Plane Thermoelastic Waves in Infinite Half-Space Caused


Decision-Making: Applications in Management and Engineering  
Vol. 3, Issue 1, 2020, pp. 92-107. 
ISSN: 2560-6018 
eISSN: 2620-0104  

 DOI: https://doi.org/10.31181/dmame2003080f 

* Corresponding author. 
 E-mail addresses: feta.sinan@unite.edu.mk (F.Sianni), zivko.erceg@sf.ues.rs.ba (Z. 
Erceg), marko.vasiljevic@sf.ues.rs.ba (M. Vasiljevic) 

An Evaluation of a Third-Party Logistics Provider: The 
Application of the Rough Dombi-Hamy Mean Operator 

Feta Sinani 1*, Zivko Erceg 2 and Marko Vasiljevic 2 

1 University of Tetova, Faculty of Applied Sciences, North Macedonia 
2 University of East Sarajevo, Faculty of Transport and Traffic Engineering Doboj, 

Bosnia and Herzegovina 
 

Received: 9 January 2020;  
Accepted: 10 March 2020;  
Available online: 14 March 2020. 

 
Original scientific paper 

Abstract: Third-party logistics (3PL) has involved a significant response 
among researchers and practitioners in the recent decade. In the global 
competitive scenario, multinational companies (MNCs) not only improve the 
quality of the service and increase efficiency, but they also decrease costs by 
means of 3PL. However, the assessment and selection of 3PL is a very critical 
decision to make, comprising intricacy due to the existence of various 
imprecisely based criteria. Also, uncertainty is an unavoidable part of 
information in the decision-making process and its importance in the selection 
process is relatively high and needs to be carefully considered. Consequently, 
incomplete and inadequate data or information may occur among other 
various selection criteria, which can be termed as a multi-criteria decision-
making (MCDM) problem. Rough numbers are very flexible to model this type 
of uncertainty occurring in MCDM problems. In this paper, the Hamy Mean 
(HM) operator and Dombi operations are expanded by rough numbers (RNs) 
so as to propose the Rough Number Dombi-Hamy Mean (RNDHM) operator. 
Then, the Multi-Attribute Decision-Making (MADM) model is designed with the 
RNDHM operator. Finally, the RNDHM is employed to achieve the final ranking 
of the 3PL providers. 

Key words: MADM; rough numbers; Rough Dombi-Hamy Mean Operator; 
third-party logistics. 

1. Introduction 

The number of transport requests for the relocation of this specific type of goods 
is inevitably increasing. The basic difference in relation to the transport of the classical 
types of goods is that, in the transport of dangerous materials, participants are faced 
with additional safety requirements. This type of transport is specific because of the 

mailto:marko.vasiljevic@sf.ues.rs.ba


An Evaluation of a Third-Party Logistics Provider: The Application of the Rough Dombi-Hamy... 

93 

risks that present themselves during the implementation of this logistics process 
(Tanackov et al., 2018), which may endanger people’s life and health, the environment, 
and material resources due to the dangerous properties of the transported matter and 
potential reactions (Di Fazio et al., 2016). That is the reason why, at both the national 
and international levels, the hazardous substance domain is one of the most regulated 
areas in terms of the set directives tailored for each type of transport. 

The companies that produce or use hazardous materials are frequently unable to 
provide full transport services as they have the specific and varied capacities that must 
be used in the provision of complete transport services, which is why they often turn 
to 3PL services.  Also, numerous extra security requirements specific to dangerous 
materials must be met in organizing transport. When systematically viewed in the 
economic context, there are companies which find it more economically viable to 
outsource transport to third-party logistics providers (3PL), rather than possess and 
use their own fleets. When assessing the logistics sector, a number of companies will 
abolish or reduce their own fleet and hire 3PL service providers to handle transport, 
with the goal of reducing total costs and increasing profits (Azadi & Saen, 2013). In 
certain cases in practice, this solution has proven to be correct (Lien & Day, 2017), 
which is especially interesting as being relevant to the dangerous goods transport 
field, where the cost is not the only criterion. The evaluation of 3PL providers is a 
critical step for any manufacturer seeking to select a suitable 3PL provider as a 
business partner (Jung, 2017). Today, the evaluation of 3PL providers is most often 
performed by using multi-criteria decision-making methods. 

