https://doi.org/10.31449/inf.v46i2.3452 Informatica 46 (2022) 235–242 235 
 
A Hesitant Fuzzy Multiplicative Base-criterion Multi-criteria Group 
Decision Making Method 
Monika Narang
1*
, M.C. Joshi
1
 and A.K. Pal
2
 
E-mail: monikanarang5@gmail.com, mcjoshi69@gmail.com, arun_pal1969@yahoo.in 
1
Department of Mathematics, D.S.B. Campus, Kumaun University, Nainital, India 
2
Department of Mathematics, Statistics & Computer Science, College of Basic Sciences & Humanities 
Govind Ballabh Pant University of Agriculture & Technology, Pantnagar, India 
Keywords: multi-criteria group decision-making, hesitant fuzzy sets, base-criterion method, pairwise comparisons 
Received: February 22, 2021 
Hesitant fuzzy sets have a unique characteristic that its basic element could manifest the assessment values 
of different decision makers on the same option under a certain criterion. Base-criterion method is a very 
significant tool for calculating the weights of the criteria in multiple criteria decision-making. In this 
paper, we developed a novel approach hesitant fuzzy BCM based on hesitant fuzzy multiplicative 
preference relation for multiple criteria group decision making. The base-comparison of the preferential 
criterion relative to other criteria is expressed as linguistic terms, which might be converted into hesitant 
multiplicative elements (HMEs). HMEs are extended along the same length according to the attitude of 
the decision makers. Then normalized optimal hesitant fuzzy weights are calculated. The normalized 
optimal hesitant fuzzy weights of criteria may be transferred to crisp values by employing score function. 
To illustrate the applicability and suitability of hesitant fuzzy BCM, we analyse the optimal transportation 
mode selection problem and car selection problem under hesitant fuzzy environment. The outcomes of the 
proposed model indicate that the hesitant fuzzy BCM is highly consistent and can yield appreciable 
preference ranking of criteria and alternatives.  
Povzetek: Predstavljena je metoda odločanja v skupinah s pomočjo obotavljivih mehkih kriterijev.  
 
1 Introduction
Decision making is the process of selecting an optimal 
alternative from a set of alternatives. Multi-criteria 
decision-making is an important tool of decision-making 
process that explicitly evaluates multiple conflicting 
criteria in decision making (both in daily life and in 
settings such as business, government and medicine) [1]. 
MCDM methods are divided into two main categories: 
Multi-Objective Decision-Making (MODM) and Multi-
Attribute Decision-Making (MADM). The fundamental 
difference between MODM and MADM is that MODM 
have no predetermined alternatives and MADM have 
limited number of alternatives [2,3]. MODM methods are 
employed to handle continuous problems, on the other 
hand, multi-attribute decision making (MADM) methods 
are used to solve discrete problems. MCDM is commonly 
used to describe the discrete MADM.  
Over the previous years, many MCDM methods have 
been introduced by researchers such as VIKOR [4], 
ELECTRE (Elimination and Choice Expressing Reality) 
[5], TOPSIS (Technique for order preference by similarity 
to an ideal solution) [6,7], COPRAS (Complex 
Proportional Assessment) [8,9], SWARA (step-wise 
weight assessment ratio analysis) [10], ANP (Analytic 
Network Process) [11,12], AHP (Analytic Hierarchy 
Process) [13], BWM (Best-Worst Method) [14,15]. 
 
 
*
 Corresponding author 
In a practical problem, MCDM consists of two parts: 
(a) obtaining decision information, including criterion 
weight, (b) ranking the alternatives by a certain approach. 
The most important part is how we calculate the criterion 
weights, which has been the foundation for the 
introduction of many MCDM methods. Recently, Haseli 
et al [16] developed a novel Base-Criterion Method 
(BCM) which is a better route to determine the weight of 
the criteria. First, the decision maker selects the base-
criterion (preferential, selective) and then a pairwise 
comparison is made between the base-criterion and other 
criteria. This technique is much clearer and more accurate 
because the execution of secondary comparisons is not 
necessary. This can achieve the weight of the criterion 
with less pairwise comparisons than existing MCDM 
methods. The final weights determined through BCM are 
very authentic as the comparison is completely consistent 
while other traditional MCDM techniques such as BWM 
and AHP have low inconsistency ratio. 
It is hard to recognize all the facets of a decision-
making problem for a single decision maker. The 
decisions made by groups are mostly different from those 
made by individuals. So, it is essential to have opinions 
from group of experts/decision makers. When more than 
one expert evaluates an option, it is very possible for them 

