New view of fuzzy aggregations. Part III: extensions of the FPOWA operator in the problem of political management

Corresponding Author: gia.sirbiladze@tsu.ge
http://dx.doi.org/10.22105/jfea.2021.275097.1082
E-ISSN: 2717-3453 | P-ISSN: 2783-1442
|
Abstract
1 | Introduction
In this paper we continue the research concerned with quantitative-information analysis of the
complex uncertainty and its use for modelling of more precise decisions with minimal decision risks
from the point of view of systems approach. We continue the construction of new generalizations of
OWA-type operators in fuzzy-probabilistic uncertainty [13], [14], which condense both characteristics
of incomplete information - an uncertainty measure and an imprecision variable in the scalar ranking
values of possible alternatives in the decision-making system. In the Part I of this work the definition
of the OWA operator ([15]-[17], [25]-[30], [34]-[36] and others) and some of its extensions–POWA
and FPOWA operators were presented. In this work our focus is directed to the construction of new
generalizations of the FPOWA operator described in the Section 1 of Part I (Definition 5).
Journal of Fuzzy Extension and Applications
www.journal-fea.com
J. Fuzzy. Ext. Appl. Vol. 2, No. 4 (2021) 321–333.
Paper Type: Research Paper
New View of Fuzzy Aggregations. Part III: Extensions of
the FPOWA Operator in the Problem of Political Management
Gia Sirbiladze*
Department of Computer Sciences, Tbilisi State University, Georgia; gia.sirbiladze@tsu.ge.
Citation:
Sirbiladze, G. (2021). New view of fuzzy aggregations. Part III: extensions of the FPOWA operator
in the problem of political management. Journal of fuzzy extension and application, 2 (4), 321-333.
Accept: 27/05/2021
Revised: 09/04/2021
Reviewed: 13/04/2021
Received: 25/02/2021
The Ordered Weighted Averaging (OWA) operator was introduced by Yager [34] to provide a method for aggregating
inputs that lie between the max and min operators. In this article we continue to present some extensions of OWA-type
aggregation operators. Several variants of the generalizations of the fuzzy-probabilistic OWA operator-FPOWA
(introduced by Merigo [13], [14]) are presented in the environment of fuzzy uncertainty, where different monotone
measures (fuzzy measure) are used as uncertainty measures. The considered monotone measures are: possibility measure,
Sugeno 
λ additive measure, monotone measure associated with Belief Structure and Choquet capacity of order two.
New aggregation operators are introduced: AsFPOWA and SA-AsFPOWA. Some properties of new aggregation
operators and their information measures are proved. Concrete faces of new operators are presented with respect to
different monotone measures and mean operators. Concrete operators are induced by the Monotone Expectation
(Choquet integral) or Fuzzy Expected Value (Sugeno Integral) and the Associated Probability Class (APC) of a monotone
measure. New aggregation operators belong to the Information Structure I6 (see Part I, Section 3). For the illustration
of new constructions of AsFPOWA and SA-AsFPOWA operators an example of a fuzzy decision-making problem
regarding the political management with possibility uncertainty is considered. Several aggregation operators (“classic”
and new operators) are used for the comparing of the results of decision making.
Keywords: Mean aggregation operators, Fuzzy aggregations, Fuzzy measure, Fuzzy numbers, Fuzzy decision making.
Licensee Journal
of Fuzzy Extension and
Applications. This article
is an open access article
distributed under the
terms and conditions of
the Creative Commons
Attribution (CC BY)
license
(http://creativecommons.
org/licenses/by/4.0).