To deal with uncertainties and determine the values of qualitative attributes, the 
majority of authors use fuzzy sets (Zadeh, 1965) or various extensions of fuzzy theory, 
such as: interval-valued fuzzy sets (Sizong and Tao, 2016; Zywica, 2016), intuitionistic 
fuzzy sets (Atanassov, 1986), interval intuitionistic fuzzy sets (Nguyen, 2016), hesitant 
fuzzy sets (Ngan, 2017), as well as other extensions. Fuzzy sets are a very powerful 
and commonly used tool for dealing with imprecision. However, when selecting an 
appropriate membership function for fuzzy sets, subjectivity may affect the final 
decision, for which reason particular care needs to be taken with that respect 
(Pamucar et al., 2017; Pamucar et al., 2018). In addition to fuzzy theory, rough set 
theory, originally introduced by Pawlak (1991), is another suitable tool for treating 
imprecision. In recent years, rough set theory has been successfully implemented in 
various fields of human activities. Its application can be said to be adequate and 
usually irreplaceable when handling uncertainty and inaccuracy analyses. Knowing 
the advantages of rough set theory (Pawlak 1991), the application of rough sets and 
rough numbers is fully justified in today’s modern practice in the decision-making 
process when it includes imprecision in data.  

Although RNs have been effectively used in some fields, all the existing approaches 
are unsuitable to depict the interrelationships among any number of RNs assigned by 
a variable vector. The Hamy mean (HM) operator (Li et al., 2018) is a famous operator 
able to show interrelationships among any number of arguments assigned by a 
variable vector. Therefore, the HM operator can assign a robust and flexible 
mechanism to solve an information fusion in MADM problems. Thus, the HM operator 
is proposed in this paper in order to overcome these limits. How to aggregate RNs by 
applying traditional HM operators while simultaneously using the Dombi T-norms and 
co-norms (Sremac et al., 2018) is an interesting issue. So, the purpose of this paper is 
to propose the HM operator in order to solve a MADM problem of the evaluation of 
managers in the human resource department with RNs. In order to do that, the rest of 
the paper is organized as follows: in Section 2, IRNs are introduced; in Section 3, the 
HM operator with RNs is developed based on the Dombi operations; in Section 4, an 



 Sinani et al./Decis-Mak. Appl. Manag. Eng. 3 (1) (2020) 92-107 

94 

example of the evaluation of managers in a human resource department with RNs is 
presented, and finally in Section 5, comments are given at the end of the paper. 

2. Rough Numbers and Operations 

Rough numbers (RNs), consisting of the upper, lower and boundary intervals, 
respectively, determine the intervals of multiple experts’ evaluations without 
requiring any additional information, only relying on original data (Stevic et al., 2018). 
Hence, the obtained experts’ preferences objectively represent and improve the 
decision-making process. The definition of RNs according to Song et al. (2013) is given 
below. 

Let U be a universe containing all the objects and let X be a random object from U. 

Then, a set of k  classes is assumed to exist, which represents the DM’s preferences,

1 2( , ,..., )kR J J J , with the condition 1 2 ,..., kJ J J   . Then, for every

, , 1qX U J R q k    , the lower approximation ( )qApr J , the upper approximation 

( )qApr J  and the boundary interval ( )qBnd J  are determined as follows: 

 ( ) / ( )q qApr J X U R X J     (1) 

 ( ) / ( )q qApr J X U R X J       (2) 

     ( ) / ( ) / ( ) / ( )q q q qBnd J X U R X J X U R X J X U R X J         (3) 

The object can be represented by a rough number with the lower limit ( )qLim J  

and the upper limit ( )qLim J  in Eqs. (4)-(5). 

1
( ) ( ) ( )q q

L

Lim J R X X Apr J
M

    (4) 

1
( ) ( ) ( )q q

U

Lim J R X X Apr J
M

 
 

 (5) 

where LM  and UM  represent the sum of the objects given in the lower and upper 

object approximations of qJ , respectively. For the object qJ , the rough boundary 

interval ( ( )qIRBnd J ) is the interval between the lower and upper limits (Pamucar et 

al., 2018). The rough boundary interval is a measure of uncertainty. A bigger 
( )qIRBnd J  value shows that there are variations in experts’ preferences, while smaller 

values show that experts’ opinions do not considerably differ. All the objects between 

the lower limit ( )qLim J  and the upper limit ( )qLim J  of the rough number ( )qRN J  

are included in ( )qIRBnd J . Since RNs belong to a group of interval numbers, arithmetic 

operations applied to interval numbers are also appropriate for RNs. 



An Evaluation of a Third-Party Logistics Provider: The Application of the Rough Dombi-Hamy... 

95 

3. The Dombi Operations of RNs and the Dombi-Hamy Mean Operator 
With Rough Numbers 

Definition 1. Let p and q be any two real numbers. Then, the Dombi T-norm and T-
conorm between p and q are defined as follows (Dombi, 1982): 

1/

1
( , )

1 1
1

DO p q

p q

p q


 


      

     
     

 

 (6) 

1/

1
( , ) 1

1
1 1

c
DO p q

p q

p q


 

 
     

     
      

 

 (7) 

where 0   and  ( , ) 0,1p q  . 
In accordance with the Dombi T-norm and the T-conorm, Dombi operations are 

defined by rough numbers.  

