A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems

The concept of a single valued neutrosophic number (SVN-number) is of importance for quantifying an ill-known quantity and the ranking of SVN-numbers is a very difficult problem in multi-attribute decision making problems. The aim of this paper is to present a methodology for solving multi-attribute decision making problems with SVN-numbers. Therefore, we firstly defined the concepts of cut sets of SVN-numbers and then applied to single valued trapezoidal neutrosophic numbers (SVTN-numbers) and triangular neutrosophic numbers (SVTrN-numbers). Then, we proposed the values and ambiguities of the truth-membership function, indeterminacy-membership function and falsity-membership function for a SVN-numbers and studied some desired properties. Also, we developed a ranking method by using the concept of values and ambiguities, and applied to multi-attribute decision making problems in which the ratings of alternatives on attributes are expressed with SVTN-numbers.

Multi-attribute decision making (MADM) which is an important part of decision science is to find an optimal alternative, which are characterized in terms of multiple attributes, from alternative sets.In some practical applications, the decision makers may be not able to evaluate exactly the values of the MADM problems due to uncertain and asymmetric information between decision makers.As a result, values of the MADM problems are not measured by accurate numbers.It is feasible for some sets which contain uncertainty such as; a fuzzy set, intuitionistic set and neutrosophic set to represent an uncertainty of values of the MADM problems.Intuitionistic fuzzy numbers, intuitionistic triangular fuzzy numbers and intuitionistic trapezoidal fuzzy numbers is introduced by Mahapatra and Roy [27], Liang [26] and Jianqiang [20], respectively.Li [23] gave a ranking method of intuitionistic fuzzy numbers and application to multiattribute decision-making problems in which the attribute ratings are expressed with intuitionistic fuzzy numbers in management.Therefore, he defined the notation of cut sets of intuitionistic fuzzy numbers and their values and ambiguities of membership and nonmembership functions.Also, the notions of intuitionistic fuzzy numbers were studied in [4,6,10,12,17,18,28,35,39,40,49] and applied to multi-attribute decision making problems in [6,24,37,38,40,43,46,48].
Ye [45] presented the notations of simplified neutrosophic sets and gave the operational laws of simplified neutrosophic sets.Then, he introduced some aggregation operators are called simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator.Also, he developed ranking method by using cosine similarity measure between an alternative and the ideal alternative.Peng et al. [30] introduced the concept of multi-valued neutrosophic set with the operations.Then, they gave two multi-valued neutrosophic power aggregation operators and applied to multi-criteria group decision-making problems.A novel concept of expected values of fuzzy variables, which is essentially a type of Choquet integral and coincides with that of random variables is introduced by Liu et al. [22].Li [25] developed a new methodology for ranking TIFNs and cut sets of intuitionistic trapezoidal fuzzy numbers as well as arithmetical operations are developed.Then, he gave the values and ambiguities of the membership function and the non-membership function for a intuitionistic trapezoidal fuzzy number and a new ranking method with applications.Kumar and Kaur [21] and Nehi [29] presented a new ranking approach by modifying an existing ranking approach for comparing intuitionistic fuzzy numbers.Zhang [51] proposed a methodology for intuitionistic trapezoidal fuzzy multiple and give a numerical example by using similarity measure.Zeng et al. [52] and De and Das [11] developed a method for ranking trapezoidal intuitionistic fuzzy numbers and gave cut sets over intuitionistic triangular fuzzy numbers.Then, they presented the values and ambiguities of the membership degree and the nonmembership degree for intuitionistic triangular fuzzy numbers and developed a method.Ye [44] proposed the expected values for intuitionistic trapezoidal fuzzy numbers and presented a handling method for intuitionistic trapezoidal fuzzy multicriteria decision-making problems, in which the preference values of an alternative on criteria and the weight values of criteria take the form of intuitionistic trapezoidal fuzzy numbers.
In a multiple-attribute decision-making problem the decision makers need to rank the given alternatives and the ranking of alternatives with neutrosophic numbers is many difficult because neutrosophic numbers are not ranked by ordinary methods as real numbers.However it is possible with score functions [16], aggregation operators [16], distance measures [13], and so on.Therefore; in this study we extend the ranking method as well as applications of fuzzy numbers by given [53] and intuitionistic fuzzy numbers by given [23,25,28] to SVN-numbers for solving MAGDM problems in which the ratings of alternatives with respect to each attribute are represented by SVN-numbers.To do so, the rest of this paper is organized as: In the next section, we will present some basic definitions and operations of SVN-numbers.In Sect.3, we introduce the concepts of cut sets of N-numbers and applied to single valued trapezoidal neutrosophic numbers (SVTN-numbers) and triangular neutrosophic numbers (SVTrN-numbers).Meanwhile, we also describe the values and ambiguities of the truthmembership function, indeterminacy-membership function and falsity-membership function for a SVN-numbers and studied some desired properties.In Sect.4, we develop a novel ranking method is called SVTrN-multi-attribute decision-making method.Afterwards, we present a decision algorithm and applied to multi-attribute decision making problems in which the ratings of alternatives on attributes are expressed with SVTrN-numbers.In Sect.5, the method is compared with the other methods that were outlined in Refs.[13], [16] and [47] using SVTrN-numbers.In last section, a short conclusion are given.The present expository paper is a condensation of part of the dissertation [13].

