Communications on Applied Nonlinear Analysis 

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Decision–Making Problem for Triangular Hesitant Fuzzy Set 
  

E. Fany Helena  

Assistant Professor in P.G. and Research Department of Mathematics 

T.B.M.L. College (Affiliated to Annamalai University, Chidambaram) 

Porayar-609307, Tamil Nadu, India. 

E-mail: fanyhelena3@gmail.com 

 

Article History: 

Received: 12-05-2024 

Revised: 24-06-2024 

Accepted: 06-07-2024 

Abstract:  

In this paper, we proposed two new algorithms to address the task of multicriteria decision-

making problems in which the criteria weights which have been unidentified, and alternatives 

are provided by triangular fuzzy numbers' linguistic values. A decision-making problem with 

assessments has been proposed on the basis of expected values. The expected values for two 

algorithms are calculated with two different 'Ranking Functions'. The weights are obtained 

by calculating the variance and mean of expected values with the aid of the triangular hesitant 

fuzzy decision matrix. The all alternatives ranking order is determined, and the maximum 

one is the best that can be identified easily. Lastly, a representative example is provided 

regarding the health issues of a community. 

Keywords: Expected Value, Triangular Hesitant Fuzzy Set, Variance, Mean, Multicriteria 

Decision Making. 

 

1. Introduction 

The idea of a fuzzy set is unable to accurately model unpredictability, inaccurate, and vague 

information when numerous sources of vagueness occur at the same time. To overcome this limitation, 

many fuzzy set extensions had been proposed in literature. A demonstration of interval-valued dual 

hesitant fuzzy information aggregation to multiple attribute decision-making has been presented by 

Peng X, Liu L, and Dai J [2]. A few noteworthy extensions are the interval-valued fuzzy set, which 

assigns a closed subinterval of [0, 1] as the membership degree for each component; the IFS 

(Intuitionistic Fuzzy Set), which takes in to the considerations both the degree of membership and non-

membership degree of each element; the type-2 fuzzy set, which uses a fuzzy set over [0, 1] to 

incorporate uncertainty into the membership function definition; and the fuzzy multiset, on the basis 

of a multiset with possibly repeated elements. Nevertheless, hesitant fuzzy sets are a more popular and 

often utilized concept of fuzzy extensions. They are used to model a decision-making scenario where 

an expert may weigh the relative merits of each element in a set. In [1], they studied the speed 

forecasting system with novel defuzzification using the hesitant number. Humble fuzzy sets are 

thought to be the most comprehensive set because they support a flexible approach while decision-

makers make their own decisions. This is because hesitant fuzzy sets have been formed by top 

researchers who has been given a broad conceptualization along with the application of the concept, 

which was later adopted by other researchers in the field. That's why hesitant fuzzy sets are more 

extensively used in many areas. In addition to researching the similarity metric with sign distance, 

Stephen Dinagar and Fany Helena [3,4] also presented a novel approach for computing the Centroids 

of both Vertical and Horizontal Axes as well as the Value and Ambiguity Indices using trapezoidal 



Communications on Applied Nonlinear Analysis 

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intuitionistic fuzzy numbers. Similarity Measures with Vector-Length under Fuzzy Environment were 

proposed in [5]. In this paper, we proposed a new expected value using different ranking procedures 

and the ranking functions in [3,4,5]. The order in which all options are ranked is determined, and the 

maximum one is the best, which can be identified easily through these ranking procedures.  

2. Objectives 

Definition:2.1 

Let X be a fixed reference set. A HFS on X is defined in terms of a function from X to a subset of 

[0,1], that has been considered by 

        A ={〈𝑥, ℎ̃𝐴(𝑥)〉/𝑥 ∈ 𝑋},  

in which ℎ̃𝐴(𝑥) as hesitant fuzzy element (H.F.E.), and it represents a set of some values in [0,1]. 

Furthermore, the H.F.E. ℎ̃𝐴(𝑥) stands for the possible membership degrees of the element x ∈ X to the 

set A. 

Definition:2.2 

 For a hesitant, fuzzy element, h 

   𝑆′(ℎ̃) =
1

∗ℎ̃
∑ ∆∆∈∗ℎ̃  

is called the score function of ℎ̃, where *ℎ̃ represents the number of elements. 

