x0 ≤ 6000
otherwise
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31
Figure 13: Example of simplified fuzzy reasoning.
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It is easy to see that the relationship
A1 (x0 ) + A2 (x0 ) = 1
holds for all x0 ≥ 2000.
It means that our system output can be written in the form.
z0 = α1 z1 + α2 z2 = A1 (x0 )z1 + A2 (x0 )z2
that is,
z0 =
6000 − x0
x0 − 2000
z1 + 1 −
z2
1−
4000
4000
if 2000 ≤ x0 ≤ 6000.
And z0 = 1 if x0 ≥ 6000. And z0 = 0 if x0 ≤ 2000.
The (linear) input/oputput relationship is illustrated in figure 14.
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Figure 14: Input/output function derived from fuzzy rules.
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Section 8: Tsukamoto’s and Sugeno’s fuzzy reasoning scheme
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8. Tsukamoto’s and Sugeno’s fuzzy reasoning scheme
Tsukamoto’s reasoning scheme
<1 :
also
<2 :
fact :
if x is A1 and y is B1 then
z is C1
if x is A2 and y is B2 then
z is C2
x is x̄0 and y is ȳ0
cons. :
z is z0
Sugeno and Takagi use the following architecture
<1 :
if x is A1 and y is B1 then
z1 = a1 x + b1 y
<2 :
if x is A2 and y is B2 then
z2 = a2 x + b2 y
also
fact :
x is x̄0 and y is ȳ0
cons. :
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Section 8: Tsukamoto’s and Sugeno’s fuzzy reasoning scheme
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Figure 15: An illustration of Tsukamoto’s inference mechanism. The firing
level of the first rule: α1 = min{A1 (x0 ), B1 (y0 )} = min{0.7, 0.3} =
0.3, The firing level of the second rule: α2 = min{A2 (x0 ), B2 (y0 )} =
min{0.6, 0.8} = 0.6, The crisp inference: z0 = (8 × 0.3 + 4 × 0.6)/(0.3 +
0.6) = 6.
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Section 8: Tsukamoto’s and Sugeno’s fuzzy reasoning scheme
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Figure 16: Example of Sugeno’s inference mechanism. The overall system
output is computed as the firing-level-weighted average of the individual
rule outputs: z0 = (5 × 0.2 + 4 × 0.6)/(0.2 + 0.6) = 4.25.
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9. Bellman and Zadeh’s principle to fuzzy decision making
A classical MADM problem can be expressed in a matrix format. The
decision matrix is an m × n matrix whose element xij indicates the
performance rating of the i-th alternative, ai , with respect to the j-th
attribute, Xj ,


a1
x11 x12 . . . x1n



a2 
 x21 x22 . . . x2n 


..  ..
..
.. 

. 
.
.
.