236 Informatica 46 (2022) 235–242 M. Narang et al. 
 
to have different opinions. In today's environment, group 
decision making (GDM) methods pay more attention to 
the fuzzy information contained in group decision making 
problems. 
Due to the hesitant of the purpose as well as the 
fuzziness of human mind, human decisions generally 
collide the characteristics of opacity.  Prof. Zadeh in 1965 
proposed the fuzzy set theory [17]. Fuzzy mention the 
things which are obscure. Working system of fuzzy sets is 
retraced uncertainty and lack of clarity in daily life. After 
that, several tools have been introduced to handle the 
fuzzy information such as interval valued fuzzy set 
[18,19], type-2 fuzzy set [20], intuitionistic fuzzy set [21], 
and hesitant fuzzy sets [22]. 
People usually hesitate about one thing or the other 
while making decisions, which makes it difficult to reach 
a final agreement. Hesitant Fuzzy Set (HFS) [22,23,24] is 
very powerful tool for obtaining the optimum alternative 
in a decision-making process with multi-criteria and 
multiple individuals. HFS allows the membership having 
a set of few values. Generally, the grades of preference are 
not symmetric, but distributed asymmetrically around 
some value. Saaty's 1/9-9 scale is a useful tool to deal with 
such a situation, especially in expressing the 
multiplicative preference relation applied in many areas. 
A hesitant fuzzy multiplicative preference relation 
(HFMPR), which is developed based on the fuzzy 
preference relation [25,26], is the most usual tool for 
expressing DMs' preferences over decision-making 
alternatives.  
In the present study, using hesitant fuzzy 
multiplicative numbers the reference comparisons of 
BCM are executed from group decision making process to 
have more real and convincing ranking results. So, a novel 
approach, including BCM under hesitant fuzzy 
environment has been developed and employed for the 
first time. We expended the BCM to more authentic HF-
BCM.  
We systematize the rest of paper as follows: Section 2 
demonstrates the basic concepts. Section 3 introduces the 
proposed methodology. Section 4 describes the 
application of our proposed method by considering simple 
example of decision-making problems. Section 5 shows 
findings and conclusion. 
2 Preliminaries 
Definition 2.1 ([22],[24]) Let 𝑋 be a fixed set, HFS on 𝑋 
is in terms of a function that when applied to 𝑋 returns a 
subset of [0,1]. 
Xia and Xu [27] expressed the HFS by a mathematical 
symbol: 
𝐴 ={〈𝑥 ,ℎ
𝐴 (𝑥 )〉|𝑥 ∈𝑋 }  (1) 
Where ℎ
𝐴 (𝑥 )  is a set of some values in [0,1], denoting 
the possible membership degrees of the element 𝑥 ∈𝑋 to 
the set 𝐴 and ℎ=ℎ
𝐴 (𝑥 ) is a hesitant fuzzy element (HFE). 
Definition 2.2 [27] For a HFE ℎ,𝑠 (ℎ)=
1
𝑙 ℎ
∑ 𝑝 𝑝 ∈ℎ
, is 
called the score of  ℎ,  where 𝑙 ℎ
 is the number of the 
elements in ℎ. 
For two HFEs, ℎ
1
 and ℎ
2
, if 𝑠 (ℎ
1
)>𝑠 (ℎ
2
), then ℎ
1
>ℎ
2
. 
Definition 2.3 [27] Let ℎ
1
 and ℎ
2
 be two HFEs, then basic 
operations on HFEs are as follows: 
ℎ
1
⊕ℎ
2
=⋃ {𝑝 1
+𝑝 2
}
𝑝 ∈ℎ
1
,𝑝 ∈ℎ
2
; 
ℎ
1
⊖ℎ
2
=⋃ {𝑝 1
−𝑝 2
}
𝑝 1
∈ℎ
1
,𝑝 2
∈ℎ
2
; 
ℎ
1
⊗ℎ
2
=⋃ {𝑝 1
.𝑝 2
}
𝑝 1
∈ℎ
1
,𝑝 2
∈ℎ
2
; 
ℎ
1
⦼ℎ
2
=⋃ {
𝑝 1
𝑝 2
}
𝑝 1
∈ℎ
1
,𝑝 2
∈ℎ
2
  
(2) 
Definition 2.4 [13] Let 𝐴 ={𝐴 1
,𝐴 2
,..,𝐴 𝑛 } be a set of 𝑛 
alternatives, then 𝐵 =(𝑏 𝑖𝑗
)
𝑛 ×𝑛 is called a multiplicative 
preference relation on 𝐴 ×𝐴 , whose element 𝑏 𝑖𝑗
 estimates 
the preference of the alternative 𝐴 𝑖 over 𝐴 𝑗 , and is 
characterized by a ratio scale such as Saaty’s ratio scale 
such that 𝑏 𝑖𝑗
∈[
1
9
,9], and 𝑏 𝑖𝑗
.𝑏 𝑗𝑖
=1,𝑖 ,𝑗 =1,2,..,𝑛 .  
Where 𝑏 𝑖𝑗
 unfolds indifference between 𝐴 𝑗 and 𝐴 𝑖 
; 𝑏 𝑖𝑗
>1 unfolds that 𝐴 𝑖 is preferred to 𝐴 𝑗 , 
especially, 𝑏 𝑖𝑗
=9 unfolds that 𝐴 𝑖 is absolutely preferred 
to 𝐴 𝑗 ; 𝑏 𝑖𝑗
<1 unfolds that 𝐴 𝑗 is preferred to 𝐴 𝑖 , 
especially, 𝑏 𝑖𝑗
=
1
9
 unfolds that 𝐴 𝑗 is absolutely preferred 
to 𝐴 𝑖 . 
Definition 2.5 [28] Let 𝑋 ={𝑥 1
,𝑥 2
,….𝑥 𝑛 } be a fixed set, 
then a hesitant fuzzy multiplicative preference relation on 
the set 𝑋 is represented by the matrix  𝐻 =(ℎ
𝑖𝑗
)
𝑛 ×𝑛 ∈
𝑋 ×𝑋 , where ℎ
𝑖𝑗
={𝑝 𝑖𝑗
𝑙 , 𝑙 =1,2,…|ℎ
𝑖𝑗
|} is a HME 
which expresses all possible preference degrees of the 
alternative 𝑥 𝑖 over 𝑥 𝑗 given by the DMs. ℎ
𝑖𝑗
 (HME) should 
satisfies the conditions: 
𝑝 𝑖𝑗
𝜎 (𝑙 )
.𝑝 𝑗𝑖
𝜎 (|ℎ
𝑖𝑗
|−𝑙 +1)
=1,    𝑝 𝑖𝑖
=1,   |ℎ
𝑖𝑗
|=|ℎ
𝑗𝑖
|,  
𝑖 ,𝑗 =1,2,…,𝑛 .   (3) 
Where all 𝑝 𝑖𝑗
 are ranked in ascending order, 𝑝 𝑖𝑗
𝜎 (𝑙 )
 