Preliminary
In this section, we recall some basic notions of fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, single valued neutrosophic sets and single valued neutrosophic numbers.
From now on we use I n ¼ f1; 2; . ..; ng and I m ¼ f1; 2; . ..; mg as an index set for n 2 N and m 2 N, respectively.
Definition 2.1 [3] Let E be a universe.An intuitionistic fuzzy set K over E is defined by where l K : E ! ½0; 1 and c K : E ! ½0; 1 such that 0 l K ðxÞ þ c K ðxÞ 1 for any x 2 E. For each x 2 E, the values l K ðxÞ and c K ðxÞ are the degree of membership and degree of non-membership of x, respectively.Definition 2.2 [32] Let E be a universe.A neutrosophic sets A over E is defined by where T A ðxÞ, I A ðxÞ and F A ðxÞ are called truth-membership function, indeterminacy-membership function and falsitymembership function, respectively.They are respectively defined by Definition 2.3 [36] Let E be a universe.An single valued neutrosophic set (SVN-set) over E is a neutrosophic set over E, but the truth-membership function, indeterminacymembership function and falsity-membership function are respectively defined by Definition 2.4 [13] Let w ã; u ã; y ã 2 ½0; 1 be any real numbers, a i ; b i ; c i ; d i 2 R and a i b i c i d i (i = 1, 2, 3).Then a single valued neutrosophic number (SVN-number) ã ¼hðða 1 ; b 1 ; c 1 ; d 1 Þ; w ãÞ; ðða 2 ; b 2 ; c 2 ; d 2 Þ; u ãÞ; where the functions f l l : ½a 1 ; b 1 !½0; w ã, f r m : ½c 2 ; d 2 !½u ã; 1 f r k : ½c 3 ; d 3 !½y ã; 1 are continuous and nondecreasing, and satisfy the conditions: 1 are continuous and nonincreasing, and satisfy the conditions: d 1 are called the mean interval and the lower and upper limits of the general neutrosophic number ã for the truth-membership function, respectively.½b 2 ; c 2 ; a 2 and d 2 are called the mean interval and the lower and upper limits of the general neutrosophic number ã for the indeterminacymembership function, respectively.½b 3 ; c 3 ; a 3 and d 3 are called the mean interval and the lower and upper limits of the general neutrosophic number ã for the falsity-membership function, respectively.w ã, u ã and y ã are called the maximum truthmembership degree, minimum indeterminacy-membership degree and minimum falsity-membership degree, respectively.
y bi be two SVTN-numbers and c 6 ¼ 0 be any real number.Then, y bi be two SVTrN-numbers and c 6 ¼ 0 be any real number.Then,