Definition 2. Assuming that 1 1 1( ) [ ( ), ( )]RN Lim Lim   i

2 2 2( ) [ ( ), ( )]RN Lim Lim    are two rough numbers, , 0    and let 

     
1 1

( ) ( )
( ) ( ) , ( ) ,

( ) ( )

i i
i i i n n

i i
i i

Lim Lim
f RN f Lim f Lim

Lim Lim

 
  

 
 

 
   
  
   

 be a rough 

function, then some operational laws of rough numbers based on the Dombi T-norm 
and T-conorm can be defined as follows: 

(1) Addition “+” 

 
 

 
 

 
 

 
 

2

1

1/

1 2

1 2

2

1

1 2

1 2

( )2
,

1
( ) ( )

1
1 ( ) 1 ( )

1 2
( )2

1
( ) ( )

1
1 ( ) 1 ( )

( )

( ) ( )

( )

jj

jj

Lim

j
j

f Lim f Lim

f Lim f Lim

Lim

j
j

f Lim f Lim

f Lim f Lim

Lim

RN RN

Lim


 

 



 

 



 

 



 







      
              

    
           
   



 









 1/




 


 
 
 
 
 
 
 
 
 
  

  (8) 

(2) Multiplication “×” 

 
 

 
 

 
 

 
 

2

1

1/

1 2

1 2

2

1

1/

1 2

1 2

( )

,

1 ( ) 1 ( )
1

( ) ( )

1 2
( )

1 ( ) 1 ( )
1

( ) ( )

( ) ( )

jj

jj

Lim

f Lim f Lim

f Lim f Lim

Lim

f Lim f Lim

f Lim f Lim

RN RN


 


 



 

 



 

 

 





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

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

 
 
 
 
 














 





   (9) 

 
(3) Scalar multiplication, where 0   



 Sinani et al./Decis-Mak. Appl. Manag. Eng. 3 (1) (2020) 92-107 

96 

1

1/

1

1

1
1/

1

1

( )
,1

( )
1

1 ( )

1
( )

1
( )

1
1 ( )

( )

( )
( )

Lim

Lim

Lim

Lim

Lim

Lim

Lim

RN
Lim























 


   
   

   

   
       

 
 
 
 


 

 
 
  

   (10) 

 
(4) Power, where 0   

  1 11/ 1/
1 1

1 1

( ) ( )
,1

1 ( ) 1 ( )
1 1

( ) ( )

( )
Lim Lim

Lim Lim

Lim Lim

RN
 

 

  

 
 

 


         

        
        

 
 
 
 
 
  

   (11) 

 
The Hamy mean (HM) (Hara et al., 1998) is used to aggregate values, 

simultaneously including mutual correlations among multiple arguments, and is 
defined in the following way:    

Definition 3 (Hara et al., 1998). Assume that  ( 1, 2,..., )ix i n  
represents a set of non-

negative real numbers and the parameter 1, 2,...,k n , then the HM is defined as: 

  

1

1/

( )
1 2

1 ... 1

1
( , ,..., )

j

k

k
k

k
n i

i i n j

HM x x x x
n

k
    

 
 
  
  

 

     (12) 

where 1 2( , ,..., )ki i i  includes all the k-tuple combinations of (1, 2,..., )n  and 
n

k

 
 
 

 

represents a binomial coefficient calculated as: 

!

!( )!

n n

k k n k

 
 

 
   (13) 

 
Based on the RN operators (8)-(11), the RN Dombi-Hamy mean (RNDHM) operator 

is proposed. 

Theorem 1. Let ( ) [ ( ), ( )]j j jRN Lim Lim   ;  1, 2,...,j n , the collection of RNs 

in R, then the RNDHM operator is defined as follows: 



An Evaluation of a Third-Party Logistics Provider: The Application of the Rough Dombi-Hamy... 