represents the 𝑙 𝑡 ℎ
 smallest value in ℎ
𝑖𝑗
 and |ℎ
𝑖𝑗
| unfolds 
the number of elements in ℎ
𝑖𝑗
. In particular, 𝑝 𝑖𝑖
=1 
unfolds the indifference between 𝑥 𝑖 and 𝑥 𝑗 , 𝑝 𝑖𝑗
>1 
indicates that 𝑥 𝑖 is preferred to 𝑥 𝑗 , and 𝑝 𝑖𝑗
<1 
indicates that 𝑥 𝑖 is not preferred to 𝑥 𝑗 or 𝑥 𝑗 is preferred to 
𝑥 𝑖 .  
In fact, if the preference degree of the alternative 
𝑥 𝑖 over 𝑥 𝑗 is 𝑝 , then the preference degree of the 
alternative 𝑥 𝑗 over 𝑥 𝑖 should be 
1
𝑝 ⁄ . Thus, in the hesitant 
multiplicative circumstance, the product of the 𝑙 𝑡 ℎ
 
smallest value in ℎ
𝑖𝑗
 and the  𝑙 𝑡 ℎ
 largest value in ℎ
𝑗𝑖
 should 
be 1. The second condition defines that the preference 
degree of the alternative 𝑥 𝑖 over itself should be 1. The 
third one states that the lengths of ℎ
𝑖𝑗
 and ℎ
𝑗𝑖
 should be the 
same. 
Example 2.1 If a group of decision makers is asked to give 
the estimation of the degree to which 𝐴 𝑖 is preferred to 𝐴 𝑗 
(𝑖 ≠𝑗 ) , some DMs give 𝑝 𝑖𝑗
1
, some give 𝑝 𝑖𝑗
2
 and others give 
𝑝 𝑖𝑗
3
, where 𝑝 𝑖𝑗
1
,𝑝 𝑖𝑗
2
,𝑝 𝑖𝑗
3
∈(
1
9
,9) . Then the preference 
information ℎ
𝑖𝑗
 that 𝐴 𝑖 is preferred to 𝐴 𝑗 is given by ℎ
𝑖𝑗
=
{𝑝 𝑖𝑗
1
,𝑝 𝑖𝑗
2
,𝑝 𝑖𝑗
3
}. For alternatives 𝐴 𝑖 and 𝐴 𝑘 (𝑖 ≠𝑗 ≠𝑘 ) , 
some DMs in a group may give 𝑝 𝑖𝑘
1
 and the others give 𝑝 𝑖𝑘
2
. 

A Hesitant Fuzzy Multiplicative Base-criterion... Informatica 46 (2022) 235–242 237 
 
Then the preference information ℎ
𝑖𝑗
 that 𝐴 𝑖 is preferred to 
𝐴 𝑘 is given by ℎ
𝑖𝑗
={𝑝 𝑖𝑘
1
,𝑝 𝑖𝑘
2
}. 
Definition 2.6 [29] According to the definition of HME, 
it can be seen that the number of assessment values in 
different HMEs may vary and the assessment values in 
each HME are usually out of order.  
To guarantee that the number of values in different 
HMEs be equal, we can add elements by the linear 
combination of the maximal and minimal element in ℎ
𝑖𝑗
 
with a parameter 𝜆 , shown as:  
ℎ
𝑖𝑗
⃗⃗⃗⃗⃗ 
  =  𝜆 ℎ
𝑖𝑗
𝑚𝑎𝑥 +(1−𝜆 )ℎ
𝑖𝑗
𝑚𝑖𝑛 (4) 
where 𝜆 =1 and 𝜆 =0 means the optimistic attitude 
and pessimistic attitude of DMs, respectively. 𝜆 =
1
2
  indicates the neutral attitude of the DMs and the adding 
element ℎ
𝑖𝑗
 is the average value of ℎ
𝑖𝑗
. 
3 Hesitant fuzzy BCM (HF-BCM)  
Because of the hesitancy of the purpose as well as the 
fuzziness of human mind, human decisions generally 
clasp the characteristics of opacity. To elaborate hesitancy 
and vagueness involved in decision making, Hesitant 
fuzzy sets (HFS) have been introduced by Torra and 
Narukawa [22,24] as an extension of fuzzy sets [17]. HFS 
allows the membership having a set of several possible 
values to deal with uncertain information. when the 
assessment values given by experts are different, we could 
unify them into an HFE. The features of HFE are very 
compatible with GDM problems. BCM [16] is a novel 
MCDM method to calculate the weights of the criteria and 
alternatives. In the BCM, First, the decision maker selects 
the base-criterion (preferential) and then a pairwise 
comparison is made between the base-criterion and other 
criteria. This technique is much clearer and more accurate 
because the execution of secondary comparisons is not 
necessary.  
In our proposed methodology, hesitant fuzzy BCM 
method is developed based on hesitant fuzzy 
multiplicative preference relation for multi-criteria group 
decision making (MCGDM). 
3.1 Hesitant fuzzy BCM with hesitant 
fuzzy multiplicative preference 
relations 
In a MCGDM, it is too difficult to determine the weights 
of the criteria because DMs often have different 
preferences for criteria and also uses natural language. 
Linguistic terms such as “Equally important”, “Extremely 
important” and “Strongly important” are used to make the 
hesitant pairwise comparisons. The decision makers 
provide the hesitant fuzzy multiplicative preference 
relations via pairwise comparison on the n criteria by 
using the Saaty’s 1/9-9 [11] scale. The rules of 
transformation are listed in Table 1. Suppose an HFMPRs 
be 
𝐻 = (
ℎ
11
ℎ
12
⋯ ℎ
1𝑛 ℎ
21
ℎ
22
⋯ ℎ
2𝑛 ⋮ ⋮ ⋱ ⋮
ℎ
𝑛 1
ℎ
𝑛 2
⋯ ℎ
𝑛 1
)   (5) 
Whereℎ
𝑖𝑗
={𝑝 𝑖𝑗
𝑙 ,𝑙 =1,2,…|ℎ
𝑖𝑗
|} (hesitant multiplicative 
element (HME)) unfolds the relative hesitant fuzzy 
preference (HFP) of criterion 𝑖 to the criterion 𝑗 given by 
the decision makers; 𝑝 𝑖𝑖
=1, when 𝑖 =𝑗 . 
In hesitant fuzzy BCM, Hesitant fuzzy pairwise 
comparisons are divided into two parts: 
Definition 3.1 If 𝑖 is the base-criterion, then ℎ
𝑖𝑗
 is called 
hesitant fuzzy base comparison. 
Definition 3.2 If 𝑖 and 𝑗 are not base-criterion, then ℎ
𝑖𝑗
 is 
called hesitant fuzzy final comparison. 
     The basic principle of BCM [16] tells us that not 
all hesitant fuzzy pairwise comparisons are required to 
obtain a complete matrix. There is total 𝑛 2
 HFMNs in the 
matrix  𝐻 =(ℎ
𝑖𝑗
)
𝑛 ×𝑛 . But we only require 𝑛 −1 hesitant 
fuzzy pairwise comparisons (hesitant fuzzy base-
comparison). The hesitant fuzzy final comparisons are 
taken from the hesitant fuzzy base comparisons. Without 
making the hesitant fuzzy final comparisons, optimal 
hesitant fuzzy weight values are obtained. The HFWs of 
criteria and alternatives with respect to various criteria 
could be derived using HF-BCM. 
Now, we elaborate the steps of hesitant fuzzy BCM to 
determine the optimal hesitant fuzzy weights. 
Step 1 Determine the decision criteria set and group of 
experts 
Determine a set of decision criteria {𝐶 1
,𝐶 2
,𝐶 3,
….,𝐶 𝑛 } 
and group of experts {𝐷𝑀
1
,𝐷𝑀
2
,..,𝐷𝑀
𝑘 } on the basis of 
which decision is taken. 
Step 2 Determine the base-criteria (preferential, 
selective). 
Decision makers select one of the criteria as a base-
criteria (preferential) from a set of decision criteria 
{𝐶 1
,𝐶 2
,𝐶 3,
….,𝐶 𝑛 } but no comparison is performed in this 
step. 
Step 3 Execute the hesitant fuzzy base-comparisons. 
Based on Table 1, the relative hesitant fuzzy 
preference of the base criteria at the other criteria is 
derived. The linguistic terms are transformed into HMEs.  
The resulting vector of hesitant fuzzy base-comparisons as 
follows. 
𝐻 𝐵 =(ℎ
𝐵 1
,ℎ
𝐵 2
,ℎ
𝐵 3
,….,ℎ
𝐵𝑛
) 
Table 1: The Saaty’s Scale. 
1/9-9 scale 0.1-0.9 scale Linguistic terms 
1/9 0.1 Extremely not important 
1/7 0.2 Very strongly not important 
1/5 0.3 Strongly not important 
1/3 0.4 Moderately not important 
1 0.5 Equally important 
3 0.6 Moderately important 
5 0.7 Strongly important 
7 0.8 Very strongly important 
9 0.9 Extremely important 
 Other 
values 
b/w 1/9-9 
Other values 
b/w 0-1 
Intermediate values used to 
present compromise 
 

