12037


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 21, No 3, Special Issue, 2023, pp. 339 - 357  

https://doi.org/10.22190/FUME230805029W 

© 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

COMPLEX INTUITIONISTIC FUZZY DOMBI PRIORITIZED 

AGGREGATION OPERATORS AND THEIR APPLICATION FOR 

RESILIENT GREEN SUPPLIER SELECTION 

Peng Wang1, Baoying Zhu1, Yang Yu1, Zeeshan Ali2, 3,  

Bandar Almohsen4 

1School of Management Science and Engineering,  

Shandong University of Finance Economics, Jinan, China 
2Department of Mathematics and Statistics, Riphah International University Islamabad, 

Pakistan 
3Department of Mathematics and Statistics, International Islamic University Islamabad, 

Pakistan 
4Mathematics Department, College of Science, King Saud University, Saudi Arabia 

Abstract. One of the main problems faced by resilient supply chain management is how 

to solve the problem of supplier selection, which is a typical multi-attribute 

decision-making (MADM) problem. Given the complexity of the current decision-making 

environment, the primary influence of this paper is to propose the theory of Dombi 

operational laws based on complex intuitionistic fuzzy (CIF) information. Moreover, we 

examined the theory of CIF Dombi prioritized averaging (CIFDPA) and CIF weighted 

Dombi prioritized averaging (CIFWDPA), where these operators are the modified 

version of the prioritized aggregation operators and Dombi aggregation operators for 

fuzzy, intuitionistic fuzzy, complex fuzzy and complex intuitionistic fuzzy information. 

Some reliable properties for the above operators are also established. Furthermore, to 

state the art of the proposed operators, an application example in the presence of the 

invented operators is evaluated for managing resilient green supplier selection 

problems. Finally, through comparative analysis with mainstream technologies, we 

provide some mechanism explanations for the proposed method to show the supremacy 

and worth of the invented theory. 

Key words: Complex intuitionistic fuzzy sets, Prioritized aggregation operators, 

multi-attribute decision-making problem, Dombi t-norm and t-conorm, 

Resilient Green Supplier Selection 

 
Received: August 05, 2023 / Accepted September 16, 2023  

Corresponding author: Peng Wang  

School of Management Science and Engineering, Shandong University of Finance Economics, Jinan 250014, 
China  

E-mail: pengwang@sdufe.edu.cn 



340 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

  1. INTRODUCTION 

Supply chain activity refers to planning, controlling material, data, logistics management, 

and internal and external financial flow among institutions to meet the investor’s needs. 

Valuable and dominant business strategies or policies are needed to cope with internal and 

external problems and challenges in supply chains and their valuable and essential stages. In 

the last few years, especially during COVID-19, the supply chain has faced many pressures 

for sustainable business evaluation, including global sourcing, environment, short time to 

market, and demand ambiguity or uncertainties. The lifetime or long-term theme or goal to 

survive during the new COVID-19 outbreak is to thrive in the “new normal”. Moreover, the 

virus of COVID-19 is not limited to control in the supply chain. Therefore, coping with all 

stakeholders in the supply chain is very awkward and challenging. To manage the above 

problems, the companies must select the right supplier to help reduce costs and remain 

competitive by meeting stakeholders’ expectations effectively and accurately. Various 

well-known supplier selection techniques have been employed, including the Markov chain, 

Bayesian network, and deterministic and stochastic optimization. However, the past existing 

information rarely evaluates sustainable supplier considerations from a resilience and 

sustainability point of view. Therefore, this information aims to study the best supply chain 

selection in the presence of some suitable supplier chain management, which is a typical 

multi-attribute decision-making (MADM) problem. 

Decision-making procedures, clustering analysis, and many other valuable techniques 

are used for evaluating the finest optimal under particular criteria. The MADM technique is 

one of the most valuable and reliable techniques and subparts of the decision-making 

procedure, which is also used for selecting the optimal alternative from the family of 

preferences based on some classical set theory. Moreover, in the case of classical set 

theory, we have only two options, such as zero or one; in many real-life cases, we need 

more options accepted as zero and one. Therefore, to accommodating the above problem, 

the fuzzy set (FS) was proposed by Zadeh [1], where the range of FS is more extended than 

the crisp set because the function defined in FS is defined from universal set to unit interval 

instead of {0, 1}. Many generalizations of the FS and some applications have been 

proposed by different scholars, such as fuzzy superior Mandelbrot set [2], fuzzy N-soft 

information [3], multi-fuzzy N-soft set theory [4]. 

The main idea of FS is very famous and reliable because they have received a lot of 

attention from different scholars. Furthermore, the FS has only talked about the grade of 

membership but has not considered the theory of falsity information, where the grade of 

non-membership also plays a very critical and valuable role in decision-making analysis. 

In many situations, we not only face positive information about anyone in real life because 

many people have given negative information about their enemies. Therefore, the theory of 

intuitionistic fuzzy set (IFS) was invented by Atanassov [5]. IFS theory handles two types 

of information, truth and falsity information, with a valuable and effective option, such as 

the sum of the duplet, which will be covered in the unit interval. The traditional theory of 

FS is a special case of the IFS theory because if we remove the falsity information from 

IFS, then the theory of IFS will be converted to FS theory, where the range of both grades is 

the same, such as unit interval. Many applications and different generalizations of the IFS 

have been proposed by scholars, such as Schweizer-Sklar power aggregation operators [6], 

Maclaurin symmetric aggregation operators [7], Einstein interactive aggregation operators 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 341 

 

[8], MAIRCA technique [9], twin support vector machine [10], analysis and application of 

evolutionary procedure [11]. 

The term “periodic” was critical and essential in complex set theory. In various cases, we 

failed to cope with the problem without periodic function because of the two-dimensional 

information in every field of life. In the presence of the FS theory, we can only cope with 

one-dimensional information at a time. Therefore, the concept of complex fuzzy set (CFS) 

theory was invented by Ramot et al. [12] by including the periodic term in the structure of 

truth grade and given a new shape in the form of the complex number, where the range of the 

real and unreal parts are unit interval. After the construction of CFS, various applications and 

generalizations of CFS have been proposed by different scholars, such as distance and 

entropy measures in the presence of the CFSs [13], complex fuzzy N-soft set theory [14], 

complex multi-fuzzy soft sets [15], and another overview of complex fuzzy logic and their 

application in decision-making problems [16]. Furthermore, the CFS has only talked about 

the grade of membership but has not considered the theory of falsity information, where the 

grade of non-membership also plays a very critical and valuable role in decision-making 

analysis. In many situations, we not only face positive information about anyone in real life 

because many people have given negative information about their enemies. Therefore, the 

complex intuitionistic fuzzy set (CIFS) was invented by Alkouri and Salleh [17], which 

covers two types of information, truth and falsity information, with a valuable and effective 

option such as the sum of the real part (same rule for the imaginary part) of the duplet, which 

will be covered in the unit interval. The traditional theory of CFS is a special case of the CIFS 

theory because if we remove the falsity of information from CIFS, then the theory of CIFS 

will be converted to CFS theory. Numerous applications and generalizations of the CIFS 

have been proposed by different scholars, such as prioritized aggregation operators [18], 

simple aggregation operators [19], novel aggregation operators [20], generalized geometric 

aggregation operators [21], and robust averaging\geometric aggregation operators [22]. 

