In this paper, we introduce a distance measure on single-valued neutrosophic sets by sine function which is a generalization of intuitionistic fuzzy sine distance measure. The axiom of metric on single-valued neutrosophic sets is verified and shows that the difference of distance measure from unity is a similarity measure. A new methodology for multi-attribute decision-making problems (MADM) is developed for the most common decision by the smallest measure value of the proposed single-valued neutrosophic distance measure. We further apply this distance measure to a multi-attribute decision-making problem (MADM) for student career determination in a neutrosophic environment to find the best career for suitable students. Finally, the comparison is made between the proposed distance measure and the other distance measures for the final decision chosen from the most common decisions of them.
Hinweise
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Introduction
In this world, we are facing many uncertain concepts in our real-life situations and such situations cannot be handled by the classical set theory. In this connection, Zadeh [1] developed the concept of fuzzy set theory to analyze imprecise mathematical information which is characterized by the membership degree of its element to a real number in the standard interval [0, 1]. This theory is intensely applied in many fields such as artificial intelligence, control systems, decision analysis, medical diagnosis, operations research and robotics, pattern recognition problems, etc. By the marvelous growth of fuzziness, the similarity measure in fuzzy sets is a measure that plays an important key role in finding the relations and the similarities of two different fuzzy sets. In this growth, Pappis and Karacapilidis [2] defined some similarity measures in fuzzy sets with their applications. Pramanik and Mondal [3] introduced weighted fuzzy tangent similarity measures and framed a medical application using these fuzzy tangent similarity measures in this fuzzy world. In the next milestone of Zadeh’s fuzzy world, Voxman [4] insisted on some distances for the fuzzy modeling to handle real-life problems by defining fuzzy numbers. As a generalization of the fuzzy set, an intuitionistic fuzzy set is drawn by Atanassov [5] for a better depiction of uncertainty by considering both membership and non-membership degree at the same time in the standard interval [0, 1], such that their summation should be less than 1. Atanassov’s intuitionistic fuzzy set theory deviated into many application areas in the uncertainty situation by taking both membership degrees. Recently, Dutta and Goala [6] introduced intuitionistic fuzzy distance measure and gave its application in medical diagnosis. Szmidt and Kacprzyk [7] also introduced distance measures on intuitionistic fuzzy sets and they [8] applied further these in medical diagnosis as a real-life application of mankind.
The picture fuzzy set is an extension of the intuitionistic fuzzy set to deal with the uncertainty concept which is developed by Cuong [9]. The hierarchical picture fuzzy clustering and generalized picture distance measure are proposed by Son [10]. Dutta [11] established an application in medical diagnosis based on distance measures between picture fuzzy sets. Si et al. [12] developed an approach to rank picture fuzzy numbers for decision-making problems. Tolga et al. [13] introduced finite-interval-valued type-2 Gaussian fuzzy numbers applied in a healthcare problem by fuzzy TODIM. Amit K. Shukla et al. [14] explained type-2 intuitionistic fuzzy TODIM. Muhammet Deveci et al. [15] discussed advantage prioritization of digital carbon footprint awareness using fuzzy Aczel Alsina-based decision-making. Ilgin Gokasar et al. [16] further introduced Metaverse integration alternatives of connected autonomous vehicles with self-powered sensors. Xiao-hui Wu and Lin Yang [17] defined the hesitant picture fuzzy linguistic prospects theory for digital transformation solution. Tan et al. [18] combined the intuitionistic fuzzy sets with rough set to attribute subset selection. De et al. [19] formulated multiple attribute decision making problems on probabilistic interval-valued intuitionistic hesitant fuzzy set and on the TOPSIS method. Rekha Sahu et al. [20] presented a picture fuzzy set-based approach that used the hybridized distance measure for choosing a suitable and appropriate career for a student, where the rough set is used to avoid any kind of confusion in choosing a stream. Ejegwa et al. [21] gave a new method for career determination by introducing normalized Euclidean distance measures in Atanassov’s intuitionistic fuzzy sets. Tugrul et al. [22] proposed an application of school career selection in intuitionistic fuzzy sets using normalized Euclidean distance measures. Here the solution is determined by measuring the smallest distance between each student and each school. Citil [23] interpreted the relationship between the student’s pilot tests and the official test in Turkey Kahramanmara’s city school. He assumed the solution by measuring the shortest distance between each student and each school via distance measure.
