Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

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Application of Neutrosophic Sets in Multi Attribute Decision Making 

 

R. Ramesh1, K. Rajupillai2, K. Kalaiselvi3, R. Nishanthi4 and G. Deepa5* 

1,3,4Department of Mathematics, Dr. Mahalingam College of Engineering and Technology, India 

2Department of Mathematics, Government College of Engineering, Thanjavur, India 

5*Department of Computer science and Engineering, Dr. Mahalingam College of Engineering and Technology, India. 

*Correspondence: deepaganesh2011@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: The selection of an optimal hospital for healthcare services is a complex decision-

making problem due to the involvement of multiple, often conflicting, criteria such as quality 

of care, infrastructure, staff expertise, cost, and patient satisfaction. This paper presents an 

innovative approach for hospital selection using a hybrid decision-making framework 

combining neutrosophic sets, entropy weight, and the Multi-Attribute Decision-Making 

(MADM) method. Neutrosophic sets are employed to handle the inherent uncertainty and 

indeterminacy in hospital evaluation criteria, allowing for more flexible and accurate 

representation of data that may not be fully precise or certain. The entropy weight method is 

applied to objectively determine the significance of each evaluation criterion, ensuring that 

the decision process is not biased by subjective judgments. Finally, the MADM technique 

integrates the weighted criteria into a comprehensive decision model to rank hospitals based 

on their overall performance. The proposed model is tested with a real-world dataset of 

hospitals, demonstrating its effectiveness in selecting the most suitable healthcare facility 

while considering diverse factors. The results highlight the potential of combining 

neutrosophic sets with advanced decision-making techniques for improving decision 

accuracy and reliability in complex healthcare decisions. 

Keywords: Neutrosophic Sets; Entropy; MADM. 

1. Introduction 

Atanassov (1986, 2000) first proposed the use of intuitionistic fuzzy sets as a means of representation 

of data information that makes use of ordered pairs, i.e., 2-tuples, which may account for the 

membership and the non-membership degrees. According to Atanassov (1989), the sum of the 

membership and non-membership degrees of each 2-tuple in an IFS must be less than or equal to 1. 

Although fuzzy sets (FSs), first proposed by Zadeh in 1965, have shown promise in various domains, 

accurately evaluating their membership functions in practice can be difficult, since they are constrained 

to real values between zero and one. To overcome this difficulty, Atanassov and Gargov (1989) 

introduced interval-valued intuitionistic fuzzy sets (IVIFS) by extending the concept to represent the 

membership and non-membership degrees using intervals instead of precise values. This way is very 

useful in the complex environment, owing to a lot of facilitation in improving the flexibility of fuzzy 

sets, which was proposed to deal with ambiguity and uncertainty (Zhou et al., 2014a; Meng and Chen, 

2015; Wu and Chiclana, 2014; Onar et al., 2015; Hu et al., 2015; Zhou et al., 2016). 

IFSs and IVIFSs can deal with incomplete information. However, the information at the hands of a 

decision-maker is often fuzzy and inconsistent, and IFSs and IVIFSs are incapable of dealing with it; 

mailto:deepaganesh2011@gmail.com


Communications on Applied Nonlinear Analysis 

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therefore, Smarandache (1999, 2003) developed the neutrosophic sets (NSs), primarily by a 

philosophical perspective. NSs simultaneously consider truth membership, indeterminacy 

membership, and falsity membership, each independent of the others. Despite their theoretical 

attractiveness, NSs remain challenging to be applied in practical science and engineering applications. 

There are three significant branches of fuzzy theory research: entropy, similarity measures, and cross-

entropy. They have a lot of broad applications in medical diagnosis, information fusion systems, and 

so on. Entropy is one of the important elements to quantify uncertain information, which attracts much 

attention since its emergence (Wei et al., 2011). The concept of fuzzy entropy was first proposed by 

Zadeh in 1968 to quantify fuzziness in information for decision-making. Luca and Termini (1972) 

afterwards gave a formal definition of the entropy of a FS and pointed out the axioms that fuzzy entropy 

should fulfill. Szmidt and Kacprzyk (2001) proposed a non-probabilistic entropy measure for IFSs. 

The idea was to use the ratio of intuitionistic fuzzy cardinalities in defining the axiomatic conditions 

for intuitionistic fuzzy entropy. Ye created an entropy-weighted approach to calculate entropy weights 

and suggested two entropy measures for interval-valued intuitionistic fuzzy sets (IVIFSs) in 2010. Jin 

et al. (2014) investigated an interval-valued intuitionistic fuzzy continuous weighted entropy to solve 

multi-criteria decision-making (MADM) problems using the continuous ordered weighted averaging 

(COWA) operator. An entropy function was created by Majumdar and Samanta (2014) to quantify the 

degree of uncertainty in a single-valued neutrosophic set (SVNV). 

Similarity measures and cross-entropy are mainly used to evaluate the discrimination information. 

Great amounts of work have been carried out for such topics (Grzegorzewski, 2004; Zhou and Chen, 

2013; Hung and Yang, 2007; Zhou et al., 2014b; Ye, 2017). Liu (1992) proposed axiomatic definitions 

for entropy, distance measures and similarity measures of FSs, and revealed their basic interrelations 

deeply. Vlachos and Sergiadis (2007) firstly proposed the concept of intuitionistic fuzzy cross-entropy 

and investigated the relationships of cross-entropy and entropy. Beliakov et al. (2014) proposed a novel 

method to define the similarity measures for IFSs, under which each similarity measure consists of 

two parts: one representing the similarity aspect and another handling hesitancy. Based on Jaccard, 