For any 2 H.F.E 

(i) If 𝑆′(ℎ̃1) > 𝑆′(ℎ̃2) then ℎ̃1 > ℎ̃2 

(ii) If 𝑆′(ℎ̃1) = 𝑆′(ℎ̃2) then ℎ̃1 = ℎ̃2 

Definition:2.3 

  Let ℎ̃𝐴1and ℎ̃𝐴1  Be two H.F.E.s, their union & intersection are, respectively, explained 

below: 

                               ℎ̃𝐴1 ∪ ℎ̃𝐴2 = ⋃ max⁡{∆𝐴1 , ∆𝐴2⁡∆𝐴1∈ℎ̃𝐴1 ,∆𝐴2∈ℎ̃𝐴2
} 

                                ℎ̃𝐴1 ∩ ℎ̃𝐴2 = ⋃ min⁡{∆𝐴1 , ∆𝐴2 ⁡∆𝐴1∈ℎ̃𝐴1 ,∆𝐴2∈ℎ̃𝐴2
}   

From a mathematical perspective, for any x ∈ X, an H.F.E. can be thought of as the other well-known 

extensions of fuzzy sets. 

Definition:2.4 

 Let x be a finite set. A triangular hesitant fuzzy set on X has been explained as  

       �̃� = {{𝑥, ℎ̃Ƞ(𝑥)/𝑥 ∈ 𝑋} 

where ⁡ℎ̃Ƞ(𝑥) represents as a set of some triangular fuzzy numbers in the set of real numbers R, 

representing the possible membership function of the element ⁡𝑥 ∈ 𝑋, ℎ̃Ƞ(𝑥) . It is called a T.H.F.E .  

 



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The TFN in ℎ̃Ƞ(𝑥) has been represented as  Ƕ by and given by 

                                         χ = (𝑎𝑖, 𝑏𝑗 , 𝑐𝑘), where χ∈ Ƕ 

2.5. Some Basic Operations of Triangular Hesitant Fuzzy Numbers (T.H.F.N.s) 

Suppose Ƕ, Ƕ1, Ƕ2 be three T.H.F.E and 𝝀 > 0, then 

1. 𝝀 Ƕ = ⋃ ⁡{(λ𝑎𝑖 + λ⁡𝑏𝑗 + λ𝑐𝑘)⁡χ∈⁡Ƕ } 

2. Ƕ1 ⊕Ƕ2 = ⋃ ⁡{⁡χ1∈Ƕ1,χ2∈Ƕ2
𝑎1
𝑖 + 𝑎2

𝑖 , 𝑏1
𝑗
+ 𝑏2

𝑗
, 𝑐1

𝑘 + 𝑐2
𝑘} 

3. Ƕ1 ⊗Ƕ2 = ⋃ ⁡{⁡χ1∈Ƕ1,χ2∈Ƕ2
𝑎1
𝑖𝑎2

𝑖 , 𝑏1
𝑗
𝑏2
𝑗
, 𝑐1

𝑘𝑐2
𝑘} 

4. Ƕ𝑐 = ⋃ ⁡{(1 − 𝑎𝑖, 1 −⁡𝑏𝑗 , 1 − λ𝑐𝑘)⁡χ∈⁡Ƕ } 

5. Ƕ1 ∪Ƕ2 = ⋃ {𝑚𝑎𝑥⁡{𝑎1
𝑖 , 𝑎2

𝑖 ⁡χ1∈Ƕ1,χ2∈Ƕ2
},𝑚𝑎𝑥⁡{𝑏1

𝑗
, 𝑏2

𝑗
},𝑚𝑎𝑥⁡{𝑐1

𝑘, 𝑐2
𝑘}} 

6. Ƕ1 ∩Ƕ2 = ⋃ {𝑚𝑖𝑛⁡{𝑎1
𝑖 , 𝑎2

𝑖 ⁡χ1∈Ƕ1,χ2∈Ƕ2
},𝑚𝑖𝑛⁡{𝑏1

𝑗
, 𝑏2

𝑗
},𝑚𝑖𝑛⁡{𝑐1

𝑘, 𝑐2
𝑘}} 

3. Methods  

Ranking Approach of Triangular Hesitant Fuzzy Set 

Definition:3.1 (Existing Approach) 

 The T.H.F.E is denoted as Ƕ, and the expected value is defined as E(Ƕ) = ⁡
1

3∗Ƕ
∑ (𝜒∗Ƕ 𝑎𝑖+𝑏𝑗 +

⁡𝑐𝑘), where Ƕ as  number of TFN in Ƕ. 