97 

 

 
 

1

1

1/

1 ...1

1

1/

,( )
1 2

1 ... 1

( )

,

1

1

1 ( )

( )

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

( )

(

j

k

n

i

i

i i nk k
i j

j i j

k
k

kk
n i

i i n j

n Lim

i

i

n k

k
f Lim

f Lim

i

RNDHM RN RN RN RN
n

k

Lim

Lim













   







   



    

 
  

 
 
  

  
    

  
  
  
  

 
 
  
  

 











 



 
 

1

1/

1 ...1

1

( )

1

1

1 ( )

( )

)

n

i

i

i i nk k
i j

j i j

n Lim

i

n k

k
f Lim

f Lim













   



 
  

 
 
  

  
    

  
  
  
  

 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 









            

 (14) 

where  
 

 
1

1

( ) ( ) ( ) ;
( )

( ) ( ) ( ).

n

i i i
i

i n

i i i
i

f Lim Lim Lim
f RN

f Lim Lim Lim

  


  









 
 





 represents a rough 

function. 
The proof. Based on the Dombi operations of the RN, the following is obtained: 

a) 

 
 

 
 

1/ 1/

1 1

( ) ( )
,

1 ( ) 1 ( )1
1 1

( ) ( )

( )
i ij j

j

k k
i ij j

j ji ij j

k
Lim Lim

i

f Lim f Limj

f Lim f Lim

RN
 

 

 

 

 



 

                 
       
      

         

 
 
 
 
 
 
 
  

 

  

Thus,  

 
 

 
 

1/ 1/

1 1

1/

( ) ( )
,

1 ( ) 1 ( )1
1 1

1 1
( ) ( )

( )
i ij j

j

k k
i ij j

j ji ij j

k
k

Lim Lim

i

f Lim f Limj

k kf Lim f Lim

RN
 

 

 

 

 



 

                 
       
      

         

 
 
  
   
  

   
 
  

 

  

 

 

 

 

 

 

 



 Sinani et al./Decis-Mak. Appl. Manag. Eng. 3 (1) (2020) 92-107 

98 

 
b) Thereafter, 

 
 

 
 

1/

1 ...1

1

1

1 ...1

1

( )
,

1

1 ( )

1/
( )

( )
1 ... 1

1

1 ( )

( )

( )

( )

( )

i j

j

i i nk k
i j

j i j

j
i j

k
j

i i k
i j

j i j

Lim

i

k

f Lim

k
f Limk

i
Lim

i i n j
i

k

f Lim

f Lim

Lim

RN

Lim

























   



  



 
 
 
 
 

 
  
  
  
  
  

    


 
 
 
 
 



 
  
 

 







 
1/

nk





 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



 

c) Therefore, 

 

 
 

1

1

1/

1 ...1

1

1/

,( )
1 2

1 ... 1

( )

,

1

1

1 ( )

( )

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

( )

(

j

k

n

i

i

i i nk k
i j

j i j

k
k

kk
n i

i i n j

n Lim

i

i

n k

k
f Lim

f Lim

i

RNDHM RN RN RN RN
n

k

Lim

Lim













   







   



    

 
  

 
 
  

  
    

  
  
  
  

 
 
  
  

 











 



 
 

1

1/

1 ...1

1

( )

1

1

1 ( )

( )

)

n

i

i

i i nk k
i j

j i j

n Lim

i

n k

k
f Lim

f Lim













   



 
  

 
 
  

  
    

  
  
  
  

 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 









 

Besides, since 



An Evaluation of a Third-Party Logistics Provider: The Application of the Rough Dombi-Hamy... 

99 

 
 

 
 

1

1/

1 ...1

1

1

( )

1

1

1 ( )

( )

( )

1

1

1 ( )

( )

0 ( ) ( ),

( ) ( )

n

i

i

i i nk k
i j

j i j

n

i

i

i j

i j

n Lim

i RNGHM

i

n k

k
f Lim

f Lim

n Lim

RNGHM i

i

n k

k
f Lim

f Lim

Lim Lim

Lim Lim

















 

 



   





 
  

 
 
  

  
    

  
  
  
  



 
  

   





  

 











 1/

1 ...1

1

( )

i i nk k

j

RNGHMLim





   



 
 
 
 
 
 
 

 
 
 
 







,  

then 

 
 

 
 

1

1/

1 ...1

1

1

1 ...1

1

( )

,

1

1

1 ( )

( )

( )

1

1

1 ( )

( )

( )

( )

n

i

i

i i nk k
i j

j i j

n

i

i

i i nk k
i j

j i j

n Lim

i

i

n k

k
f Lim

f Lim

n Lim

i

i

n k

k
f Lim

f Lim

Lim

Lim

























   





   



 
  

 
 
  

  
    

  
  
  
  




 
 

   
 
 
 
 



















 1/


 
 
 
 


 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 represents the RN, so that 

Theorem 1 has been proven and Equation (14) is correct. 
The RNDHM operator also contains the following properties: Idempotency, 

Boundedness and Monotonicity.  

Theorem 2. (Idempotency). If ( ) ( ) [ ( ), ( )]jRN RN Lim Lim      for all 

 1, 2,...,j n  then  ( ), ( ),.., ( ) ( ) [ ( ), ( )]RNHM RN RN RN RN Lim Lim       . 