322
Sirbiladze|J.
Fuzzy.
Ext.
Appl.
2(4)
(2021)
321-333
In Section 2 new generalizations of the FPOWA operator are presented with respect to different
monotone measures (instead of the probability measure) and different mean operators. New versions
of the FPOWA operator are defined: AsFPOWA operators are induced by the Monotone Expectation
(ME) ([1], [2], [5], [10], [11], [20]-[23] and others) and SA-AsFPOWA operators are induced by the Fuzzy
Expected Value (FEV) ([3], [5], [11], [12], [18], [19], [21], [23], [24] and others). All generalizations are
constructed with respect to different monotone measures ([1]-[12], [18]-[24], [31]-[33] and others). Some
properties of new operators and their information measures [13], [14] are proved. For the illustration of
the applicability of the new generalizations of the FPOWA operator an example of the fuzzy decision-
making problem regarding political management is considered (Section 3), where we study a country
that is planning its fiscal policy for the next year analogously to the example considered by Merigo [14].
But we use the possibility distribution (possibility uncertainty) on the states of nature of decision-making
system instead of probability distribution (probability uncertainty) as considered in [14]. We think our
approach is more natural and applicable then the case presented in [14]. In this example several
aggregation operators are used for the comparing of the results in decision making:
 SEV (Shapely Expected Value) operator, introduced be Yager [29].
 A new operator SEV-FOWA as a weighted combination of SEV and FOWA operators.
 New operators – AsFPOWAmin, AsFPOWAmean, AsFPOWAmax, SA-AsFPOWAmin, SA-AsFPOWAmean
and Sa-AsFPOWAmax operators introduced in Section 2. The resulting table (see Table 8) is presented for
ordering of the policies. The values of Orness parameter are calculated for all presented aggregation operators.
2| Associated Probabilities’ Aggregations in the FPOWA Operator
In this section we construct new aggregations in the FPOWA operator (Definition 5, Part I) by
monotone measure’s associated probabilities (Definition 3, Part II [37]) when the imprecision variable
is presented by the fuzzy triangular numbers, FTNs, (Definition 2, Part I). So, we consider the
Information Structure I6 (Definition 7, Part I).
Let on the states of nature of General Decision-Making System (Definition 7, Part I) be given some
monotone measure as a uncertainty measure of incomplete information and on defined some payoffs
(utilities and so on) which are presented by triangular fuzzy numbers as expert reflections on possible
alternatives. i.e., for every alternative and for every state of nature there exists- positive triangular fuzzy
number as some payoff. So, vector is imprecision values of expert reflections on states of nature with
respect to alternatives.
Using the arithmetic operations on the triangular fuzzy numbers [6], [8], presented in Section 1, Part I,
we may define new aggregations in the FPOWA operator with respect to monotone measures’
associated probabilities.
2.1| AsFPOWA Operators Induced by the ME
Let  
 
 
k
M :Ψ Ψ k m! be some deterministic mean aggregation function with symmetricity,
boundedness, monotonicity and idempotency properties ([29] and Section 1, Part I), where 
Ψ denotes
the set of all positive TNF.
Definition 1. An associated FPOWA operator AsFPOWA of dimension m is mapping
 

m
AsFPOWA :Ψ Ψ , that has an associated objective weighted vector W of dimension m such that

323
New
view
of
fuzzy
aggregations.
Part
III:
extensions
of
the
FPOWA
operator
in
the
problem
of
political
management

j
w (0,1) and



m
j
j 1
w 1 , and some uncertainty measure – monotone measure
 
 
 
 
 
 
S
g : 2 0,1 with
associated probability class    m
σ σ S
P and is defined according to the following formula:
where j
b
% is the jth largest of the 
i
{a },i 1,...,m
% .
Now we consider concrete AsFPOWA operators for concrete mean functions M and induced by the ME.
Definition 2.
I. Let be the-operator dimension of k=m! then
II. Let be the -operator dimension of k=m! then
III. Let be the averaging operator dimension of k=m!, then
IV. Let be the -averaging operator dimension of k=m!, then
The propositions analogous to Propositions 9-12, Part II [37], are true (we omitted this proposition here).
Now we define concrete AsFPOWA operators for concrete monotone measures analogously to Section 3,
Part II [37]. Consider AsFPOWAmaxfor Sugeno λ -additive monotone measure - λ
g . Analogously to Eq.
(37), Part II [37], we have:
 
     
σ σ σ
1 2 k
1 2 m
m m
j j i σ i m
j 1 i 1
m
j j P P P
j 1
AsFPOWA(a ,a ,...,a )
β w b (1 β)M a P s σ S
β w b (1 β)M E a ,E a ,...,E a
 


 
 
 
    
 
 
 
 
 
 
    
 
 
 
 

% % %
% %
% % % %
(1)
 
m
1 2 m
m m
j j i σ i
σ S
j 1 i 1
AsFPOWAmin(a ,a ,...,a )
β w b (1 β)Min a P s .

 

 
 
    
 
 
 
% % %
% %
(2)
 
m
1 2 m
m m
j j i σ i
σ S
j 1 i 1
AsFPOWAmax(a ,a ,...,a )
β w b (1 β)Max a P s

 

 
 
    
 
 
 
% % %
% %
. (3)
 
m
1 2 m
m m
j j i σ i
j 1 σ S i 1
AsFPOWAmean(a ,a ,...,a )
1
β w b (1 β) a P s .
m!
  

 
 
    
 
 
 
% % %
% %
(4)
m
1 2 m
1
m m α
α
j j i σ i
j 1 σ S i 1
AsFPOWAmeanα(a ,a ,...,a )
1
β w b (1 β) { a P (s )} .
m!
  