Definition:3.2 (Mid Value Ranking) 

 The T.H.F.E is denoted as Ƕ, and the expected value is defined as E(Ƕ) =

⁡
1

3∗Ƕ
∑ (

𝑎𝑖+𝑏𝑗+𝑐𝑘⁡

3
)𝜒∗Ƕ , where Ƕ as number of TFN in Ƕ. This ranking is called mid-value ranking.  

Theorem:3.3 

Let Ƕ be any hesitant triangular fuzzy element, and their alpha cut be denoted as Ƕ1𝑙 = 𝑎1 + 𝛼(𝑎2 −

𝑎1) and Ƕ1𝑟 = 𝑎3 − 𝛼(𝑎3 − 𝑎2) then E(Ƕ) = ⁡
1

3∗Ƕ
∑ (

𝑎𝑖+𝑏𝑗+𝑐𝑘⁡

3
)𝜒∗Ƕ . 

Proof: 

 Let us consider 

                      E(Ƕ) =⁡
1

3∗Ƕ
∑ 𝑀𝜇(𝜒∗Ƕ ћ)         - (1) 

Where Ƕ = Number of Triangular Hesitant Fuzzy Set  

i=1,2,….,n; j = 1,2,…..,n; k = 1,2,….,n 

Where   𝑀𝜇(ћ) =
1

2
∫ {[
1

0
𝑎𝑖 + (𝛼(𝑏𝑗 − 𝑎𝑖))𝛼] + [𝑐𝑘 − (𝛼(𝑐𝑘 − 𝑏𝑗))𝛼} 𝑑𝛼 

 After simplification, we get 

               𝑀𝜇(ћ) = 
𝑎𝑖+𝑏𝑗+𝑐𝑘⁡

3
                         - (2) 

 



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Now substitute equation (2) in (1) we get  

E(Ƕ) =⁡
1

3∗Ƕ
∑ (

𝑎𝑖+𝑏𝑗+𝑐𝑘⁡

3
)𝜒∗Ƕ  

Definition:3.4 (Ambiguity Ranking) 

 The T.H.F.E is denoted as Ƕ, and the expected value explained as E(Ƕ) =

⁡
1

3∗Ƕ
∑ (

𝑎𝑖+4𝑏𝑗+𝑐𝑘⁡

6
)𝜒∗Ƕ , where Ƕ is the number of TFN in Ƕ. The above ranking is called ambiguity 

ranking. 

Theorem:3.5 

Let Ƕ be any hesitant triangular fuzzy element, and their alpha cut be denoted as Ƕ1𝑙 = 𝑎1 + 𝛼(𝑎2 −

𝑎1) and Ƕ1𝑟 = 𝑎3 − 𝛼(𝑎3 − 𝑎2) then E(Ƕ) =
1

3∗Ƕ
∑ (

𝑎𝑖+4𝑏𝑗+𝑐𝑘⁡

6
)𝜒∗Ƕ . 

Proof: 

 Let us consider 

                      E(Ƕ) =⁡
1

3∗Ƕ
∑ 𝐴𝜇(𝜒∗Ƕ ћ)               - (1) 

Where Ƕ = Number of Triangular Hesitant Fuzzy Set  

i=1,2,….,n; j = 1,2,…..,n; k = 1,2,….,n 

Where   𝑀𝜇(ћ) = ∫ {[Ƕ1𝑙 −Ƕ1𝑟]
1

0
⁡𝑓(𝛼)𝑑𝛼 

                         = ∫ {[
1

0
𝑎𝑖 + 𝛼(𝑏𝑗 − 𝑎𝑖)] + [𝑐𝑘 − 𝛼(𝑐𝑘 − 𝑏𝑗)} 𝛼𝑑𝛼 

 After simplification, we get 

               𝑀𝜇(ћ) = 
𝑎𝑖+4𝑏𝑗+𝑐𝑘⁡

6
                         - (2) 