Dice, and cosine similarity measures in vector space, Ye (2014a) presented three vector similarity 

measures for SVNSs to determine the ranking of alternatives in MADM problems. With a view to 

overcoming the limitation of the similarity measure in Ye (2014a), Ye (2015) developed modified 

cosine similarity measures for SVNSs based on the cosine function. Based on the distance between 

two SVNSs, Majumdar and Samanta (2014) proposed some similarity measures for SVNSs and 

discussed their properties. In a single-valued neutrosophic environment, Ye (2014b) initiated a cross-

entropy method to determine MADM methods. The relation among entropy, similarity measures, and 

cross-entropy has gained much attention. Zhang et al. (2009) and Zeng and Li (2006) proved that the 

entropy and similarity measures of interval-valued fuzzy sets (IVFSs) can be transformed into each 

other 

The process of selecting the best hospital is a multifaceted decision that demands careful consideration 

of various factors, such as medical expertise, infrastructure, patient care quality, and operational 

efficiency. Given the complexities involved in hospital assessment, decision-makers often face 

significant challenges in making an informed choice. Traditional methods of hospital selection 

primarily rely on qualitative assessments or limited quantitative data, which may not fully capture the 

underlying uncertainties and subjectivity inherent in the evaluation process. 



Communications on Applied Nonlinear Analysis 

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To address these challenges, this paper proposes an advanced decision-making framework that 

combines Neutrosophic Set Theory, Entropy Weighting, and Multi-Attribute Decision-Making 

(MADM) techniques to facilitate a more accurate and robust hospital selection process. Neutrosophic 

Set Theory offers a flexible and powerful way to model and manage imprecision, inconsistency, and 

indeterminacy in hospital performance data. The Entropy Weighting method is introduced to 

objectively assign weights to different selection criteria based on their importance, ensuring that the 

decision-making process aligns with real-world priorities. Finally, the MADM technique is applied to 

evaluate and rank hospitals based on the aggregated performance across multiple dimensions. This 

integrated approach is designed to provide healthcare decision-makers with a reliable tool for choosing 

the best hospital, offering a more comprehensive and systematic analysis than traditional methods. 

The significance of this approach lies in its ability to combine theoretical robustness with practical 

applicability, addressing key limitations of existing methods in hospital selection. By incorporating 

these advanced techniques, we aim to deliver a framework that can be applied to various healthcare 

settings, ensuring that patients and stakeholders make well-informed decisions in selecting healthcare 

providers. 

Entropy Weight Method: 

The Entropy Weight method is a well-established technique used to assign weights to different criteria 

based on their level of information or uncertainty. The concept of entropy comes from information 

theory and measures the level of disorder or uncertainty in a system. In the entropy weighting method, 

the entropy of each criterion is computed, and the weights are assigned inversely proportional to the 

entropy values. Criteria with higher uncertainty or variability are assigned lower weights, while those 

with more stable or consistent data receive higher weights. This method is particularly useful for 

ensuring that the most informative and significant criteria are given more emphasis in the decision-

making process. 

Multi-Attribute Decision-Making (MADM) Methods: 

A collection of decision-making strategies known as MADM approaches seeks to assess options 

according to a number of factors. These techniques are especially helpful when decision-makers are 

faced with a range of options, each of which performs differently across a number of qualities. Among 

the popular MADM techniques are the Analytical Hierarchy Process (AHP), the Weighted Sum Model 

(WSM), and the Weighted Product Model (WPM). Using entropy-based weights and neutrosophic set 

representations, we employ a MADM technique in this study to aggregate hospital performance across 

multiple parameters, offering a thorough assessment and ranking of options. 

Together, these techniques create a robust decision-making framework that is capable of handling the 

uncertainty, imprecision, and complexity inherent in hospital selection, while offering a systematic 

approach to identify the best hospital based on a set of predefined criteria. 

Definition 1.1.1. In 1998, Smarandache published the single-valued Neutrosophic Set (NS) [38, 39, 

40]. Assume that U is a discourse universe and that ANS⸦U. Then, ANS={ <x, 𝒯A(x)), ℐA(x), ℱA(x)>| 



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x ∊ U}, where 𝒯A(x), ℐA(x) and ℱA(x): U → [0, 1] stand for the degrees of membership, indeterminacy, 

and non-membership, respectively, with 0 ≤ 𝒯A(x) (x) + ℐA(x)  + ℱA(x) ≤ 3.  

2. Operators in Neutrosophic Sets 

Definition 2.1:  Let 𝛾, 𝛿 are two NSs. Then operational relations are defined as,  

1. 𝛾𝛿 = <𝒯𝛾(𝑥) +  𝒯𝛿(𝑧) - 𝒯𝛾(𝑥) . 𝒯𝛿(𝑧), ℐ𝛾(𝑥) . ℐ𝛿(𝑧), ℱ𝛾(𝑥) . ℱ𝛿(𝑧)>. 

2. 𝛾𝛿 =< 𝒯𝛾(𝑥). 𝒯𝛿(𝑧), ℐ𝛾(𝑥) +  ℐ𝛿(𝑧) − ℐ𝛾(𝑥) . ℐ𝛿(𝑧), ℱ𝛾(𝑥) + ℱ𝛿(𝑧) − ℱ𝛾(𝑥) . ℱ𝛿(𝑧) >  

3.  . 𝛾 = < 1 − (1 − 𝒯𝛾(𝑥))


,  (ℐ𝛾(𝑥))


, (ℱ𝛾(𝑥))


>, where  >  0. 

4. 𝛾    = < (𝒯𝛾(𝑥)) , 1 − (1 − ℐ𝛾(𝑥)), 1 − (1 − ℱ𝛾(𝑧))


>, where  >  0. 

