Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  

Series: Mechanical Engineering Vol. 16, N
o
 2, 2018, pp. 171 - 191 

https://doi.org/10.22190/FUME180503018P  
 

© 2018 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original research article 

NORMALIZED WEIGHTED GEOMETRIC BONFERRONI MEAN 

OPERATOR OF INTERVAL ROUGH NUMBERS – APPLICATION 

IN INTERVAL ROUGH DEMATEL-COPRAS MODEL 

UDC 519.8:656 

Dragan Pamučar, Darko Božanić, Vesko Lukovac, Nenad Komazec  

University of Defence in Belgrade, Military Academy, Serbia 

Abstract. This paper presents a new approach to the treatment of uncertainty and 

imprecision in the multi-criteria decision-making based on interval rough numbers (IRN). 

The IRN-based approach provides decision-making using only internal knowledge for the 

data and operational information of the decision-maker. A new normalized weighted 

geometric Bonferroni mean operator is developed on the basis of the IRN for the 

aggregation of the IRN (IRNWGBM). Testing of the IRNWGBM operator is performed 

through the application in a hybrid IR-DEMATEL-COPRAS multi-criteria model which is 

tested on the real case of selecting an optimal direction for the creation of a temporary 

military route. The first part of the hybrid model is the IRN DEMATEL model, which 

provides objective expert evaluation of criteria under the conditions of uncertainty and 

imprecision. In the second part of the model, the evaluation is carried out by using the 

new interval rough COPRAS technique. 

Key Words: Interval Rough Numbers, DEMATEL, COPRAS, Bonferroni Mean Operator 

1. INTRODUCTION 

The decision-making theory comprises many multi-criteria decision-making models 

(MCDM) that support solving of various problems such as those in management science, 

urban planning issues, problems in natural sciences and military affairs, etc. According to 

Triantaphyllou and Mann [1], MCDM plays an important role in real-life problems, 

considering that there are many everyday decisions to be taken which include a number of 

criteria, while according to Chen et al. [2], the multi-criteria decision making is an 

                                                           
Received May 03, 2018 / Accepted June 07, 2018  

Corresponding author: Dragan Pamuĉar  

University of Defence in Belgrade, Military Academy, Pavla Jurišica Šturma 33, 11000 Belgrade, Serbia  

E-mail: dpamucar@gmail.com 



172 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

efficient systematic and quantitative manner of solving vital real-life problems in the 

presence of a large number of alternatives and several (opposing) criteria. 

The MCDM area is an area that has experienced remarkable advances in the last two 

decades, as demonstrated by numerous models developed in this area: the AHP (Analytical 

Hierarchical Process) method [3, 4], the TOPSIS (Technique for Order of Preference by 

Similarity to the Ideal Solution method) method [5], the VIKOR (VlseKriterijumska 

Optimizacija I Kompromisno Resenje) method [6], the DEMATEL (Decision Making Trial 

and Evaluation Laboratory) method [7], the ELECTRE (ELimination and Choice 

Expressing REALITY) method [8], the COPRAS (Complex Proportional Assessment) 

method [9], the MABAC (Multi-Attributive Border Approximation area Comparison) [10, 

11], the EDAS (Evaluation Based on Distance from Average Solution) method [12,13], the 

CODAS (COmbinative Distance-based Assessment) method [14, 15], MAIRCA (Multi-

Attributive Ideal-Real Comparative Analysis) method [16,17]. 

As already mentioned, the MCDM models are used to solve many problems. In 

complex MCDM models, a large number of experts participate in order to find the most 

objective solution [18]. Such models require the application of mathematical aggregators 

to obtain an aggregated initial decision-making matrix. There are many traditional 

aggregators used in group MCDM models, such as Dombi aggregators [19], Bonferroni 

aggregators [20], Einstein and Hamacher operators [21], Heronian aggregation operators 

[22]. These aggregation operators have been widely used in theories of uncertainty such 

as fuzzy MCDM models [23-26], single-valued neutrosophic MCDM models [27-29], 

linguistic neutrosophic models [30, 31], etc. 

In this paper, a new approach in the theory of rough sets is applied to the treatment of 

uncertainty and imprecision contained in the data in group decision-making, namely, an 

approach based on interval rough numbers (IRN). Since this is a new approach, only 

traditional arithmetic aggregators have been used so far in the MCDM models based on 

rough numbers [34-36]. This paper presents the application and development of a new 

normalized weighted geometric Bonferroni mean operator for the IRN aggregation 

(IRNWGBM). The application of the new IRNWGBM operator is shown in hybrid IR-

DEMATEL-COPRAS model. In the literature, there are numerous examples of using the 

DEMATEL model for determining weight coefficients [17, 37], as well as the COPRAS 

model for evaluating alternatives [9]. However, so far in the literature the DEMATEL and 

COPRAS models based on interval rough numbers are not familiar. To the best of this 

author’s knowledge, there is no hybrid IR-DEMATEL-COPRAS model in the field of 

MCDM, which in this way takes into consideration mutual dependence of criteria, evaluates 

alternatives and treats imprecision and uncertainty with the IRN. One of the goals of this 

paper is the development of a new IRNWGBM operator for the IRN aggregation. The 

second goal of this paper is the improvement of the MCDM area through the development of 

a new hybrid IR-DEMATEL-COPRAS model based on the IRN. 

The rest of the paper is organized as follows. The second chapter presents a mathematical 

analysis of interval rough numbers and the development of new IRNWGBM operator. The 

third chapter presents the algorithm of hybrid IR-DEMATEL-COPRAS model, which is later 

tested in the fourth chapter using a real example of selecting an optimal direction for the 

creation of a temporary military route. In the fifth chapter, the concluding observations are 

presented with a special emphasis on the directions for future research. 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 173 

2. INTERVAL ROUGH NUMBERS AND NORMALIZED WEIGHTED GEOMETRIC BONFERRONI 

MEAN OPERATOR 

If we suppose that there is a set of k classes which present the preferences of a DM, 

R=(J1,J2,...,Jk), provided that these belong to the series which meets the condition where 

J1<J2<,...,<Jk and another set of m classes which also present the preferences of a DM, 

R
*
=(I1,I2,...,Ik). All the objects are defined in the universe and related to the preferences of 

a DM. In R
*
 every object class is presented in the interval Ii={Ili,Iui}, meeting the 

condition where Ili≤Iui (1≤i≤m), as well as the condition where Ili,IuiR. Then, Ili presents 

the lower limit of the interval, while  Iui presents the upper limit of the interval of the i-th 

class of objects. If both upper and lower limits of the class of objects are sorted so that 

I
*
l1< I

*
l2<…< I

*
lj, I

*
u1<I

*
2u<…<I

*
uk (1≤j,k≤m), respectively, then we can define the two 

new sets containing the lower class of objects Rl
*
=(I

*
l1,I

*
l2,…,I

*
lj) and upper class of 

objects Ru
*
=(I

*
u1,I

*
u2,…,I

*
uk), respectively. Then, for any class of the objects I

*
liR  

(1≤i≤j) and I
*

ui  R  (1≤i≤k) we can define the lower approximation I
*

li and I
*
ui as follows 

[38]: 

  * * *( ) / ( )  li l liApr I Y U R Y I  (1) 

  * * *( ) / ( )  ui u uiApr I Y U R Y I  (2) 

The upper approximations I
*
li 

and I
*
ui 

are defined by applying the following expressions: 

  * * *( ) / ( )li l liApr I Y U R Y I    (3) 

  * * *( ) / ( )ui u uiApr I Y U R Y I    (4) 

Both classes of objects (upper and lower class of the objects I
*
li and I

*
ui) are defined by their 

lower limits
*

( )
li

Lim I  and 
*

( )
ui

Lim I  and upper limits 
*

( )
li

Lim I  and 
*

( )
ui

Lim I , respectively 

 
* * *1

( ) ( ) ( )
li l li

L

Lim I R Y Y Apr I
M

   (5) 

 
* * *

*

1
( ) ( ) ( )

ui u ui

L

Lim I R Y Y Apr I
M

   (6) 

where ML and M
*

L present the sum of objects contained in the lower approximation of the 

classes of objects I
*
li  and I

*
ui, respectively. The upper limits 

*
( )

li
Lim I  and 

*
( )

ui
Lim I  are 

defined by Eqs. (7) and (8) 

 
* * *1

( ) ( ) ( )
li l li

U

Lim I R Y Y Apr I
M

   (7) 

 
* * *

*

1
( ) ( ) ( )

ui u ui

U

Lim I R Y Y Apr I
M

   (8) 



174 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

where MU and M
*

U present the sum of objects contained in the upper approximation of the 

classes of objects I
*

li and I
*

ui, respectively. Then, the uncertain class of objects I
*
li and I

*
ui 

can be shown with their lower and upper limit 

 * * *( ) ( ), ( )
li li li

RN I Lim I Lim I 
 

 (9) 

 * * *( ) ( ), ( )
ui ui ui

RN I Lim I Lim I 
 

 (10) 

As can be seen, every class of objects is defined by its upper and lower limits, which 

consist of interval rough numbers, defined as follows 

 
* * *

( ) ( ), ( )
i li ui

IRN I RN I RN I   
 (11) 

Interval rough numbers are characterized by specific arithmetic operations differing 

from the arithmetic operations with classic rough numbers. Detailed arithmetic operations 

with the IRN and mutual comparison of the IRN are presented in Pamuĉar et al. [34].  