am
xm1 xm2 . . . xmn
The classical maximin method is defined as: choose ak such that
sk = max si = max min xij .
i=1,...,m
i
j
where si the security level of ai , i.e. ai guarantees the decision maker a
return of at least si .
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Section 9: Bellman and Zadeh’s principle to fuzzy decision making
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Example 9.1.
M athematics
English
History
John
5
8
10
M ary
6
6
7
Kate
10
7
5
Mary is selected if we use the maximin method, because her minimal performance is the maximal.
The overall performance of an alternative is determined by its weakest
performance:
A chain is only as strong as its weakest
link.
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Section 9: Bellman and Zadeh’s principle to fuzzy decision making
39
The classical maximax return criterion is: choose ak such that
ok = max oi = max max xij .
i=1,...,m
i
j
where
oi = max xij .
j=1,...,n
is the best performance of ai .
Example 9.2.
M ath
English
History
John
5
8
10
M ary
6
6
7
Kate
10
7
5
John or Kate are selected if we use the maximax method, because their
maximal performances provide the global maximum.
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Section 9: Bellman and Zadeh’s principle to fuzzy decision making
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In the fuzzy case the values of the decision matrix are given as degrees
of ”how an alternative satisfies a certain attribute”. For each attribute
Xj we are given a fuzzy set µj measuring the degrees of satisfaction to
the j-th attribute.
(
1 − (8 − t)/8 if t ≤ 8
µM ath (t) =
1
otherwise
Figure 17: Membership function for attribute ’Math’.
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Section 9: Bellman and Zadeh’s principle to fuzzy decision making
µEnglish (t) =
(
1 − (7 − t)/7
1
41
if t ≤ 7
otherwise
Figure 18: Membership function for attribute ’English’.
µHistory (t) =
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1 − (6 − t)/6
if t ≤ 6
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Section 9: Bellman and Zadeh’s principle to fuzzy decision making
42
Figure 19: Membership function for attribute ’History’.
John
M ary
Kate
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M ath
English
History
5/8
1
1
6/8
6/7
1
1
1
5/6
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Section 9: Bellman and Zadeh’s principle to fuzzy decision making
43
Bellman and Zadeh’s principle to fuzzy decision making chooses the
”best compromise” alternative using the maximin method:
score(John) = min{5/8, 1, 1} = 5/8
score(Mary) = min{6/8, 6/7, 1} = 6/8
score(Kate) = min{1, 1, 5/6} = 5/6
Since
score(Kate) = max{5/8, 6/8, 5/6}.
Kate is chosen as the best student.
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10. The continuous case
If we have a continuous problem, which means that we have infinite
number of alternatives then suppose that
C1 , C2 , . . . , Cn
are the membership functions corresponding to attributes X1 , X2 , . . . ,
Xn , respectively.
Let x be an object such that for any criterion Cj , Cj (x) ∈ [0, 1] indicates
the degree to which this criterion is satisfied by x.
If we want to find out the degree to which x satisfies
’all the criteria’
denoting this by D(x), we get following
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Section 10: The continuous case
45
D(x) = min{C1 (x), . . . , Cn (x)}.
In this case we are essentially requiring x to satisfy
C1 and C2 and . . . and Cn .
The best alternative, denoted by x∗ , is determined from the relationship
D(x∗ ) = max D(x)
x∈X
This method is called Bellman and Zadeh’s principle to fuzzy decision
making.
The first attribute called ”x should be close to 3” is represented by the
fuzzy set C1 defined by
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Section 10: The continuous case
46
Figure 20: The fuzzy attributes.
C1 (x) =
(
1 − |3 − x|/2
0
if |3 − x| ≤ 2
otherwise
The second attribute called ”x should be close to 5” is represented by
the fuzzy set C2 defined by
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Section 10: The continuous case
C2 (x) =
47
(
1 − |5 − x|/2
0
if |5 − x| ≤ 2
otherwise
The fuzzy decision D is defined by the intersection of C1 and C2 and
the membership function of D is
D(x) =
(
1/2(1 − |4 − x|) if |4 − x| ≤ 1
0
otherwise
The optimal solution is
x∗ = 4
since
D(4) = 1/2 = max D(x).
x
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11. Aggregation operators
In a decision process the idea of trade-offs corresponds to viewing the
global evaluation of an action as lying between the worst and the best
local ratings. This occurs in the presence of conflicting goals, when a
compensation between the corresponding performances is allowed. Averaging operators realize trade-offs between objectives, by allowing a
positive compensation between ratings. An averaging (or mean) operator M is a function M : [0, 1] × [0, 1] → [0, 1], satisfying the following
properties
M (x, x) = x, ∀x ∈ [0, 1], (idempotency)
M (x, y) = M (y, x), ∀x, y ∈ [0, 1], (commutativity)
M (0, 0) = 0, M (1, 1) = 1, (extremal conditions)
M (x, y) ≤ M (x0 , y 0 ) if x ≤ x0 and y ≤ y 0 (monotonicity)
M is continuous (continuity).
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49
Lemma 1. If M is an averaging operator then
min{x, y} ≤ M (x, y) ≤ max{x, y},
for all x, y ∈ [0, 1].
Proof. From idempotency and monotonicity of M it follows that
min{x, y} = M (min{x, y}, min{x, y})
≤ M (x, y)
and
M (x, y) ≤ M (max{x, y}, max{x, y})
= max{x, y}.
Which ends the proof.
The most often used mean operators.
2xy
x+y
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harmonic mean:
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Section 11: Aggregation operators
√
geometric mean:
arithmetic mean:
xy
x+y
2
dual of geometric mean: 1 −
dual of harmonic mean:
median
50
p
(1 − x)(1 − y)
x + y − 2xy
2−x−y