238 Informatica 46 (2022) 235–242 M. Narang et al. 
 
where 𝐻 𝐵 unfolds the hesitant fuzzy base-criteria at the 
other criteria vector, and ℎ
𝐵𝑗
 unfolds the hesitant fuzzy 
preference (HFP) of the base-criteria over the 𝑗 criteria.  
Step 4 Normalization of the hesitant fuzzy base-
comparison vector. 
According to the definition of HME, it can be seen 
that the number of assessment values in different HMEs 
may vary and the values in each HME are usually out of 
order.  
Using Equation 4, the hesitant fuzzy base-comparison 
vector is normalized according to decision-makers’ 
attitude. 
Step 5 Derive the normalized hesitant fuzzy optimal 
weights. 
The normalized optimal hesitant fuzzy weights for 
each 
ℎ
𝑤 𝐵 ℎ
𝑤 𝑗 will be equal to ℎ
𝐵𝑗
 for all 𝑗 . The optimal hesitant 
fuzzy weight values can be determined by absolute 
differences |
ℎ
𝑤 𝐵 ℎ
𝑤 𝑗 −ℎ
𝐵𝑗
| for all 𝑗 . Regarding the weight 
values are non-negative, the normalized optimal hesitant 
fuzzy weights can be determined by deriving the problem 
as follows: 
 
Min max|
ℎ
𝑤 𝐵 ℎ
𝑤 𝑗 −ℎ
𝐵𝑗
|≤𝜉 
Such that     {
∑ 𝑅 (ℎ
𝑤 𝑗 )=1
𝑛 𝑗 =1
𝑝 𝑗 𝑙 ≥0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 (6) 
Where ℎ
𝑤 𝐵 ={𝑝 𝐵 𝑙 ,𝑙 =1,2,..,|ℎ
𝑖𝑗
|},  ℎ
𝑤 𝑗 ={𝑝 𝑗 𝑙 ,𝑙 =
1,2,..,|ℎ
𝑖𝑗
|}, 𝜉 ={𝑝 𝑙𝜉
,𝑙 =1,2,..,|ℎ
𝑖𝑗
|} 
The Equation 6 can be rewritten as the nonlinearly 
constrained problem.            
min𝜉 
Such that 
 
                
{
 
 
 
 
|
ℎ
𝑤 𝐵 ℎ
𝑤 𝑗 −ℎ
𝐵𝑗
|≤𝜉 ∑ 𝑅 (ℎ
𝑤 𝑗 )=1
𝑛 𝑗 =1
𝑝 𝑗 𝑙 >0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ;𝑙 =1,2,..,|ℎ
𝑖𝑗
|
  (7) 
       
Regarding the HMEs and 𝜉 ={𝑘 ∗
}, Equation 7 can be   
rewritten as: 
min𝜉 
Such that 
 