Theorem 2.2:  Let 𝛾𝑖 = (𝒯𝛾(𝑥𝑖), ℐ𝛾(𝑥𝑖), ℱ𝛾(𝑥𝑖)), 𝑖 = 1,2,3. . . . 𝑛 be a collection of NS, then 

Neutrosophic Weighted Aggregation (NWA) operator value is also a NS and 

𝑁𝑆WA(𝛾1, 𝛾2, 𝛾3 .  .  . 𝛾𝑛) = (1 − ∏ (1 − 𝒯𝛾(𝑥𝑖))
𝑤𝑗

,   𝑛
𝑖,𝑗=1 ∏ (ℐ𝛾(𝑥𝑖))

𝑤𝑗
,𝑛

𝑖,𝑗=1  ∏ (ℱ𝛾(𝑥𝑖))
𝑤𝑗𝑛

𝑖,𝑗=1 )   

……………..  (1)  where 𝑤 = (𝑤1, 𝑤2, 𝑤3, … 𝑤𝑛 )𝑇  be the weight vector of 𝛾𝑗(𝑗 = 1,2,3, … . 𝑛) and 

𝑤𝑗 > 0, ∑ 𝑤𝑗 = 1.𝑛
𝑗=1  

Proof. We prove Equation (1) by mathematical induction on n. 

If n = 2, then we have 𝑁WA(𝛾1, 𝛾2) = 𝑤1𝛾1𝑤2𝛾2 

By Definition 2.1, we can see that both 𝑤1𝛾1 𝑎𝑛𝑑 𝑤2𝛾2 are NS and the value of 𝑤1𝛾1𝑤2𝛾2 is also a 

NS. From the operational laws of NS, we have 

𝑤1𝛾1 = (1 − (1 − 𝒯𝛾(x1))
𝑤1

, (ℐ𝛾(x1))
𝑤1

 , (ℱ𝛾(x1))
𝑤1

); 

𝑤2𝛾2 = (1 − (1 − 𝒯𝛾(x2))
𝑤2

, (ℐ𝛾(x2))
𝑤2

 , (ℱ𝛾(x2))
𝑤2

) 

Then, 𝑁WA(𝛾1, 𝛾2)  = 𝑤1𝛾1𝑤2𝛾2 =< 2 − (1 − 𝒯𝛾(x1))
𝑤1

− (1 − 𝒯𝛾(x2))
𝑤2

−

(1 − (1 − 𝒯𝛾(x1))
𝑤1

) (1 − (1 − 𝒯𝛾(x2))
𝑤2

) ,  (ℐ𝛾(x1))
𝑤1

(ℐ𝛾(x2))
𝑤2

, (ℱ𝛾(x1))
𝑤1

(ℱ𝛾)(x2))
𝑤2  

> 

=< 1 − (1 − 𝒯𝛾(x1))
𝑤1

(1 − 𝒯𝛾(x2))
𝑤2

, (ℐ𝛾(x1))
𝑤1

(ℐ𝛾(x2))
𝑤2

,                                

(ℱ𝛾(x1))
𝑤1

(ℱ𝛾(x2))
𝑤2

> . 

If n = k, then Equation (1) holds, i.e., 

𝑁WA(𝛾1, 𝛾2, 𝛾3 … 𝛾n) = 𝑤1𝛾1𝑤2𝛾2 … … …𝑤𝑘𝛾𝑘         



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            =< 1 − ∏ (1 − 𝒯𝛾(x𝑖))
𝑤𝑗

,

𝑘

𝑗=1

∏ (ℐ𝛾(x𝑖))
𝑤𝑗

,

𝑘

𝑗=1

∏ (ℱ𝛾(x𝑖))
𝑤𝑗

𝑘

𝑗=1

> 

and the aggregated value is a NS, then when n = k + 1, by the operational laws of NS, we have 

𝑁WA(𝛾1, 𝛾2, 𝛾3 … 𝛾k+1) = 𝑤1𝛾1𝑤2𝛾2 … … …𝑤𝑘𝛾𝑘𝑤𝑘+1𝛾𝑘+1 

= 〈1 − ∏ (1 − 𝒯𝛾(x𝑖))
𝑤𝑗

+ (1 − (1 − 𝒯𝛾(x𝑘+1))
𝑤𝑘+1

) − (1 − ∏ (1 − 𝒯𝛾(x𝑖))
𝑤𝑗

𝑘
𝑗=1 ) (1 −𝑘

𝑗=1

(1 − 𝒯𝛾(x𝑘+1))
𝑤𝑘+1

) , ∏ (ℐ𝛾(x𝑖))
𝑤𝑗

,𝑘+1
𝑗=1 ∏ (ℱ𝛾)(x𝑖))

𝑤𝑗𝑘+1
𝑗=1 〉  

=< 1 − ∏ (1 − 𝒯𝛾(x𝑖))
𝑤𝑗

,𝑘+1
𝑗=1 ∏ (ℐ𝛾(x𝑖))

𝑤𝑗

,𝑘+1
𝑗=1 ∏ (ℱ𝛾(x𝑖))

𝑤𝑗𝑘+1
𝑗=1 >. 

By which aggregated value is also a NS, Therefore, when n = k + 1, Equation (1) holds. Thus, by 

steps 1 and 2, we recognize that Equation (1) sustain for all n.  

 

Theorem 2.3:  Let 𝛾𝑖 = (𝒯𝛾(𝑥𝑖), ℐ𝛾(𝑥𝑖), ℱ𝛾(𝑥𝑖)), 𝑖 = 1,2,3. . . . 𝑛 be a collection of NS, then 

Neutrosophic Weighted geometric (NWG) operator value is also a NS and  

𝑁WG(𝛾1, 𝛾2, 𝛾3 .  .  . 𝛾𝑛) = 

 (∏ (𝒯𝛾(𝑥𝑖))
𝑤𝑗

,   1 −𝑛
𝑖,𝑗=1 ∏ (1 − ℐ𝛾(𝑥𝑖))

𝑤𝑗
,𝑛

𝑖,𝑗=1  ∏ (1 − ℱ𝛾(𝑥𝑖))
𝑤𝑗𝑛

𝑖,𝑗=1 )   ……………..  (2)  

where 𝑤 = (𝑤1, 𝑤2, 𝑤3, … 𝑤𝑛 )𝑇  be the weight vector of 𝛾𝑗(𝑗 = 1,2,3, … . 𝑛) and 𝑤𝑗 > 0,

∑ 𝑤𝑗 = 1.𝑛
𝑗=1  

Proof. We prove Equation (1) by mathematical induction on n. 