Definition 1 [20]. Let (a1,a2,…,an) be a set of non-negative numbers, the function 

NWGBM: R
n
→R, wi (i=1,2,…,n) be the relative weight of ai  (i=1,2,…,n), wi  [0,1] and 

1
1

n

ii
w


 . If p,q≥0 and normalized weighted geometric Bonferroni mean operator satisfies: 

 

 , 11 2
, 1

1
( , ,..., )

i j

i

w wn
p q w

n i j

i j

NWGBM a a a pa qa
p q





 



 (12) 

Then NWGBM
p,q

 is called a normalized weighted geometric Bonferroni mean 

(NWGBM) operator. 

 

Definition 2 Set IRN(ξi)=[RN(ξi
L
),RN(ξi

U’
)]=([ξi

L
,ξi

U
],[ξi

’L
,ξi

’U
]) (i=1,2,..,n) as a collection 

of interval rough numbers (IRNs) in Ψ, then the IRNWBM can be defined as follows 

 , 11 2
, 1

1
( ( ), ( ),..., ( )) ( ) ( )

i j

i

w wn
p q w

n i j

i j
i j

IRNWGBM IRN IRN IRN pIRN qIRN
p q

     




 

 (13) 

where 
i

w   is the relative weight of IRN(ξi), wi
 
 [0,1]  and 

1
1

n

ii
w


 , wj is the relative 

weight of IRN(ξj), wj
 
 [0,1] and 

1
1

n

jj
w


 . 

According to the arithmetic operations applied in interval numbers and Definition 2, 

we can obtain the following theorems: 

Theorem 1 Set IRN(ξi)=[RN(ξi
L
),RN(ξi

U’
)]=([ξi

L
,ξi

U
],[ξi

’L
,ξi

’U
]) (i=1,2,..,n) as a collection 

of IRNs in  , then according to Eq. (12), aggregation results obtained is still RN, and we 

can get the following aggregation formula 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 175 

 

   

 

, 1
1 2

, 1

' ' 11

, 1 , 1

1

, 1

1
( ( ), ( ),..., ( )) ( ) ( )

1 1
( ) ( ) , ( ) ( )

1

    

   

 








 
 





  


 
   
  
  








 

i j

i

i ji j

ii

i j

i

w wn
p q w

n i j

i j
i j

w ww wn n
L L U U ww
i j i j

i j i j
i j i j

w w

L L w
i j

i j

IRNWGBM IRN IRN IRN pIRN qIRN
p q

pRN qRN pRN q Lim
p q p q

p q
p q

 

   

1

, 1

' ' ' '1 1

, 1 , 1

1
, ,

1 1
,

 

   




 

 

 
 

  
  
  
   
 
  
   
       

 

 

i j

i

i j i j

i i

w wn n
U U w
i j

i j
i j i j

w w w wn n
L L U Uw w

i j i j

i j i j
i j i j

p q
p q

p q p q
p q p q

 (14) 

Proof. 

 ' ' '( ) ( ), ( ) , , ,L U L U L Ui i i i i i iIRN RN RN                  ; 

 ' ' '( ) ( ), ( ) , , ,L U L U L Ui i i i i i ipIRN pRN pRN p p p p                  ; 

 ' ' '( ) ( ), ( ) , , ,L U L U L Ui i i i i i iqIRN qRN qRN q q q q                  ; 

 ' ' ' '( ) ( ) , , ,L L U U L L U Ui j i j i j i j i jpIRN qIRN p q p q p q p q                      

  ' ' ' '1 11( ) ( ) , , ,
i j i ji j

i ii

w w w ww w

L L U U L L U Uw ww
i j i j i j i j i j

pIRN qIRN p q p q p q p q          
 
            
  

 

 

   

   

1

, 1

1 1

, 1 , 1

' ' ' '1 1

, 1 , 1

1
( ) ( )

1 1
, ,

1 1
,

 

   

   






 

 
 

 

 
 

 


 
  
  
  

 
 
   
  
  



 

 

i j

i

i j i j

i i

i j i j

i i

w wn

w
i j

i j
i j

w w w wn n
L L U Uw w
i j i j

i j i j
i j i j

w w w wn n
L L U Uw w

i j i j

i j i j
i j i j

pIRN qIRN
p q

p q p q
p q p q

p q p q
p q p q









 
 



 

 

So, Theorem 1 is true. 

Theorem 2 (Idempotency). Set IRN(ξi)=[RN(ξi
L
),RN(ξi

U’
)]=([ξi

L
,ξi

U
],[ξi

’L
,ξi

’U
]) (i=1,2,..,n)  

as a collection of IRNs in Ψ, if IRN(ξi)= IRN(ξ), then  

, ,

1 2
( ( ), ( ),.., ( )) ( ( ), ( ),.., ( )).     

p q p q

n
IRNWGBM IRN IRN IRN IRNWGBM IRN IRN IRN

   



176 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

Proof. Since IRN(ξi)= IRN(ξ), i.e. ξi
L
=ξ

L
, ξi

U
=ξ

U
, ξi

’L
=ξ

’L

 
 and  ξi

’U
=ξ

’U
 for  i=1,2,..,n, then  

 

     

, 1
1 2

, 1

' ' '1 1 1

, 1 , 1 , 1

1
( ( ), ( ),..., ( )) ( ) ( )

1 1 1 1
, , ,

i j

i

i j i j i j

i i i

w wn
p q w

n i j

i j
i j

w w w w w wn n n
L L U U L L Uw w w
i j i j i j i

i j i j i j
i j i j i j

IRNWGBM IRN IRN IRN pIRN qIRN
p q

p q p q p q p q
p q p q p q p q

    

      






  

  
  

  


 
     
    
  



    

       

' 1

, 1

' ' ' '1 1 1 1

, 1 , 1 , 1 , 1

1 1 1 1
, , ,

1

i j

i

i j i j i j i j

i i i i

w wn
U w
j

i j
i j

w w w w w w w wn n n n
L L U U L L U Uw w w w

i j i j i j i j
i j i j i j i j

p q p q p q p q
p q p q p q p q

p



       






   

   
   

  
  
  
    

    
        
               






   

       

   

' '1 1 1 1

, 1 , 1 , 1 , 1

1 1

, 1 , 1

1 1 1
( ) , ( ) , ( ) , ( )

,

i j i j i j i j

i i i i

i j i j

i i

w w w w w w w wn n n n
L U L Uw w w w

i j i j i j i j
i j i j i j i j

w w w wn n
L Uw w

i j i j
i j i j

p q p q p q p q
q p q p q p q

   

 

   

   
   

 

 
 

    
       
              

 
 
 
  

   

     

 

' '1 1

, 1 , 1

' '

, ,

, , , ( ), ( ) ( )

i j i j

i i

w w w wn n
L Uw w

i j i j
i j i j

L U L U L U
RN RN IRN

 

      

 

 
 

  
  
  
    

            

 

 

The proof of Theorem 2 is completed.  