The proof. Since ( ) [ ( ), ( )] ( )j j jRN Lim Lim RN     ;  1, 2,...,j n , then the 

following calculations are obtained by using Theorem 1: 



 Sinani et al./Decis-Mak. Appl. Manag. Eng. 3 (1) (2020) 92-107 

100 

 

 
 

1

1/

1 ...1

1

1

( )

( )

,

1

1

1 ( )

( )

( )

1

1

1

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

( )

( )

n

i

i

i i nk k
i j

j i j

n

i

i

k

n Lim

i

i

n k

k
f Lim

f Lim

n Lim

i

i

n k

k
f L

RNDHM RN RN RN Lim Lim

Lim

Lim













    







   





 
  

 
 
  

  
    

  
  
  
  



 
  

 


  
 



















 
 

 

1/

1 ...1

1

1 1

1/

( )

( )

( ) ( )

,

1 11 ( ) 1

( ) ( )

i i nk k
i j

j i j

n n

i i

i i

i j

im

f Lim

n Lim n Lim

i i
n n

i iLim
k k

Lim Lim











 



 

   



 

 
 
 
 
 
 

  
  
  
  
  

    
    

    

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

  





 
 

 
1/

( )

( ), ( ) ( )

i j
Lim

Lim Lim RN





  

  
  
  

 
 
 
 
 

  
 

 

Theorem 3. (Commutativity). Let the RN rough set ' ' '1 2( ( ), ( ),.., ( )nRN RN RN    be 

any permutation of 1 2( ( ), ( ),.., ( ))nRN RN RN   . Then, there is 

   ' ' '1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nRNDHM RN RN RN RNDHM RN RN RN      . 
The proof. Based on Definition 2, the conclusion is obvious: 
 

 

 
 

 

' '

1 1

1/

' '1 ...1

'
1

( ) ' ' '
1 2

( ) ( )
'

,

1 1

1 1

1 ( ) 1 ( )

( )

( ), ( ),.., ( )

( ) ( )

n n

i i

i i

i i nk k
i ij j

j i j

k
n

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim

RNDHM RN RN RN

Lim Lim




 

 



  

  

   



 
  

 
 
    

     
      

  
  
  
  

  
 





 

 

1/

1 ...1

'
1 ( )

i i nk k

j i j
f Lim







   



 
 
 
 
 
 

  
  
  
  
  

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 





 



An Evaluation of a Third-Party Logistics Provider: The Application of the Rough Dombi-Hamy... 

101 

 
 

 
 

1 1

1/

1 ... 1 ...1 1

1 1

( ) ( )

,

1 1

1 1

1 ( ) 1 ( )

( ) ( )

( ) ( )

n n

i i

i i

i i n i i nk kk k
i ij j

j ji ij j

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim f Lim

Lim Lim


 

 

 

 

  

       

 

  
  

 
 
    

     
        

    
    
        

  
 

 

 

 

 

1/

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

j j j

k
i i iRNDHM RN RN RN



  


 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
  



 

Theorem 4. (Boundedness). Let ( ) [ ( ), ( )]j j jRN Lim Lim   ;  1, 2,...,j n , tte 

collection of RNs in R, let    ( ) min ( ) , min ( )j j jRN Lim Lim        and 

   ( ) max ( ) , max ( )j j jRN Lim Lim       , then

 1 2( ) ( ), ( ),.., ( ) ( )j n jRN RNHM RN RN RN RN    
 
  . 

The proof. Let 
 1 2( ) min ( ), ( ),..., ( ) ( ), ( )j n j jRN RN RN RN Lim Lim     

    
    

and 
 1 2( ) max ( ), ( ),..., ( ) ( ), ( )j n j jRN RN RN RN Lim Lim     

    
  . Then, 

 ( ) min ( )j j
j

Lim Lim 

 ,  ( ) min ( )j j

j
Lim Lim 


 ,  ( ) max ( )j j

j
Lim Lim 


  and 

 ( ) max ( )j j
j

Lim Lim 

 . Based on that, it follows:   

 
 

 
 

1 1

1/

1 ... 1 ...1 1

1 1

( ) ( )

1 1

1 1

1 ( ) 1 ( )

( ) ( )

( ) ( )

n n

i i

i i

i i n i ik kk k
i ij j

j ji ij j

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim f Lim

Lim Lim


 

 

 

 

 



 

       


 




 
  

 
 
    

     
        

    
    
        

 
 



 

  1/

n


 
 
 
 
 
 
 
 
 
 
 



  

 
 