Concepts of values and ambiguities for SVNnumbers
In this section, we first define the concept of cut (or level) sets, values, ambiguities, weighted values and weighted ambiguities of SVN-numbers and give some desired properties.Also we developed a ranking method of SVNnumbers.In the following, some definitions and operations on intuitionistic sets defined in [23-25, 37, 48], we extend these definitions and operations to single valued neutrosophic sets [36].
Clearly, any a-cut set of a SVN-number ã is a crisp subset of the real number set R.
In here, any a-cut set of a SVN-number ã for truthmembership function is a closed interval, denoted by ãa ¼ ½L ãðaÞ; R ãðaÞ: Clearly, any c-cut set of a SVN-number ã is a crisp subset of the real number set R.
Proof It is easily derived from Eq. ( 1) that Noticing that the assumption condition: w ã ¼ w b, we have Combining with the assumption condition: a 1 [ c 2 , we can prove that V l ðãÞ [ V l ð bÞ.
It easily follows from Eq. ( 2) that Due to the assumption condition: Combining with the assumption condition: Likewise, it easily follows from Eq. ( 3) that Due to the assumption condition: y ã ¼ y b, it directly follows that Z 1 Combining with the assumption condition: a 1 [ c 2 , we have: According to Definition 3.10, for any h 2 ½0; 1, we have Proof According to Eq. ( 1), we have where w c is the truth-membership of the SVN-number c.
Noticing that the assumption conditions: ã [ b and w ã ¼ w b, we have Likewise, it is derived from Eq. ( 2) that where u c is the indeterminacy-membership of the SVNnumber c. Noticing that the assumption conditions: ã [ b and u ã ¼ u b, we have Z 1 Hereby, we have Therefore, we have Likewise, it is derived from Eq. ( 3) that where y c is the falsity-membership of the SVN-number c.
Noticing that the assumption conditions: ã [ b and y ã ¼ y b, we have Hereby, we have According to Definition 3.10, and combining with Eqs. ( 4), ( 5) and ( 6) the following inequality is always valid for any h 2 ½0; 1 : Therefore, it is easy to see from the case (1) of Definition 3.

A multi-attribute decision-making method with SVTrN-numbers
In this section, we define a multi-attribute decision making method, so called SVTrN-multi-attribute decision-making method.Its adopted from [23][24][25].
From now on we use the weight of each attribute u i ði 2 I m Þ is x i , which should satisfy the normalized conditions: x i 2 ½0; 1 ði 2 I m Þ and P m i¼1 x i ¼ 1. Definition 4.1 Let X ¼ ðx 1 ; x 2 ; . ..; x n Þ be a set of alternatives, U ¼ ðu 1 ; u 2 ; . ..; u m Þ be the set of attributes and ½ Ãij ¼ hða ij ; b ij ; c ij Þ; w ãij ; u ãij ; y ãij i(for i 2 I m ; j 2 I n ) be a SVTrN-numbers.Then, s called an SVTrN-multi-attribute decision making matrix of the decision maker.Now, we can give an algorithm of the SVTrN multiattribute decision-making method as follows; Algorithm: Step 1. Construct the decision-making matrix A ¼ ð Ãij Þ mÂn ; for decision; Step 2. Compute the normalized decision-making matrix Step 5. Determine the nonincreasing order of Sj ðj 2 I n Þ Step 6. Rank the alternatives x j according to Sj ðj 2 I n Þ and select the best alternative.
Since humans might feel more comfortable using words by means of linguistic labels or terms to articulate their preferences, the ratings of each alternative with respect to each attribute are given as linguistic variables characterized by SVTrN-numbers in the evaluation process.(seeTable 1, also the values can be replace by experts).
Example 4.2 (Its adopted from [23,24]) Suppose that a software company desires to hire a system analyst.After preliminary screening, three candidates (i.e., alternatives) x 1 ; x 2 and x 3 remain for further evaluation.The panel (or decision making committee) assesses the three candidates according to the five attributes (or criteria, factors), which are emotional steadiness u 1 ; oral communication skill u 2 , personality u 3 ; past experience u 4 and self-confidence u 5 ; respectively.Also, the weight vector of the five attributes may be given as x ¼ ð0:15; 0:25; 0:20; 0:25; 0:15Þ T : The four possible candidates (or alternatives) are to be evaluated under the above five attributes by corresponding to linguistic values of SVTrN-numbers for linguistic terms (adapted from [13]), as shown in Table 1.
Step 5. Determine the nonincreasing order of Sj ðj 2 I n Þ The values of the S1 ; S2 and S3 respectively, as follows: Then we have the weighted values of the SVTrN-numbers S1 , S2 and S3 respectively, as follows: It is depicted as in Fig. 1 It is easy to see from Fig. 1 that the weighted values of the SVTrN-numbers S1 and S2 are identical if h ¼ 0:268 Also, we have the weighted ambiguities of S1 and S2 can be calculated, respectively, as follows: Therefore, the ranking order of S1 and S2 is S1 [ S2 .
Further, it is easy to see from Fig. 1 that for any given weight h 2 ½0; 0:268Þ, we have Step 6. Rank the alternatives x j according to Sj ðj 2 I n Þ as; and the best candidate is x 3 : However, for any given h ¼ ð0:268; 1; we have which infers that the ranking order of the three candidates is x 3 1 x 2 1 x 1 and the best candidate is x 3 :