Theorem 3 (Boundedness). Let IRN(ξi)=[RN(ξi
L
),RN(ξi

U’
)]=([ξi

L
,ξi

U
],[ξi

’L
,ξi

’U
]) (i=1,2,..,n) 

as a collection of IRNs in Ψ, let  

 ' '( ) min , min , min , minL U L UIRN               and 

 ' '( ) max , max , max , maxL U L UIRN              , then 
,

1 2
( ) ( ( ), ( ),.., ( ) ( )

p q

n
IRN IRNWGBM IRN IRN IRN IRN    

 
  . 

Proof. Let  

IRN(ξ)=min(IRN(ξ1),IRN(ξ2),…,IRN(ξn))=([minξi
L
,minξi

U
],[minξi

’L
,minξi

’U
]) 

and  

IRN(ξ
+
)=max(IRN(ξ1),IRN(ξ2),…,IRN(ξn))=([maxξi

L
,maxξi

U
],[maxξi

’L
,maxξi

’U
]). 

then we have minξi
L
=min(ξi

L
), minξi

U
=min(ξi

U
), minξi

’L
=min(ξi

’L
), minξi

’U
=min(ξi

’U
), 

maxξi
L
=max(ξi

L
), maxξi

U
=max(ξi

U
), maxξi

’L
=max(ξi

’L
) and maxξi

’U
=max(ξi

’U
). Based on 

that we have  

   

   

   

   

' ' '

' ' '

( ) ( ) ( );

min max ;

min max ;

min max ;

min max ;

i

L L L

i i i

U U U

i i i

L L L

i i i

U U U

i i i

IRN IRN IRN  

  

  

  

  

 
 

 

 

 

 

 

According to the inequalities showed above, we can conclude that IRN(ξ
-
) ≤ 

IRNWGBM
p,q

(IRN(ξ1),IRN(ξ2),…,IRN(ξn)) ≤ IRN(ξ
+
)
 
 holds.  



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 177 

Theorem 4 (Commutativity). Let rough set (IRN(ξ1
’
),IRN(ξ2

’
),…,IRN(ξn

’
)) be any 

permutation of (IRN(ξ1),IRN(ξ2),…,IRN(ξn)). Then there is 

IRNWGBM
p,q

(IRN(ξ1),IRN(ξ2),…,IRN(ξn)) =IRNWGBM
p,q

(IRN(ξ1
’
),IRN(ξ2

’
),…,IRN(ξn

’
)). 

Proof. The property is obvious. 

3. IRN DEMATEL-COPRAS MODEL 

3.1. Extension of the DEMATEL method based on interval rough numbers 

Using the DEMATEL method, the dependent factors are considered and the degree of 

dependency between them is determined [32]. The method is based on the graph theory and 

enables visual planning and problem solving. This method allows better understanding of the 

relationship between factors, the relationship between the level of structure and the strength 

of factor influence [33, 34]. As the result of the method application, total direct and indirect 

effects of every factor upon other factors as well as those received from other factors are 

obtained. 

In order to comprehensively consider imprecision and uncertainty existing in group 

decision-making, in this paper the modification of the DEMATEL method is performed by 

using interval rough numbers. Their use eliminates the need for additional information in 

order to determine uncertain number intervals [34]. So far, in the literature the modification 

of the DEMATEL method by applying interval rough numbers (IR'DEMATEL) for 

determining interval rough coefficients of weight criteria has not been considered. In the 

following part, the steps of the IR'DEMATEL method are elaborated: 

Step 1: Expert analysis of factors. Assuming that there are m experts and n factors 

which are considered, every expert should determine the degree of influence of factor i on 

factor j. A comparative analysis of the pair of i-th and j-th factor by e-th expert is marked 

with xij
e
, where: i=1,...,n; j=1,...,n. The value of every pair xij

e
 has one whole number 

value with the following meaning: 0 – no influence; 1 – low influence; 2 – middle 

influence; 3 – high influence; 4 – very high influence. The response of the e-th expert is 

shown with nonnegative matrix of the range n×n, and every element of the e-th matrix in 

the expression X
e
=[x

e
ij]n×n marks the whole nonnegative number x

e
ij, where 1 ≤ e ≤ b.  

 

' '

12 12 1 1

' '

21 21 2 2

' '

1 1 2 2

0 ; ;

; 0 ;
;   1 , ;   1

; ; 0

e e e e

n n

e e e e

e n n

e e e e

n n n n nxn

x x x x

x x x x
X i j n e b

x x x x

 
 
     
 
 
  

 (15) 

where xij
e
  and xij

e’
 present linguistic expressions from the predefined linguistic scale by 

which expert e presents his comparison in the pairs of criteria.  

Therefore, matrices X
1
, X

2
, …, X

m
 are those of the response of every of b experts. The 

diagonal matrix elements of the responses of all experts have the value zero because the 

same factors have no influence.  

If expert e has uncertainty during a pair comparison of criteria (i,j), that is, if expert e 

cannot decide between two values from the linguistic scale, then both values from the 



178 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

scale are stated in matrix X
e
. Then, at the position (i,j) in matrix X

e
 we have different 

values of x
e
ij, that is, x

e
ij ≠ x

e’
ij. If there is no uncertainty, expert e unequivocally selects 

one value. Then, in the matrix of the comparison of criteria (X
e
) is entered the same value 

at the position ( ,i j ), that is x
e
ij= x

e’
ij.  

Step 2: Determination of the matrix of average responses of experts. Based on the 

matrices of the responses X
e
=[x

e
ij]n×n of all m experts, by applying Eqs. (1-11) are 

determined classes of objects and defined interval rough numbers in matrices X
1
, X

2
, …, 

X
b
. That is how interval rough matrices X

e
 (i=1,2,...,b) are obtained and presented in the 

form  

 

0 ( ) ( )

( ) 0 ( )

( ) ( ) 0

e e

ij ij

e e

ij ij

e

e e

ij ij

IRN x IRN x

IRN x IRN x
X

IRN x IRN x

 
 
 
 
 
  

 (16) 

where e (e=1,2,...,b) presents the mark (number) of the expert, and IRN(x
e
ij) presents the 

interval rough number presented in the form IRN(x
e
ij)=[RN(xij

eL
),RN(xij

e’U
)]=([xij

eL
, xij

eU
], 

[xij
e’L

,xij
e’U

]). 

By applying the IRNWGBM operator, Eq. (14), we obtain averaged interval rough 

number IRN(x
e
ij)=[RN(xij

eL
),RN(xij

e’U
)]. 

In this way we obtain averaged rough matrix of average responses Z  

 

12 1

21 2

1 2

0 ( ) ( )

( ) 0 ( )

( ) ( ) 0

n

n

n n

IRN z IRN z

IRN z IRN z
Z

IRN z IRN z

 
 
 
 
 
 

 (17) 

Step 3: Based on matrix Z initial direct-relation matrix D=[IRN(dij)]n×n is calculated, 

see Eq. (18). By normalization every element of matrix D takes the value between zero 

and one. Matrix D is obtained when every element IRN(zij) of matrix Z is divided by 

rough number IRN(s), see Eqs. (18)-(20) 

 

12 1

21 2

1 2

0 ( ) ( )

( ) 0 ( )

( ) ( ) 0

n

n

n n

IRN d IRN d

IRN d IRN d
D

IRN d IRN d

 
 
 
 
 
 

 (18) 

where IRN(dij) is obtained by applying Eq. (18) 

 

' '

' '

( )
( ) , , ,

( )

L U L U

ij ij ij ij ij

ij U L U L

ij ij ij ij

IRN z z z z z
IRN d IRN

IRN s s s s z

    
      
        

 (19) 

The value of interval rough number IRN(s) is obtained by applying Eq. (20) 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 179 

        ' '1 1 1 1( ) max max , max , max , max
n n n nL U L U

ij ij ij ijj j j j
IRN s x x x x

   

   
      

   
 (20) 

Step 4: The total relation matrix (T=[IRN(tij)]n×n) of the range n×n is calculated, 

according to Eq. (21). Element IRN(tij) presents direct influence of factor i on factor j, and 

matrix T shows total relations between every pair of factors. 

 

2

1
lim( ... )

m i

mm
T D D D D




     

 (21) 

where 

 

2

1

1 2 1

1 1 2 1

1

1

...