1

1/

1 ...1

1

( )

1

1

1 ( )

( )

( )

n

i

i

i i n kk
i j

j i j

n Lim

i

i

n k

k
f Lim

f Lim

Lim
















   





 

  
 
 
  

  
    

  
  
    








  



 Sinani et al./Decis-Mak. Appl. Manag. Eng. 3 (1) (2020) 92-107 

102 

 
 

 
 

1 1

1/

1 ... 1 ...1 1

1 1

( ) ( )

1 1

1 1

1 ( ) 1 ( )

( ) ( )

( ) ( )

n n

i i

i i

i i n i ik kk k
i ij j

j ji ij j

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim f Lim

Lim Lim


 

 

 

 

 



 

       


 




 
  

 
 
    

     
        

    
    
        

 
 



 

  1/

n


 
 
 
 
 
 
 
 
 
 
 



 

 
 

1

1/

1 ...1

1

( )

1

1

1 ( )

( )

( )

n

i

i

i i n kk
i j

j i j

n Lim

i

i

n k

k
f Lim

f Lim

Lim
















   







 
  

 
 
  

  
    

  
  
    








  

According to the foregoing inequalities, it can be concluded that 

 1 2( ) ( ), ( ),.., ( ) ( )j n jRN RNHM RN RN RN RN    
 
  . 

Theorem 5. (Monotonicity). Let ( ) [ ( ), ( )]j j jRN Lim Lim   , 

( ) [ ( ), ( )]j j jRN Lim Lim   ;  1, 2,...,j n , be the two collections of RNs, if 

( ) ( )j jLim Lim  , ( ) ( )j jLim Lim   for all j  , then 

   1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nRNHM RN RN RN RNHM RN RN RN      . 

The proof. Since 0 ( ) ( )j jLim Lim   , 0 ( ) ( )j jLim Lim   , then, based on 

Theorem 1, the following is obtained: 

 
 

 
 

1 1

1/

1 ... 1 ...1 1

1 1

( ) ( )

1 1

1 1

1 ( ) 1 ( )

( ) ( )

( ) ( )

n n

i i

i i

i i n i i nk kk k
i ij j

j ji ij j

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim f Lim

Lim Lim


 

 

 

 

  

       

 


  

  
 
 
    

      
        

    
    
        

 
 

 

 

  1/


 
 
 
 


 
 
 
 
    

 
 

 
 

1 1

1/

1 ... 1 ...1 1

1 1

( ) ( )

1 1

1 1

1 ( ) 1 ( )

( ) ( )

( ) ( )

n n

i i

i i

i i n i i nk kk k
i ij j

j ji ij j

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim f Lim

Lim Lim


 

 

 

 

  

       

 


  

  
 
 
    

      
        

    
    
        

 
 

 

 

  1/


 
 
 
 


 
 
 
 
    

Let 
 1 2( ) ( ), ( ),.., ( )nRN RNHM RN RN RN     and 

 1 2( ) ( ), ( ),.., ( )nRN RNHM RN RN RN     be two RNs, then: 
(1) If ( ) ( )Lim Lim   and ( ) ( )Lim Lim  , then the following is obtained: 

   1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nRNHM RN RN RN RNHM RN RN RN      ; 



An Evaluation of a Third-Party Logistics Provider: The Application of the Rough Dombi-Hamy... 

103 

(2) If ( ) ( )Lim Lim   and ( ) ( )Lim Lim  , then it can be concluded that there 

are the following equalities: 

 
 

 
 

1 1

1/

1 ... 1 ...1 1

1 1

( ) ( )

1 1

1 1

1 ( ) 1 ( )

( ) ( )

( ) ( )

n n

i i

i i

i i n i i nk kk k
i ij j

j ji ij j

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim f Lim

Lim Lim


 

 

 

 

  

       

 


  

  
 
 
    

      
        

    
    
        

 
 

 

 

  1/


 
 
 
 


 
 
 
 
 

  

 
 

 
 

1 1

1/

1 ... 1 ...1 1

1 1

( ) ( )

1 1

1 1

1 ( ) 1 ( )

( ) ( )

( ) ( )

n n

i i

i i

i i n i i nk kk k
i ij j

j ji ij j

n Lim n Lim

i i

i i

n nk k

k k
f Lim f Lim

f Lim f Lim

Lim Lim


 

 

 

 

  

       

 


  

  
 
 
    

      
        

    
    
        

 
 

 

 

  1/


 
 
 
 


 
 
 
 
 

  

Finally, it can be concluded that there is the following inequality 

   1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nRNHM RN RN RN RNHM RN RN RN      . 

