Neutrosophic Soft Sets with Medical Decisin-Making Applications

Abstract

Keywords: Neutrosophic soft sets; soft sets; decision making; comparison matrix.

1. Introduction

Fuzziness has revolutionized many areas such as mathematics, science, engineering, medicine. This concept was initiated by Zadeh [13]. Not only Zadeh discovered this concept, but he also developed the infrastructure of today's popular forms of use such as relations of similarity, decision making, and fuzzy programming in a short time. Just like in the theory of sets, fuzzy sets(FS) also have led to the emergence of new mathematical concepts, research topics, and the design of engineering applications. Therefore, the nature of the classical set theory must be well known and understood. In particular, consider the two fundamental laws of Boolean algebra the law of excluded middle and law of contradiction. In logic, the proposition every proposition is either true or false excludes any third, or middle, possibility, which gave this principle the name of the law of excluded middle. When look at the principles of Boolean algebra, there are two items as prediction: "True" or "False". Whether classical, Boolean or crisp, set theory can be defined as a characteristic function of the membership of an element x in a set A. For each elements of universal set 𝑋, the function that generates the values 0 and 1 is called the characteristic function. In some real life problems in expert system, belief system, information fusion and so on, we must consider the truth-membership as well as the falsity-membership for proper description of an object in uncertain, ambiguous environment. Neither the fuzzy sets nor the interval valued fuzzy sets is appropriate for such a situation. Intuitionistic fuzzy sets introduced by Atanassov [1] is appropriate for such a situation. The intuitionistic fuzzy sets can only handle the incomplete information considering both the truth-membership (or simply membership ) and falsity*

Corresponding Author: e-mail: mkirisci@hotmail.com, tel: (212) 440 00 00 / 11993 231

M. Kirişci, N. Şimşek / Sigma J Eng & Nat Sci 10 (2), 231-235, 2019

membership (or non-membership ) values. It does not handle the indeterminate and inconsistent information which exists in belief system. Smarandache [7, 8, 9, 10] introduced the concept of neutrosophic set which is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. Neutrosophy is a branch of philosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with di_erent ideational spectra [8]. Neutrosophic set is a powerful general formal framework which generalizes the concept of the classic set, fuzzy set [13] intervalvalued fuzzy set [11], intuitionistic fuzzy set [1], interval-valued intuitionistic fuzzy set [2], paraconsistent set, dialetheist set, paradoxist set, and tautological set [8]. In [3], a relation on neutrosophic soft sets which allows composing two neutrosophic soft sets is defined. We firstt define a relation on neutrosophic soft sets which allows composing two neutrosophic soft sets. In this paper, we will use a real datasaet [4] and make medical decision making using this data set in neutrosophic soft sets. We will use Maji's method for this.

2. Preliminaries

Let’s consider that 𝑋 is a space of points (objects), with 𝑥 ∈ 𝑋. For 𝐴 is a set in 𝑋, truthmembership, indeterminacy-membership, falsity-membership functions are denoted by 𝐺𝐴 (𝑥), 𝐵𝐴 (𝑥), 𝐻𝐴 (𝑥), respectively. Then, 𝐺𝐴 (𝑥): 𝑋 → ]0− , 1+ [,

𝐻𝐴 (𝑥): 𝑋 → ]0− , 1+ [. There is no restriction on the sum of 𝐺𝐴 (𝑥), 𝐵𝐴 (𝑥), 𝐻𝐴 (𝑥). Therefore, 0− ≤ sup 𝐺𝐴 (𝑥) + sup 𝐵𝐴 (𝑥) + sup 𝐻𝐴 (𝑥) ≤ 3+ . The set 𝐴 which consist of with 𝐺𝐴 (𝑥), 𝐵𝐴 (𝑥), 𝐻𝐴 (𝑥) in 𝑋 is called a neutrosophic sets (NS) and can be written as 𝐴 = {〈𝑥, 𝐺𝐴 (𝑥), 𝐵𝐴 (𝑥), 𝐻𝐴 (𝑥) 〉: 𝑥 ∈ 𝑋, 𝐺𝐴 (𝑥), 𝐵𝐴 (𝑥), 𝐻𝐴 (𝑥) ∈ ]0− , 1+ [ }. In the NS definition, 0− = 0 − 𝜀 and 1+ = 1 + 𝜀, where 0 and 1 are standard parts and 𝜀 is non-standard part. From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ]0− , 1+ [. So instead of ]0− , 1+ [ we need to take the interval [0,1] for technical applications, because ]0− , 1+ [ will be difficult to apply in the real applications such as in the scientific and engineering problems. Because of the NSs are difficult to apply in the realsituations, single-valued neutrosophic sets were proposed. Additionally, Ye [12] also introduced the concept of simplified neutrosophic sets (SNSs), which can be described by three real numbers in the real unit interval [0,1]. The definition of SNSs given by Ye [12] is as follows: Let’s consider that 𝑋 is a space of points (objects), with 𝑥 ∈ 𝑋. A NS 𝐴 in 𝑋 is characterized by 𝐺𝐴 (𝑥), 𝐵𝐴 (𝑥), 𝐻𝐴 (𝑥), which are subintervals/subsets in the standard interval [0,1], that is 𝐺𝐴 (𝑥): 𝑋 → [0,1], 𝐵𝐴 (𝑥): 𝑋 → [0,1],, 𝐻𝐴 (𝑥): 𝑋 → [0,1]. Then, a simplification of 𝐴 is denoted by 𝐴 = {〈𝑥, 𝐺𝐴 (𝑥), 𝐵𝐴 (𝑥), 𝐻𝐴 (𝑥) 〉: 𝑥 ∈ 𝑋} which is called an SNS. Clearly, SNSs are a subclass of NSs. Take an initial universe set 𝑁. The power set of 𝑁 and the set of parameters are denoted by 𝑃(𝑁) and 𝑀, respectively. For a non-empty set 𝐴 ⊂ 𝑀, if 𝐹: 𝐴 → 𝑃(𝑁) is a mapping then, the pair (𝐹, 𝐴) is called a soft set (SS) over 𝑁 [6]. Following definitions are given by Maji [5]: Take an initial universe set 𝑁. The all neutrosophic sets on 𝑁 and the set of parameters are denoted by 𝑃(𝑁) and 𝑀, respectively. For a non-empty set 𝐴 ⊂ 𝑀, if 𝐹: 𝐴 → 𝑃(𝑁) is a mapping then the collection (𝐹, 𝐴) is termed to be the neutrosophic soft set (NSS) over 𝑁. 232