 
 
    
 
 
  
% % %
% %
(5)

324
Sirbiladze|J.
Fuzzy.
Ext.
Appl.
2(4)
(2021)
321-333
Analogously we may construct the face of the AsFPOWAmin:
Analogously to Section 3, Part II [37] (Eqs. (38)-(39)) we may construct AsFPOWAmin and
AsFPOWAmax operators induced by the belief structure’s associated monotone measure (omitted
here). We also may define some other combinations of different monotone measures and averaging
operator M . So, there exist many cases of Information Structures on the level I6 for the constructions
of the AsFPOWA operator. For example - AsFPOWAmeanα with respect to the belief structure:
Note the information measures of the AsFPOWA operator - Orness, Entropy, Div and Bal ([13], [14],
[26] and others) are defined analogously to Subsection 3.3, Part II [37] (omitted here). We may add the
proposition concerning the dual monotone measures *
g and *
g [1], [20], [23] which is general for the
AsPOWA (see definition 7, Part II [37]) and AsFPOWA operators.
Proposition 1. Let *
g and *
g be dual monotone measures on
 
 
  
 
 
 
S
2 0,1 ; let AsPOWA* and
AsPOWA* (or AsFPOWA* and AsFPOWA*) be AsFPOWA (or AsFPOWA) operators constructed
on the basis of the measures *
g and 
g respectively. Then corresponding information measures
coincide:
  
* * *
* * *
α α ; H H ; Div Div ; and  *
*
Bal Bal .
Proof. We prove the equality 

*
α α . Other proofs are analogous.
 
   
 
m
1 2 m
m
j j
j 1
m i 1
λ σ(i) λ σ(j) σ(i)
σ S i 1 j 1
AsFPOWAmax a ,a ,...,a
β b w (1 β)
Max g s (1 λg s ) a .


  
 
 

 
 
 
 
   
 
 
 
 
   
 
 
 
 
 
 

 
% % %
%
%
(6)
 
   
 
m
1 2 m
m
j j
j 1
m i 1
σ S i 1 j 1
AsFPOWAmin a ,a ,...,a
β b w (1 β)
min g s (1 λg s ) a .


  
 
 

 
 
 
 
   
 
 
 
 
  
 
 
 
 
 
 

 
% % %
%
%
(7)
 
   
 
m
1 2 m
m
j j
j 1
m i 1
σ S i 1 j 1
AsFPOWAmin a ,a ,...,a
β b w (1 β)
min g s (1 λg s ) a .


  
 
 

 
 
 
 
   
 
 
 
 
  
 
 
 
 
 
 

 
% % %
%
%
(8)

325
New
view
of
fuzzy
aggregations.
Part
III:
extensions
of
the
FPOWA
operator
in
the
problem
of
political
management
Consider
In this proof we use the property of symmetry of the function M ; the fact, that Associated Probability
Classes of *
g and 
g coincide    


 

m m
*
*σ σ
σ S σ S
P P (see Proposition 2, Part II [37]) and

 
 *
*σ( j ) σ ( m j 1)
P P
, where σ and 
σ are dual permutations (Section 2, Part II [37]).
2.2| AsFPOWA Operators Induced by the FEV
Now we define new generalizations of the FPOWA operator induced by the  
P
FEV . The values of
imprecision of the incomplete information on S are presented by the fuzzy variable
 
i i
a TFN, a : S Ψ ,
(or a a s Ψ for every i 1,2,..,m).


 
  
% %
% %
Definition 3. A Sugeno Averaging FPOWA operator SA-FPOWA of dimension m is mapping
 
 
m
SA FPOWA : Ψ Ψ , that has an associated weighting vectorW of dimension m such that
 
 
 
 
 
 
j
w 0,1 ,



m
j
j 1
w 1 and is defined according to the following formula:
Where j
b
% is the jth largest of the   
i
a , i 1,...,m
% ;
m
* j
j 1
m
σ(j) m
j 1
m
j
j 1
m
*
σ (m j 1) * m
j 1
m
j
j 1
m j
α β w
m 1
m σ(j)
(1 β)M P σ S
m 1
m j
β w (1 β)
m 1
m σ (m j 1)
M P σ S
m 1
m j
β w
m 1






 


 

 
 
 
 

 
 
 

 
 
   
 
 
 
 

 
 
 
 

 
   
 
 

 
 
 
  
 
 
 
  
 
 
 
 

 
 
 
 

 
  
 

 




m
* *
σ (j) * m
j 1
(1 β)
m σ (j)
M P σ S α .
m 1



  
 
 

 
 
 
  
 
 
 
 

 
 
 


 
1 2 m
m
j j P 1 2 m
j 1
m
P
j j l j j
l 1,m j 1,m
j 1
SA FPOWA(a ,a ,...,a )
β w b (1 β)FEV a ,a ,...,a
β w b (1 β)max a max[min{b ,w }].