4. A Numerical Example: An Evaluation of a Third-Party Logistics 
Provider 

The practical application of the RNDHM aggregator is a logical conclusive phase of 
the subject-matter research. The multi-criteria model consists of a total of the five 
logistics providers who represent the alternatives having been evaluated based on the 
four criteria presented in detail in Table 1. The research study involved five experts, 
who evaluated the significance of the criteria, i.e. the alternatives according to the 
given criteria. The experts used a 1-9 scale for the evaluation of 3PL. 

Table 1. The criteria based on which the logistics provider was evaluated 

No Criteria Description of the criteria 

1. Financial stability 

The logistics provider delivers a financial report on 
positive business practices and a certificate of the 
settlement of tax liabilities, i.e. a certificate issued by 
the tax authorities, which confirms no extant debts in 
connection with the payment of taxes, fees, etc.  

2. 
The 

professionalization of 
drivers 

The logistics provider employs the drivers who have 
a certificate of approval for dangerous goods 
transport. The ongoing education of drivers is 
organized through seminars, additional and 
specialist training programs, safe and eco-friendly 
driving programs, etc.  

3. The cost of transport 
Covers all costs incurred by the carrier during the 
performance of services. 

4. 
The application of risk 
mitigation measures 

The logistics provider has one or more designated 
security advisors for the transport of dangerous 



 Sinani et al./Decis-Mak. Appl. Manag. Eng. 3 (1) (2020) 92-107 

104 

No Criteria Description of the criteria 

goods. At points of loading/unloading, the property 
of the contracting authority or third parties is treated 
with special care. The carrier acquaints his workers 
with the rules of conduct, movement, safety, fire 
protection, and all other measures and procedures.  

 
The five logistics providers Ai (i=1, 2, 3, 4, 5) were evaluated: 

1

2

( )
3

4

5

1           2          3          4

4, 5, 6, 8 8, 8, 9, 8 8, 8, 9, 7 8, 8, 9, 9

5, 5, 5, 5 7,8, 6, 9 8, 8, 8, 7 9, 9, 6, 9

7, 5, 8, 4 9, 9, 9, 7 7, 7,8, 9 6, 6, 7, 6

6, 5, 8, 3 7, 5, 6, 8 5, 5, 5, 6 9, 9, 8, 7

5, 5, 6, 4 4, 4, 5, 8 9, 9, 9, 8 6, 6, 8, 6

e

C C C C

A

A

AX

A

A









;1 e m




  


 
 



  

where m (in our case m=4) is the number of experts. By applying the rules for the 

transformation of crisp numbers into rough numbers, the elements of the matrix ( )eX
are transformed into RNs and presented in the matrix ( )e

RN
X  (Table 2).  

Table 2. The expert rough decision matrix ( ( )e
RN

X ) 

Alt./Crit. C1 C2 
A1 [4.0,5.75];[4.5,6.33];[5,7];[5.75,8] [8,8.25];[8,8.25];[8.25,9];[8,8.25] 
A2 [5.0,5.0];[5.0,5.0];[5.0,5.0];[5.0,5.0] [6.5,8.0];[7,8.5];[6.0,7.5];[7.5,9.0] 
A3 [5.33,7.5];[4.5,6.67];[6,8];[4.0,6.0] [8.5,9.0];[8.5,9.0];[8.5,9.0];[7.0,8.5] 
A4 [4.67,7];[4,6.33];[5.5,8];[3,5.5] [6.0,7.5];[5.0,6.5];[5.5,7.0];[6.5,8.0] 
A5 [4.67,5.33];[4.67,5.33];[5,6];[4,5] [4,5.25];[4,5.25];[4.33,6.5];[5.25,8] 

Alt./Crit. C3 C4 
A1 [8,8.25];[8,8.25];[8.25,9];[8,8.25] [8,8.5];[8,8.5];[8.5,9];[8.5,9] 
A2 [7.75,8];[7.75,8];[7.75,8];[7,7.75] [8.25,9];[8.25,9];[6,8.25];[8.25,9] 
A3 [7,7.75];[7,7.75];[7.33,8.5];[7.75,9] [6,6.25];[6,6.25];[6.25,7];[6,6.25] 
A4 [5,5.25];[5,5.25];[5,5.25];[5.25,6] [8.25,9];[8.25,9];[7.5,8.67];[7,8.25] 
A5 [8.75,9];[8.75,9];[8.75,9];[8,8.75] [6.0,6.5];[6,6.5];[6.5,8.0];[6.0,6.5] 

The aggregated initial rough decision matrix ( RNX ) is obtained by using the 

RNDHM aggregator. Thus, based on Eq. (14), the element 11( )RN x  in the matrix 

( )