However, most of the existing aggregation operators are based on CIFS’s algebraic operation 

rules, which lack some flexibility in modeling logic operations, such as dealing with 

information aggregation and decision-making problems in complex situations. 

The main idea of Dombi t-norm and t-conorm was invented by Dombi [23] in 1982. 

Dombi t-norm and t-conorm have been used in constructing various aggregation operators. 

Because of parameter , the theory of Hamacher (algebraic) t-norm and t-conorm are the 

special cases of the Dombi t-norm and t-conorm. Furthermore, evaluating or aggregating 

information collection into a singleton set is a very challenging task for scholars. Hence, 

the prioritized aggregation operators were established by Yager [24] in 2008, which is the 

modified version of the averaging and geometric aggregation operators. The prioritized 

aggregation operator is used for aggregating information collection into a singleton set by 

considering the priority relationship between the aggregated variables. Thus, Yu and Xu 

[25] presented a novel theory of prioritized aggregation operators for IFSs. Seikh and 

Mandal [26] exposed the Dombi aggregation operators for IFS in the presence of the 

Dombi t-norm and t-conorm. To our knowledge, no scholars have derived the theories of 

Dombi aggregation operators, prioritized aggregation operators, and Dombi aggregation 

operators for CIFS theory. However, such operators can effectively improve the flexibility 

of information integration and the adaptability of decision scenarios.  

Given the unique advantages of CIFS in representing decision information and the 

flexibility brought by the Dombi t-norm and t-conorm for aggregation operations, how to 

propose a set of flexible operation rules around advanced CIF information and then 



342 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

develop a class of powerful information aggregation operators to solve decision-making 

problems in complex reality is a topic worth studying. Thus, out of the motives mentioned 

above, this paper mainly focuses on carrying out the following innovative work: 

(1) It proposes the theory of Dombi operational laws based on CIF information and 

improves further the flexibility of CIF information aggregation modeling. 

(2) It presents the CIFDPA and CIFWDPA operators, which aggregate information 

flexibly and consider the priority relationship between integrated parameters. Meanwhile, 

some reliable properties for the above mentioned operators are also established. 

(3) It constructs an MADM technique based on the proposed operators and applies it to 

the resilient green supplier selection problem. 

(4) It implements a comparative analysis with various prevailing operators and 

discusses the operating mechanism and advantages of the proposed method. 

2. PRELIMINARIES 

In this section, we collected the basic information about Dtn, Dtcn, and CIFSs and their 

valuable information, such as SV and AV. 

Definition 1: [23] For evaluating the theory of Dtn and Dtcn, we have used any two real 

numbers ux  and uy ( , [0,1]u ux y ), such that 

 
1

1
, ,

1 1
1

( ) 0Dtn u u

u u

u u

D x y

x y

x y

=
    − − 

+ +    
     

  (1) 

 
1

1

( ) 0
1

, 1 ,

1
1

Dtcn u u

u u

u u

D x y

x y

x y

= −
     

+ +    
− −   



 

 (2) 

Definition 2: [17] Assume that 
u

 be the family of fixed set with generic elements, 

then the model of CIFS is characterized by: 

 {( , ) : }( ) ( )CIF

m u n u u ux x x  =   (3) 

Furthermore, we explained the complex form of truthful information and the complex form 

of false information, such as ( ,( ) ( ) ( ))rp ip

m u m u m ux x x =    and ( )n ux =  ( , ))( ) (rp ip

n u n ux x   

with a specific condition: 0 1( ) ( )rp rp

m u n ux x +   for real parts and ( ) ( )0 ip ip

m u n ux x +  

1 for imaginary parts. Moreover, the neutral information is stated by: ( )m ux =  

)( ),( ( )rp ip

m u m ux x  = (1 ( ),1 ( ))( ) ( ) ( ) ( )rp rp ip ip

m u n u m u n ux x x x−  + −  +  and the final and 

simplest form of CIF value (CIFV) is stated by: (( , ),( , )),
j j j j

CIF rp ip rp ip

j m m n n =      

1,2,3, ,j z=  . For example, there exists a CIFV whose real part is evaluated as an 

intuitionistic fuzzy form of (0.3, 0.4) and whose imaginary part is evaluated as an intuitionistic 

fuzzy form of (0.5, 0.1), then its presentation form ((0.3, 0.5), (0.4, 0.1)). 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 343 

 

Definition 3: [19] Assume that 
u

 be the family of fixed set with generic elements, 

then in the presence of any two CIFVs (( , ),( , )), 1,2
j j j j

CIF rp ip rp ip

j m m n n j =     = , we have: 

 
1 2 1 2 1 2 1 2 1 2 1 21 2 (( , ), ( , ))CIF CIF rp rp rp rp ip ip ip ip rp rp ip ip

m m m m m m m m n n n n  =  + −   + −       (4) 

 
1 2 1 2 1 2 1 2 1 2 1 21 2 (( , ), ( , ))CIF CIF rp rp ip ip rp rp rp rp ip ip ip ip

m m m m n n n n n n n n  =     +  −    +  −    (5) 

 ( )(1 (1 ) ,1 0(1 ) ),(( ) ,( ) ) ,
j j j j

CIF rp ip rp ip

j m m n n = − − −  −   (6) 

 0( ) ((( ) ,( ) ),(1 (1 ) ,1 (1 ) )),
j j j j

CIF rp ip rp ip

j m m n n =   − − − −  (7) 

Definition 4: [19] Assume that 
u

 be the family of fixed set with generic elements, 

then in the presence of any CIFVs (( , ),( , )), 1,2
j j j j

CIF rp ip rp ip

j m m n n j =     = , we have: 

 
1

(( ) ( )) [ 1,1]
2

( )
j j j j

CIF rp ip rp ip

j m m n nSV  +  = − +  −  (8) 

 
1

(( ) ( )) [0,1]
2

( )
j j j j

CIF rp ip rp ip

j m m n nAV  = + +    +  (9) 

To evaluate any two CIFVs, we described some characteristics in the presence of the 

information in Eq. (8) and Eq. (9); we have if 
1 2( ) ( )CIF CIFSV SV   , thus 

1 2 CIF CIF ; if 

1 2( )( )CIF CIFS SVV    , thus 
1 2 CIF CIF ; if 

1 2( )( )CIF CIFS SVV  =  , then if 
1( )CIFAV    

2( )CIFAV  , thus 
1 2 CIF CIF ; if 1 2( ) ( )CIF CIFAV AV   , thus 

1 2 CIF CIF . 