 

 
 
 
   
 
 
 
 

  


% % %
% % % %
% %
(9)

326
Sirbiladze|J.
Fuzzy.
Ext.
Appl.
2(4)
(2021)
321-333
 
j
j
l
l
b
b
max{a }
%
%
%
; on the S there exist a probability distribution  
 
i i
p P s , i 1,...,m with

  

m
i i
i 1
p 1, 0 p 1 and

 
 
  
 
 
 
 


Δ j
P
j i( 1) i( j ) i( j )
i 1
w P s ,...,s p .
On the basis of Definition 8, Part II [37], and analogously to definition 1 we present a definition of the
AsFPOWA operator induced by the FEV with respect to some monotone measure
 
 
  
 
 
 
S
g : 2 0,1 .
Definition 4. A Sugeno Averaging AsFPOWA operator SA-AsFPOWA of dimension m is mapping
 
 
m
SA AsFPOWA : Ψ Ψ , that has an associated objective weighted vector W of dimension m
such that
 
 
 
 
 
 
j
w 0,1 and



m
j
j 1
w 1 ; some uncertain measure – monotone measure
 
 
  
 
 
 
S
g : 2 0,1
with associated probability class    m
σ σ S
P defined according the following formula:
Where
   
 
σ σ
σ
P P 1 m
P
l i(j) j
l 1,m j 1,m
FEV a FEV a ,...,a
max a max min a ;w .
 
 
 
 
 
 
 

 
  
 
 
 
 
 
 
% % %
% %
 
i(j)
i(j)
l
l 1,m
a
a .
max a

 
%
%
And
 
   
σ
j
P
j σ i(1) i( j) σ i(l) m
l 1
w P s ,...,s P s , σ S , j 1, 2,..,m

    

M is some averaging operator.
Analogously to Subjection 3.2, Part II [37] (Formulas (44)-(45)) we may define new SA-AsFPOWA
operators induced by the FEV with respect to concrete monotone measures: Sugeno λ -additive
measure, possibility measure, believe structure’s associated monotone measure and others (but these
procedures are omitted here).
   
 
σ σ
1 k
1 2 m
m
j i(j)
j 1
P P
SA AsFPOWA(a ,a ,...,a )
β w a
(1 β)M FEV a ,...,FEV a .

 
 
 
 

% % %
%
% %
(10)

327
New
view
of
fuzzy
aggregations.
Part
III:
extensions
of
the
FPOWA
operator
in
the
problem
of
political
management
3| Example
Analogously to [14] we analyze an illustrative example on the use of new AsFPOWA and SA-AsFPOWA
operators in a fuzzy decision-making problem regarding political management. We study a country that is
planning its fiscal policy for the next year.
Assume that government of a country has to decide on the type of optimal fiscal policy for the next year.
They consider five alternatives:
 d1: “Development a strong expansive fiscal policy”.
 d2: “Development an expansive fiscal policy”.
 d3: “Do not make any changes in the fiscal policy”.
 d4: “Development of a contractive fiscal policy”.
 d5: “Development a strong contractive fiscal policy”.
In order to analyze these fiscal policies, the government has brought together a group of experts. This
group considers that the key factors are the economic situations of the world (external) and country
(internal) economy for the next period. They consider 3 possible states of nature that in whole could occur
in the future.
 s1: “Bad economic situation”.
 s2: “Regular economic situation”.
 s3: “Good economic situation”.
As a result, the group of experts gives us their opinions and results. The results depending on the state of
nature i
s and alternative k
d that the government selects are presented in the Table 1:
Table 1. Expert’s valuations in TFNs.
Following the expert’s knowledge on the world economy for the next period, experts decided that the
objective weights (as an external factor) of states of nature must be
 
 
 

 
 
 
W 0,5;0,3;0,2 , while for the
economy of the country for the next period the occurrence of presented states of nature is defined by some
possibilities (as an internal factor). So, there exist some possibilities (internal levels), as an uncertainty
measure, of the occurrence of states of nature in the country. This decision-making model (Information
Structure I6) is more detailed than the model (Information Structure I4) presented in [14]. In another
words in decision model, we cannot define the objective probabilities 
i i
p P(s ) for the future events, but
we can define subjective possibilities 
i i
π Pos(s ) based on the experts’ knowledge ([4], [8], [20] and Section
2, Part II [37]). Based on some fuzzy terms of internal factor – country economy experts define the
possibility levels of states of nature:
S
D 1
s 2
s 3
s
1
d (60,70,80) (40,50,60) (50,60,70)
2
d (30,40,50) (60,70,80) (70,80,90)
3
d (50,60,70) (50,60,70) (60,70,80)
4
d (70,80,90) (40,50,60) (40,50,60)
5
d (60,70,80) (70,80,90) (50,60,70)