Now substitute equation (2) in (1) we get  

E(Ƕ) =⁡
1

3∗Ƕ
∑ (

𝑎𝑖+𝑏𝑗+𝑐𝑘⁡

6
)𝜒∗Ƕ  

Algorithms of Triangular Hesitant Fuzzy Set for Multicriteria Decision-Making Problems 

Algorithm: 3.6 (Existing Method) 

Step 1: Construct the T.H.F. decision matrix. Suppose we have 'm' alternatives and 'n' criteria. The 

triangular hesitant matrix is H = {h𝑖𝑗} is an 𝑚 × 𝑛 matrix. Let  two or more decisions are the same. 

Then the number should be considered as once in h𝑖𝑗. 

Step 2: Expected value is calculated. 

Step 3: Obtain the criteria weight by using the standard deviation. 

Step 4: Acquire the anticipated value that is weighted.  

Step 5: Rank the alternatives. 



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Algorithm: 3.7 (New Method using Mid Value Ranking and Variance) 

Let �̃� = { �̃�1, �̃�2, … . , �̃�𝑢} be an alternative to a set. Let �̃� = { �̃�1, �̃�2, … . , �̃�𝑣} be a criterion of a set. The 

decision-makers' suggestions are obtained as  

Step 1: A triangular, hesitant, fuzzy decision matrix is constructed with 'u' alternatives and 'v' criteria. 

The matrix is denoted as  �̃� = {Ƕ𝑖𝑗} is an 𝑢 × 𝑣 matrix. Let 2 or more decision-makers' suggestions 

are the same. Then, the number should be considered as once. Ƕ𝑖𝑗. 

                              

                                            �̃� =     

 

Step 2: Use the following formula to determine the expected value. 

     𝐸𝑖𝑗 =⁡
1

3∗Ƕ𝑖𝑗
∑ (

⁡𝑎𝑖𝑗+𝑏𝑖𝑗+𝑐𝑖𝑗

3
)𝜒𝑖𝑗∗Ƕ𝑖𝑗
, where the middle value is only taken from the T.H.F.s. Because 

with defuzzification, we will get the middle value.  

In Ƕ𝑖𝑗, where i = 1,2,…,u and j = 1,2,…..,v 

                                       

Step 3: Obtain the criteria weight vector �̃� = {𝑤1, 𝑤2, … , 𝑤𝑣} by using the below formula 

                                        �̃�𝑗 =
𝑔𝑗(𝑐𝑗)

∑ 𝑔𝑗(𝑐𝑗)
𝑣
𝑗=𝑛

 

Where �̃�𝑗 ≥ 0 and ∑ �̃�𝑗 = 1𝑣
𝑗=𝑛 . Then 𝑔𝑗(𝑐𝑗) is the variance of the expected values of various 

alternatives w.r.t criterion. 

                             𝑔𝑗(𝑐𝑗) = 
1

𝑢
∑ (𝐸𝑖𝑗 −

1

𝑢
⁡∑ 𝐸𝑖𝑗

𝑣
𝑖=1 )2𝑢

𝑗=1      

Step 4: Obtain the weighted expected value for each alternative  �̃�𝑖 where i = 1,2,…,u 

                                   �̃�(�̃�𝑖) = ∑ �̃�𝑗
𝑣
𝑗=1 𝐸𝑖𝑗 

Step 5: Rank the alternatives as per the values. 

Algorithm: 3.8 (New Method using Ambiguity Ranking and Mean) 

Let �̃� = { �̃�1, �̃�2, … . , �̃�𝑢} be an alternative to a set. Let �̃� = { �̃�1, �̃�2, … . , �̃�𝑣} be a criterion of a set. The 

decision-makers' suggestions are obtained as  

Step 1: A triangular hesitant fuzzy decision matrix is constructed with 'u' alternatives and 'v' criteria. 

The matrix is denoted as  �̃� = {Ƕ𝑖𝑗} is an 𝑢 × 𝑣 matrix. Suppose 2 or more decision-makers' 

suggestions are the same. Then the number should be considered as once in Ƕ𝑖𝑗. 