( ... )

( ) ( )( ... )

( ) ( )

( )

i m

m

m

m

m

D D D D

D I D D D

D I D I D I D D D

D I D I D

D I D







 





    

    

      

  

 



 (22) 

where I  is the unit matrix of the range n×n.  

Based on Eqs. (21) and (22), the total relation matrix is obtained: 

 

11 12 1

21 22 2

1 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

n

n

n n nn

IRN t IRN t IRN t

IRN t IRN t IRN t
T

IRN t IRN t IRN t

 
 
 
 
 
 

 (23) 

where IRN(tij)=[RN(tij
L
),RN(tij

’U
)] is interval rough number by which indirect effects of 

factor i on factor j are expressed. Then, matrix T shows mutual dependence of every pair 

of factors. 

Step 5: Calculation of the sum of rows and columns of total relation matrix T. In total 

relation matrix T the sum of rows and sum of columns is presented by vectors R and C 

with the range n×1: 

  ' '1 1 1 1
11

1

( ) ( ) , , ,
n

n n n nL U L U

i ij ij ij ij ijj j j j
nj

n

IRN R IRN t t t t t
   




               
      (24) 

  ' '1 1 1 1
11 1

( ) ( ) , , ,
n

n n n nL U L U

i ij ij ij ij iji i i i
ni n

IRN C IRN t t t t t
   

 

               
      (25) 

Value Ri presents the sum of the i-th row of matrix T, and shows total direct and indirect 

effects which criterion i provided to other criteria. Value Ci presents the sum of the j-th 

column of matrix T, and shows total direct and indirect effects that criterion j received 

from other criteria [10].  

Step 6: Determination of weight coefficients of criteria (wi). The calculation of weight 

coefficients of criteria is performed based on the values obtained in step 5, see Eq. (26)  



180 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

 

2 2

2 2

'

' ' ' 2 ' ' 2

' ' ' 2 ' ' 2

( ) ( )

( ) ( )
( ) ( ), ( )

( ) ( )

( ) ( )

L L L L L

j i i i i

U U U U U

j i i i iL U

j j j
L L L L L

j i i i i

U U U U U

j i i i i

W R C R C

W R C R C
IRN W RN W RN W

W R C R C

W R C R C

    

    

    
   


    

 (26) 

where the values Ri + Ci and Ri  Ci are obtained by applying Eqs. (27) and (28)  

 
1 1 1 1

' ' ' '

1 1 1 1

, ,

( ) ( )

,

   

   

   
  

 
 
     

   

   

n n n nL L U U

ij ij ij ijj i j i

i i
n n n nL L U U

ij ij ij ijj i j i

t t t t

IRN R IRN C

t t t t

 (27) 

 

' '

1 1 1 1

' '

1 1 1 1

, ,

( ) ( )

,

   

   

   
  

 
 
     

   

   

n n n nL U U L

ij ij ij ijj i j i

i i
n n n nL U U L

ij ij ij ijj i j i

t t t t

IRN R IRN C

t t t t

 (28) 

The normalization of weight coefficients is performed by applying Eq. (29) 

 

' '

' '

1 1 1 1 1

( )
( ) , , ,

( )

L U L U

j j j j j

j n n n n nU L U L

j j j j jj j j j j

IRN W W W W W
IRN w

IRN W W W W W
    

    
     
     
        

 (29) 

where n denotes the number of the evaluation criteria, IRN(wi) final values of weight 

coefficients which are used in the decision-making process. 

3.2. Extension of the COPRAS method based on interval rough numbers 

Every MCDM method is characterized by specific mathematical apparatus. The 

COPRAS method is characterized by a somewhat more complex aggregation process of 

the values of criteria functions and a simplified procedure for data normalization (the 

character of criteria is not considered - min/max). In the following part, the mathematical 

apparatus of the COPRAS method is briefly presented. 

Step 1: A group of experts (e=1,2,…,b) is formed where b presents the number of 

experts who select the criteria and define the elements of the initial decision-making 

matrix. The problem is formally presented by the selection of one of m options 

(alternatives), Ai, i=1,2,…,m, which are evaluated and compared mutually based on n 

criteria (Xj, j=1,2,…,n) whose values we know. The alternatives are shown with vectors 

x
e
ij; x

e*
ij, where x

e
ij; x

e*
ij presents the value of the i-th alternative by j-th criterion.  

Based on Eqs. (1)-(11), the evaluations of experts by vectors xij are transformed 

into interval rough vectors Ai=(IRN(xi1), IRN(xi2),…, IRN(xin)), where IRN(xij)= 

[RN(xij
L
),RN(xij

’U
)]=([xij

L
,xij

U
],[xij

’L
,xij

’U
]) presents the value of the i-th alternative by j-th 

criterion (i=1,2,…,m
 
; j=1,2,…,n). Since the criteria affect differently final values of 

alternatives, to every criterion is attributed weight coefficient wj, j=1,2,…,n which reflects 

its relative significance in the evaluation of alternatives. 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 181 

In that way matrices Xe=X1,X2,...,Xb  (e=1,2,…,b) are obtained in which b experts 

performed the comparison in pairs of criteria. 

 

1 2

1 11 12 1

2 21 22 2

1 2

                ...      

( ) ( ) ... ( )

( ) ( ) ... ( )

... ... ... ...

( ) ( ) ... ( )

n

n

n

e

m m m mn m n

C C C

A IRN x IRN x IRN x

A IRN x IRN x IRN x
X

A IRN x IRN x IRN x


 
 
 
 
 
 

 (30) 

where m is the number of alternatives, and n is total number of criteria. 

Step 2: Normalization of the initial decision-making matrix (Xe). Basic objective of the 

normalization of criteria values is the transformation of different values of criteria (benefit or 

cost) into the values allowing mutual comparison. By applying the IRNWGBM operator, 

from Eq. (14), averaged interval rough number IRN(xij)=[RN(xij
L
),RN(xij

’U
)] is obtained 

which is further normalized in matrix D. Normalized values are shown in matrix D 

 

1 2

1 11 12 1

2 21 22 2

1 2

                ...      

( ) ( ) ... ( )

( ) ( ) ... ( )

... ... ... ...

( ) ( ) ... ( )

n

n

n

m m m mn m n

C C C

A IRN d IRN d IRN d

A IRN d IRN d IRN d
D

A IRN d IRN d IRN d


 
 
 
 
 
 

 (31) 

The elements of normalized matrix IRN(dij) are obtained by applying the expression  

 
 ' '

' '

' '

1 1 1 1 1

, , ,( )
( ) , , ,

( ) , , ,

L U L U

ij ij ij ijijL U L U

ij ij ij ij ij m m m m m
L U L U

ij ij ij ij ij
i i i i i

x x x xIRN x
IRN d d d d d

IRN x x x x x
    

   
   

        
    
    
    

    
 (32) 

where
 
IRN(xij) presents the elements of the initial decision-making matrix (Xe), IRN(dij) 

presents normalized values of the elements of the initial decision-making matrix, m 

presents total number of alternatives. 

Step 3: In the third step the weighted normalized matrix (Z) is formed in which the 

elements of normalized matrix (D) are multiplied by weights of criteria (IRN(wj)) 

 

11 12 1

21 22 2

1 2

( ) ( ) ... ( )

( ) ( ) ... ( )

... ... ...

( ) ( ) ... ( )

n

n

m m mn

IRN z IRN z IRN z

IRN z IRN z IRN z
Z

IRN z IRN z IRN z

 
 
 
 
 
 

 (33) 

where the elements of matrix Z are obtained by multiplying normalized elements of the 

matrix given in Eq. (31) by weight coefficients of criteria IRN(wj), respectively, IRN(zij)= 

IRN(dij)* IRN(wj). 



182 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

Step 4: In the following or fourth step, are summed the values of matrix Z by columns. 

The values are summed up depending on the group of criteria they belong to (min or 

max). The values of max criteria (higher values of criteria are desirable) are obtained by 

the application of Eq. (34), respectively, Eq. (35) 

 ( ) ( )
i

i ij

z

IRN S IRN z




   (34) 

where zi=+  presents the set of max criteria, respectively,  

 
1

( ) ( ) ( )
i

k

ij j

i

IRN S IRN d IRN w




   (35) 

where k presents the total number of max criteria. 