328
Sirbiladze|J.
Fuzzy.
Ext.
Appl.
2(4)
(2021)
321-333
1 1
2 2
3 3
poss(s ) π 0,7;
poss(s ) π 1;
poss(s ) π 0,5.
 
 
 
So, we have the Information Structure I6 of general decision-making system (Definition 7, Part I), where
S
g : Pos(.): 2 0,1
 
 
   
 
 
 
,
i
i
s A
Pos(A) maxπ , A S

   ;
(a monotone measure is a possibility measure).
In this model as in [14] 
β 0,3 . Decision procedure is equivalent to the detalization of GDMS as the
Information Structure I6 (but in [14] the author had the IS as I4). So, for every decision d payoffs’
values are the column from Table 2.
 
g : Pos; W (0,5;0,3;0,2) ; 
I I6 ; 
F AsFPOWA or  
F SA AsFPOWA and others.
Im is the quadruple structure (Definition 7, Part I). For ranking of alternatives  
1 5
d ,..,d we must
calculate its AsFOWA or other operators. For
 
 
 

 
 
 
1 2 3
a a ,a ,a
% % % % we have:
     
σ σ σ
1 2 6
3
1 2 3 j j
j 1
P P P
AsFPOWA a ,a , a β b w (1 β)
M E a ,E a ,...,E a .

 
 
   
 
 
 
 
 
 
 
 
 
 
%
% % %
% % %
It is clear that k=m! =3!=6 and for calculation of the AsFPOWA operator we firstly define
the associated probability class    3
σ σ S
P for the
 
 
  
 
 
 
S
Pos :2 0,1 . For every
 
 
 
 
 
 
 
 
3
σ σ(1),σ( 2),σ(3) S
   
σ σ
3
P P σ(i) σ(i)
i 1
E d E a P a ,

  

% %
where
 
σ σ(i)
σ(1) σ(i) σ(1) σ(i 1)
σ(j) σ(j) σ(0)
j 1,i j 1,i 1
P s
Poss( s ,...,s ) Poss( s ,...,s )
maxπ maxπ , π 0.

  

   
   
   
 
   
   
   
   
  
The results are presented in the Table 2.

329
New
view
of
fuzzy
aggregations.
Part
III:
extensions
of
the
FPOWA
operator
in
the
problem
of
political
management
Table 2. Associated probability class - 3
S
{P }
  .
Following the Table 2 we calculate Mathematical Expectations -  
  

σ
3
P
σ S
E (Table 3) and Fuzzy Expected
Values- 
σ m
P σ S
{FEV (.)) (Table 4).
Table 3. Mathematical expectations -  
  3
P
S
E  
 .
Table 4. Fuzzy expected values - m
P S
{FEV (.))
  .
     
 
σ
σ 1 , σ 2 ,σ 3

  
σ 1
P  
σ 2
P  
σ 3
P
1
1,2,3 σ
 
 
  
 
 
 
1
P 0,7
 2
P 0,3

3
P 0

2
1,3,2 σ
 
 
  
 
 
 
1
P 0,7

3
P 0
 2
P 0,3

3
2, 1,3 σ
 
 
  
 
 
 
2
P 1
 1
P 0
 3
P 0

4
2,3, 1 σ
 
 
  
 
 
 
2
P 1
 3
P 0
 1
P 0

5
3, 1,2 σ
 
 
  
 
 
 
3
P 0,5
 1
P 0,2
 2
P 0,3

6
3,2, 1 σ
 
 
  
 
 
 
3
P 0,5
 2
P 0,5

1
P 0

 
σ
P
E  σ 1
σ 2
σ 3
σ 4
σ 5
σ 6
σ
 
σ
P 1
E d (54,64,74) (54,64,74) (40,50,60) (40,50,60) (49,59,69) (45,55,65)
 
σ
P 2
E d (39,49,59) (39,49,59) (60,70,80) (60,70,80) (59,69,79) (65,75,85)
 
σ
P 3
E d (50,60,70) (50,60,70) (50,60,70) (50,60,70) (55,65,75) (55,65,75)
 
σ
P 4
E d (61,71,81) (61,71,81) (40,50,60) (40,50,60) (46,56,66) (40,50,60)
 
σ
P 5
E d (63,73,83) (63,73,83) (70,80,90) (70,80,90) (58,68,78) (60,70,80)
 