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                                            �̃� =     

 

                                  

Step 2: compute the expected value by utilizing the mentioned formula 

     𝐸𝑖𝑗
′ =⁡

1

3∗Ƕ𝑖𝑗
∑ (

⁡𝑎𝑖𝑗+4𝑏𝑖𝑗+𝑐𝑖𝑗

6
)𝜒𝑖𝑗∗Ƕ𝑖𝑗
 

In Ƕ𝑖𝑗, where i = 1,2,…,u and j = 1,2,…..,v 

                                

Step 3: Obtain the criteria weight vector �̃� = {𝑤1, 𝑤2, … , 𝑤𝑣} by using the below formula 

                                        �̃�𝑗 =
𝑔𝑗

′(𝑐𝑗)

∑ 𝑔𝑗
′(𝑐𝑗)

𝑣
𝑗=𝑛

 

Where �̃�𝑗 ≥ 0 and ∑ �̃�𝑗 = 1𝑣
𝑗=𝑛 . Then 𝑔𝑗(𝑐𝑗) is the mean of the expected values of various alternatives 

w.r.t criterion. 

                             𝑔𝑗
′(𝑐𝑗) = 

1

𝑢
∑ (|𝐸𝑖𝑗

′ −
1

𝑢
⁡∑ 𝐸𝑖𝑗

′𝑣
𝑖=1 |)𝑢

𝑗=1      

Step 4: Obtain the weighted expected value for each alternative  �̃�𝑖 where i = 1,2,…,u 

                                   �̃�(�̃�𝑖) = ∑ �̃�𝑗
𝑣
𝑗=1 𝐸𝑖𝑗

′ 

Step 5: Rank the alternatives according to the values. 

4. Results 

    The World Health Organization (WHO) was created in 1948. At the time of Creation, health was 

outlined as being "a state of complete physical, mental and social wellbeing and not merely the absence 

of disease or infirmity" Comprehensive preventive health sources that enable a significant portion of 

curative and rehabilitative services to be included in occupational health services. Subsequent research 

concentrated on the physiological and psychological characteristics of workers and their management.  

Here, we considered Police officers to be the high-risk category for mental health development 

disturbances due to numerous serious incidents as well as potentially traumatic incidents that occurred 

during their careers. These include things like seeing children die, interacting with sexual harassment 

victims, experiencing major traffic mishaps, suicide, and violent incidents. These are referred to as 

operational aggravation. The likelihood of experiencing symptoms of hostility, anxiety, as well as 

fatigue may arise due to these operational stressors. Disorders like depression and post-traumatic stress 

disorder may strike some people. 



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As a result of psychological issues, the majority of the police officers experience the 10 diseases as 

mentioned below: D1- Irritable bowel syndrome / D2- Constipation / D3- High blood pressure / D4 

Headaches/ D5- cardiovascular diseases/ D6- Insomnia/ D7Nausea/ D8- Diabetic/ D9- Arthritis/ D10- 

Mental Fatigue. The alternatives with respect to the criterion are 𝐴1- Tightness; 𝐴2- Tensity; 𝐴3- 

Worry; 𝐴4- Strain; 𝐴5- Anger 

Let's look at the 7 linguistic variable scale Chen (2000) 

Table:1 

“Linguistic Variable Linguistic Value 

Very Low (𝑘1) (0,0,0.1) 

Low (𝑘2) (0,0.1,0.3) 

Medium Low (𝑘3) (0.1,0.3,0.5) 

Medium (𝑘4) (0.3,0.5,0.7) 

Medium-High (𝑘5) (0.5,0.7,0.9) 

High (𝑘6) (0.7,0.9,1) 

Very High (𝑘7) (0.9,1,1)” 

 

These substitutes are given by 5 experts as a linguistic variable Chen (2000), as shown in the table 