3. DOMBI PRIORITIZED AGGREGATION OPERATORS FOR CIF INFORMATION 

Before building the Dombi prioritized aggregation operators, we must present the basic 

Dombi operational laws in the presence of the CIFVs. Then, by using the Dombi 

operational laws for CIF values, we aim to derive the theory of CIFDPA and CIFWDPA 

operators. Moreover, we must examine the proposed operators’ basic properties, such as 

idempotency, monotonicity, and boundedness. 

Definition 5: Assume that 
u

 be the family of fixed set with generic elements, in the 

presence of any two CIFVs (( , ),( , )), 1,2
j j j j

CIF rp ip rp ip

j m m n n j =     = , we have: 

 

1 1

1 2 1 2

1 2

1 1

1 2 1 2

1 1
1 ,1 ,

1 ( ) 1 ( )

1 1
,

1 ( ) 1 ( )

CIF CIF
a a b b

c c d d

  
− −  

+ + + +  
  =  

  
  + + + +  

 (10) 



344 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

 

1 1

1 2 1 2

1 2

1 1

1 2 1 2

1 1
, ,

1 ( ) 1 ( )

1 1
1 ,1

1 ( ) 1 ( )

CIF CIF
a a b b

c c d d

  
  

+ + + +  
  =  

  − −  + + + +  

 (11) 

 
1 1 1 1 1

1 1 1 1

1 1 1 1
1 ,1 , ,

1 ( ) 1 ( ) 1 ( ) 1 ( )

CIF

a b c d


   

    
 = − −     + + + +    

 (12) 

 ( )1 1 1 1 1

1 1 1 1

1 1 1 1
, , 1 ,1

1 ( ) 1 ( ) 1 ( ) 1 ( )

CIF

a b c d



   

    
 = − −     + + + +    

 (13) 

where   and  are constants greater than 0; let
1

j

j

rp

m

j rp

m

a
 
 =
 −
 

, 
1

j

j

m

ip

ip

j

m

b
 
 =
 −
 

, 

1
j

j

rp

n

j rp

n

c
 − 
 =
 
 

, and 
1

j

j

ip

n

j ip

n

d
 − 
 =
 
 

, 1,2=j , and these symbols also apply to the following 

text. 

 Definition 6: Assume that 
u

 be the family of fixed set with generic elements, then in 

the presence of any two CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , then: 

 

1 2

1 2 1 2

1 1 1

1

1

Η Η Η
( , , , )

Η Η Η

Η

Η

z
CIF CIF CIF CIF CIF CIF

z z
z z zj j j

j j j

j
z CIF

j j
z j

j

IFDPA

= = =

=

=

    =     

 
 

=   
 
 

  



 (14) 

where, 1Η 1= , and ( )
1

1

Η
−

=

= 
j

j CIF

k

k

SV . 

Theorem 1: Assume that 
u

 is a family of fixed set with generic elements, then in the 

presence of any two CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , and in the presence 

of the information in Eq. (14), we derive that the below theory, such as: 

 

1 2

1 1 1 1

1 1 1 1

( , , , )

1 1 1 1
1 ,1 , ,

1 ( ) 1 ( ) 1 ( ) 1 ( )

CIF CIF CIF

z

z z z z

j j j jj j j j

CIFDPA

A B C D
= = = =

    =

    
    − −
    + + + + 
       

 (15) 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 345 

 

where 

1

Η

Η

j

j j
z j

j

A a

=

=


, 

1

Η

Η

j

j j
z j

j

B b

=

=


, 

1

Η

Η

j

j j
z j

j

C c

=

=


, and 

1

Η

Η

j

j j
z j

j

D d

=

=


, and 

these symbols also apply to the following text. 

Proof: Considering the primary procedure of the mathematical induction, we focus on 

evaluating the data in Eq. (15). Considering 2=z , we have 

 
1

1 1 1 1 1
2

1 1 1 1
1

Η 1 1 1 1
1 ,1 , ,

1 1 1 1Η

CIF

j

j

A B C D
=

    
 = − −     + + + +    

 

 
2

2 1 1 1 1
2

2 2 2 2
1

Η 1 1 1 1
1 ,1 , ,

1 1 1 1Η

CIF

j

j

A B C D
=

    
 = − −     + + + +    

 

Thus,  

 
1 2

1 2 1 2
2 2

1 1

Η Η
( , )

Η Η

CIF CIF CIF CIF

j j

j j

CIFDPA

= =

  =   

 
 

 
1 1 1 1

1 1 1 1

1 1 1 1
1 ,1 , ,

1 1 1 1A B C D

    
= − −     + + + +    

 

 
1 1 1 1

2 2 2 2

1 1 1 1
1 ,1 , ,

1 1 1 1A B C D

    
 − −     + + + +    

 

 
2 2 2 21 1 1 1

1 1 1 1

1 1 1 1
1 ,1 , ,

1 ( ) 1 ( ) 1 ( ) 1 ( )j j j jj j j j
A B C D

= = = =

    
    = − −
    + + + + 
       

 

For 2=z , we hold our required information; furthermore, we assume that the data in 

Eq. (15) is also held for =n k , such as 

 

( )1 2

1 1 1 1

1 1 1 1

, , ,

1 1 1 1
1 ,1 , ,

1 ( ) 1 ( ) 1 ( ) 1 ( )

CIF CIF CIF

k

z z z z

j j j jj j j j

CIFDPA

A B C D
= = = =

    =

    
    − −
    + + + + 
       

 

Then, we derive it for 1= +n k , such as 



346 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

 

1 2

1 2 1 1 2

1 1 1

1 1

1 1 1
1 1

1 1 1

Η Η Η
( , , , )