The values of min criteria (lower values of criteria are desirable) are obtained by 

applying Eq. (36), respectively, Eq. (37) 

 ( ) ( )
i

i ij

z

IRN S IRN z




   (36) 

where zi=   presents the set of min criteria, respectively,  

 
1

( ) ( ) ( )
i

p

ij j

i

IRN S IRN d IRN w




   (37) 

where p presents total number of min criteria. 

Step 5: In the fifth step by applying Eq. (38) the relevance (influence) of every 

observed alternative from the set of alternatives being compared is determined.  

 

min 1

min

1

1

1

( ) ( )
( ) ( )

( )
( )

( )

( )
( )

1
( )

( )

 

 








 





  
 
 
 











m

ii

i i
m

i i
i

m

ii

i
m

i i
i

IRN S IRN S
IRN Q IRN S

IRN S
IRN S

IRN S

IRN S
IRN S

IRN S
IRN S

 (38) 

Step 6: In the last or sixth step, the alternatives are ranged based on the values of 

criteria functions which are assigned to every alternative, where as the most desirable 

alternative is selected the one with the highest value of criteria function. 

4. APPLICATION OF THE NWGBM OPERATOR IN THE IR’DEMATEL-COPRAS MODEL 

The IR'DEMATEL-COPRAS model with the NWGBM operator was tested on the 

problem of selecting an optimal direction for making a temporary military route. The 

temporary military route represents a type of route with limited duration [39]. These 

routes are mostly used for a short time, usually during combat operations, sometimes for 

disposable use. They are built on the directions of the movement of units in situations 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 183 

where the existing roads are insufficient or highly damaged [39]. Such roads are built for 

taking position, supply, when the existing roads or road network is to be circumvented or 

because of certain objects - settlements located on the existing roads, etc. The existing 

literature elaborates the methodology for defining direction of the temporary military 

route taking into account primarily the length of route and the scope of works. Other 

segments, which have a significant influence, are usually not elaborated. For this reason, 

the criteria that influence the selection of the temporary route direction are further 

elaborated in Table 1. 

Table 1 Criteria for selecting the temporary route direction 

Name of criterion Description of criterion 

Scope of works (C1) This criterion defines the scope of works necessary for the construction of a 

particular road section. The scope of works depends on the type of soil and 

its carrying capacity, in relation to the maximum type of load planned for 

transport via the route considered. The criterion is presented through 

qualitative parameters and belongs to the group of min criteria. 

Critical points (C2) Through this criterion a number of potential regions is defined, where it is 

possible for an enemy with significant prospects of success to set an ambush. The 

criterion is of quantitative character and belongs to the group of min criteria. 

Length of route (C3) This criterion defines the length of route, which further affects the time 

when the units are retained on it. This increases or decreases the security of 

the people and means using a temporary military route. The criterion is of 

quantitative character and belongs to the group of min criteria. 

Masking the 

movement (C4) 

In this criterion, through linguistic descriptions are defined the possibilities of 

masking the movement of units while moving on a temporary military route. The 

criterion is described by linguistic values and belongs to the group of max criteria. 

Capacities for 

reparation and 

reconstruction of 

route (C5) 

Capacities necessary for reparation and reconstruction of the route. For the 

purpose of quantification of this criterion, a working group of components is 

defined including: grader, dozer, roller, loader and two self-loaders. The 

evaluation of the criteria is based on the required number of working groups 

it and belongs to the group of min criteria. 

Capacities for 

providing supply, 

respectively, the 

movement of units on 

the route made (C6) 

These units monitor the movement of own forces, as well as the activities of 

the enemy. With their presence, they should prevent attacks on the vehicles 

moving along the way. The basic unit that quantifies this criterion is the 

shooting unit. The evaluation of the criteria is based on the required number 

of shooting units and it belongs to the group of min criteria. 



184 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

The application of the hybrid IR'DEMATEL-COPRAS model with the NWGBM 

operator is shown on the example of the evaluation of six routes for the construction of a 

temporary military route in southern Serbia. The routes considered are marked with A1 to 

A6. In the first phase of the IR'DEMATEL-COPRAS model, the weight coefficients of 

the criteria are calculated using the IR'DEMATEL model.In the first step of the 

IR'DEMATEL model, an expert analysis of the factors is performed. In this research, 

three experts took part in the evaluation of the criteria using the scale: 0 – no influence; 

1 – low influence; 2 – middle influence; 3 – high influence; 4 – very high influence;  

5 – extremely high influence. The weight coefficients of the experts were determined 

(0.337, 0.314, 0.349)
T
. After the expert evaluation, three matrices of comparisons were 

obtained in pairs of criteria with the dimension 66, (Table 2). 

 

Table 2 Expert evaluation of criteria 

Expert 1 

 C1 C2 C3 C4 C5 C6 

C1 0;0 3;4 4;5 5;5 3;4 4;4 

C2 4;5 0;0 3;4 4;4 5;5 5;5 

C3 2;3 1;2 0;0 4;4 3;4 4;5 

C4 2;2 3;3 4;5 0;0 4;4 3;4 

C5 2;3 2;2 2;3 1;2 0;0 5;5 

C6 3;4 2;3 2;3 3;4 2;3 0;0 

Expert 2 

 C1 C2 C3 C4 C5 C6 

C1 0;0 4;5 5;5 5;5 5;5 5;5 

C2 5;5 0;0 4;5 4;4 4;4 5;5 

C3 2;3 3;4 0;0 4;5 4;5 4;4 

C4 3;4 3;4 3;4 0;0 4;5 4;5 

C5 2;3 1;2 3;4 2;3 0;0 4;5 

C6 4;5 3;4 2;2 2;3 3;4 0;0 

Expert 3 

 C1 C2 C3 C4 C5 C6 

C1 0;0 5;5 4;5 3;4 5;5 4;4 

C2 5;5 0;0 3;4 4;5 5;5 4;5 

C3 3;4 2;3 0;0 5;5 4;5 5;5 

C4 3;4 3;4 3;4 0;0 3;3 4;4 

C5 2;3 1;2 2;3 1;2 0;0 4;5 

C6 3;4 4;5 3;4 3;4 3;4 0;0 

In accordance with the procedure for implementing the IR-DEMATEL model, the 

initial matrices of comparison in pairs of criteria are transformed into the interval rough 

matrices by means of Eqs. (1-11). Thus, we obtain three interval rough matrices of the 

criteria, Table 3. 

 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 185 

Table 3 Interval rough matrices of comparisons in criteria pairs 

Expert 1 

 C1 C2 ... C6 

C1 [(0, 0),(0, 0)] [(3, 3.67),(4, 4.67)] ... [(4, 4.33),(4, 4.33)] 

C2 [(4, 4.67),(5, 5)] [(0, 0),(0, 0)] ... [(4.67, 5),(5, 5)] 

C3 [(2, 2.33),(3, 3.33)] [(1, 2.33),(2, 3.33)] ... [(4, 4.33),(4.67, 5)] 

C4 [(2, 2.67),(2, 3.33)] [(3, 3),(3, 3.67)] ... [(3, 3.67),(4, 4.33)] 

C5 [(2, 2),(3, 3)] [(1.67, 2),(2, 2.33)] ... [(4.33, 5),(5, 5)] 

C6 [(3, 3.33),(4, 4.33)] [(2, 2.67),(3, 3.67)] ... [(0, 0),(0, 0)] 

Expert 2 

 C1 C2 ... C6 

C1 [(0, 0),(0, 0)] [(3.67, 4),(4.67, 5)] ... [(4.33, 5),(4.33, 5)] 

C2 [(4.67, 5),(5, 5)] [(0, 0),(0, 0)] ... [(4.67, 5),(5, 5)] 

C3 [(2, 2.33),(3, 3.33)] [(2.33, 3),(3.33, 4)] ... [(4, 4.33),(4, 4.67)] 

C4 [(2.67, 3),(3.33, 4)] [(3, 3),(3.67, 4)] ... [(3.67, 4),(4.33, 5)] 

C5 [(2, 2),(3, 3)] [(1, 1.67),(2, 2.33)] ... [(4, 4.33),(5, 5)] 

C6 [(3.33, 4),(4.33, 5)] [(2.67, 3),(3.67, 4)] ... [(0, 0),(0, 0)] 

Expert 3 

 C1 C2 ... C6 

C1 [(0, 0),(0, 0)] [(3.67, 4),(4.67, 5)] ... [(4, 4.33),(4, 4.33)] 

C2 [(4.67, 5),(5, 5)] [(0, 0),(0, 0)] ... [(4, 4.67),(5, 5)] 

C3 [(2.33, 3),(3.33, 4)] [(2.33, 3),(3.33, 4)] ... [(4.33, 5),(4.67, 5)] 

C4 [(2.67, 3),(3.33, 4)] [(3, 3),(3.67, 4)] ... [(3.67, 4),(4, 4.33)] 

C5 [(2, 2),(3, 3)] [(1.67, 2),(2.33, 3)] ... [(4, 4.33),(5, 5)] 

C6 [(3, 3.33),(4, 4.33)] [(2.67, 3),(3.67, 4)] ... [(0, 0),(0, 0)] 

 

In the second step of the IR-DEMATEL model, using NWGBM, Eq. (14), the aggregation 

(averaging) of the interval rough matrices of the experts’ responses is carried out. Thus, we 

obtain a centralized interval rough matrix of the average responses of the criteria experts, 

Table 4. 