5 4
( )

e
ijRNX RN x 

 
 

 is obtained as follows: 

          

 
1 1 1 1 1 1 1 1 1 1 1 1

(1) (2) (3) (4)
11 11 11 11

19.25
11

2 1 1 1 1 1 1
1

4 3.81 3.28 3.81 2.85 3.81 2.35 3.28 2.85 3.28 2.35 2.85 2.35
2

4,5.75 ; 4.5, 6.33 ; 5, 7 ; 5.( ), ( ), ( ), ( )

( ) 19.25

75,8RNDHMA RN x RN x RN x RN x

Lim x 
  
   
         

        
 





 

1/1

1/1

1 1 1 1 1 1 1 1 1 1 1 1

4.76

27.08
6.7111

2 1 1 1 1 1 1
1

4 3.71 3.28 3.71 2.87 3.71 2.39 3.28 2.87 3.28 2.39 2.87 2.39
2

( ) 27.08Lim x







 



  
   
         

         
  







  





 



An Evaluation of a Third-Party Logistics Provider: The Application of the Rough Dombi-Hamy... 

105 

where the values  ( )11( )ef RN x ; 1 4e   represent the rough functions. In a similar 
way, the other aggregated elements of the matrix RNX  are obtained. 

       
       
       
   

1

2

3

4

5

4.76; 6.71 8.06;8.43 8.06;8.43 8.25;8.75

5.00; 5.00 6.72;8.23 7.55; 7.94 7.62;8.81

4.89; 7.00 8.10;8.87 7.26;8.23 6.06; 6.43

4.18; 6.64 5.72; 7.23 5.

1                  2                 3               4

RN

K K K K

A

A

AX

A

A



   
       

06; 5.43 7.73;8.72

4.57; 5.4 4.36; 6.14 8.56;8.94 6.12; 6.84

 
 
 
 
 
 
 
 

 

In the following steps, the final values of the criteria functions of the alternatives 
were obtained and ranked by applying additive normalization and weighting the 

normalized matrix using the following equation  
1

n

i ij ji
Q A x w


  , where xij 

represents the elements of the normalized matrix and wj = (0.2, 0.22, 0.28, 0.30)T 
represents the criteria weights. Thus, the alternative score functions: 
Q(A1)=[0.2;0.25], Q(A2)=[0.18;0.23], Q(A3)=[0.18;0.24], Q(A4)=[0.15;0.22] and 
Q(A5)=[0.16;0.21] were obtained. It is desirable for an alternative to have the highest 
possible value of its criteria function, so A1 > A2 > A3 > A5 > A4. 

5. Conclusion 

Dangerous goods transport and its potential consequences arouse the attention of 
the public because of the detrimental effect of these materials on the environment, as 
well as people, and possible accidents, too. In order to meet the complex demands of 
today’s market, the hazardous substance increased consumption trend has appeared, 
thereby increasing the volume of the production and transport of these goods. 
Transport is a mandatory logistic activity for supplying users with these materials, 
regardless of whether they are in a raw, semi-processed, or fully-processed form.  

In this paper, MADM problems with RNs were subjected to investigation, which 
was only followed by the utilization of the HM operator and the Dombi operations in 
order to design an HM operator with RNs, i.e.  the RNDHM operator. After that, the 
RNDHM was used so as to propose a model for MADM problems with RNs. Finally, a 
real example of the evaluation of 3PL was used in order to show the developed 
approach. 

The procedure for selecting 3PL carriers is particularly specific in the field of 
dangerous substances due to high risks and additional security requirements. This 
reflects in the criteria defined for the evaluation of the logistics providers. One of the 
benefits of using 3PL carriers’ services is that they have a wider range of operations 
and are able to satisfy the clients who have a high frequency of transport needs on a 
weekly basis. Specialized carriers better understand the market and customers’ needs. 
They also have well-designed strategies and business models to continue improving 
their offers, regional coverage, and specialization in all industry sectors. 

A direction for a further research study implies supporting the environmental 
sustainability of 3PL dangerous material providers, particularly in the areas marked 
as the five topical areas: influencing factors, green actions, an impact on performance, 
information and communication technology (ICT) tools supporting green actions, 
energy efficiency in road freight transport, and shippers’ perspective and 



 Sinani et al./Decis-Mak. Appl. Manag. Eng. 3 (1) (2020) 92-107 

106 

collaboration. In subsequent studies, the extension and application of RNs needs to be 
studied in many other uncertain environments and other applications. 

Author Contributions: Each author has participated and contributed sufficiently to 
take public responsibility for appropriate portions of the content.  

Funding: This research received no external funding. 

Conflicts of Interest: The authors declare no conflicts of interest. 

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