Η Η Η

Η Η Η

Η Η Η

k
CIF CIF CIF CIF CIF CIF

k k
k k kj j j

j j j

k j k
CIF k CIF CIF

k j j k
k k kj j j

j j j

CIFDPA +

= = =

+ +

+ = +
+ +

= = =

    =  

=

  

 
 

  =     
 
 

  

  

 

 
1 1 1 1

1 1 1 1

1 1 1 1
1 ,1 , ,

1 ( ) 1 ( ) 1 ( ) 1 ( )
k k k k

j j j jj j j j
A B C D

= = = =

    
    = − −
    + + + + 
       

 

 
1 1 1 1

1 1 1 1

1 1 1 1
1 ,1 , ,

1 1 1 1k k k kA B C D+ + + +

    
 − −        + + + +    

 

 
1 1 1 11 1 1 1

1 1 1 1

1 1 1 1
1 ,1 , ,

1 ( ) 1 ( ) 1 ( ) 1 ( )
k k k k

j j j jj j j j
A B C D

+ + + +

= = = =

    
    = − −
    + + + + 
       

 

Hence, our final result proved the possible values of z. Moreover, we justify the 

significant properties of the evaluated theory in Eq. (15), such as idempotency, 

monotonicity, and boundedness. 

Property 1: (Idempotency) Assume that u  be the family of fixed set with generic 

elements, then in the presence of any z CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , 

if (( , ), ( , ))CIF CIF rp ip rp ip

j m m n n = =     , then 

 
1 2( , , , )CIF CIF CIF CIF

zCIFDPA     =  (16) 

Proof: When (( , ), ( , ))CIF CIF rp ip rp ip

j m m n n = =     , then 

1 2( , , , )CIF CIF CIF

zCIFDPA      

 

1 1

1 1

1 1

1 1

1 1

1 1

1 1
1 ,1 ,

Η Η
1 1

Η Η

1 1
,

Η Η
1 1

Η Η

j j
z z

m mj jz zj j

j j

j j
z z

n nj jz zj j

j j

a b

c d

= =

= =

= =

= =

  
  
  
  
  − −
     
    
 + +    
    

    
= 

 
 
 
 
 
    
     + +    
          

 
 

 
 
















 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 347 

 

 
1 1 1 1

1

1

1 1 1 1 Η
1 ,1 , , , 1

1 1 1 1 Η

jz

z jjm m n n
j

a b c d =

=

    
= − − =        + + + +    




 

 
1 1 1 1

1 ,1 , ,
1 1 1 1

1 1

    
    
    = − −
    − + − +  + −   + − 
        − −      

rp rp ip ip rp rp ip ip

m m m m n n n n

rp ip rp ip

m m n n

 

 ((1 (1 ),1 (1 )),( , ))rp ip rp ip

m m n n= − − − −    

 (( , ),( , )) .rp ip rp ip CIF

m m n n=     =  

Property: (Monotonicity) Assume that 
u

 be the family of fixed set with generic 

elements, then in the presence of any z CIFVs (( , ),( , ))
j j j j

CIF rp ip rp ip

j m m n n =     , j = 1, 2, … z, 

if * CIF CIF

j j
, then 

 * * *

1 2 1 2( , , , ) ( , , , )CIF CIF CIF CIF CIF CIF

z zCIFDPA CIFDPA          (17) 

Proof: When 
* CIF CIF

j j , thus * *,    
j j j j

rp rp ip ip

m m m m
 and ** **,     

j j j j

rp rp ip ip

n n n n
, then 

*

*

*1 1

 
   

− −

j j

j j

j j

rp rp

m mrp rp

m m rp rp

m m

*

*1 1

    
    
   − −
   

j j

j j

rp rp

m m

rp rp

m m

 

*

*
1 1

1 1

Η Η

1 1Η Η= =

= =

    
    
   − −
   

 
 

j j

j j

rp rpj jz z
m m

rp rp
z zj jj jm m
j j

 

1 1

*

*
1 1

1 1

Η Η
1 1

1 1Η Η= =

= =

   
       
    +  +   
   − −      

   

 
 

j j

j j

rp rpj jz z
m m

rp rp
z zj jj jm m
j j

 

1 1

*

*1 1

1 1

1 1

Η Η
1 1

1 1Η Η
= =

= =

 
   

       
   + +   
   − −      

   

 
 

j j

j j

rp rpj j
m mz z

rp rpj jz zj jm m
j j

 

1 1

*

*1 1

1 1

1 1
1 1

Η Η
1 1

1 1Η Η
= =

= =

 −  −
   

       
   + +   
   − −      

   

 
 

j j

j j

rp rpj j
m mz z

rp rpj jz zj jm m
j j

 



348 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

Where, by using the same procedure, we get 

 

1 1

*

*1 1

1 1

1 1
1 1

Η Η
1 1

1 1Η Η
= =

= =

   −  −
   

       
   + +   
   − −      

   

 
 

j j

j j

j j

ip ip

m m

ip ipj j
m mz z

ip i pj jz zj jm m
j j

Furthermore, we can derive the information of falsity data, such as: 
* *1 1−      −

j j j j

rp rp rp rp

n n n n
 

* *

* *
1 1

1 1

1 1 1 1Η Η

Η Η= =

= =

       − − − −
          
       
   

   

   
   

 
 

j j j j

j j j j

rp rp rp rpj jz z
n n n n

rp rp rp rp
z zj jj jn n n n
j j

 

1 1

*

*
1 1

1 1

11Η Η

Η Η= =

= =

   
   − −    
       
          

   

 
 

j

j j

rprpj jz z
nj

rp rp
z zj jj jn n
j j

 

1 1

*

*
1 1

1 1

1 1Η Η
1 1

Η Η= =

= =

   
   −  −    
    +  +   
          

   

 
 

j j

j j

rp rpj jz z
n n

rp rp
z zj jj jn n
j j

 

1 1

*

*1 1

1 1

1 1

1 1Η Η
1 1

Η Η
= =

= =

 
   

   −  −    
   + +   
          

   

 
 

j j

j j

rp rpj j
n nz z

rp rpj jz zj jn n
j j

 

Where, by using the same procedure, we get 

*

1 1

*

*1 1

1 1

1 1

1 1Η Η
1 1

Η Η
= =

= =

    
   

   −  −    
   + +   
          

   

 
 

j j

j j

j j

ip ip

n n

ip ipj j
n nz z

ip ipj jz zj jn n
j j

 

Then, in the presence of the data in Eq. (8) and Eq. (9), we have 
* * *

1 2 1 2( , , , ) ( , , , )CIF CIF CIF CIF CIF CIF

z zCIFDPA CIFDPA         . 