below 

Table:2 

 �̃�1 �̃�2 �̃�3 �̃�4 �̃�5 �̃�6 �̃�7 �̃�8 �̃�9 �̃�10 

�̃�1 

𝑘6 

𝑘4 

𝑘5 

𝑘3 

𝑘6 

𝑘4 

𝑘5 

𝑘3 

𝑘6 

𝑘4 

𝑘6 

𝑘6 

𝑘7 

𝑘6 

𝑘6 

𝑘5 

𝑘6 

𝑘6 

𝑘5 

𝑘6 

𝑘6 

𝑘7 

𝑘5 

𝑘6 

𝑘6 

𝑘5 

𝑘3 

𝑘6 

𝑘5 

𝑘4 

𝑘4 

𝑘5 

𝑘2 

𝑘1 

𝑘1 

𝑘2 

𝑘7 

𝑘6 

𝑘6 

𝑘5 

�̃�2 

𝑘6 

𝑘5 

𝑘4 

𝑘6 

𝑘6 

𝑘5 

𝑘4 

𝑘5 

𝑘6 

𝑘7 

𝑘6 

𝑘6 

𝑘7 

𝑘5 

𝑘6 

𝑘4 

𝑘6 

𝑘4 

𝑘5 

𝑘4 

𝑘4 

𝑘4 

𝑘3 

𝑘4 

𝑘3 

𝑘2 

𝑘3 

𝑘2 

𝑘4 

𝑘3 

𝑘2 

𝑘3 

𝑘2 

𝑘1 

𝑘1 

𝑘1 

𝑘6 

𝑘6 

𝑘5 

𝑘6 

�̃�3 

𝑘6 

𝑘6 

𝑘6 

𝑘6 

𝑘6 

𝑘7 

𝑘6 

𝑘5 

𝑘7 

𝑘6 

𝑘7 

𝑘6 

𝑘7 

𝑘6 

𝑘7 

𝑘7 

𝑘6 

𝑘5 

𝑘6 

𝑘6 

𝑘7 

𝑘6 

𝑘7 

𝑘7 

𝑘5 

𝑘6 

𝑘3 

𝑘6 

𝑘3 

𝑘4 

𝑘2 

𝑘5 

𝑘3 

𝑘2 

𝑘2 

𝑘4 

𝑘7 

𝑘7 

𝑘6 

𝑘7 

�̃�4 

𝑘3 

𝑘6 

𝑘6 

𝑘5 

𝑘2 

𝑘1 

𝑘2 

𝑘3 

𝑘6 

𝑘5 

𝑘3 

𝑘4 

𝑘1 

𝑘4 

𝑘3 

𝑘4 

𝑘2 

𝑘1 

𝑘1 

𝑘2 

𝑘6 

𝑘7 

𝑘6 

𝑘6 

𝑘5 

𝑘6 

𝑘3 

𝑘4 

𝑘1 

𝑘2 

𝑘1 

𝑘2 

𝑘2 

𝑘1 

𝑘2 

𝑘1 

𝑘6 

𝑘5 

𝑘7 

𝑘6 

�̃�5 

𝑘3 

𝑘5 

𝑘4 

𝑘2 

𝑘2 

𝑘3 

𝑘1 

𝑘2 

𝑘6 

𝑘5 

𝑘7 

𝑘6 

𝑘6 

𝑘7 

𝑘6 

𝑘5 

𝑘6 

𝑘5 

𝑘3 

𝑘4 

𝑘2 

𝑘1 

𝑘3 

𝑘4 

𝑘1 

𝑘3 

𝑘4 

𝑘2 

𝑘2 

𝑘1 

𝑘4 

𝑘3 

𝑘2 

𝑘1 

𝑘2 

𝑘2 

𝑘4 

𝑘3 

𝑘4 

𝑘5 

 

 

 

Table:3 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X  

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285 
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 �̃�1 �̃�2 �̃�3 �̃�4 �̃�5 �̃�6 �̃�7 �̃�8 �̃�9 �̃�10 

�̃�1 

(0.7,0.9,

1) 

(0.3,0.5,

0.7) 

(0.5,0.7,

0.9) 

(0.1,0.3,

0.5) 

(0.7,0.9,

1) 

(0.3,0.5,

0.7) 

(0.5,0.7,

0.9) 

(0.1,0.3,

0.5) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.1,0.3,

0.5) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.3,0.5,

0.7) 

(0.3,0.5,

0.7) 

(0.5,0.7,

0.9) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0,0,0.1) 

(0,0.1,0.

3) 

 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

�̃�2 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.3,0.5,

0.7) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.3,0.5,

0.7) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.7,0.9,

1) 

 

(0.9,1,1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.3,0.5,

0.7) 

 

(0.7,0.9,

1) 

(0.3,0.5,

0.7) 

(0.5,0.7,

0.9) 

(0.3,0.5,

0.7) 

(0.3,0.5,

0.7) 

(0.3,0.5,

0.7) 

(0.1,0.3,

0.5) 

(0.3,0.5,

0.7) 

(0.1,0.3,

0.5) 

(0,0.1,0.