 y
α
med (x, y, α) =


x
if x ≤ y ≤ α
if x ≤ α ≤ y
if α ≤ x ≤ y
generalized p-mean:
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Section 11: Aggregation operators
51
xp + y p
2
1/p
, p ≥ 1.
The process of information aggregation appears in many applications
related to the development of intelligent systems.
One sees aggregation in neural networks, fuzzy logic controllers, vision
systems, expert systems and multi-criteria decision aids. In
• R.R.Yager, Ordered weighted averaging aggregation operators in
multi-criteria decision making, IEEE Trans. on Systems, Man and
Cybernetics, 18(1988) 183-190.
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52
12. OWA Operators
Yager introduced a new aggregation technique based on the ordered
weighted averaging (OWA) operators.
Definition 12.1. An OWA operator of dimension n is a mapping
F : Rn → R,
that has an associated n vector
W = (w1 , w2 , . . . , wn )T ,
such as wi ∈ [0, 1], 1 ≤ i ≤ n,
n
X
wi = 1.
i=1
Furthermore
F (a1 , . . . , an ) =
n
X
wj bj
j=1
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Section 12: OWA Operators
53
where bj is the j-th largest element of the bag
< a1 , . . . , an > .
Example 12.1. Assume
W = (0.4, 0.3, 0.2, 0.1)T
then
F (0.7, 1, 0.2, 0.6) = 0.4 × 1
+0.3 × 0.7 + 0.2 × 0.6 + 0.1 × 0.2 = 0.75.
A fundamental aspect of this operator is the re-ordering step, in particular an aggregate ai is not associated with a particular weight wi
but rather a weight is associated with a particular ordered position of
aggregate.
When we view the OWA weights as a column vector we shall find it
convenient to refer to the weights with the low indices as weights at the
top and those with the higher indices with weights at the bottom.
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It is noted that different OWA operators are distinguished by their
weighting function.
Yager pointed out three important special cases of OWA aggregations:
• F ∗ : In this case W = W ∗ = (1, 0 . . . , 0)T and
F ∗ (a1 , . . . , an ) = max{a1 , . . . , an },
• F∗ : In this case W = W∗ = (0, 0 . . . , 1)T and
F∗ (a1 , . . . , an ) = min{a1 , . . . , an },
• FA : In this case
W = WA = (1/n, . . . , 1/n)T
and
n
FA (a1 , . . . , an ) =
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ai .
n i=1
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Section 12: OWA Operators
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A number of important properties can be associated with the OWA operators. We shall now discuss some of these.
For any OWA operator F
F∗ (a1 , . . . , an ) ≤ F (a1 , . . . , an ) ≤ F ∗ (a1 , . . . , an ).
Thus the upper an lower star OWA operator are its boundaries.
From the above it becomes clear that for any F
max{a1 , . . . , an } ≤ F (a1 , . . . , an ) ≤ max{a1 , . . . , an }.
The OWA operator can be seen to be commutative.
Let {a1 , . . . , an } be a bag of aggregates and let
{d1 , . . . , dn }
be any permutation of the ai . Then for any OWA operator
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Section 12: OWA Operators
56
F (a1 , . . . , an ) = F (d1 , . . . , dn ).
A second characteristic associated with these operators is monotonicity.
Assume ai and ci are a collection of aggregates, i = 1, . . . , n such that
for each i, ai ≥ ci .
Then
F (a1 , . . . , an ) ≥ F (c1 , c2 , . . . , cn )
where F is some fixed weight OWA operator.
Another characteristic associated with these operators is idempotency.
If ai = a for all i then for any OWA operator
F (a1 , . . . , an ) = F (a, . . . , a) = a.
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Section 12: OWA Operators
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From the above we can see the OWA operators have the basic properties
associated with an averaging operator.
Example 12.2. A window type OWA operator takes the average of the m
arguments about the center. For this class of operators we have