σ
P
E  σ 1
σ 2
σ 3
σ 4
σ 5
σ 6
σ
 
σ
P 1
E d (70,70,70) (70,70,70) (50,60,70) (50,60,70) (40,50,60) (40,50,60)
 
σ
P 2
E d (30,40,50) (30,40,50) (60,70,80) (60,70,80) (60,70,80) (60,70,80)
 
σ
P 3
E d (50,60,70) (50,60,70) (50,60,70) (50,60,70) (50,60,70) (50,60,70)
 
σ
P 4
E d (40,50,60) (40,50,60) (40,50,60) (40,50,60) (40,50,60) (40,50,60)
 
σ
P 5
E d (60,70,80) (60,70,80) (70,80,90) (70,80,90) (40,50,60) (40,50,60)

330
Sirbiladze|J.
Fuzzy.
Ext.
Appl.
2(4)
(2021)
321-333
Now we may calculate the values of different variants of the AsFPOWA and SA-AsFPOWA operators
with respect to different averaging operators M (Tables 5 and 6):
Table 5. Aggregation results.
Table 6. Aggregation results.
For possibility distribution 
m
i i 1
π and payoff vector
 
 
 

 
 
 
1 m
a a ,...,a
% % % Yager in [29] defined the
aggregation mean operator-Shapely Expected Value (SEV) for possibility uncertainty:
where  
m
π
σ( i ) i 1
P is the probability distribution on
 
 
 

 
 
 
1 m
S s ,...,s induced by possibility distribution
 
m
i i 1
π :
σ(i)
σ(j) σ(j 1)
π
σ(i)
j 1
π π
P .
m 1 j




 

D/Ag.
Op.
FOWA
SEV
SEV-FOWA
AsFPOWAmin
AsFPOWAmax
AsFPOWAmean
d1 (53,63,73) (46,57,68) (48,59,70) (44,54,64) (54,64,74) (49,59,69)
d2 (59,69,73) (53,64,75) (55,66,77) (45,55,65) (64,74,84) (57,66,75)
d3 (55,65,75) (51,62,73) (52,63,74) (52,62,72) (56,66,76) (53,63,73)
d4 (63,73,83) (47,58,69) (52,63,74) (45,55,65) (60,70,80) (51,61,71)
d5 (63,73,83) (63,74,85) (63,74,85) (60,70,80) (68,78,88) (64,74,84)
D/Ag.
Op.
SA-AsFPOWAmin
SA-AsFPOWAmax
SA-AsFPOWAmean
d1 (44,54,64) (65,68,71) (55,61,69)
d2 (37,47,57) (58,68,77) (51,61,70)
d3 (51,61,71) (51,61,71) (51,61,71)
d4 (44,54,64) (44,54,64) (44,54,64)
d5 (44,54,64) (65,75,85) (56,66,76)
m
π
1 m σ(i) σ(i)
i 1
SEV a ,...,a a P .

 
 

 
 
 
 

% % (11)

331
New
view
of
fuzzy
aggregations.
Part
III:
extensions
of
the
FPOWA
operator
in
the
problem
of
political
management
And  
 

σ σ 1 ,...,σ(m) is some permutation from m
S form which     
σ( 0 ) σ( 1) σ( m)
0 π π ... π 1.
On the other hand these values are Shapley Indexes of a possibility measure with a possibility distribution
 
m
i i 1
π .
It was proved [29] that the  

SEV coincides with the ME for possibility measure:
1 m Poss 1 m
1
i
0
SEV a ,...,a ME a ,...,a
Poss a α i 1,...,m dα
   
   
 
   
   
   
   
 
 
  
 
 
 
 

% % % %
%
.
On the basis of definition SEV we connect the SEV operator to the OWA operator as weighted sum. So
we consider new generalization of the FOWA operator in Information Structure I6:
m
1 2 m j j
j 1
m σ(i)
σ(j) σ(j 1)
σ(i)
i 1 j 1
SEV FOWA(a ,a ,...,a ) β w b
π π
(1 β) a .
m 1 j


 
  
 

 
 
   
 
 
 