If n = 2, then we have 𝑁WG(𝛾1, 𝛾2) = 𝑤1𝛾1𝑤2𝛾2 

By Definition 2.1, we can see that both 𝑤1𝛾1 𝑎𝑛𝑑 𝑤2𝛾2 are NS and the value of 𝑤1𝛾1𝑤2𝛾2 is also a 

NS. From the operational laws of NS, we have 

𝑤1𝛾1 = ((𝒯𝛾(x1))
𝑤1

, 1 − (1 − ℐ𝛾(x1))
𝑤1

 ,1 − (1 − ℱ𝛾(x1))
𝑤1

); 

𝑤2𝛾2 = ((𝒯𝛾(x2))
𝑤2

, 1 − (1 − ℐ𝛾(x2))
𝑤2

 , 1 − (1 − ℱ𝛾(x2))
𝑤2

) 

Then, 𝑁WG(𝛾1, 𝛾2)  = 𝑤1𝛾1𝑤2𝛾2 =< (𝒯𝛾(x1))
𝑤1

(𝒯𝛾)(x2))
𝑤2  

,  2 − (1 − ℐ𝛾(x1))
𝑤1

−

(1 − ℐ𝛾(x2))
𝑤2

− (1 − (1 − ℐ𝛾(x1))
𝑤1

) (1 − (1 − ℐ𝛾(x2))
𝑤2

) , 2 − (1 − ℱ𝛾(x1))
𝑤1

−

(1 − ℱ𝛾(x2))
𝑤2

− (1 − (1 − ℱ𝛾(x1))
𝑤1

) (1 − (1 − ℱ𝛾(x2))
𝑤2

)
.

> 

=< (𝒯𝛾(x1))
𝑤1

(𝒯𝛾(x2))
𝑤2

, 1 − (1 − 𝒯𝛾(x1))
𝑤1

(1 − 𝒯𝛾(x2))
𝑤2

, 1 − (1 − 𝒯𝛾(x2))
𝑤1

(1 −

𝒯𝛾(x2))
𝑤2

>                 . 



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If n = k, then Equation (1) holds, i.e., 

𝑁WG(𝛾1, 𝛾2, 𝛾3 … 𝛾n) = 𝑤1𝛾1𝑤2𝛾2 … … …𝑤𝑘𝛾𝑘         

            =< ∏ (𝒯𝛾(x𝑖))
𝑤𝑗

𝑘

𝑗=1

, 1 − ∏ (1 − ℐ𝛾(x𝑖))
𝑤𝑗

, 1 − ∏ (1 − ℱ𝛾(x𝑖))
𝑤𝑗

𝑘

𝑗=1

𝑘

𝑗=1

> 

and the aggregated value is a NSHSS, then when n = k + 1, by the operational laws of NSHSS, we 

have 

𝑁WA(𝛾1, 𝛾2, 𝛾3 … 𝛾k+1) = 𝑤1𝛾1𝑤2𝛾2 … … …𝑤𝑘𝛾𝑘𝑤𝑘+1𝛾𝑘+1 

= 〈∏ (𝒯𝛾(x𝑖))
𝑤𝑗

,𝑘+1
𝑗=1 1 − ∏ (1 − ℐ𝛾(x𝑖))

𝑤𝑗

+ (1 − (1 − ℐ𝛾(x𝑘+1))
𝑤𝑘+1

) − (1 −𝑘
𝑗=1

∏ (1 − ℐ𝛾(x𝑖))
𝑤𝑗

𝑘
𝑗=1 ) (1 − (1 − ℐ𝛾(x𝑘+1))

𝑤𝑘+1

) , 1 − ∏ (1 − ℱ𝛾(x𝑖))
𝑤𝑗

+ (1 −𝑘
𝑗=1

(1 − ℱ𝛾(x𝑘+1))
𝑤𝑘+1

) − (1 − ∏ (1 − ℱ𝛾(x𝑖))
𝑤𝑗

𝑘
𝑗=1 ) (1 − (1 − ℱ𝛾(x𝑘+1))

𝑤𝑘+1

)〉  

=< ∏ (𝒯𝛾(x𝑖))
𝑤𝑗

,𝑘+1
𝑗=1 1 − ∏ (1 − ℐ𝛾(x𝑖))

𝑤𝑗

,𝑘+1
𝑗=1 1 − ∏ (1 − ℱ𝛾(x𝑖))

𝑤𝑗
𝑘+1
𝑗=1 >. 

By which aggregated value is also a NS, Therefore, when n = k + 1, Equation (1) holds. Thus, by 

steps 1 and 2, we recognize that Equation (1) sustain for all n.  