Table 4 Averaged interval rough matrix of criteria 

 C1 C2 ... C6 

C1 [(0.0,0.0),(0.0,0.0)] [(3.4,3.9),(4.4,4.9)] ... [(4.1,4.5),(4.1,4.5)] 

C2 [(4.4,4.9),(5.0,5.0)] [(0.0,0.0),(0.0,0.0)] ... [(4.4,4.9),(5.0,5.0)] 

C3 [(2.1,2.6),(3.1,3.6)] [(1.9,2.8),(2.9,3.8)] ... [(4.1,4.6),(4.5,4.9)] 

C4 [(2.4,2.9),(2.9,3.8)] [(3.0,3.0),(3.4,3.9)] ... [(3.4,3.9),(4.1,4.5)] 

C5 [(2.0,2.0),(3.0,3.0)] [(1.4,1.9),(2.1,2.6)] ... [(4.1,4.6),(5.0,5.0)] 

C6 [(3.1,3.5),(4.1,4.5)] [(2.4,2.9),(3.4,3.9)] ... [(0.0,0.0),(0.0,0.0)] 



186 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

Averaging of the elements of interval rough matrix of comparison in pairs of criteria at 

the C3-C1 position is performed by applying Eq. (14): 

            

       

 

1, 1

0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349

1 0.337 1 0.337 1 0.314 1 0.314

0.349 0.337

1 0.349

2,  2.33 , 3,  3.33 , 2,  2.33 , 3,  3.33 , 2.33,  3 , 3.33,  4

2 2 2 2.33 2 2 2 2.331

2
2.33 2 2.33

 

   

   





          

      

  



p q
IRNWGBM

 

       

   

0.349 0.314

1 0.349

0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349

1 0.337 1 0.337 1 0.314 1 0.314

0.349 0.337 0.349 0.314

1 0.349 1 0.349

2.11

2

2.3 2.3 2.3 3 2.3 2.3 2.3 31
2.55

2
3 2.3 3 2.3

1





   

   

 

 

 
 

 
  

 
       

 
     

       

   

   

0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349

1 0.337 1 0.337 1 0.314 1 0.314

0.349 0.337 0.349 0.314

1 0.349 1 0.349

0.337 0.314 0.337 0.349

1 0.337 1

3 3 3 3.3 3 3 3 3.3
3.11

2
3.3 3 3.3 3

3.3 3.3 3.3 41

2

   

   

 

 

 



 
       

 
     

      

   

   

0.314 0.337 0.314 0.349

0.337 1 0.314 1 0.314

0.349 0.337 0.349 0.314

1 0.349 1 0.349

3.3 3.3 3.3 4
3.56

4 3.3 4 3.3

2.11, 2.55 , 3.11, 3.56

 

  

 

 

















      
  
       

   

 

After determining the averaged matrix of criteria (Table 4), using Eqs. (18-20), the 

third step of the IR-DEMATEL model is carried out, which assumes the determination of 

the initial direct-relation matrix. In the next step, using Eqs. (21-23), the initial direct-

relation matrix is transformed into total relation matrix of the criteria. On the basis of total 

relation matrix, direct and indirect effects are determined (Table 5), which criterion i 

provided to other criteria and received from other criteria, see Eqs. (24) and (25). 

Table 5 Direct and indirect effects of the criteria 

Criteria IRN(Ri) IRN(Ci) IRN(wj) Rank 

C1 [(1.7,2.6),(2.9,5.0)] [(1.3,1.9),(2.4,4.2)] [(0.056,0.153),(0.205,0.548)] 2 

C2 [(1.8,2.8),(2.9,5.4)] [(1.0,1.8),(2.0,4.3)] [(0.054,0.158),(0.193,0.577)] 1 

C3 [(1.3,2.0),(2.3,4.4)] [(1.3,2.0),(2.5,4.4)] [(0.047,0.135),(0.182,0.522)] 5 

C4 [(1.4,2.0),(2.4,4.5)] [(1.4,2.1),(2.4,4.4)] [(0.051,0.138),(0.186,0.521)] 6 

C5 [(1.0,1.5),(2.1,3.6)] [(1.6,2.4),(2.6,4.8)] [(0.048,0.133),(0.183,0.500)] 4 

C6 [(1.2,1.8),(2.3,4.2)] [(1.8,2.6),(2.9,5.0)] [(0.055,0.151),(0.203,0.545)] 3 

In the last step, using Eqs. (26-29), we obtain final interval rough weight coefficients 

of the criteria, Table 5. 

After determining the weight coefficients of the criteria, the evaluation of the alternatives 

using the IR-COPRAS method is carried out. As with the IR-DEMATEL model, three 

experts evaluated six ways for the construction of a temporary military route. Experts 

evaluated the alternatives by assigning a certain value from the scale 1-9: 1  very low 

influence; 2  medium low influence; 3  low influence; ...; 8  high influence; 9  very high 

influence. The results of the expert evaluation of the alternatives are shown in Table 6. 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 187 

Table 6 Expert evaluation of the alternative 

Expert 1 

Alt./Crit. C1 C2 C3 C4 C5 C6 

A1 (7;8) (7;8) (3;4) (1;2) (5;6) (3;4) 

A2 (9;9) (7;8) (5;6) (9;9) (7;8) (9;9) 

A3 (7;8) (3;4) (5;6) (5;6) (7;8) (8;9) 

A4 (9;9) (7;8) (7;7) (9;9) (9;9) (8;9) 

A5 (7;8) (5;6) (5;5) (7;8) (5;6) (5;6) 

A6 (7;8) (7;8) (7;8) (5;6) (7;8) (7;8) 

Expert 2 

Alt./Crit. C1 C2 C3 C4 C5 C6 

A1 (3;4) (5;6) (3;4) (5;6) (3;4) (5;5) 

A2 (9;9) (8;9) (5;5) (7;8) (7;8) (9;9) 

A3 (8;9) (7;8) (5;6) (5;6) (5;6) (8;9) 

A4 (9;9) (9;9) (7;8) (9;9) (8;9) (8;9) 

A5 (7;8) (8;9) (5;6) (7;8) (7;8) (7;8) 

A6 (6;7) (7;8) (7;8) (5;6) (7;8) (5;6) 

Expert 3 

Alt./Crit. C1 C2 C3 C4 C5 C6 

A1 (5;6) (9;9) (7;8) (7;8) (1;2) (3;3) 

A2 (9;9) (9;9) (9;9) (8;9) (9;9) (9;9) 

A3 (7;8) (3;4) (5;6) (7;8) (8;9) (8;9) 

A4 (9;9) (8;9) (8;9) (9;9) (8;9) (8;9) 

A5 (7;8) (7;8) (5;6) (7;8) (8;9) (5;6) 

A6 (5;5) (3;4) (5;5) (7;8) (8;9) (5;5) 

Using Eqs. (1-11), the elements from Table 6 are transformed into interval rough 

numbers, which using Eq. (14) are aggregated into the initial decision-making matrix, 

Table 7. 