3) 

(0.1,0.3,

0.5) 

(0,0.1,0.

3) 

(0.3,0.5,

0.7) 

(0.1,0.3,

0.5) 

(0,0.1,0.

3) 

(0.1,0.3,

0.5) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0,0,0.1) 

(0,0,0.1) 

 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

�̃�3 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.9,1,1) 

 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.9,1,1) 

 

 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.9,1,1) 

 

(0.5,0.7,

0.9) 

(0.1,0.3,

0.5) 

(0.1,0.3,

0.5) 

(0.9,1,1) 

(0.7.0.9.

1) 

(0.3,0.5,

0.7) 

(0,0.1.0.

3) 

(0.9,1,1) 

(0.1.0.3,

0.5) 

(0,0.1,0.

3) 

(0,0.1,0.

3) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.3,0.5,

0.7) 

(0.9,1,1) 

�̃�4 

(0.1,0.3,

0.5) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0,0.1.0.

3) 

(0,0,0.1) 

(0,0.1,0.

3) 

(0.1,0.3,

0.5) 

(0.7.0.9,

1) 

(0.5,0.7,

0.9) 

(0.1,0.3,

0.5) 

(0.0,0.1) 

(0.3,0.5,

0.7) 

(0.1,0.3,

0.5) 

(0.3,0.5,

0.7) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0,0,0.1) 

(0,0.1,0.

3) 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.1,0.3,

0.5) 

(0.3,0.5,

0.7) 

(0,0,0.1) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0,0.1,0.

3) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.9,1,1) 

(0.7,0.9,

1) 

�̃�5 

(-

0.1,0.3,0

.5) 

(0.5,0.7,

0.9) 

(0.3,0.5,

0.7) 

(0,0.1,0.

3) 

(0,0.1,0.

3) 

(0.1,0.3,

0.5) 

(0,0,0.1) 

(0,0.1,0.

3) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.9,1,1 

(0.7,0.9,

1) 

(0.9,1,1) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.7,0.9,

1) 

(0.5,0.7,

0.9) 

(0.1,0.3,

0.5) 

(0.3,0.5,

0.7) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0.1,0.3,

0.2) 

(0.3,0.5,

0.7) 

(0,0,0.1) 

(0.1,0.3,

0.5) 

(0.3,0.5,

0.7)  

(0,0.1,0.

3) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0.3,0.5,

0.7) 

(0.1,0.3,

0.5) 

(0,0.1,0.

3) 

(0,0,0.1) 

(0,0.1,0.

3) 

(0,0.1,0.

3) 

(0.3,0.5,

0.7)  

(0.1,0.3,

0.5) 

(0.3,0.5,

0.7)  

(0.5,0.7,

0.9) 

 

 

 

 

Triangular hesitant fuzzy decision matrix 



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Calculate the anticipated value for every T.H.F.E in the decision matrix �̃� Utilizing the algorithms 

listed above: 

Algorithm: 4.1 (Existing Method) 

By using all the steps in the existing method algorithm, we get the ranking as  

                            �̃�3 > �̃�1 > �̃�2 > �̃�4 > �̃�5 

This leads us to the conclusion that worry is the primary cause of illness. 

Algorithm: 4.2 (New Method using Mid Value Ranking and Variance) 

The expected value is calculated by using step 2 in the algorithm (2) 

We get  

  

Next, the weight of each criterion is obtained from �̃� By using step 3 in an algorithm (2), We have  

 �̃�1 =0.2602; �̃�2 = 0.08943; �̃�3 =0.0163; �̃�4 =0.0569; �̃�5 =0.0732 

�̃�6 =0.0894; �̃�7 =0.0407; �̃�8 =0.3415; �̃�9 =0.0081;�̃�10 =0.0244 

Also, the weighted expected value for each alternative �̃�𝑖 

  �̃�𝑊(�̃�1) = 0.0734 

  �̃�𝑊(�̃�2) = 0.0510 

  �̃�𝑊(�̃�3) = 0.0792 

  �̃�𝑊(�̃�4) = 0.0392 

  �̃�𝑊(�̃�5) = 0.0380 

                 Now rank the alternatives as per the expected value 

                                         �̃�3 > �̃�1 > �̃�2 > �̃�4 > �̃�5 

This leads us to the conclusion that worry is the primary cause of the group of illnesses. 