{
 
 
 
 |
(𝑝 𝐵 𝑙 )
(𝑝 𝑗 𝑙 )
−(𝑝 𝐵𝑗
𝑙 )|≤(𝑘 ∗
)
∑ 𝑅 (ℎ
𝑤 𝑗 )=1
𝑛 𝑗 =1
𝑝 𝑗 𝑙 >0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ;𝑙 =1,2,..,|ℎ
𝑖𝑗
|
        (8) 
 
Solution of the Equation 8 gives the normalized optimal 
hesitant fuzzy weights{ℎ
𝑤 1
,ℎ
𝑤 2
,…,ℎ
𝑤 𝑛 }, then these 
normalized optimal HFWs can be converted to crisp 
numbers by employing score function. Figure 1 represents 
the flowchart of the proposed methodology. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 1: Flowchart of the proposed methodology. 
3.2 Consistency for HF-BCM 
The hesitant fuzzy pairwise comparison is fully consistent 
if  
ℎ
𝐵𝑎𝑠𝑒 .𝑖 ×ℎ
𝑖𝑗
=ℎ
𝐵𝑎𝑠𝑒 .𝑗     for all 𝑖 and 𝑗 . 
The decision maker should pursuance the following 
principle in entrusting the hesitant fuzzy multiplicative 
numbers for hesitant fuzzy base-comparisons. 
(
1
9
,
1
8
,…)≤{𝑝 𝑖𝑗
𝑙 ,𝑙 =1,2,…|ℎ
𝑖𝑗
|}≤(8,9,…) 
    (
1
9
,
1
8
,…)≤
(𝑝 𝐵𝑗
𝑙 ,𝑙 =1,2,..,|ℎ
𝑖𝑗
|)
(𝑝 𝐵𝑖
𝑙 ,𝑙 =1,2…,|ℎ
𝑖𝑗
|)
≤(8,9,…)     (9) 
4 Case study 
In this section, we describe the application of hesitant 
fuzzy BCM by considering simple examples of decision-
making problems.  
4.1 Case study 1 
A company wants to select the best transportation mode to 
deliver the product to the market place. As a case study, 
we adopted the example of mode of transport described in 
[16] and deal the problem by using our proposed hesitant 
fuzzy BCM method from group decision making process. 
There are three criteria chosen for the optimal 
transport mode selection issue: (a) load flexibility (b) 
accessibility (c) cost. The group of decision makers 
chooses cost criterion as the base-criterion. DMs executes 
hesitant fuzzy base-comparisons based on HMEs using 
Table 1 group of decision makers provide the estimation 
of the degree to which cost is preferred to load flexibility, 
some DMs provide the preference value 6, and others 
   Start 
Create the criteria 
set 
Determine the group 
of DMs 
Select the preferential 
criteria 
Execute the base 
comparisons of 
preferential criteria 
relative to other 
criteria 
Convert preference 
of different decision 
makers into one 
HME 
Normalized the base-vector as per DMs attitude 
Solve the model, calculate the weight of criteria  

A Hesitant Fuzzy Multiplicative Base-criterion... Informatica 46 (2022) 235–242 239 
 
provide the preference value 8. Group of decision makers 
provide the estimation of the degree to which cost is 
preferred to accessibility, some DMs provide the 
preference value 2, some DMs provides the preference 
value 4, some DMs provides the preference value 1 and 
other provides 3. On the basis which the hesitant fuzzy 
base-comparison vector can be obtained as: 
𝐻 𝐵 =[(6,8),(1,2,3,4),(1,1,1,1)] 
Now, the normalized hesitant fuzzy base-comparison 
vector can be computed by using Equation 4 as follows: 
Case-1 Optimistic attitude of DMs ( 𝜆 =1). 
𝐻 𝐵 =[(6,8,8,8),(1,2,3,4),(1,1,1,1)] 
Case-2 Neutral attitude of DMs ( 𝜆 =
1
2
 ). 
𝐻 𝐵 =[(6,7,7,8),(1,2,3,4),(1,1,1,1)] 
Case-3 Pessimistic attitude of DMs ( 𝜆 =0). 
𝐻 𝐵 =[(6,6,6,8),(1,2,3,4),(1,1,1,1)] 
The maximum length in the normalized hesitant fuzzy 
preference is 4. Thus, the number of elements in the 
normalized hesitant fuzzy preference are 4, and the 
number of several possible values in the normalized 
hesitant fuzzy optimal weights is four as well. 
Based on Table 2 and Case-1, the normalized optimal 
hesitant fuzzy weight for each criterion can be obtained by 
solving the non-linear optimization problem as follows:
 min𝜉 
Such that 
    
{
 
 
 
 
 
 
 
 
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
,𝑝 3
4
)
(𝑝 1
1
,𝑝 1
2
,𝑝 1
3
,𝑝 1
4
)
−(𝑝 31
1
,𝑝 31
2
,𝑝 31
3
,𝑝 31
4
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
,𝑝 3
4
)
(𝑝 2
1
,𝑝 2
2
,𝑝 2
3
,𝑝 2
4
)
−(𝑝 32
1
,𝑝 32
2
,𝑝 32
3
,𝑝 32
4
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
,𝑝 3
4
)
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
,𝑝 3
4
)
−(𝑝 33
1
,𝑝 33
2
,𝑝 33
3
,𝑝 33
4
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
∑ 𝑅 (ℎ
𝑤 𝑗 )=1
𝑛 𝑗 =1
𝑝 𝑗 𝑙 >0 𝑓𝑜𝑟 𝑎 𝑙𝑙 𝑗 .
 