Neutrosophic Soft Sets with Medical Decisin- … / Sigma J Eng & Nat Sci 10 (2), 231-235, 2019

The value-class of the NSS is define d as the class of all value sets of a NSS (𝐹, 𝑀). It is denoted by 𝐶(𝐹,𝑀) . Clearly, 𝐶(𝐹,𝑀) ⊂ 𝑃(𝑁). Choose two neutrosophic soft sets (𝐹, 𝐴) and (𝐽, 𝐵) over 𝑁. If 𝐴 ⊂ 𝐵, then (𝐹, 𝐴) is said to be neutrosophic soft subset of (𝐽, 𝐵). ∀𝑒 ∈ 𝐴, 𝑥 ∈ 𝑁, 𝐺𝐹(𝑒) (𝑥) ≤ 𝐺𝐽(𝑒) (𝑥),

3. Methods

In this work, the dataset which are given by Kirisci et.al.[4] used for independent variables and T2DM dependent variable are considered. As the method, the Maji’s neutrosophic soft set theory approach [5] and also the algorithm given by Maji will be used. Let the 𝑛 number of participants are 𝑝1 , 𝑝2 , ⋯ , 𝑝𝑛 and the 𝑚 numbers of choice parameters are 𝑒1 , 𝑒2 , ⋯ , 𝑒𝑚 . We also assume that corresponding to the parameter 𝑒𝑗 (𝑗 = 1,2, ⋯ , 𝑚) the rating or performance value of the participants 𝑝𝑖 (𝑖 = 1,2, ⋯ , 𝑛) is a tuple 𝑡𝑖𝑗 = {𝐺𝑝(𝑒𝑗 ) (𝑝𝑖 ), 𝐵𝑝(𝑒𝑗 ) (𝑝𝑖 ), 𝐻𝑝(𝑒𝑗 ) (𝑝𝑖 )}, such that for a fixed 𝑖 that values 𝑡𝑖𝑗 (𝑗 = 1,2, ⋯ , 𝑚) represents a NSS of all the 𝑛 objects. Thus the performance values could be arranged in the form of a matrix called the ‘criteria matrix’. If the criteria values are more, preferability of the corresponding object is also more. Our problem is to select the most suitable object i.e. the object which dominates each of the objects of the spectrum of the parameters 𝑒𝑗 . Select the participants 𝑝1 , 𝑝2 , 𝑝3 , 𝑝4 , 𝑝5 . The problem is that according to the variables studied, it is one of these participants to find out which one has T2Dm. The 𝑝1 participant will suffer from T2DM while the other two 𝑝2 and 𝑝3 participants will not suffer, as the selection is dependent on the choice parameters of each participants. We use the technique to calculate the score for the objects. Comparison Matrix: It is a matrix whose rows are labelled by the object names 𝑝1 , 𝑝2 , ⋯ , 𝑝𝑛 and columns are labelled by the parameters 𝑒1 , 𝑒2 , ⋯ , 𝑒𝑚 . The entries 𝑒𝑖𝑗 are calculated by 𝑐𝑖𝑗 = 𝑎 + 𝑏 − 𝑐, where  ′𝑎′ is the integer calculated as ‘how many times 𝐺𝑝𝑖 (𝑒𝑗 ) exceeds or equal to 𝐺𝑝𝑘 (𝑒𝑗 )’ for 𝑝𝑖 ≠ 𝑝𝑘 , ∀𝑝𝑘 ∈ 𝑈,  ′𝑏′ is the integer calculated as ‘how many times 𝐵𝑝𝑖 (𝑒𝑗 ) exceeds or equal to 𝐵𝑝𝑘 (𝑒𝑗 )’ for 𝑝𝑖 ≠ 𝑝𝑘 , ∀𝑝𝑘 ∈ 𝑈,  ′𝑐′ is the integer ‘how many times 𝐻𝑝𝑖 (𝑒𝑗 ) exceeds or equal to 𝐻𝑝𝑘 (𝑒𝑗 )’ for 𝑝𝑖 ≠ 𝑝𝑘 , ∀𝑝𝑘 ∈ 𝑈. Score of an Object: The score of an object 𝑝𝑖 is 𝜎𝑖 and is calculated as 𝜎𝑖 = ∑𝑗 𝑐𝑖𝑗 . Algorithm:       