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As a generalization of both fuzzy sets and intuitionistic fuzzy sets, Smarandache [24, 25] initiated the concept of neutrosophic set theory as a mathematical tool to deals with problems containing inconsistent, indeterminate, and incomplete information in the real world. These sets elements are characterized by three independent components such as truth-membership, indeterminacy-membership, and falsity membership whose values are philosophically taken within the real standard or non-standard unit interval\({]}^{-}0, {1}^{+}[\). In real-life applications, scientific or engineering problems face difficulties in using the neutrosophic set whose values are from real standard or non-standard interval\({]}^{-}0, {1}^{+}[\), but it needs the specified neutrosophic set operators. For applying the neutrosophic set more conveniently in a real-life situation, Wang et al. [26] defined single-valued neutrosophic sets (shortly SVNSs) in which the components are single-valued numbers and which take the value from the unit interval [0, 1]. Thus a single-valued neutrosophic set is a special case of the neutrosophic set. The different properties of neutrosophic sets and single-valued neutrosophic sets are extensively studied and applied to various fields. Majumdar and Samanta [27] defined Hamming and Euclidean distance measures on single-valued neutrosophic sets. Biswas et al. [28] discussed the cosine similarity measure in multi-attribute decision-making problems. Pramanik and Mondal [29] introduced the cotangent similarity measure and the tangent similarity measure was introduced by Mondal and Pramanik [30] with its application. Biswas et al. [31] again defined a variety of distance measures for single-valued neutrosophic sets and compared their method with other existing methods for solving multi-attribute decision-making problems. Shahzadi et al. [32] established an application of single-valued neutrosophic sets in medical diagnosis. Ye and Zhang [33] applied the single-valued neutrosophic similarity measures and single-valued neutrosophic cross-entropy [34] in multi-criterion decision-making problems. Jayaparthasarathy et al. [35] introduced neutrosophic supra topology with the application in the data mining process. Recently Chai et al. [36] defined new distance and similarity measures for single-valued neutrosophic sets with applications in pattern recognition and medical diagnosis problems. Broumi and Smarandache [37] established the properties of distance and similarity measures on single-valued neutrosophic sets. Karaaslan and Khizar [38] developed a multi-criteria group decision-making method based on some operations on neutrosophic matrices. Karaaslan [39] also developed a medical diagnosis decision-making algorithm by defining Gaussian numbers and α-cut in single-valued neutrosophic numbers. Karaaslan [40] presented a multi-criteria decision-making problem by introducing the correlation coefficient measures between the neutrosophic sets, the interval-neutrosophic sets, and the neutrosophic refined sets. Gulistan et al. [41] introduced a multiple attribute group decision-making method by introducing the Heronian mean operator, geometric Heronian mean operator, neutrosophic cubic number–improved generalized weighted Heronian mean operator, neutrosophic cubic number–improved generalized weighted geometric Heronian mean operator. Karaaslan and Hunu [42] defined type-2 single-valued neutrosophic sets and developed a multi-criteria group decision-making method with an illustrative example based on the TOPSIS approach. Jana et al. [43] introduced a multi-attribute decision-making method with a numerical example by introducing the interval trapezoidal neutrosophic set, some operations, the score, and the accuracy functions of interval trapezoidal neutrosophic numbers. Jana and Karaaslan [44] further defined Dice, Jaccard, weighted Dice, and weighted Jaccard similarity measures between two trapezoidal neutrosophic fuzzy numbers and established a multi-criteria decision-making method under a trapezoidal neutrosophic fuzzy environment developed. Arockia Dasan et al. [45] developed a method to solve decision-making problems in plant hybridization by using score functions on single-valued neutrosophic sets. Smarandache and Muhammad Aslam [46] introduced some advances to neutrosophic probability and statistical applications in bioinformatics as well as in other fields. On this occasion, Amna Riaz et al. [47] systematically reviewed both the neutrosophic statistics and the medical data. Arockia Dasan et al. [48] also introduced one new neutrosophic probabilistic distance measure to discuss multi-attribute decision-making problems.
The motivation of this paper is as follows: On neutrosophic sets and single-valued neutrosophic sets, there are many distance measures and similarity measures [18‐30, 36, 37, 47, 48] introduced and applied in medical diagnosis, data analysis, and pattern recognition to deal with multi-decision-making problems. The cosine, cotangent and tangent [28‐30] functions are used only in neutrosophic similarity measures, but not used the sine function. In addition, the obtained results by these measures cannot be compared to another whenever data are in the same neutrosophic form. That is, previously defined distance measures are not used sine functions on a neutrosophic environment, and none of these methods demonstrated the course selection problem as a part of it. Due to these research gaps, this paper has motivated us to define a novel distance measure on single-valued neutrosophic sets in terms of sine functions without using previous trigonometric functions and without their identities. A method is formulated to solve multi-attribute decision-making problems (MADM) of the course selection for the student’s career determination and to find the most common decision of MADM problems by comparing the decision of sine metric single-valued neutrosophic distance measure with existing single-valued neutrosophic distance measures [27, 36].