 

Definition 2.3. Let 𝛾i = { (𝒯(x𝑖), ℐ(x𝑖), ℱ(x𝑖))|𝑥𝑖 ∈ 𝑋} be a NS set on X.  Then the entropy of 𝛾i is 

defined as,  

𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((𝒯2

𝛾(x𝑖) + ℐ2
𝛾(x𝑖))𝑚

𝑖=1 |(4ℱ2
𝛾(x𝑖) − 1)| + (𝒯2

𝛾(x𝑖) + ℱ2
𝛾(x𝑖))|(4ℐ2

𝛾(x𝑖) −

1)| + ( ℱ2
𝛾(x𝑖) + ℐ2

𝛾(x𝑖))|(4𝒯2
𝛾(x𝑖) − 1)|)          . . . . . . (2)                                                                                                                                                        

Theorem 2.4.  The proposed entropy on NS(X) fulfilling the conditions: 

1.  𝐸(𝛾i) = 0, if 𝛾 is a crisp set ie., 𝛾i = (𝒯𝛾(𝑦)(x𝑖), ℐ𝛾(𝑦)(x𝑖), ℱ𝛾(𝑦)(x𝑖)) = (1,0,0) or 

 𝛾i = (𝒯𝛾(x𝑖), ℐ𝛾(x𝑖), ℱ𝛾(x𝑖)) = (0,0,1) for all 𝑥𝑖 ∈ 𝑋. 

2.  𝐸(𝛾i) = 1, if  𝛾i  =  {(𝑥𝑖 ,
1

2
,

1

2
,

1

2
) |𝑥𝑖 ∈ 𝑋} 

3.  𝐸(𝛾𝑖)  =  𝐸(𝛾𝑖
𝐶), for all 𝛾𝑖 ∈ 𝑁𝑆𝐻𝑆𝑆(𝑋); 

4. 𝐸(𝛾𝑖)  ≤  𝐸(𝛿𝑖) if either 𝒯𝛾(𝑥𝑖) ≤  𝒯𝛿(𝑥𝑖), ℐ𝛾(𝑥𝑖) ≤ ℐ𝛿(𝑥𝑖), ℱ𝛾(𝑥𝑖) ≤ ℱ𝛿(𝑥𝑖) when  

𝑚𝑎𝑥( 𝒯𝛿(𝑥𝑖), ℐ𝛿(𝑥𝑖), ℱ𝛿)(𝑥𝑖))  ≤
1

2
  or 𝒯𝛾(𝑥𝑖)  ≥  𝒯𝛿(𝑥𝑖), ℐ𝛾(𝑥𝑖)  ≥ ℐ𝛿(𝑥𝑖), ℱ𝛾(𝑥𝑖) ≥  ℱ𝛿(𝑥𝑖)  when 

𝑚𝑖𝑛(𝒯𝛿(𝑥𝑖), ℐ𝛿(𝑥𝑖), ℱ𝛿(𝑥𝑖))  ≥  
1

2
. 

Proof: 

1. Let 𝛾i = (𝒯𝛾(𝑦)(x𝑖), ℐ𝛾(𝑦)(x𝑖), ℱ𝛾(𝑦)(x𝑖)) = (1,0,0) for all 𝑥𝑖 ∈ 𝑋. Then,  



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𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((𝒯2

𝛾(x𝑖) + ℐ2
𝛾(x𝑖))𝑚

𝑖=1 |(4ℱ2
𝛾(x𝑖) − 1)| + (𝒯2

𝛾(x𝑖) + ℱ2
𝛾(x𝑖))|(4ℐ2

𝛾(x𝑖) −

1)| + ( ℱ2
𝛾(x𝑖) + ℐ2

𝛾(x𝑖))|(4𝒯2
𝛾(x𝑖) − 1)|)  

⇒ 𝐸(𝛾i) = 1 −
1

2𝑚
∑((12 + 0

𝑚

𝑖=1

)|(40 − 1)| + (12 + 0)|(4 0 − 1)| + (0 + 0)|(4 1 − 1)|)  

⇒ 𝐸(𝛾i) = 1 −
1

2𝑚
∑ 2

𝑚

𝑖=1

= 1 −
1

2𝑚
(2𝑚) = 1 − 1 = 0 

Similarly, If 𝛾i = (𝒯𝛾(x𝑖), ℐ𝛾(x𝑖), ℱ𝛾(x𝑖)) = (0,1,0) and (𝒯𝛾(x𝑖), ℐ𝛾(x𝑖), ℱ𝛾(x𝑖)) = (0,0,1) 

 ∀𝑥𝑖 ∈ 𝑋, then  𝐸(𝛾i) = 0. 

2. Let  𝛾i  =  {(𝑥𝑖 ,
1

2
,

1

2
,

1

2
) |𝑥𝑖 ∈ 𝑋}. Then  

𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((𝒯2

𝛾(x𝑖) + ℐ2
𝛾(x𝑖))𝑚

𝑖=1 |(4ℱ2
𝛾(x𝑖) − 1)| + (𝒯2

𝛾(x𝑖) + ℱ2
𝛾(x𝑖))|(4ℐ2

𝛾(x𝑖) −

1)| + ( ℱ2
𝛾(x𝑖) + ℐ2

𝛾(x𝑖))|(4𝒯2
𝛾(x𝑖) − 1)|)   

⇒ 𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((

1

22
+

1

22
)) |(4 (

1

22
) − 1)|

𝑚

𝑖=1

+ (
1

22
+

1

22
) |(4 (

1

22
) − 1)|

+ (
1

22
+

1

22
) |(4 (

1

22
) − 1)| 

⇒ 𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((

1

4
+

1

4
) |(1 − 1)| + (

1

4
+

1

4
) |(1 − 1)| + (

1

4
+

1

4
) |(1 − 1)|)

𝑚

𝑖=1

 

                 = 1 − 0 = 1.  

3. Let 𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((𝒯2

𝛾(x𝑖) + ℐ2
𝛾(x𝑖))𝑚

𝑖=1 |(4ℱ2
𝛾(x𝑖) − 1)| + (𝒯2

𝛾(x𝑖) +

ℱ2
𝛾(x𝑖))|(4ℐ2

𝛾(x𝑖) − 1)| + ( ℱ2
𝛾(x𝑖) + ℐ2

𝛾(x𝑖))|(4𝒯2
𝛾(x𝑖) − 1)|)   

⇒ 𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((ℱ2

𝛾(x𝑖) + ℐ2
𝛾(x𝑖)𝑚

𝑖=1 )|(4𝒯2
𝛾(x𝑖) − 1)| + (ℱ2

𝛾(x𝑖) +

𝒯2
𝛾(x𝑖))|(4ℐ2

𝛾(x𝑖) − 1)| + (𝒯2
𝛾(x𝑖) + ℐ2

𝛾(x𝑖))|(4ℱ2
𝛾(x𝑖) − 1)|) = 𝐸(𝛾i

𝐶).  