Comparison analysis and discussion
In order to show the feasibility of the introduced method, a comparative study with other methods was conducted.The proposed method is compared to the methods that were outlined in Refs.[13,16] and [47] using SVTrNnumbers.With regard to the method in Ref. [13], the score values were firstly found and used to determine the final ranking order of all the alternatives, and then arithmetic and geometric aggregation operators were developed in order to aggregate the SVTrN-numbers which is used in Method 1 and Method 2, respectively.Also, a SV-trapezoidal neutrosophic number is a SVTrNnumbers (In Definition, 2.8 if b 1 ¼ c 1 then the SVtrapezoidal neutrosophic number is a SVTrN-number).Therefore, the method in Ref. [16] and [47] were used to determine the final ranking order of all the alternatives Method 3 and Method 4, respectively.The results from the different methods used to resolve the MCDM problem in Example 4.2 are shown in Table 2.
From the results presented in Table 2, the best alternatives is x 3 and the worst one is x 1 in all methods.Firstly, method 1-4 use distance measure, score function, and aggregation operator and it is very difficult for decision makers to confirm their judgments when using operators and measures that have similar characteristics.Secondly, the proposed method in this paper pays more attention to the impact that uncertainty has on the alternatives and also takes into h-weighted value of the SVN-numbers by using the concepts of cut sets of SVN-numbers.By comparison, the proposed method in this paper focuses on the hweighted value of the SVN-numbers, the ranking of the proposed method is the same as that of the other results.Therefore, the proposed method is effective and feasible.
Table 2 The results of the methods

Methods
The final ranking The best alternative(s) The worst alternative(s) The proposed method Fig. 1 The weighted values of the S1 , S2 and S3 The ranking method which developed by Li [23,24] for intuitionistic numbers plays an important role in solving multipleattribute decision-making problems and successfully applied in many fields.Therefore, developing a method, by using the methods by given Li [23,24], in order to solve SVN-numbers is seen as a valuable research topic.This paper gives two characteristics of a SVN-number is called the value and ambiguity.Then, a ratio ranking method is developed for the ordering of SVN-numbers and applied to solve multi-attribute decision making problem with SVN-numbers.It is easily seen that the proposed ratio ranking method can be extended to rank more general SVN-numbers in a straightforward manner.Due to the fact that a SVN-numbers is a generalization of a fuzzy number and intuitionistic fuzzy number, the other existing methods of ranking fuzzy numbers and intuitionistic fuzzy number may be extended to SVN-numbers.More effective ranking methods of SVN-numbers will be investigated in the near future and applied this concepts to game theory, algebraic structure, optimization and so on.