Property 3: (Boundedness) Assume that 
u

 be the family of fixed set with generic 

elements, then in the presence of any z CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , 

if ((min ,min ), (max ,max ))
j j j j

rp ip rp ip

j m m n n
j j j j

−     = and ((max ,max ),
j j

rp ip

j m m
j j

+   =

(min ,min ))
j j

rp ip

n n
j j
  , then 

 1 2( , , , )CIF CIF CIF

j z jCIFDPA− +        (18) 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 349 

 

Proof: Select the theory in Proposition 1 and Proposition 2; we have 

1 2 1 2( , , , ) ( , , , )CIF CIF CIF

z z jCIFDPA CIFDPA + + + +         =  

1 2 1 2( , , , ) ( , , , )CIF CIF CIF

z z jCIFDPA CIFDPA − − − −         =  

Thus, we have 

1 2( , , , ) .CIF CIF CIF

j z jCIFDPA− +        

Definition 7: Assume that 
u

 be the family of fixed set with generic elements, then in 

the presence of any z CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , then 

 

1 2

1 2
1 2 1 2

1 1

1

1 1

Η Η
( , , , )

Η Η

ΗΗ

Η Η

CIF CIF CIF CIF CIF

z
z zj j

j jj j

jz
jCIF z CIFz

z j j
z zj j

j jj j

CIFWDPA

= =

=

= =

    =    

 
 

  =   
 
 

 

 

 

 



 

 (19) 

Here, 1Η 1=  and 
1

1

Η ( )
j

j CIF

k

k

sv
−

=

=    with weight vector [0,1]j   such that 
1

1
=

=
z

j

j

 . 

Theorem 2: Assume that 
u

 be the family of fixed set with generic elements, then in 

the presence of any two CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , then in the 

presence of the information in Eq. (19), we derive that the below theory, such as: 

 

1 2

1 1

1 1

1 1

1

1 1

1 1

( , , , )

1 1
1 ,1 ,

Η Η
1 ( ) 1 ( )

Η Η

1 1
,

Η Η
1 ( ) 1

Η

j j

j

CIF CIF CIF

z

j j
z zj jrp ip

m mj jz zj j

j jj j

j j
z zj jrp

nj jz zj

jj j

CIFWDPA

= =

= =

= =

= =

    =

 
 
 
 
 − −
    
    

+ +    
    

    

 
 

+  + 
 
 

 
 

 
 

 

 

 



1

( )

Η
j

ip

n
j

j

 
 
 
 
 
 
 
 
 
 
 

  
  
  
  
  
   
      
      



 (20) 

Proof: Similar to Theorem 1, omitted. 



350 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

Property 4: (Idempotency) Assume that 
u

 be the family of fixed set with generic 

elements, then in the presence of any z CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , 

if (( , ), ( , ))CIF CIF rp ip rp ip

j m m n n = =     , then 

 
1 2( , , , )CIF CIF CIF CIF

zCIFWDPA     =  (21) 

Property 5: (Monotonicity) Assume that 
u

 be the family of fixed set with generic 

elements, then in the presence of any two CIFVs (( , ),( , )),
j j j j

CIF rp ip rp ip

j m m n n =      

1,2, ,j z=  , if 
* CIF CIF

j j , then 

 * * *

1 2 1 2( , , , ) ( , , , )CIF CIF CIF CIF CIF CIF

z zCIFWDPA CIFWDPA          (22) 

Property 6: (Boundedness) Assume that 
u

 be the family of fixed set with generic 

elements, then in the presence of any z CIFVs (( , ),( , )), 1,2, ,
j j j j

CIF rp ip rp ip

j m m n n j z =     =  , 

if ((min ,min ), (max ,max ))
j j j j

rp ip rp ip

j m m n n
j j j j

−     = and m ),(( ax ,max
j j

rp ip

j m m
j j

+   =

(min ,min ))
j j

rp ip

n n
j j
  , then 

 1 2( , , , )CIF CIF CIF

j z jCIFWDPA− +        (23) 

4. AN APPROACH FOR RESILIENT GREEN SUPPLIER SELECTION -BASED PROPOSED THEORY 

In this section, we will construct a MADM method based on the proposed CIFWDPA 

operator to solve the practical problem of resilient green supplier selection. 

4.1 Method Description 

To accommodate the above problem, we select the collection of finite values of 

alternatives, that is 
1 2, , ,   Al AlAl

g
. For every alternative, we consider the finite values of 

attributes such as 
1 2, , ,   At At At

z
 with weight vector [0,1]j   such that 

1
1

=
=

z

jj
 . 

In the presence of the above information, such as g-alternatives and z-attributes, we 

concentrate on computing the decision matrix by using the theory of information of CIF 

values; we explained the complex form of truthful information and the complex form of 

falsity information such as ( ,( ) ( ) ( ))rp ip

m u m u m ux x x =    and ( ,( ) ( ) ( ))rp ip

n u n u n ux x x =    

with a specific condition: 0 1( ) ( )rp rp

m u n ux x +   for the real part and 

0 1( ) ( )ip ip

m u n ux x +   for imaginary parts. Moreover, the neutral information is stated 

by: ( , ) (1 ( ),1 ( ))( ) ( ) ( ) ( ) ( ) ( ) ( )rp ip rp rp ip ip

m u m u m u m u n u m u n ux x x x x x x =   = −  + −  +  and the 

final and simplest form of CIFV is stated: (( , ),( , ))
hjhj hj hj

CIF rp ip rp ip

hj m m n n =     , 1,2,3, , ;h g=   

1,2,3, ,j z=  . Therefore, to address some real-life problems, we compute the procedure 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 351 

 

of decision-making for evaluating the finest optimal form of the collection of preferences, 

such as:  

Step 1: Arrange the CIFVs as a decision matrix. 

Step 2: Normalize the decision matrix if possible (in the case of cost of data) 

 
for benfit attribu(( , ), ( , ))

(( , ), ( ,

te

for cost attrib)) ute

hj hj hj hj

hj hj hj hj

rp ip rp ip

m m n nCIF

hj rp ip rp ip

n n m m

    
 = 

   

 

Step 3: Aggregate the data in the decision matrix with the help of the CIFWDPA 

operator as Eq. (20), and obtain the final aggregated value )1,2, ,( 3 , = CIF

h h g for each 

alternative. 

Step 4: Determine the SVs 2( )( ) 1, ,3, ,CIF

hSV h g =   of the aggregated data with the 

help of data in Eq. (8). If necessary, the AV is also required. 