if i < k
 0
1/m if k ≤ i < k + m
wi =

0
if i ≥ k + m
In order to classify OWA operators in regard to their location between
and and or, a measure of orness, associated with any vector W is introduce by Yager as follows
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Section 12: OWA Operators
58
n
orness(W ) =
1
n−1
1 X
(n − i)wi
n − 1 i=1
orness(W ) =
× (n − 1)w1 + · · · + wn−1
orness(W ) =
w1 +
n−2
1
× w2 + · · · +
× wn−1
n−1
n−1
It is easy to see that for any W the orness(W ) is always in the unit
interval.
Furthermore, note that the nearer W is to an or, the closer its measure
is to one; while the nearer it is to an and, the closer is to zero.
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Lemma 2. Let us consider the the vectors
W ∗ = (1, 0 . . . , 0)T ,
W∗ = (0, 0 . . . , 1)T ,
WA = (1/n, . . . , 1/n)T .
Then
orness(W ∗ ) = 1,
orness(W∗ ) = 0
orness(WA ) = 0.5
Proof.
orness(W ∗ ) =
1
(n − 1)w1 + · · · + wn−1 =
n−1
1
(n − 1) + · · · + 0 = 1.
n−1
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Section 12: OWA Operators
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orness(W∗ ) =
=
1
(n − 1)w1 + · · · + wn−1
n−1
1
0 + · · · + 0 = 0.
n−1
orness(WA ) =
1
n−1
1
+ ··· +
=
n−1
n
n
n(n − 1)
= 0.5.
2n(n − 1)
A measure of andness is defined as
andness(W ) = 1 − orness(W )
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Generally, an OWA opeartor with much of nonzero weights near the
top will be an orlike operator,
orness(W ) ≥ 0.5
and when much of the weights are nonzero near the bottom, the OWA
operator will be andlike operator
andness(W ) ≥ 0.5.
Example 12.3. Let W = (0.8, 0.2, 0.0)T . Then
1
orness(W ) = (2 × 0.8 + 0.2)
2
= 0.8 + 1/2 × 0.2 = 0.9
and
andness(W ) = 1 − orness(W )
= 1 − 0.9 = 0.1.
This means that the OWA operator, defined by
F (a1 , a2 , a3 ) = 0.8b1 + 0.2b2 + 0.0b3
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Section 12: OWA Operators
62
= 0.8b1 + 0.2b2
where bj is the j-th largest element of the bag
< a1 , a2 , a3 >,
is an orlike aggregation.
Suppose we have n appliccants for a Ph.D. program. Each application is
evaluated by experts, who provides ratings on each of the criteria from
the set
• 3
(high)
• 2
(medium)
• 1
(low)
Compensative connectives have the property that a higher degree of
satisfaction of one of the criteria can compensate for a lower degree of
satisfaction of another criteria.
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Oring the criteria means full compensation and Anding the criteria means
no compensation.
We illustrate the effect of compensation rate on the overall rating:
Let us have the following ratings
(3, 2, 1)
If w = (w1 , w2 , w3 ) is an OWA weight then
1
orness(w1 , w2 , w3 ) = w1 + w2 .
2
Min operator: the overalling rating is
min{3, 2, 1} = 1
in this case there is no compensation, because
orness(0, 0, 1) = 0.
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64
Max operator: the overalling rating is
max{3, 2, 1} = 3
in this case there is full compensation, because
orness(1, 0, 0) = 1.
Mean operator: the overalling rating is
1
(3 + 2 + 1) = 2
3
in this case the measure of compensation is 0.5, because
orness(1/3, 1/3, 1/3) = 1/3 + 1/2 × 1/3 = 1/2.
An andlike operator: the overalling rating is
0.2 × 3 + 0.1 × 2 + 0.7 × 1 = 1.5
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65
in this case the measure of compensation is 0.25, because
orness(0.2, 0.1, 0.7) = 0.2 + 1/2 × 0.1 = 0.25
An orlike operator: the overalling rating is
0.6 × 3 + 0.3 × 2 + 0.1 × 1 = 2.5
in this case the measure of compensation is 0.75, because
orness(0.6, 0.3, 0.1) = 0.6 + 1/2 × 0.3 = 0.75
Yager defined the measure of dispersion (or entropy) of an OWA vector
by
disp(W ) = −
X
wi ln wi .
i
We can see when using the OWA operator as an averaging operator
Disp(W ) measures the degree to which we use all the aggregates equally.
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