Table 7 Initial decision-making matrix  

Alt. C1 C2 ... C6 

A1 [(3.99,6),(5,7)] [(6.01,8.01),(6.9,8.4)] ... [(3.21,4.08),(3.48,4.48)] 

A2 [(9,9),(9,9)] [(7.5,8.5),(8.44,8.89)] ... [(9,9),(9,9)] 

A3 [(7.11,7.54),(8.11,8.54)] [(3.41,5.15),(4.42,6.15)] ... [(8,8),(9,9)] 

A4 [(9,9),(9,9)] [(7.49,8.49),(8.44,8.89)] ... [(8,8),(9,9)] 

A5 [(7,7),(8,8)] [(5.87,7.38),(6.87,8.38)] ... [(5.21,6.09),(6.21,7.09)] 

A6 [(5.49,6.49),(5.87,7.38)] [(4.72,6.54),(5.72,7.54)] ... [(5.22,6.11),(5.6,7.1)] 

Averaging of the elements of the evaluation matrices of alternatives at the A1-C2 

position is carried out using Eq. (14): 



188 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

 

            

       

   

1, 1

0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349

1 0.337 1 0.337 1 0.314 1 0.314

0.349 0.337 0.349 0.314

1 0.349 1 0.349

6, 8 , 7, 8.5 , 5, 7 , 6, 7.67 , 7, 9 , 7.67, 9

6 5 6 7 5 6 5 71

2
7 6 7 5

 

   

   

 

 

          

 
       

 
    



p q
RNWGBM

       

   

   

0.337 0.314 0.337 0.349 0.314 0.337 0.314 0.349

1 0.337 1 0.337 1 0.314 1 0.314

0.349 0.337 0.349 0.314

1 0.349 1 0.349

0.337 0.314 0.337 0.349

1 0.337 1 0.3

6.01

8 7 8 9 7 8 7 91
8.01

2
9 8 9 7

7 6 7 7.671

2

   

   

 

 

 

 




 
       

 
     

      

   

     

 

0.314 0.337 0.314 0.349

37 1 0.314 1 0.314

0.349 0.337 0.349 0.314

1 0.349 1 0.349

0.337 0.314 0.337 0.349 0.314 0.337

1 0.337 1 0.337 1 0.314

0

6 7 6 7.67
6.9

7.67 7 7.67 6

8.5 7.67 8.5 9 7.67 8.51

2
7.67 9

 

 

 

 

  

  

 
    

 
     

    

     

   

.314 0.349 0.349 0.337 0.349 0.314

1 0.314 1 0.349 1 0.349

8.4

9 8.5 9 7.67

6.01, 8.01 , 6.90, 8.40

  

  

















 
 

  
       

   

 

After normalizing the initial matrix using Eq. (32) and summing the elements of the 

normalized matrix using Eqs. (34-38), we obtain final rank of the alternative shown in 

Table 9. 

Table 9 Values of criteria functions of alternatives and their ranking 

Alt. IRN(Qi) IRN(Pi) Rank 

A1 [(0.06,0.17),(0.24,0.79)] [(100.0,100.0),(100.0,100.0)] 1 

A2 [(0.03,0.13),(0.17,0.63)] [(60.98,75.12),(67.99,80.04)] 5 

A3 [(0.04,0.15),(0.18,0.68)] [(70.18,86.84),(72.43,86.29)] 4 

A4 [(0.03,0.13),(0.16,0.62)] [(62.31,76.44),(65.62,78.41)] 6 

A5 [(0.04,0.15),(0.19,0.70)] [(75.09,88.55),(78.12,87.79)] 2 

A6 [(0.04,0.15),(0.19,0.67)] [(73.14,85.35),(78.57,85.00)] 3 

In the following section, the analysis of the influence of parameters p and q from the 

IRNWBM operator to final ranges from the Table 9 was performed. The analysis assumes 

taking different values of parameters p and q and their impact on final values of the IRN 

weight coefficient of criteria, as well as the influence on the averaging of the value of the 

initial decision-making matrix or ranks from the Table 9. The considered values of 

parameters p and q and their influence on changing the alternatives rank are shown in 

Table 10. 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 189 

Table 10 Ranking order by different parameters p and q 

Parameters 

p and q 
Ranking order 

Parameters 

p and q 
Ranking order 

p=1 

q=1 
A1>A5>A6>A3>A2>A4 

p=5 

q=0 
A1>A5>A6>A3>A2>A4 

p=0 

q=1 
A1>A5>A6>A3>A2>A4 

p=10 

q=10 
A1>A5>A6>A3>A2>A4 

p=1 

q=0 
A1>A5>A6>A3>A2>A4 

p=0 

q=10 
A1>A5>A6>A3>A2>A4 

p=2 

q=2 
A1>A5>A6>A3>A2>A4 

p=10 

q=0 
A1>A5>A6>A3>A2>A4 

p=0 

q=2 
A1>A5>A6>A3>A2>A4 

p=50 

q=10 
A1>A5>A6>A3>A2>A4 

p=2 

q=0 
A1>A5>A6>A3>A2>A4 

p=10 

q=50 
A1>A5>A6>A3>A2>A4 

p=5 

q=5 
A1>A5>A6>A3>A2>A4 

p=50 

q=50 
A1>A5>A6>A3>A2>A4 

p=0 

q=5 
A1>A5>A6>A3>A2>A4 

p=100 

q=100 
A1>A5>A6>A3>A2>A4 

Changes in the values of parameters p and q lead to certain changes of the values of 

the criteria functions of alternatives. However, the values of the criteria functions are such 

that they do not lead to changes in final ranges of alternatives, as shown in Table 10. 

Table 10 shows the influence of randomly selected values of parameters p and q on final 

ranges of alternatives in the IR-DEMATEL-COPRAS model. On the basis of the obtained 

results we can conclude that in the considered multi-criterion problem, changes of 

parameters p and q have no influence on the final rank of alternatives. 

5. CONCLUSION 

The recognition of imprecision and uncertainty in the multi-criteria decision-making is 

a very important aspect of an objective and impartial decision-making. There are often 

difficulties in presenting information about decision attributes by accurate (precise) 

numerical values. These difficulties are the result of doubts in the decision-making 

process just as they are due to the complexity and uncertainty of many real indicators. 

This paper presents a new approach to the exploitation of imprecision and uncertainty in 

group decision-making, which is based on interval rough numbers. 

The application of interval rough numbers in the multi-criteria decision-making is presented 

through a hybrid model consisting of the IR-DEMATEL model and the IR-COPRAS method. 

In addition to the modification of the DEMATEL and the COPRAS models, the IRNWGBM 

operator for interval rough numbers is developed in this paper. The application of the IR-

DEMATEL-COPRAS model and the IRNNWGBM operator is presented through a case study 

in which the evaluation of alternatives for the construction of a temporary military route is 

performed. This study shows that the IRNNWGBM operator can be effectively applied in 



190 D. PAMUĈAR, D. BOŢANIĆ, N. KOMAZEC, V. LUKOVAC 

group decision-making models, respecting imprecision and uncertainty. Since this is a new IRN 

aggregator, which has not been applied as yet in the MCDM, the direction of future research 

should focus on the application of the IRNNWGBM in other models based on the IRN 

approach. 

REFERENCES  

1. Triantaphyllou, E., Mann, S.H., 1995, Using the analytic hierarchy process for decision making in 

engineering applications: some challenges, International Journal of Industrial Engineering: Applications 

and Practice, 2(1), pp. 35-44.  

2. Chen, N., Xu, Z., Xia, M., 2015, The ELECTRE I Multi-Criteria Decision-Making Method Based on 

Hesitant Fuzzy Sets, International Journal of Information Technology & Decision Making, 14(3), pp. 

621–657. 

3. Saaty, T. L., 1980, The Analytic Hierarchy Process, McGraw-Hill, NewYork.  

4. Boţanić, D., Pamuĉar, D., Bojanić, D., 2015, Modification of the Analytic Hierarchy Proces (AHP) Method 

using fuzzy logic: fuzzy AHP approach as a support to the decision making process concerning engagement 

of the Group for Additional Hindering, Serbian Journal of Management, 10(2), pp. 151-171. 

5. Yoon, K., 1980, System Selection by Multiple Attribute Decision Making, PhD Thesis, Kansas State 

University, Manhattan, Kansas. 