Algorithm:4. 3 (New Method using Ambiguity Ranking and Mean) 

The expected value is calculated by using step 2 in an algorithm (3) 

 

 



Communications on Applied Nonlinear Analysis 

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We get  

  

Next, the weight of each criterion is obtained from �̃�′ By using step 3 in an algorithm (3), We have  

   �̃�1 = 0.0746 ; �̃�2 = 0.1729; �̃�3 = 0.0648; �̃�4 = 0.1172⁡; �̃�5 = 0.1316 

   �̃�6 = 0.1790; �̃�7 = 0.1201; �̃�8 = 0.0909; �̃�9 = 0.0483; �̃�10 = 0.0006 

Also, the weighted expected value for each alternative �̃�𝑖 

  �̃�′
𝑊(�̃�1) = 0.2289 

  �̃�′
𝑊(�̃�2) = 0.1370 

  �̃�′
𝑊(�̃�3) = 0.2601 

  �̃�′
𝑊(�̃�4) = 0.1311 

  �̃�′
𝑊(�̃�5) = 0.1246 

                 Now rank the alternatives according to the expected value 

                                         �̃�3 > �̃�1 > �̃�2 > �̃�4 > �̃�5 

This leads us to the conclusion that worry is the primary cause of illness. 

5. Discussion and Conclusion 

Comparison Table of the Existing and Proposed Methods 

Table:4 

Existing Method 

 

Proposed Method – 1 

Mid value Ranking and variance 

Proposed Method – II 

Ambiguity Ranking and Mean 

The rank according 

to the expected value is  

 �̃�3 > �̃�1 > �̃�2 > �̃�4 > �̃�5 

The rank according to the  

expected value is    

�̃�3 > �̃�1 > �̃�2 > �̃�4 > �̃�5 

The rank according to the 

 expected value is  

   �̃�3 > �̃�1 > �̃�2 > �̃�4 > �̃�5 

 

In this article, two algorithms for calculating the expected value are proposed to overcome the 

limitations and shortcomings of the existing method. This paper mainly concentrates on focusing on 

the newly emerging fuzzy set theory "Hesitant Fuzzy Set Theory". Here, the Triangular Hesitant Fuzzy 

Sets, which are made up of Triangular Hesitant Fuzzy Elements, were utilized. Under the triangular 

hesitant fuzzy environment, we provided the decision-making issue with linguistic evaluations and 

fully unknown criteria weights. Comparison is done between the algorithms. The "mid value ranking 

variance" of these two proposed methods is a good ranking principle that is also simple to apply to 



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MCDM problems with uncertain, imprecise, or vague circumstances. This approach in particular takes 

a lot of time.  

References 

[1] Jianzhou Wang, Hongmin Li, Ying Wang, Haiyan Lu, A hesitant fuzzy wind speed forecasting system with novel 

defuzzification method and multi-objective optimization algorithm, Expert Systems with Applications 68 

(2021),114364. 

[2] X. Peng, J. Dai, L. Liu, Interval-valued dual hesitant fuzzy information aggregation and its application in multiple 

attribute decision making, International Journal of Uncertainty Quantification 8 (2018), 361–382.  

[3] D. Stephen Dinagar, E. Fany Helena, Similarity Measure using Sign Distance, Advances in Mathematics: Scientific 

Journal 10:1 (2021), 193-197.  

[4] D. Stephen Dinagar, E. Fany Helena, Similarity Measures of Intuitionistic Trapezoidal Fuzzy Number using Centroids 

of Horizontal & Vertical Axes and Value & Ambiguity Indices, Advances and Applications in Mathematical Sciences 

20 : 5 (2021), 875-885.  

[5] D. Stephen Dinagar, E. Fany Helena, Similarity Measures with Vector – Length under Fuzzy Environment, Advances 

and Applications in Mathematical Sciences 20:8 (2021),1425- 1432.