The main contributions of the present study, which are originally different from previous distance measures, numerical approach, are as follows: A novel sine distance measure on single-valued neutrosophic sets is introduced and their detailed proofs are discussed with the properties of similarity measure also. This distance measure is the first distance measure using the sine function without trigonometric identities. Then a new methodological approach is formulated using the proposed sine distance measure. A numerical problem is constructed to find the decision of multi-attribute decision-making problems (MADM) for the student’s course selection of their career. By comparing the decision, the decision of the proposed methodology of the proposed sine distance measure coincides with most of the decisions of other existing single-valued neutrosophic distance measures, which leads to the conclusion of the most common decision produced by our method.
The organization of this paper starts from the preliminary section, which presents some basic preliminaries about fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, and some distance measures. The next section introduces a new distance measure on single-valued neutrosophic sets and shows the proposed distance measure satisfies the metric axioms. In the next section, a new methodology for multi-attribute decision-making problems is developed by using the proposed single-valued neutrosophic distance measure. Then the continuing section introduces an illustrative example as a real-life application to find the student’s suitable career in a neutrosophic environment. The result and discussion of the work are presented in the previous section of stimulation and comparison study. Some advantages and limitations of the present work are respectively offered in the next two sections. The previous section of the final states some assumptions of the proposed method. The final section states the conclusion and the future work of the present work.
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Preliminaries
In this section, we review some basic definitions and properties of the neutrosophic sets, single-valued neutrosophic sets, and neutrosophic distance measures.
Definition 1
[1] Let \(X\) be a non-empty set and a fuzzy set \(A\) on \(X\) is of the form\(A=\{\left(x, {\mu }_{A}\left(x\right)\right) : x\in X\}\), where \(0\le {\mu }_{A}\left(x\right)\le 1\) represents the degree of membership function of each \(x\in X\) to the set\(A\).
Definition 2
[5] Let \(X\) be a non-empty set. An intuitionistic fuzzy set \(A\) is of the form \(A=\left\{\left(x, {\mu }_{A}\left(x\right), {\gamma }_{A}\left(x\right)\right):x\in X\right\},\) where \({\mu }_{A}\left(x\right)\) and \({\gamma }_{A}\left(x\right)\) represents the degree of membership and non-membership function of each \(x\in X\) to the set \(A\) and \(0\le {\mu }_{A}\left(x\right)+ {\gamma }_{A}\left(x\right)\le 1\) for all \(x\in X\), respectively. The set of all intuitionistic fuzzy sets of \(X\) is denoted by \(IFS(X)\).
Definition 3
[24] Let \(X\) be a non-empty set. A neutrosophic set \(A\) having the form\(A=\{(x, {\mu }_{A}\left(x\right), {\sigma }_{A}\left(x\right), {\gamma }_{A}\left(x\right)) :x\in X\}\), where \({\mu }_{A}\left(x\right), {\sigma }_{A}\left(x\right)\) and \({\gamma }_{A}\left(x\right)\in {]}^{-}0, {1}^{+}[\) represent the degree of membership (namely\({\mu }_{A}\left(x\right))\), the degree of indeterminacy (namely\({\sigma }_{A}\left(x\right)\)) and the degree of non membership (namely\({\gamma }_{A}\left(x\right)\)) of each \(x\in X\) to the set A such that \({\mu }_{A}\left(x\right)+{\sigma }_{A}\left(x\right)+{\gamma }_{A}\left(x\right)\in {]}^{-}0, {3}^{+}[,\) for all\(x\in X\), respectively. For \(X, NS\left(X\right)\) denotes the collection of all neutrosophic sets of\(X\).
Definition 4
[24] The following statements are true for the neutrosophic sets \(A\) and \(B\) on \(X\):
(i)
\({\mu }_{A}\left(x\right)\le {\mu }_{B}\left(x\right)\)i., \({\sigma }_{A}\left(x\right)\le {\sigma }_{B}\left(x\right)\) and \({\gamma }_{A}\left(x\right)\ge {\gamma }_{B}\left(x\right)\) for all \(x\in X\) if and only if \(A\subseteq B\).
(ii)
\(A\subseteq B\)ii. and \(B\subseteq A\) if and only if \(A=B\).
[26] A single-valued neutrosophic set (SVNS)\(A\) in \(X\) is a neutrosophic set which is of the form\(A=\{(x, {\mu }_{A}\left(x\right), {\sigma }_{A}\left(x\right), {\gamma }_{A}\left(x\right)) :x\in X\}\), that is characterized by the degree of membership (namely\({\mu }_{A}\left(x\right))\), the degree of indeterminacy (namely\({\sigma }_{A}\left(x\right)\)) and the degree of non membership (namely\({\gamma }_{A}\left(x\right)\)), where \({\mu }_{A}\left(x\right), {\sigma }_{A}\left(x\right), {\gamma }_{A}\left(x\right)\in [0, 1]\) such that such that\(0\le {\mu }_{A}\left(x\right)+{\sigma }_{A}\left(x\right)+{\gamma }_{A}\left(x\right)\le 3\), for all\(x\in X\), respectively. For \(X, SVNS\left(X\right)\) denotes the collection of all single valued neutrosophic sets of\(X\).