Step 5: Evaluate or derive the ranking values in the presence of the values of the score 

data and examine the finest optimal from the collection of preferences.  

Furthermore, we simplify the above procedure of decision-making technique with the 

help of numerical examples in the environment of the real-life problem and try to evaluate 

it in the presence of the presented operators. 

4.2 Application Example 

Smart green resilient, where the short form is SGR, refers to an emerging planning 

technique or approach that themes to enable pragmatic concerns to be evaluated 

continuously with main environmental problems and to compute long-life resilience in 

urban planning strategies, sustainability, masterplans, and frameworks for an extensive 

range of contexts. With the global economic recovery in the post-pandemic era, the 

selection of resilient green suppliers has become particularly important in the enterprise 

supply chain management, as it can significantly improve the economic benefits of 

enterprises. This problem is taken from Ref. [27], computed based on sustainable supply 

chain management. This problem aims to study supplier selection by integrating lean, 

green agile, and sustainable aspects of the supply chains and resilience. Therefore, in this 

problem, we consider the finest supplier chain for future consideration. For this, we 

consider five majors from the supplier chain which are used as the alternatives such as: 

1
Al : Sustainable supply chain: The green supply chain is becoming an essential and 

major part of the strategic thrust in business companies. 

2Al : Lean supply chain: Lean supply chain depended on the lean management 

principles at Toyota Motor Corporation, Japan, concentrating on waste lessening and 

attaining higher earnings. 

3Al : Resilient supply chain: The capability of a supply chain to cope with surprising 

interruption is described as a resilient supply chain. 

4Al : Green supply chain: It includes efforts to improve environmental performance, 

cut costs and waste in internal supply chains, external supply chains, and reverse logistics 

to eliminate negative environmental effects 

5Al : Agile supply chain: In this procedure, flexibility and quick reconfigurability are 

the main concentrates. 



352 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

To select the best alternative, we give the following features as attribute information: 

1
At : Response rate, 

2At : Capability for unexpected disruptions, 
3At : Capability for 

reverse logistics, 
4At : Sustainable product design, and 

5At : Long-term relationship. 

Therefore, to address the above real-life problem, we use the procedure of decision-making 

for evaluating the finest optimal form of the collection of preferences, such as: 

Step 1: Arrange the CIF values as a decision matrix in Table 1. 

Step 2: Normalize the decision matrix. It’s worth noting that the data in Table 1 is not 

required to be normalized. 

Table 1 Mathematical representation of CIF decision matrix  

 1
At

 2 At
 3 At

 4 At
 5 At

 

1
Al

 
(0.7,0.7),

(0.1,0.2)

 
 
 

 
(0.71,0.71),

(0.11,0.21)

 
 
 

 
(0.72,0.72),

(0.12,0.22)

 
 
 

 
(0.73,0.73),

(0.13,0.23)

 
 
 

 
(0.74,0.74),

(0.14,0.24)

 
 
 

 

2 Al
 

(0.5,0.4),

(0.2,0.3)

 
 
 

 
(0.51,0.41),

(0.21,0.31)

 
 
 

 
(0.52,0.42),

(0.22,0.32)

 
 
 

 
(0.53,0.43),

(0.23,0.33)

 
 
 

 
(0.54,0.44),

(0.24,0.34)

 
 
 

 

3 Al
 

(0.5,0.5),

(0.2,0.4)

 
 
 

 
(0.51,0.51),

(0.21,0.41)

 
 
 

 
(0.52,0.52),

(0.22,0.42)

 
 
 

 
(0.53,0.53),

(0.23,0.43)

 
 
 

 
(0.54,0.54),

(0.24,0.44)

 
 
 

 

4 Al
 

(0.4,0.6),

(0.3,0.1)

 
 
 

 
(0.41,0.61),

(0.31,0.11)

 
 
 

 
(0.42,0.62),

(0.32,0.12)

 
 
 

 
(0.43,0.63),

(0.33,0.13)

 
 
 

 
(0.44,0.64),

(0.34,0.14)

 
 
 

 

5 Al
 

(0.4,0.3),

(0.2,0.1)

 
 
 

 
(0.41,0.31),

(0.21,0.11)

 
 
 

 
(0.42,0.32),

(0.22,0.12)

 
 
 

 
(0.43,0.33),

(0.23,0.13)

 
 
 

 
(0.44,0.34),

(0.24,0.14)

 
 
 

 

Step 3: Aggregate the data in the decision matrix with the help of the Eq. (20) (where 

3= ), and obtain the aggregated value 
1 ((0.8161,0.8161),(0.0012,0.0431))CIF = , 

2 ((0.0048,0.0006),(0.3540,0.8769))CIF = ,
3 ((0.0048,0.0048),(0.3540,0.9840))CIF = , 

4 ((0.0020,0.1040),(0.6965,0.0045))CIF = ,
5 ((0.0006,0.0001),(0.3540,0.0139))CIF = . 

Step 4: Determine the score values of the aggregate data with the help of data in Eq. (8), 

and get 
1( ) 0.7940CIFSV  = , 

2( ) 0.6127CIFSV  = − , 
3( ) 0.6642CIFSV  = − , 

4( )CIFSV  =  

0.2976− , 
5( ) 0.1836CIFSV  = − . 

Step 5: Evaluate or derive the ranking values in the presence of the values of the score 

data and examine the finest optimal from the collection of preferences. Thus, we get the 

rank result of 
1 5 4 2 3    Al Al Al Al Al , and 

1

Al  is the best decision. 

5. PARAMETRIC ANALYSIS AND COMPARATIVE ANALYSIS 

In this section, we compare the proposed operators with some prevailing techniques to 

show the worth and supremacy of the invented techniques. Before making the comparison, 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 353 

 

we check the sensitivity of the parameter  by using its different values in the presence of 

the data in Table 1. The experimental results are shown in Table 2 and Fig. 1. 

Table 2 Stability of the parameters for the CIFWDPA operator  

Parameters Ranking results 

1=  1 4 5 2 3
Al Al Al Al Al       =   

2=  1 4 5 2 3
Al Al Al Al Al         

3=  1 5 4 2 3
Al Al Al Al Al         

4=  1 5 4 2 3
Al Al Al Al Al         

5=  1 4 5 2 3
Al Al Al Al Al         

6=  1 4 5 2 3
Al Al Al Al Al         

7=  1 4 5 2 3
Al Al Al Al Al         

8=  1 4 5 2 3
Al Al Al Al Al         

9=  1 4 2 5 3
Al Al Al Al Al         

10=  1 4 2 5 3
Al Al Al Al Al         

 

Fig. 1 Score values of alternatives under different  values 



354 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

According to Table 2, although the ranking results for each parameter are not precisely 

the same, the optimal results are the same, they are
1

Al . Meanwhile, as shown in Fig.1, as 

parameter  increases, the score value of each alternative increases. From the above 

parameter analysis, it can be seen that the parameter  can reflect the preferences of 

different decision-makers (DMs) so that DMs can choose flexibly according to different 

decision environments and risk preferences. For example, DMs who are risk-averse can 

choose a smaller  value. Of course, this is also based on the premise that the CIFWDPA 

operator-based MADM method can maintain a specific stability of the result. 