 
%
% % %
%
Calculating numerical values of the FOWA ([13], [14] and Definition 3, Part I), SEV, SEV-FOWA,
AsFPOWAmin, AsFPOWAmax, AsFPOWAmean, SA-AsFPOWAmin, SA-AsFPOWAmax, SA-
AsFPOWAmean operators we constructed the Decision Comparing Matrix (Table 8). Firstly, we calculated
Shapely Indexes -   
π
i
P , j 1,3 for the possibility measure (Table 7).
Table 7. Shapley Indexes of the possibility distribution.
According to the information received in this Section, we can rank the alternatives from the most preferred
to the less preferred. The results are shown in Table 8.
Table 8. Ordering of the policies.
π
i
P 4/15 17/30 1/6
i
s 1
s 2
s 3
s
N Aggreg. Operator Ordering Information Structure
1 FOWA 5 4 2 3 1
d d d d d
 f f f I2
2 SEV 5 2 3 4 1
d d d d d

f f f I6 (without weights)
3 SEV-FOWA 5 2 4 3 1
d d d d d

f f f I6
4 AsFPOWAmin 5 3 2 4 1
d d d d d

f f f I6
5 AsFPOWAmax 5 2 4 3 1
d d d d d
f f f f I6
6 AsFPOWAmean 5 2 3 4 1
d d d d d
f f f f I6
7 SA-AsFPOWAmin 3 5 4 1 2
d d d d d
 
f f I6
8 SA-AsFPOWAmax 5 2 1 3 4
d d d d d
f f f f I6
9 SA-AsFPOWAmean 5 3 2 1 4
d d d d d
f f f f I6

332
Sirbiladze|J.
Fuzzy.
Ext.
Appl.
2(4)
(2021)
321-333
We also calculated values of the Orness parameter of the aggregation operators presented in Table 9.
Table 9. Orness values.
Following Table 9 we see that for the nearer of SA-AsFPOWA operators is to on or, the closer its
measure is to one, while AsFPOWAmin operator is to on and, the closer is to zero. Calculations of other
information measures are omitted here. More on these measures of new aggregation operators we will
present in our future papers.
4| Conclusion
New generalizations of the FPOWA operator were presented with respect to monotone measure’s
associated probability class and induced by the Choquet and Sugeno integrals (finite cases). There exist
many combinatorial variants to construct faces or expressions of generalized operators: AsFPOWA and
SA-AsFPOWA for concrete mean operators (Mean, Max, Min and so on) and concrete monotone
measures (Choquet capacity of order two, monotone measure associated with belief structure, possibility
measure and Sugeno 
λ additive measure). Some properties of new operators and their information
measures (Orness, Enropy, Divergence and Balance) are proved. But only some variants
(AsFPOWAmax, AsFPOWAmin and others) are presented, the list of which may be longer than it is
presented in the paper. So, other presentations of new operators and properties of information measures
will be considered in our future research. An example was constructed for the illustration of the
properties of generalized operators in the problems of political management.
Acknowledgment
This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [FR-
18-466].
References
de Campos Ibañez, L. M., & Carmona, M. J. B. (1989). Representation of fuzzy measures through
probabilities. Fuzzy sets and systems, 31(1), 23-36.
Choquet, G. (1954). Theory of capacities. In Annales de l'institut Fourier (Vol. 5, pp. 131-295). DOI
: https://doi.org/10.5802/aif.53
Dubois, D., Marichal, J. L., Prade, H., Roubens, M., & Sabbadin, R. (2001). The use of the discrete Sugeno
integral in decision-making: A survery. International journal of uncertainty, fuzziness and knowledge-based
systems, 9(05), 539-561.
Dubois, D., & Prade, H. (2007). Possibility theory. Scholarpedia, 2(10), 2074.
Grabisch, M., Sugeno, M., & Murofushi, T. (2010). Fuzzy measures and integrals: theory and applications.
Heidelberg: Physica.
Kaufman, A., & Gupta, M. M. (1991). Introduction to fuzzy arithmetic. New York: Van Nostrand Reinhold
Company.
Klir, G. J. (2013). Architecture of systems problem solving. Springer Science & Business Media.
α Ag.Op.
FOWA
SEV
SEV-FOWA
AsFPOWA
min
AsFPOWA
max
AsFPOWA
mean
SA-AsFPOWA
min
SA-AsFPOWA
max
SA-AsFPOWA
mean
α 0,65 0,55 0,58 0,37 0,79 0,58 0,68 0,89 0,79