            (10) 
By putting HMEs of hesitant fuzzy base-comparison 
vector in the Equation 10, the nonlinearly constrained 
optimization problem as follows:   
   
min𝑘 ∗
 
Such that   
{
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
𝑝 3
1
−6∗𝑝 1
1
≤𝑘 ∗𝑝 1
1
;𝑝 3
1
−6∗𝑝 1
1
≥𝑘 ∗𝑝 1
1
;
𝑝 3
2
−8∗𝑝 1
2
≤𝑘 ∗𝑝 1
2
;𝑝 3
2
−8∗𝑝 1
2
≥𝑘 ∗𝑝 1
2
;
𝑝 3
3
−8∗𝑝 1
3
≤𝑘 ∗𝑝 1
3
;𝑝 3
3
−8∗𝑝 1
3
≥𝑘 ∗𝑝 1
3
;
𝑝 3
4
−8∗𝑝 1
4
≤𝑘 ∗𝑝 1
4
;𝑝 3
4
−8∗𝑝 1
4
≥𝑘 ∗𝑝 1
4
;
𝑝 3
1
−𝑝 2
1
≤𝑘 ∗𝑝 2
1
;𝑝 3
1
−𝑝 2
1
≥𝑘 ∗𝑝 2
1
;
𝑝 3
2
−2∗𝑝 2
2
≤𝑘 ∗𝑝 2
2
;𝑝 3
2
−2∗𝑝 2
2
≥𝑘 ∗𝑝 2
2
;
𝑝 3
3
−3∗𝑝 2
3
≤𝑘 ∗𝑝 2
3
;𝑝 3
3
−3∗𝑝 2
3
≥𝑘 ∗𝑝 2
3
;
𝑝 3
4
−4∗𝑝 2
4
≤𝑘 ∗𝑝 2
4
;𝑝 3
4
−4∗𝑝 2
4
≥𝑘 ∗𝑝 2
4
(
1
12
)∗(
𝑝 1
1
+𝑝 1
2
+𝑝 1
3
+𝑝 1
4
+𝑝 2
1
+𝑝 2
2
+
𝑝 2
3
+𝑝 2
4
+𝑝 3
1
+𝑝 3
2
+𝑝 3
3
+𝑝 3
4
)=1;
𝑝 1
1
≤𝑝 1
2
≤𝑝 1
3
≤𝑝 1
4
;
𝑝 2
1
≤𝑝 2
2
≤𝑝 2
3
≤𝑝 2
4
;
𝑝 3
1
≤𝑝 3
2
≤𝑝 3
3
≤𝑝 3
4
;
𝑝 1
1
,𝑝 2
1
,𝑝 3
1
,𝑝 4
1
>0;
𝑘 ∗
≥0;                                               (11)
       
After solving the Equation 11, the normalized optimal 
HFWs of criteria are obtained, which are: 
ℎ
𝑤 1
={0.081,0.162,0.292,0.492};  
ℎ
𝑤 2
={0.489,0.649,0.779,0.985}; 
 ℎ
𝑤 3
={0.489,1.299,2.337,3.941}; 
By employing score function, the crisp weights of 
normalized optimal HFWs are obtained. 
Load flexibility𝑆 (ℎ
𝑤 1
)=0.256;   
Accessibility 𝑆 (ℎ
𝑤 2
)=0.725; Cost  𝑆 (ℎ
𝑤 3
)=
2.016;  
Similarly, the normalized optimal HFWs of criteria 
are obtained for all the cases. Table 3 shows the 
calculations. 
It can be noted from Table 4 that for BCM [16] and 
HF-BCM criteria have the same preference order, but 
there is a slight difference in the criteria weights. Due to 
 𝜉 =0 (0.00000008) , regardless of any values for the 
consistency index, the consistency ratio is optimal. Also, 
the proposed method is even better than the HF-BWM 
[27] in terms of consistency. Since the consistency ratio is 
the closest value to zero. This example unfolds that the 
hesitant fuzzy BCM method could consider the ambiguity 
of DMs in the process of group decision making. 
Table 2: Hesitant fuzzy pairwise comparisons. 
Transportation 
criteria 
Load 
flexibility 
Accessibility Cost 
Base-criterion 
(Cost) 
(6,8) (1,2,3,4) (1,1,1,1) 
Table 3: Scores of criteria for all the three cases. 
Cases 
𝑺 (𝒉 𝒘 𝟏 )      𝑺 (𝒉 𝒘 𝟐 )        𝑺 (𝒉 𝒘 𝟑 )             𝝃 
𝜆 =1            0.256 0.725 2.016 0.0000008 
𝜆 =1/2          0.267 0.717 2.010 0.0000003 
𝜆 =0           0.290 0.711 1.997 0.0000002 
 
Table 4: Comparison of results. 
Methods Pairwise 
comparisons 
Weights 
of 
criterion 
Consistency 
HF-BCM 
(Proposed) 
HMPRs  
(1/9-9 scale) 
0.267 
0.717 
2.010 
 
0.0000 
BCM [16] 1/9-9 scale 0.076 
0.307 
0.615 
 
0.0000 
BWM [14] 1-9 scale 0.071 
0.338 
0.589 
 
0.0580 
 
HF-BWM 
[27] 
HMPRs 
(1-9 scale) 
0.413 
1.120 
1.465 
 
0.0558 
 

240 Informatica 46 (2022) 235–242 M. Narang et al. 
 
4.2 Case study 2 
We consider the example of car selection which is handled 
by using BCM method [16]. In this case study, we solve 
the same problem by our proposed method HF-BCM from 
group decision making process. As the number of criteria 
increases, it becomes difficult to specify values for relative 
preference in base-comparison. There are six criteria 
chosen for the car selection problem: (1) convenience (2) 
fuel consumption (3) safety (4) style (5) acceleration (6) 
consumer price. The group of decision makers chooses 
safety criterion as the base-criterion. DMs executes 
hesitant fuzzy base-comparisons based on HMEs using 
Table 1. On the basis which the hesitant fuzzy base-
comparison vector can be obtained as: 
𝐻 𝐵 =[(1,2),(
1
3
,
1
2
),(1,1,1),(2,4,6),(
1
2
),(
1
4
,
1
3
)] 
The maximum length in the normalized hesitant fuzzy 
preference is 3. So, the number of elements in the 
normalized hesitant fuzzy preference are 3. 
The normalized hesitant fuzzy base-comparison 
vector 
can be computed by using Equation 4 as follows: 
Case-1 Optimistic attitude of DMs ( 𝜆 =1). 
𝐻 𝐵 =[
(1,2,2),(
1
3
,
1
3
,
1
2
),(1,1,1),(2,4,6),
(
1
2
,
1
2
,
1
2
),(
1
4
,
1
3
,
1
3
)
] 
Case-2 Neutral attitude of DMs (𝜆 =
1
2
⁄ ) . 
𝐻 𝐵 =[
(1,
1
2
,2),(
1
3
,
5
12
,
1
2
),(1,1,1),(2,4,6),
(
1
2
,
1
2
,
1
2
),(
1
4
,
7
24
,
1
3
)
] 
Case-3 Pessimistic attitude of DMs (𝜆 =0) . 
𝐻 𝐵 =[
(1,1,2),(
1
3
,
1
3
,
1
2
),(1,1,1),(2,4,6),
(
1
2
,
1
2
,
1
2
),(
1
4
,
1
4
,
1
3
)
] 
Based on Table 5 and Case-1, the normalized optimal 
hesitant fuzzy weight for each criterion can be obtained by 
solving the non-linear optimization problem as follows:   
min𝜉 
Such that    
{
 