Table 3 Result representation of the comparative analysis 

Methods Ranking results 

Yu and Xu’s method [25] −−−−− −−−−−Bounded  

Seikh and Mandal’s method [26] −−−−− −−−−−Bounded  

Senapati’s method [28]  −−−−− −−−−−Bounded  

Garg and Rani’s method [19] (CIFWA operator) 1 4 5 2 3
Al Al Al Al Al     =  =   

Garg and Rani’s method [20] (CIFWA operator) 1 4 5 2 3
Al Al Al Al Al     =  =   

Garg and Rani’s method [21] (CIFAWG operator) 1 4 5 2 3
Al Al Al Al Al     =  =   

Garg and Rani’s method [22] (CIFWG operator) 1 4 5 2 3
Al Al Al Al Al     =  =   

Our method ( 3= ) 1 5 4 2 3
Al Al Al Al Al         

Our method ( 5= ) 1 4 5 2 3
Al Al Al Al Al         

Our method ( 9= ) 1 4 2 5 3
Al Al Al Al Al         

Furthermore, to demonstrate the proposed method’s superiority, we evaluate the 

comparison between the proposed work and some existing MADM methods. For this, we 

consider the following existing operators: Yu and Xu [25] presented a novel theory of 

prioritized aggregation operators for IFSs. Further, Seikh and Mandal [26] exposed the 

major information of Dombi aggregation operators for IFS in the presence of the Dombi 

t-norm and t-conorm. Senapati et al. [28] derived the Aczel-Alsina aggregation operators in 

the presence of the IFS and their application in decision-making. Furthermore, aggregation 

operators for CIF values were derived by Garg and Rani [19], novel aggregation operators 

for CIF information were presented by Garg and Rani [20], geometric aggregation 

operators for CIFS were exposed by Garg and Rani [21], and robust averaging and 

geometric aggregation operators for CIFS was presented by Garg and Rani [22]. Based on 

the data in Table 1, the comparative analysis results are listed in Table 3. 

As shown in Table 3, we obtained the same ranking results, where the optimal 

alternatives are all 
1

Al  according to the theory of the CIFWDPA operator, which is 

consistent with the results obtained by methods in [19-22]. This first indicates that the 

method proposed in this paper is effective. The selection results are not exactly the same, 

mainly because the characteristics of various methods are different. The advantages of the 

proposed method and the limitations of the existing work are stated below: 

The prioritized aggregation operators in [25], the Dombi aggregation operators in [26], 

and the Aczel-Alsina aggregation operators in [28] are proposed for IFS, and it is clear that 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 355 

 

the theory of IFS has the special case of the CIFS; therefore, these proposed operators in 

[25, 26, 28] have failed to evaluate the information in Table 1. However, the method 

proposed in this paper is based on the CIFS with stronger information representation 

ability, which can deal with more complex decision-making and evaluation problems and 

more conveniently simulate human decision-making cognition. 

Compared to the theory of Garg and Rani [19-22], the results obtained by their method 

are significantly different from those obtained by the method proposed in this paper. 

However, the optimal choice is the same. And the methods in [19-22] cannot effectively 

distinguish the priority of alternatives 2

Al , 
3

Al , and 5

Al . This is because the operators 

proposed in this paper adopt the Dombi operation rules and consider the priority 

relationship between the aggregated variables. Therefore, the constructed aggregation 

operators are more flexible and have more comprehensive functions. 

In summary, the characteristics of each method can be summarized in Table 4. 

Table 4 Comparison of characteristics of different methods 

     Characteristics 

Methods 

Stronger information 

representation ability 

Considering DM’s 

preference attitude  

Considering attributes’ 

priority relationship 

Yu and Xu’s  

method [25] 
No No Yes 

Seikh and Mandal’s 

method [26] 
No Yes No 

Senapati’s  

method [28] 
No Yes No 

Garg and Rani’s 

method [19] 
Yes No No 

Garg and Rani’s 

method [20] 
Yes No No 

Garg and Rani’s 

method [21] 
Yes Yes No 

Garg and Rani’s 

method [22] 
Yes No No 

Our  

method 
Yes Yes Yes 

6. CONCLUSIONS 

The prioritized aggregation and Dombi norms play a valuable and dominant role in the 

environment of fuzzy set theory. Inspired by the structure of the CIFS as well as the idea of 

Dombi prioritized aggregation operators, the significant influence of this paper is listed 

below: 

(1) This paper proposed the theory of Dombi operational laws based on CIF 

information.  

(2) This paper examined the theory of the CIFDPA and CIFWDPA operators, which 

are the modified versions of the prioritized aggregation operators and Dombi aggregation 



356 P. WANG, B. ZHU, Y. YU, Z. ALI, B. ALMOHSEN 

 

operators for fuzzy, intuitionistic, complex, and complex intuitionistic fuzzy information. 

Some reliable properties for the above operators are also established. 

(3) This paper derived a MADM method based on the proposed operators and applied it 

to the resilient green supplier selection problem. 

(4) This paper discusses the advantages and characteristics of the proposed method 

through detailed comparative analysis. 

In future research, we will extend CIFS to some novel MADM techniques, such as the 

RAFSI method [29]. Furthermore, we will apply it to the selection of the group of 

construction machines [30], machining process technology selection [31], evaluation of 

ecological governance [32], and other fields. 

Acknowledgement: This study was funded by the Shandong Provincial Key Research and Development 

Program (Soft Science Project) (No. 2023RKY07020), Researchers Supporting Project of King Saud 

University (RSP2023R158). 

REFERENCES  

1. Zadeh, L.A., 1965, Fuzzy sets, Information and control, 8(3), pp. 338-353. 

2. Mahmood, T., Ali, Z., 2022, Fuzzy superior mandelbrot sets, Soft Computing, 26(18), pp. 9011-9020. 

3. Akram, M., Adeel, A., Alcantud, J.C.R., 2018, Fuzzy N-soft sets: A novel model with applications, Journal of 
Intelligent & Fuzzy Systems, 35(4), pp. 4757-4771. 