4. Let 𝒯𝛾(x𝑖) ≤ 𝒯𝛿(x𝑖), ℐ𝛾(x𝑖) ≤ ℐ𝛿(x𝑖), ℱ𝛾(x𝑖) ≤ ℱ𝛿(x𝑖) and  

𝑚𝑎𝑥(𝒯𝛿(x𝑖), ℐ𝛿(x𝑖), ℱ𝛿(x𝑖)) ≤
1

2
.  

Then,  1 −
1

2𝑚
∑ ((𝒯2

𝛾(x𝑖) + ℐ2
𝛾(x𝑖)𝑚

𝑖=1 )|(4ℱ2
𝛾(x𝑖)) − 1)| + (𝒯2

𝛾(x𝑖) + ℱ2
𝛾(x𝑖))|(4ℐ2ℐ𝛾(x𝑖) −

1)| + (ℱ2
𝛾(x𝑖) + ℐ2

𝛾(x𝑖)) |(4𝒯2
𝛾(x𝑖) − 1)| ≤ 



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1 −
1

2𝑚
∑ ((𝒯2

𝛿(x𝑖) + ℐ2
𝛿(x𝑖)𝑚

𝑖=1 )|(4ℱ2
𝛿(x𝑖) − 1)| + (𝒯2

𝛿(x𝑖) + ℱ2
𝛿(x𝑖))|(4ℐ2

𝛿(x𝑖) − 1)| +

(ℱ2
𝛿(x𝑖) + ℐ2

𝛿(x𝑖))|(4𝒯2
𝛿(x𝑖) − 1)|)  

When 𝑚𝑎𝑥(𝒯𝛿(x𝑖), ℐ𝛿(x𝑖), ℱ𝛿(x𝑖)) ≤
1

2
. 

3. Neutrosophic Sets in MADM 

The selection of an optimal hospital for healthcare services is a complex decision-making problem due 

to the involvement of multiple, often conflicting, criteria such as quality of care, infrastructure, staff 

expertise, cost, and patient satisfaction. This approach neutrosophic methods in handling uncertainty 

and imprecision through the NS model. The proposed methodology benefits customers by enabling 

informed choices, making the review process both comprehensive and reliable. 

3.1 Algorithm: Multi-Attribute Decision-Making Using NS and Entropy 

Step 1: Initialize the Process 

To evaluate each criterion, it is necessary to first identify the decision-makers 

𝜅= {𝜅1, 𝜅2,. ….. 𝜅n} criteria 𝛻={𝛻1, 𝛻2,….. 𝛻n}, and alternatives 𝛺= {𝛺1, 𝛺2 …., 𝛺n}, then gather input 

data from the decision-makers. 

Step 2: Selection of Criteria and Decision Makers 

Select the decision-makers 𝜅= {𝜅1, 𝜅2,……n} who will assess the criteria  

𝛻={𝛻1, 𝛻2,….. 𝛻n} and alternatives after it has been established which criteria bear on the decision-

making process. 

Step 3: Formulate Propositions Using NSHSS 

Utilise attributes and sub-attributes to create power sets of criteria and an NSHSS architecture. The 

evaluation of each alternative is then represented as follows: 

𝛾i = (𝒯𝛾(x𝑖), ℐ𝛾(x𝑖), ℱ𝛾(x𝑖)). 

Step 4: Linguistic Evaluation of Each Criterion 

• Decision-makers' linguistic evaluations for each of the NSs criteria 𝛻={𝛻1, 𝛻2,….. 𝛻n}. To denote 

membership (𝒯), indeterminacy (ℐ) and non-membership (ℱ), propositions are converted into 

neutrosophic values. 

Step 5: Utilising the proposed entropy method, determine the entropy value, weight of the criteria and 

degree of divergence of each criterion. 

Apply the entropy formula to compute the weight wi for each criterion Ci  

𝐸(𝛾i) = 1 −
1

2𝑚
∑ ((𝒯2

𝛾(x𝑖) + ℐ2
𝛾(x𝑖))𝑚

𝑖=1 |(4ℱ2
𝛾(x𝑖) − 1)| + (𝒯2

𝛾(x𝑖) + ℱ2
𝛾(x𝑖))|(4ℐ2

𝛾(x𝑖) −

1)| + ( ℱ2
𝛾(x𝑖) + ℐ2

𝛾(x𝑖))|(4𝒯2
𝛾(x𝑖) − 1)|) such that ∑ wi

n
i=1 = 1. 

 



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Step 6: Aggregate the PFS using NWA 

Use 𝑁𝑊𝐴𝑤 and  𝑁𝑊𝐺𝑤 aggregation operator to combine the evaluations of all criteria for each 

alternative 

𝑁𝑊𝐴𝑤(𝛼1, 𝛼2 … . . 𝛼𝑛) = (1 − ∏ (1 − 𝒯(𝑥𝑖))𝑤𝑗 ,   𝑛
𝑖,𝑗=1 ∏ (ℐ(𝑥𝑖))𝑤𝑗 ,𝑛

𝑖,𝑗=1  ∏ (ℱ(𝑥𝑖))𝑤𝑗𝑛
𝑖,𝑗=1 ) ________(1)  

and  

𝑁𝑊𝐺𝑤(𝛼1, 𝛼2 … . . 𝛼𝑛) = (∏ (𝒯(𝑥𝑖))𝑤𝑗 ,𝑛
𝑖,𝑗=1 1 − ∏ (1 − ℐ(𝑥𝑖))𝑤𝑗 , 1 − ∏ (1 − ℱ(𝑥𝑖))𝑤𝑗𝑛

𝑖,𝑗=1  𝑛
𝑖,𝑗=1 ) 

________(2)   

Step 7: Evaluate Alternatives Using a Score Function 𝑆 =
2+𝒯𝑎−ℐ𝑎−ℱ𝑎

3
___________(3) 

Step 8: Rank the alternatives 𝐴 = {𝐴1, 𝐴2, . . . . 𝐴𝑛} based on their scores. 