6. Opricovic, S., Tzeng, G.-H., 2004, The Compromise solution by MCDM methods: A comparative 

analysis of VIKOR and TOPSIS, European Journal of Operational Research, 156(2), pp. 445-455. 

7. Tzeng, G.H., Chiang, C.H., Li, C.W., 2007, Evaluating intertwined effects in e-learning programs: a 

novel hybrid MCDM model based on factor analysis and DEMATEL, Expert Systems with Applications, 

32(4), pp. 1028–1044. 

8. Bernard, R., 1968, Classement et choix en présence de points de vue multiples (la méthode ELECTRE), 

La Revue d'Informatique et de Recherche Opérationelle (RIRO), 8, pp. 57-75. 

9. Kaklauskas, A., Zavadskas, E.K., Raslanas, S., Ginevicius, R., Komka, A., Malinauskas, P., 2006, 

Selection of Low-e tribute in retrofit of public buildings by applying multiple criteria method COPRAS: 

A Lithuanian case, Energy and buildings, 38, pp. 454-462. 

10. Pamuĉar, D., Ćirović G., 2015, The selection of transport and handling resources in logistics centers 

using Multi-Attributive Border Approximation area Comparison, Expert Systems with Applications, 

42(6), pp. 3016-3028. 

11. Bojanic, D., Kovaĉ, M., Bojanic, M., Ristic, V., 2018, Multi-criteria decision-making in A defensive 

operation of THE Guided anti-tank missile battery: an example of the hybrid model fuzzy AHP – 

MABAC, Decision Making: Applications in Management and Engineering, 1(1), pp. 51-66. 

12. Ghorabaee, M.K, Zavadskas, E.K., Olfat, L., Turskis, Z., 2015, Multi-Criteria Inventory Classification 

Using a New Method of Evaluation Based on Distance from Average Solution (EDAS), Informatica, 

26(3), pp. 435-51. 

13. Stević, Ţ., Pamuĉar, D, Vasiljević, M., Stojić, G., Korica, S., 2017, Novel integrated multi-criteria 

model for supplier selection: Case study construction company, Symmetry, 9(11), 279,  pp. 1-34. 

14. Keshavarz G. M., Zavadskas, E. K., Turskis, Z., Antucheviciene, J., 2016, A new combinative distance-

based assessment (CODAS) method for multi-criteria decision-making, Economic Computation & 

Economic Cybernetics Studies & Research, 50(3), pp. 25–44. 

15. Badi, I.A., Abdulshahed, A.M., Shetwan, A.G., 2018, A case study of supplier selection for a 

steelmaking company in Libya by using the Combinative Distance-based ASsessment (CODAS) model, 

Decision Making: Applications in Management and Engineering, 1(1), pp. 1-12. 

16. Pamuĉar, D., Vasin, Lj., Lukovac, L., 2014,  Selection of railway level crossings for investing in 

security equipment using hybrid DEMATEL-MARICА model, XVI International Scientific-expert 

Conference on Railway, Railcon 2014,  pp. 89-92, Niš. 

17. Chatterjee, K., Pamuĉar, D., Zavadskas, E.K., 2018, Evaluating the performance of suppliers based on 

using the R’AMATEL-MAIRCA method for green supply chain implementation in electronics industry, 

Journal of Cleaner Production, 184, pp. 101-129. 

18. Roy, J., Pamuĉar, D., Adhikary, K., Kar, S., 2018, A rough strength relational DEMATEL model for 

analysing the key success factors of hospital service quality, Decision Making: Applications in 

Management and Engineering, 1(1), pp. 121-142. 



 Normalized Weighted Geometric Bonferroni Mean Operator of Interval Rough Numbers... 191 

19. Dombi, J., 1982, A general class of fuzzy operators, the demorgan class of fuzzy operators and 

fuzziness measures induced by fuzzy operators, Fuzzy Sets Syst, 8, pp. 149–163. 

20. Bonferroni, C., 1950, Sulle medie multiple di potenze, Bollettino dell'Unione Matematica Italiana, 5, pp. 

267–270.(Only in Italian). 

21. Wang, W. Z., Liu, X.W., 2011, Intuitionistic fuzzy geometric aggregation operators based on Einstein 

operations, International Journal of Intelligent Systems, 26(11), pp. 1049-1075. 

22. Xu, Z., Yager, R.R., 2006, Some geometric aggregation operators based on intuitionistic fuzzy sets, 

International Journal of General Systems, 35(4), pp. 417-433. 

23. Herrera, F., Herrera-Viedma, E., Verdegay, L., 1996, A model of consensus in group decision making 

under linguistic assessments, Fuzzy Sets and Systems, 78(1), pp. 73–87.  

24. Liu, P.D., Teng, F., 2016, An extended TODIM method for multiple attribute group decision-making 

based on 2-dimension uncertain linguistic variable, Complexity, 21, pp. 20–30. 

25. Yang, W., Pang, Y., Shi, J., Yue, H., 2017, Linguistic hesitant intuitionistic fuzzy linear assignment 

method based on choquet integral, Journal of Intelligent & Fuzzy Systems, 32(1), pp. 767–780. 

26. Zhao, H., Xu, Z., Ni, M., Liu, S., 2010, Generalized aggregation operators for intuitionistic fuzzy sets, 

International Journal of Intelligent Systems, 25 (1), pp. 1-30. 

27. Li, Y.H., Liu, P.D., Chen, Y.B., 2016, Some single valued neutrosophic number Heronian mean operators 

and their application in multiple attribute group decision making, Informatica, 27(1), pp. 85–110. 

28. Ye, J., 2014, Multiple attribute group decision-making method with completely unknown weights based 

on similarity measures under single valued neutrosophic environment, Journal of Intelligent & Fuzzy 

Systems, 27(6), pp. 2927–2935. 

29. Ye, J., 2015, An extended TOPSIS method for multiple attribute group decision making based on single 

valued neutrosophic linguistic numbers, Journal of Intelligent & Fuzzy Systems, 28 (1), pp. 247–255. 

30. Fang, Z., Ye, J., 2017, Multiple Attribute Group Decision-Making Method Based on Linguistic 

Neutrosophic Numbers, Symmetry, 9(7), 111, pp. 1-12. 

31. Liang, W., Zhao, G., Wu, H., 2017, Evaluating Investment Risks of Metallic Mines Using an Extended 

TOPSIS Method with Linguistic Neutrosophic Numbers, Symmetry, 9(8), 149, 1-18. 

32. Ranjan, R., Chatterjee, P., Chakraborty, S., 2016, Performance evaluation of Indian railway zones using 

DEMATEL and VIKOR methods, Benchmarking: An International Journal, 23(1), pp.78 – 95. 

33. Ranjan, R., Chatterjee, P., Chakraborty, S., 2015, Evaluating performance of engineering departments 

in an Indian University using DEMATEL and compromise ranking methods, OPSEARCH, 52(2), pp. 

307-328. 

34. Pamucar, D., Mihajlovic, M., Obradovic, R., Atanaskovic, P., 2017, Novel approach to group multi-

criteria decision making based on interval rough numbers: Hybrid DEMATEL-ANP-MAIRCA model, 

Expert Systems with Applications, 88, pp. 58-80. 

35. Stević, Ţ., Pamuĉar D, Zavadskas, E.K., Ćirović, G., Prentkovskis, O., 2017, The selection of wagons 

for the internal transport of a logistics company: A novel approach based on rough BWM and rough 

SAW methods, Symmetry, 9(11), 264,  pp. 1-25. 

36. Vasiljevic, M., Fazlollahtabar, H., Stevic, Z., Veskovic, S., 2018, A rough multicriteria approach for 

evaluation of the supplier criteria in automotive industry, Decision Making: Applications in 

Management and Engineering, 1(1), pp. 82-96. 

37. Gigović, Lj., Pamuĉar, D., Bajić, Z., Milićević, M., 2016, The combination of expert judgment and GIS-

MAIRCA analysis for the selection of sites for ammunition depot, Sustainability, 8(4), 372, pp. 1-30. 

38. Wang, J., Tang, P., 2011, A rough random multiple criteria decision making method based on Interval 

rough operator, Control and decision making, 26(7), pp. 1056-1059. 

39. Španović, I., 1983, Vojni putevi, VIZ, Belgrade. (Only in Serbian).