 
 
 
 
 
 
 
 
 
 
 
 
 
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
)
(𝑝 1
1
,𝑝 1
2
,𝑝 1
3
)
−(𝑝 31
1
,𝑝 31
2
,𝑝 31
3
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
)
(𝑝 2
1
,𝑝 2
2
,𝑝 2
3
)
−(𝑝 32
1
,𝑝 32
2
,𝑝 32
3
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
)
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
)
−(𝑝 33
1
,𝑝 33
2
,𝑝 33
3
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
)
(𝑝 4
1
,𝑝 4
2
,𝑝 4
3
)
−(𝑝 34
1
,𝑝 34
2
,𝑝 34
3
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
)
(𝑝 5
1
,𝑝 5
2
,𝑝 5
3
)
−(𝑝 35
1
,𝑝 35
2
,𝑝 35
3
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
|
(𝑝 3
1
,𝑝 3
2
,𝑝 3
3
)
(𝑝 6
1
,𝑝 6
2
,𝑝 6
3
)
−(𝑝 36
1
,𝑝 36
2
,𝑝 36
3
)|≤(𝑘 ∗
,𝑘 ∗
,𝑘 ∗
)
∑ 𝑅 (ℎ
𝑤 𝑗 )=1
𝑛 𝑗 =1
𝑝 𝑗 𝑙 >0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 .                                   (12)
 
By putting HMEs of hesitant fuzzy base-comparison 
vector in the Equation 12, the nonlinearly constrained 
optimization problem as follows: 
min𝑘 ∗
 
Such that  
{
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
𝑝 3
1
−1∗𝑝 1
1
≤𝑘 ∗𝑝 1
1
;𝑝 3
1
−1∗𝑝 1
1
≥𝑘 ∗𝑝 1
1
;
𝑝 3
2
−2∗𝑝 1
2
≤𝑘 ∗𝑝 1
2
;𝑝 3
2
−2∗𝑝 1
2
≥𝑘 ∗𝑝 1
2
;
𝑝 3
3
−2∗𝑝 1
3
≤𝑘 ∗𝑝 1
3
;𝑝 3
3
−2∗𝑝 1
3
≥𝑘 ∗𝑝 1
3
;
𝑝 3
1
−
1
3
∗𝑝 2
1
≤𝑘 ∗𝑝 2
1
;𝑝 3
1
−
1
3
∗𝑝 2
1
≥𝑘 ∗𝑝 2
1
;
𝑝 3
2
−
1
2
∗𝑝 2
2
≤𝑘 ∗𝑝 2
2
;𝑝 3
2
−
1
2
∗𝑝 2
2
≥𝑘 ∗𝑝 2
2
;
𝑝 3
3
−
1
2
∗𝑝 2
3
≤𝑘 ∗𝑝 2
3
;𝑝 3
3
−
1
2
∗𝑝 2
3
≥𝑘 ∗𝑝 2
3
;
𝑝 3
1
−2∗𝑝 4
1
≤𝑘 ∗𝑝 4
1
;𝑝 3
1
−2∗𝑝 4
1
≥𝑘 ∗𝑝 4
1
;    
𝑝 3
2
−4∗𝑝 4
2
≤𝑘 ∗𝑝 4
2
;𝑝 3
2
−4∗𝑝 4
2
≥𝑘 ∗𝑝 4
2
;
𝑝 3
3
−6∗𝑝 4
3
≤𝑘 ∗𝑝 4
3
;𝑝 3
3
−6∗𝑝 4
3
≥𝑘 ∗𝑝 4
3
;
𝑝 3
1
−
1
2
∗𝑝 5
1
≤𝑘 ∗𝑝 5
1
;𝑝 3
1
−
1
2
∗𝑝 5
1
≥𝑘 ∗𝑝 5
1
;
𝑝 3
2
−
1
2
∗𝑝 5
2
≤𝑘 ∗𝑝 5
2
;𝑝 3
2
−
1
2
∗𝑝 5
2
≥𝑘 ∗𝑝 5
2
;
𝑝 3
3
−
1
2
∗𝑝 5
3
≤𝑘 ∗𝑝 5
3
;𝑝 3
3
−
1
2
∗𝑝 5
3
≥𝑘 ∗𝑝 5
3
;
𝑝 3
1
−
1
4
∗𝑝 6
1
≤𝑘 ∗𝑝 6
1
;𝑝 3
1
−
1
4
∗𝑝 6
1
≥𝑘 ∗𝑝 6
1
;
𝑝 3
2
−
1
3
∗𝑝 6
2
≤𝑘 ∗𝑝 6
2
;𝑝 3
2
−
1
3
∗𝑝 6
2
≥𝑘 ∗𝑝 6
2
;
𝑝 3
3
−
1
3
∗𝑝 6
3
≤𝑘 ∗𝑝 6
3
;𝑝 3
3
−
1
3
∗𝑝 6
3
≥𝑘 ∗𝑝 6
3
;
(
1
18
)∗(
𝑝 1
1
+𝑝 1
2
+𝑝 1
3
+𝑝 2
1
+𝑝 2
2
+𝑝 2
3
+𝑝 3
1
+𝑝 3
2
+𝑝 3
3
+𝑝 4
1
+𝑝 4
2
+𝑝 4
3
+ 𝑝 5
1
+𝑝 5
2
+𝑝 5
3
+𝑝 6
1
+𝑝 6
2
+𝑝 6
3
)=1;
𝑝 1
1
≤𝑝 1
2
≤𝑝 1
3
;
𝑝 2
1
≤𝑝 2
2
≤𝑝 2
3
;
𝑝 3
1
≤𝑝 3
2
≤𝑝 3
3
;
𝑝 4
1
≤𝑝 4
2
≤𝑝 4
3
;
𝑝 5
1
≤𝑝 5
2
≤𝑝 5
3
;
𝑝 6
1
≤𝑝 6
2
≤𝑝 6
3
;
𝑝 1
1
,𝑝 2
1
,𝑝 3
1
>0;
𝑘 ≥0                                                     (13)
         