Step 9: Select the Best Alternative(s) 

Step 10: End the Process 

3.2 Flow Chart for Multi Attributes Decision Making Using Neutrosophic Set 

 

Figure 1: Flow Chart for Multi Attributes Decision Making Using Neutrosophic Set. 

 

Evaluation Criteria: 

When selecting the best hospital, multiple factors must be considered to ensure a comprehensive 

evaluation that meets the needs of patients and healthcare stakeholders. In the context of Multi-

Attribute Decision-Making (MADM), these factors (also called criteria) serve as the key decision-

making parameters used to evaluate and rank hospitals. MADM methods enable the comparison of 

alternative hospitals based on several criteria, leading to an informed, systematic, and data-driven 

decision. 

The criteria involved in selecting the best hospital can vary depending on the specific healthcare needs, 

but some core and commonly used criteria include: 



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1. Quality of Care (ξ1) 

One of the most important criteria in selecting the best hospital is the quality of care provided. This 

encompasses a variety of factors such as the effectiveness of medical treatments, the experience of 

healthcare professionals, patient safety, and overall patient satisfaction. 

2. Cost and Affordability (ξ2) 

The cost of healthcare services is another critical criterion for hospital selection. Hospitals often 

provide a wide range of services, and their pricing can vary depending on factors such as location, 

facilities, and specialization. Cost must be balanced with the quality of services provided. 

3. Hospital Infrastructure and Facilities (ξ3) 

A hospital's infrastructure and facilities play a crucial role in determining its overall performance. 

High-quality hospitals tend to have state-of-the-art medical equipment, clean and comfortable 

facilities, and modern technology that can improve patient care. 

4. Staff Competency and Training (ξ4) 

The competency and training of the hospital’s healthcare providers are essential to ensuring that 

patients receive the best possible care. Hospitals that invest in ongoing training and professional 

development for their medical staff are likely to maintain high standards of care. 

5. Patient Services and Satisfaction (ξ5) 

The overall patient experience is another vital criterion. This encompasses the interactions between 

patients and hospital staff, waiting times, the quality of communication, and the comfort of hospital 

facilities. 

Evaluation Process: 

Table 1: The way that the DMs evaluated the alternatives through experts' opinions, in which linguistic 

variables are justifiable for assessment of the alternatives under prefixed standards. It also shows a 

way of converting linguistic inputs from experts to NSs 

Table 1: Linguistic Terms and corresponding NS values 

Linguistic Term Notation (𝓣, 𝓘, 𝓕)×10-2 

Extremely Weak Ξ1 (0, 100, 100) 

Very Weak Ξ2 (20, 90, 80) 

Weak Ξ3 (30, 80, 70) 

Slightly Weak Ξ4 (40, 70, 70) 

Below Average Ξ5 (50, 60, 60) 

Average Ξ6 (60, 60, 50) 

Above Average Ξ7 (70, 40, 40) 

Slightly Strong Ξ8 (80, 30, 30) 

Strong Ξ9 (85, 20, 30) 



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Very Strong Ξ10 (90, 20, 20) 

Extremely Strong Ξ11 (100, 10, 10) 

 

The decision matrix is presented in Table 2, where evaluations of the decision-makers on each 

alternative (ϑ1, ϑ2, ϑ3 and ϑ4) are carried out with respect to identified criteria and sub-criteria. 

Evaluations will be made using pre-defined linguistic phrases in order to express the different levels 

of performance as depicted below. 

Table 2: Decision Matrix in Linguistic Terms 

Criteria Θ1 Θ2 Θ3 Θ4 

𝒜1 Ξ1 Ξ2 Ξ4 Ξ3 

𝒜2 Ξ2 Ξ4 Ξ3 Ξ3 

𝒜3 Ξ1 Ξ2 Ξ3 Ξ4 

𝒜4 Ξ10 Ξ9 Ξ7 Ξ6 

𝒜5 Ξ2 Ξ4 Ξ3 Ξ1 

 

Table 3: Decision matrix in neutrosophic values. Here each alternative ϑ1, ϑ2, ϑ3, ϑ4 is evaluated w.r.t. 

criteria. Each evaluation will be made by using three components: Truth (𝒯), Indeterminacy (ℐ) and 

Falsity (𝓕). This will provide a complete description of the extent to which a criterion is satisfied, 

uncertain, or unsatisfied for each alternative. 

Table 3: Decision Matrix in Neutrosophic Values 

Criteria Θ1 (𝓣, 𝓘, 𝓕)×10-2 Θ2 (𝓣, 𝓘, 𝓕) ×10-2 Θ3 (𝓣, 𝓘, 𝓕) ×10-2 Θ4 (𝓣, 𝓘, 𝓕) ×10-2 

𝒜1 (0,100,100) (20, 90, 80) (40, 70, 70) (30, 80, 70) 

𝒜2 (20, 90, 80) (40, 70, 70) (30, 80, 70) (30, 80, 70) 

𝒜3 (0,100,100) (20, 90, 80) (30, 80, 70) (40, 70, 70) 

𝒜4 (90, 20, 20) (85, 20, 30) (70, 40, 40) (60, 60, 50) 

𝒜5 (20, 90, 80) (40, 70, 70) (30, 80, 70) (0,100,100) 

 

Using the proposed entropy we will get the entropy, Degree of divergence and weight of the criteria 

in Table 4. 