4. Fatimah, F., Alcantud, J.C.R., 2021, The multi-fuzzy N-soft set and its applications to decision-making, Neural 

Computing and Applications, 33(17), 11437-11446. 

5. Atanassov, K., 1986, Intuitionistic fuzzy sets, Fuzzy sets and systems, 20 (1), pp. 87-96. 

6. Liu, P., Wang, P., 2018, Some interval-valued intuitionistic fuzzy Schweizer–Sklar power aggregation operators 
and their application to supplier selection. International Journal of Systems Science, 49(6), pp. 1188-1211. 

7. Wang, P., Liu, P., 2019, Some Maclaurin symmetric mean aggregation operators based on Schweizer-Sklar 
operations for intuitionistic fuzzy numbers and their application to decision making, Journal of Intelligent & 

Fuzzy Systems, 36(4), pp. 3801-3824. 

8. Liu, P., Wang, P., 2020, Multiple attribute group decision making method based on intuitionistic fuzzy Einstein 
interactive operations, International Journal of Fuzzy Systems, 22, pp. 790-809. 

9. Ecer, F., 2022, An extended MAIRCA method using intuitionistic fuzzy sets for coronavirus vaccine selection in 
the age of COVID-19, Neural Computing and Applications, 34(7), pp. 5603-5623. 

10. Liang, Z., Zhang, L., 2022, Intuitionistic fuzzy twin support vector machines with the insensitive pinball loss, 

Applied Soft Computing, 115, 108231. 
11. Yu, D., Sheng, L., Xu, Z., 2022, Analysis of evolutionary process in intuitionistic fuzzy set theory: A dynamic 

perspective, Information Sciences, 601, pp. 175-188. 
12. Ramot, D., Milo, R., Friedman, M., Kandel, A., 2002, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems, 

10(2), pp. 171-186. 

13. Liu, P., Ali, Z., Mahmood, T., 2020, The distance measures and cross-entropy based on complex fuzzy sets and 
their application in decision making, Journal of Intelligent & Fuzzy Systems, 39(3), pp. 3351-3374. 

14. Mahmood, T., Ali, Z., 2021, A novel complex fuzzy N-soft sets and their decision-making algorithm, Complex & 
Intelligent Systems, 7(5), pp. 2255-2280. 

15. Al-Qudah, Y., Hassan, N., 2018, Complex multi-fuzzy soft set: Its entropy and similarity measure, IEEE Access, 

6, pp. 65002-65017. 
16. Tamir, D.E., Rishe, N.D., Kandel, A., 2015, Complex fuzzy sets and complex fuzzy logic an overview of theory 

and applications, Fifty years of fuzzy logic and its applications, pp. 661-681. 
17. Alkouri, A.S., Salleh, A.R., 2012, Complex intuitionistic fuzzy sets, In AIP conference proceedings American 

Institute of Physics, 1482, 1, pp. 464-470. 

18. Ali, Z., Mahmood, T., Aslam, M., Chinram, R., 2021, Another view of complex intuitionistic fuzzy soft sets based 
on prioritized aggregation operators and their applications to multiattribute decision making, Mathematics, 

9(16), 1922. 



 Complex Intuitionistic Fuzzy Dombi Prioritized Aggregation Operators and Their Application for... 357 

 

19. Garg, H., Rani, D., 2019, Some generalized complex intuitionistic fuzzy aggregation operators and their 
application to multicriteria decision-making process, Arabian Journal for Science and Engineering, 44(3), pp. 

2679-2698. 
20. Garg, H., Rani, D., 2020, Novel aggregation operators and ranking method for complex intuitionistic fuzzy sets 

and their applications to decision-making process, Artificial Intelligence Review, 53(5), pp. 3595-3620. 

21. Garg, H., Rani, D., 2020, Generalized geometric aggregation operators based on t-norm operations for complex 
intuitionistic fuzzy sets and their application to decision-making, Cognitive Computation, 12(3), pp. 679-698. 

22. Garg, H., Rani, D., 2020, Robust averaging–geometric aggregation operators for complex intuitionistic fuzzy sets 
and their applications to MCDM process, Arabian Journal for Science and Engineering, 45(3), pp. 2017-2033. 

23. Dombi, J., 1982, A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness 

measures induced by fuzzy operators, Fuzzy sets and systems, 8(2), pp. 149-163. 
24. Yager, R. R., 2008, Prioritized aggregation operators, International Journal of Approximate Reasoning, 48(1), 

pp. 263-274. 
25. Yu, X., Xu, Z., 2013, Prioritized intuitionistic fuzzy aggregation operators, Information Fusion, 14(1), pp. 

108-116. 

26. Seikh, M. R., Mandal, U., 2021, Intuitionistic fuzzy Dombi aggregation operators and their application to 
multiple attribute decision-making, Granular Computing, 6, pp. 473-488. 

27. Sonar, H., Gunasekaran, A., Agrawal, S., Roy, M., 2022, Role of lean, agile, resilient, green, and sustainable 

paradigm in supplier selection, Cleaner Logistics and Supply Chain, 4, 100059. 
28. Senapati, T., Chen, G., Yager, R. R., 2022, Aczel–Alsina aggregation operators and their application to 

intuitionistic fuzzy multiple attribute decision making, International Journal of Intelligent Systems, 37(2), pp. 
1529-1551. 

29. Deveci, M., Gokasar, I., Pamucar, D., Chen, Y., Coffman, D. M., 2023, Sustainable E-scooter parking operation 

in urban areas using fuzzy Dombi based RAFSI model, Sustainable Cities and Society, 91, 104426. 
30. Božanić, D., Milić, A., Tešić, D., Salabun, W., Pamučar, D., 2021, D numbers–FUCOM–fuzzy RAFSI model for 

selecting the group of construction machines for enabling mobility, Facta Universitatis-Series Mechanical 
Engineering, 19(3), pp. 447-471. 

31. Delice, E., Tozan, H., Karadayi, M.A., Harnicarova, M., Turan, B., 2022, An integrated framework for 

non-traditional machining process technology selection in healthcare applications, Tehnički Vjesnik, 29(6), pp- 
2137-2146.      

32. Wang, P., Fu, Y., Liu, P., Zhu, B., Wang, F., Pamučar, D., 2024, Evaluation of ecological governance in the 
Yellow River basin based on Uninorm combination weight and MULTIMOORA-Borda method, Expert Systems 

with Applications, 121227.