Input the NSS (𝐾, 𝐴) Input 𝑉, the choice parameters of 𝑝𝑖 which is a subset of 𝐴 Consider the NSS (𝐾, 𝑉) and write it in tabular form Compute the comparison matrix of the NSS (𝐾, 𝑉) Compute the score 𝜎𝑖 of 𝑝𝑖 , ∀𝑖 Find 𝜎𝑘 = max𝑖 𝜎𝑖 If 𝑘 has more than one value then any one of 𝑝𝑖 could be the preferable choice. 233

M. Kirişci, N. Şimşek / Sigma J Eng & Nat Sci 10 (2), 231-235, 2019

4. Medicine Applicaton

In this section, we will use the algorithm in previous section. The set 𝑉 is contained independent variables which used in dataset of [4], where 𝑎𝑔𝑒(𝐴) , {

For example, for BMI and age, while Table 2 is obtained, it was used the table "The International Classi_cation of adult underweight, overweight and obesity according to BMI" in web site of World Health Organization in Table 1. The value-classes of other variables in the set 𝑉 have been obtained in a similar way. Table 1. Value-class of BMI & Age BMI & Age

19-24 25-34 35-44 44-54 55-64 65+ BMI<19 BMI<20 BMI<21 BMI<22 BMI<23 BMI<24 Underweight 0.1-0.3 0.1-0.3 0.1-0.3 0.1-0.3 0.1-0.3 0.1-0.3 BMI<19-24 BMI<20-25 BMI<21-26 BMI<22-27 BMI<23-28 BMI<24-29 Normal 0.4-0.6 0.4-0.6 0.4-0.6 0.4-0.6 0.4-0.6 0.4-0.6 BMI>24 BMI>25 BMI>26 BMI>27 BMI>28 BMI>29 Overweight 0.7-0.8 0.7-0.8 0.7-0.8 0.7-0.8 0.7-0.8 0.7-0.8 0.9 0.9 0.9 0.9 0.9 0.9 Obese Follow the steps below according to the algorithm.    

Generate the NSS (𝐾, 𝑉) (Table 2) Write the comparison matrix table form (Table 3) Compute the score for each 𝑝𝑖 (Table 4) Decide according to the maximum score in Table 4. Table 2. The NSS (K,V) A (0.7,0.2,0.7) (0.5,0.1,0.5) (0.7,0.2,0.4) (0.6,0.4,0.6) (0.7,0.6,0.6)

WCR (0.6,0.2,0.5) (0.5,0.3,0.6) (0.8,0.6,0.2) (0.7,0.7,0.6) (0.5,0.6,0.7)

NH (0.7,0.4,0.3) (0.6,0.5,0.5) (0.4,0.6,0.7) (0.8,0.5,0.8) (0.6,0.7,0.8)

Act (0.8,0.3,0.6) (0.5,0.8,0.3) (0.7,0.3,0.2) (0.7,0.2,0.5) (0.7,0.6,04)

G (0.4,0.6,0.7) (0.8,0.2,0.7) (0.6,0.1,0.5) (0.8,0.2,0.5) (0.7,0.2,07)

BMI (0.6,0.3,0.8) (0.6,0.3,0.7) (0.3,0.5,0.6) (0.8,0.3,0.6) (0.5,0.6,0.8)

Neutrosophic Soft Sets with Medical Decisin- … / Sigma J Eng & Nat Sci 10 (2), 231-235, 2019

Decision: It seen that in Table 4, the maximum score is 21. This score is measured in the third patient. When the available data are calculated with the neutrosophic soft set, the 𝑝3 appears to be the patient most suffering from T2DM. After 𝑝3 , 𝑝4 , 𝑝5 , 𝑝1 , 𝑝2 are suffering from the disease, respectively.

5. Conclusion

In this paper, we applied the notion of NSS in Maji's approach for medical diagnosis. Maji's aprroach is based on NSS. Maji use this concept in soft sets considering the fact that the parameters (which are words and sentences) are mostly neutrosophic set. A case study based on real data has been taken to demonstrate the simplicity of the Maji's approach. Future work in this regard would be required to study whether the notions put forward in this paper yield a fruitful result.

Share and Cite