 

A Hesitant Fuzzy Multiplicative Base-criterion... Informatica 46 (2022) 235–242 241 
 
After solving the Equation 13, the normalized optimal 
HFWs of criteria are obtained, which are: 
ℎ
𝑤 1
={0.161,0.210,0.370};  
ℎ
𝑤 2
={0.487,0.841,1.483};  
ℎ
𝑤 3
={0.160,0.420,0.741}; 
 ℎ
𝑤 4
={0.080,0.105,0.123}; 
ℎ
𝑤 5
={0.321,0.841,1.483}; 
ℎ
𝑤 6
={0.643,1.271,2.242}; 
By employing score function, the crisp weights of 
normalized optimal HFWs are obtained. 
Convenience = 0.247, fuel consumption = 0.937, 
safety = 0.440, style = 0.102, acceleration = 0.881, price = 
1.384, ξ = 0.00000007; 
Consistency ratio is optimal because ξ = 0 
(0.00000007). 
Table 6 shows the normalized optimal HFWs of 
criteria for all the cases.  
5 Conclusion 
Multi-criteria group decision making problems often have 
strong uncertainty, which is characterized as fuzziness 
within the group decision-making. When more than one 
expert evaluates an option, it is very possible for them to 
have different opinions. In our proposed methodology, we 
have developed a unique approach hesitant fuzzy Base-
criterion method which is an extension of latest MCDM 
method BCM in the hesitant situation for multi-criteria 
group decision making. HFS has its unique advantages in 
a decision-making problem with multi-criteria and 
multiple individuals. Employing linguistic variables is 
more worthy than crisp values, in order to make a base-
comparison for criteria and alternatives in the group 
decision-making process.  
The base-comparisons values manifested by linguistic 
terms could be converted into HMEs that are used in the 
nonlinear optimization problem and then normalized 
optimal HFWs of criteria can be obtained. The normalized 
optimal hesitant fuzzy weights of criteria can be 
transferred to crisp values by employing score function. 
The suitability and applicability of the developed HF-
BCM is verified by discussing the optimal transportation 
mode selection problem and car selection problem. In 
terms of strength and direction, the results of HF-BCM are 
completely consistent if decision makers have selected 
HMEs for pairwise-comparisons based on Equation 9. The 
proposed method HF-BCM has several important features 
as follows: 
▪ Integration of Base-criterion method with 
hesitant fuzzy circumstances is novel and 
provides more reliable and accurate weights of 
criteria and alternatives for MCGDM.  
▪ Hesitant fuzzy set has its unique advantages over 
the other methods. In MCGDM, when the 
assessment values given by DMs vary, we can 
unify them into one HME. Also, BCM requires 
fewer comparisons to calculate the weights of 
criteria and alternative. In this way, the process 
of calculating lots of group decision making 
problems is effectively simplified.   
▪ Compared to the BCM method, the hesitant 
fuzzy BCM method also uses linguistic terms to 
do base-comparisons. Use of linguistic variables 
makes pairwise-comparisons more accurate and 
easier. HF-BCM can gain the more reliable 
weights. 
▪ The hesitant fuzzy BCM needs less number of 
pairwise comparisons and highly consistent 
rather than the hesitant fuzzy AHP and hesitant 
fuzzy BWM methods. 
R efer ence s  
[1] Parreiras, R., Pedrycz, W., Ekel, P. (2011). Fuzzy 
Multicriteria Decision‐Making: Models, Methods 
and Applications. John Wiley & Sons. 
http://dx.doi.org/10.1109/IFSA-
NAFIPS.2013.6608469  
[2] Yoon, K., Hwang, C.L. (1981). Multiple Attribute 
Decision Making: Methods and Applications. A 
State‐of‐the‐Art Survey. New York, Springer‐
Verlag.  
[3] Rao, R.V. (2013). Decision Making in The 
Manufacturing Environment Using Graph Theory 
and Fuzzy Multiple Attribute Decision Making. 2. 
London, Springer‐Verlag.  
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Table 5: Hesitant fuzzy pairwise comparisons. 
Criteria Base-criterion 
(Safety)  
Convenience (1,2) 
Fuel   (1/3,1/2) 
Safety (1,1,1) 
Style (2,4,6) 
Acceleration (1/2) 
Price (1/4,1/3) 
Table 6: Scores of criteria for all the three cases. 
Cases 𝝀 =𝟏 𝝀 =𝟏 /𝟐 𝝀 =𝟎 
ξ 0.00000 0.00000 0.00000 
𝑆 (ℎ
𝑤 1
)        0.247 0.343 0.283 
𝑆 (ℎ
𝑤 2
)       0.937 0.921 0.984 
𝑆 (ℎ
𝑤 3
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𝑆 (ℎ
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𝑆 (ℎ
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