333
New
view
of
fuzzy
aggregations.
Part
III:
extensions
of
the
FPOWA
operator
in
the
problem
of
political
management
Klir, G. J., & Folger, T. A. (1988). Fuzzy sets: uncertainty and information, prentice. New York: Prentice-Hall,
Englewood Cliffs.
Klir, G. J., & Wierman, M. J. (2013). Uncertainty-based information: elements of generalized information
theory (Vol. 15). Physica.
Marichal, J. L. (2000). An axiomatic approach of the discrete Choquet integral as a tool to aggregate
interacting criteria. IEEE transactions on fuzzy systems, 8(6), 800-807.
Marichal, J. L. (2000). On Choquet and Sugeno integrals as aggregation functions. Fuzzy measures and
integrals-theory and applications, 247-272.
Marichal, J. L. (2000). On Sugeno integral as an aggregation function. Fuzzy sets and systems, 114(3), 347-
365.
Merigo, J. M. (2011). The uncertain probabilistic weighted average and its application in the theory of
expertons. African journal of business management, 5(15), 6092-6102.
Merigó, J. M. (2011). Fuzzy multi-person decision making with fuzzy probabilistic aggregation
operators. International journal of fuzzy systems, 13(3), 163-174.
Merigó, J. M., & Casanovas, M. (2011). The uncertain induced quasi‐arithmetic OWA operator.
International journal of intelligent systems, 26 (1), 1-24. https://doi.org/10.1002/int.20444
Merigó, J. M., & Casanovas, M. (2010). Fuzzy generalized hybrid aggregation operators and its
application in fuzzy decision making. International journal of fuzzy systems, 12(1), 15-24.
Merigo, J. M., & Casanovas, M. (2010). The fuzzy generalized OWA operator and its application in
strategic decision making. Cybernetics and systems: an international journal, 41(5), 359-370.
Sirbiladze, G. (2012). Extremal fuzzy dynamic systems: Theory and applications (Vol. 28). Springer Science &
Business Media.
Sirbiladze, G. (2005). Modeling of extremal fuzzy dynamic systems. Part I. Extended extremal fuzzy
measures. International journal of general systems, 34(2), 107-138.
Sirbiladze, G., & Gachechiladze, T. (2005). Restored fuzzy measures in expert decision-
making. Information sciences, 169(1-2), 71-95.
Sirbiladze, G., Ghvaberidze, B., Latsabidze, T., & Matsaberidze, B. (2009). Using a minimal fuzzy
covering in decision-making problems. Information sciences, 179(12), 2022-2027.
Sirbiladze, G., & Sikharulidze, A. (2003). Weighted fuzzy averages in fuzzy environment: Part I.
Insufficient expert data and fuzzy averages. International journal of uncertainty, fuzziness and knowledge-
based systems, 11(02), 139-157.
Sirbiladze, G., Sikharulidze, A., Ghvaberidze, B., & Matsaberidze, B. (2011). Fuzzy-probabilistic
aggregations in the discrete covering problem. International journal of general systems, 40(02), 169-196.
Sugeno, M. (1974). Theory of fuzzy integrals and its applications (Doctoral Thesis, Tokyo Institute of
technology). Retrieved from https://ci.nii.ac.jp/naid/10017209011/
Torra, V. (1997). The weighted OWA operator. International journal of intelligent systems, 12(2), 153-166.
Yager, R. R. (2009). On the dispersion measure of OWA operators. Information sciences, 179(22), 3908-
3919.
Yager, R. R. (2007). Aggregation of ordinal information. Fuzzy optimization and decision making, 6(3), 199-
219.
Yager, R. R. (2004). Generalized OWA aggregation operators. Fuzzy optimization and decision making, 3(1),
93-107.
Yager, R. R. (2002). On the evaluation of uncertain courses of action. Fuzzy optimization and decision
making, 1(1), 13-41.
Yager, R. R. (2002). Heavy OWA operators. Fuzzy optimization and decision making, 1(4), 379-397.
Yager, R. R. (2002). On the cardinality index and attitudinal character of fuzzy measures. International
journal of general systems, 31(3), 303-329.
Yager, R. R. (2000). On the entropy of fuzzy measures. IEEE transactions on fuzzy systems, 8(4), 453-461.
Yager, R. R. (1999). A class of fuzzy measures generated from a Dempster–Shafer belief
structure. International journal of intelligent systems, 14(12), 1239-1247.
Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multicriteria
decisionmaking. IEEE transactions on systems, man, and cybernetics, 18(1), 183-190.
Yager, R. R., & Kacprzyk, J. (Eds.). (2012). The ordered weighted averaging operators: theory and applications.
Springer Science & Business Media.
Yager, R. R., Kacprzyk, J., & Beliakov, G. (Eds.). (2011). Recent developments in the ordered weighted
averaging operators: theory and practice (Vol. 265). Springer.
Sirbiladze, G., Badagadze, O., & Tsulaia, G. (2012). New fuzzy aggregations. Part II: associated
probabilities in the aggregations of the POWA operator. International journal of control systems and robotics,
1, 73-85.

New view of fuzzy aggregations. Part III: extensions of the FPOWA operator in the problem of political management

More Related Content

Similar to New view of fuzzy aggregations. Part III: extensions of the FPOWA operator in the problem of political management (20)

More from Journal of Fuzzy Extension and Applications (20)

Recently uploaded (20)

New view of fuzzy aggregations. Part III: extensions of the FPOWA operator in the problem of political management