Table 4: Criteria Weights  

Methods / Criteria  

 

Quality of 

Care 

Cost and 

Affordability 

 

Hospital 

Infrastructure 

and Facilities 

Staff 

Competency 

and 

Training 

Patient 

Services 

and 

Satisfaction 

Entropy value -0.9996 -0.2907 -0.9996 0.458075 -0.9996 



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Degree of 

divergence 𝑫𝒋
𝒑
 1.9996 1.2907 1.9996 0.541925 1.9996 

weight of the 

criteria 𝑾𝒋
𝒑
 0.25533 0.16481 0.25533 0.069199 0.25533 

 

Table 5 The aggregating results by the 𝑵𝑾𝑨𝒘 and 𝑵𝑾𝑮𝒘operators 

Hospital 𝑁𝑊𝐴𝑤( 𝓣, 𝓘, 𝓕)  𝑁𝑊𝐺𝑤(𝓣, 𝓘, 𝓕) 

Θ1 (0.35,0.57,0.72)  (0.34,0.77,0.73) 

Θ2 (0.44,0.65,0.66)  (0.44,0.66,0.66) 

Θ3 (0.41,0.70,064)  (0.40,0.71,0.65) 

Θ4 (0.35,0.76,0.69)  (0.32,0.78,0.72) 

  

In Table 5, The aggregating values found by using 𝑁𝑊𝐴𝑤 operator and 𝑁𝑊𝐺𝑤 operator 

Table 6 The score functions  

Hospital 𝑁𝑊𝐴𝑤 𝑁𝑊𝐺𝑤 

Θ1 0.2931 0.2810 

Θ2 0.3778 0.3733 

Θ3 0.3562 0.3475 

Θ4 0.3002 0.2739 

 

In table 6, we find the score values by using aggregating values from table 5. 

Table 7 Ordering of Hospitals 

 Ranking 

𝑁𝑊𝐴𝑤 Θ2 > Θ3 > Θ4 > Θ1 

𝑁𝑊𝐺𝑤 Θ2 > Θ3 > Θ1 > Θ4 

  

Table 7, represented the ordering of Hospitals. From this table we conclude Θ2 is the best Hospital 

comparing with others. 

 

 

 



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3.3 Visualization of Entropy and Ranks  

 

Figure 2: Flow Chart for Multi Attributes Decision Making Using Neutrosophic Set. 

 

 

Figure 3 Ranking of Hospitals by N𝑊𝐴𝑤 

 

 

Figure 4 Ranking of Hospitals by N𝑊G𝑤 

-1.00

-0.29

-1.00

0.46

-1.00

2.00

1.29

2.00

0.54

2.00

0.26 0.16 0.26 0.07 0.26

-2.00

-1.00

0.00

1.00

2.00

3.00

0 1 2 3 4 5 6

Entropy value Degree of divergence

0.2931

0.3778 0.3562

0.3002

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

0.3000

0.3500

0.4000

Θ1 Θ2 Θ3 Θ4

0.2810

0.3733
0.3475

0.2739

0.0000

0.1000

0.2000

0.3000

0.4000

Θ1 Θ2 Θ3 Θ4



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Figure 5 Comparison Ranking of Hospitals by 𝑃𝐹𝑊𝐴𝑤 and 𝑃𝐹𝑊G𝑤 

 

 

 

 

 

 

 

4. Conclusions  

In this study, we aimed to develop a robust methodology for selecting the best hospital from a set of 

available options by leveraging a combination of Neutrosophic Sets, Entropy, and Multi-Attribute 

Decision Making (MADM). The integration of these techniques provides a comprehensive and flexible 

approach to address the inherent uncertainty, vagueness, and imprecision involved in hospital 

selection, a process influenced by numerous qualitative and quantitative criteria. Neutrosophic Sets 

enabled us to model the uncertainty and indeterminacy present in the data, accommodating incomplete, 

inconsistent, or contradictory information. The inclusion of neutrosophic elements allowed for a more 

nuanced evaluation of hospital attributes, considering degrees of truth, falsity, and indeterminacy. he 

Entropy method was employed to quantify the uncertainty and variability in the decision-making 

criteria, helping to derive objective weights for each attribute. This ensured that the criteria were 

ranked appropriately based on their importance in the decision-making process, accounting for their 

inherent uncertainty. Multi-Attribute Decision Making (MADM) was utilized to combine all the 

gathered information and facilitate a well-rounded evaluation of each hospital. The MADM process 

enabled the synthesis of various criteria—such as medical expertise, infrastructure, accessibility, and 

patient satisfaction—into a single decision matrix, allowing for an optimal selection. 

By applying this integrated framework, we were able to rank hospitals more effectively, overcoming 

challenges such as missing data, conflicting opinions, and diverse performance measures. This 

methodology not only provides a structured approach to decision-making but also enhances 

transparency and robustness in hospital selection, particularly in contexts where multiple decision-

makers or stakeholders are involved. 

In conclusion, the combination of Neutrosophic Sets, Entropy, and MADM offers a powerful tool for 

hospital selection, providing decision-makers with a comprehensive, flexible, and data-driven 

framework that improves the accuracy and reliability of the selection process. Future work could 

further refine this methodology by incorporating additional factors like time and dynamic changes in 

0.293

0.378
0.356

0.300
0.281

0.373
0.347

0.274

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

Θ1 Θ2 Θ3 Θ4

𝑃𝐹𝑊𝐴𝑤 𝑃𝐹𝑊G𝑤



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hospital performance, extending its applicability to real-time decision-making scenarios in healthcare 

management. 

References 

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