Two important concepts for SVNSs are the distance measure and similarity measure, which are applying to compare the neutrosophic fuzzy information.
Definition 6
[27] Let \(X=\{{x}_{1}, {x}_{2}, . . .,{x}_{n}\}\) be a discrete confined set. A mapping \(d:NS\left(X\right)\times NS\left(X\right)\to [\mathrm{0,1}]\) is said to be a distance measure between two neutrosophic sets if it satisfies the following axioms:
(i)
\(d\left(A, B\right)\ge 0\)(i) for all \(A, B\in NS(X)\).
(ii)
\(d\left(A, B\right)=0\)(ii) if and only if \(A=B\) for all \(A, B\in NS(X)\).
(iii)
\(d\left(A, B\right)=d\left(B, A\right)\)(iii) for all \(A, B\in NS(X)\).
(iv)
If \(A\subseteq B\subseteq C\) for all \(A, B, C\in NS(X),\) then \(d\left(A, C\right)\ge d\left(A, B\right)\) and \(d\left(A, C\right)\ge d(B, C)\).
If the mapping is defined as \(d(A,B)={\text{max}}\{|{\mu }_{A}({x}_{i})-{\mu }_{B}({x}_{i})|, |{\sigma }_{A}({x}_{i})-{\sigma }_{B}({x}_{i})|, |{\gamma }_{A}({x}_{i})-{\gamma }_{B}({x}_{i})|\}, {\forall x}_{i}\in X,\) then \(d\left(A,B\right)\) satisfies axioms of distance measure and is called the extended Hausdorff distance measure between two neutrosophic sets \(A\) and \(B.\)
Definition 7
[27] The normalized Hamming distance between two single-valued neutrosophic sets \(A\) and \(B\) is defined as.
[37] A mapping \(S:NS\left(X\right)\times NS\left(X\right)\to [\mathrm{0,1}]\) is said to be a similarity measure between two neutrosophic sets if it satisfies the properties of axioms:
(i)
\(S\left(A, B\right)\ge 0\) for all \(A, B\in NS(X)\).
(ii)
\(S\left(A, B\right)=1\) if and only if \(A=B\) for all \(A, B\in NS(X)\).
(iii)
\(S\left(A, B\right)=S\left(B, A\right)\) for all \(A, B\in NS(X)\).
(iv)
If \(A\subseteq B\subseteq C\) for all \(A, B, C\in NS(X),\) then \(S\left(A, C\right)\le S\left(A, B\right)\) and S \(\left(A, C\right)\le S(B, C)\).
Distance measure on single-valued neutrosophic sets
This section defines a distance measure on single-valued neutrosophic sets which is a generalization of intuitionistic fuzzy distance measure [6] and some of its properties are derived.
Definition 11
Let \(X=\left\{{x}_{1}, {x}_{2}, . . . ,{x}_{n}\right\}\) be a universal set. Let \(A=\{({x}_{i}, {\mu }_{A}\left({x}_{i}\right), {\sigma }_{A}\left({x}_{i}\right),\)
\({\gamma }_{A}\left({x}_{i}\right)) : {x}_{i}\in X\}\) and \(B=\left\{\left({x}_{i}, {\mu }_{B}\left({x}_{i}\right), {\sigma }_{B}\left({x}_{i}\right), {\gamma }_{B}\left({x}_{i}\right)\right): {x}_{i}\in X\right\}\) be two single valued neutrosophic sets on\(X\). Then define a mapping \(d:SVNS\left(X\right)\times SVNS\left(X\right)\to [\mathrm{0,1}]\) as:
Therefore, \(d\left(A, C\right)\ge d(A, B)\) is proved. Similarly, we can prove \(d\left(A, C\right)\ge d(B, C)\) and hence the theorem is proved.
Hence we can observe that the proposed mapping \(d\left(A, B\right)\) satisfies the distance measure axioms [27] and so it is a distance measure on single-valued neutrosophic sets.
Theorem 2
\(d\left(A, C\right)\le d\left(A, B\right)+d\left(B, C\right)\) is true for \(A, B, C\in SVNS\left(X\right)\).
Proof
Let \(A, B, C\in SVNS\left(X\right)\) then for all \({x}_{i}\in X ,\) the following inequalities are true for the real numbers
Therefore,
\(d\left(A, C\right)\le d\left(A,
B\right)+d\left(B, C\right)\)
is true for \(A, B, C\in
SVNS\left(X\right)\).
Hence from the above theorem, the distance measure \(d\left(A, B\right)\) satisfies all the metric axioms [49] for single-valued neutrosophic sets and we call it a sine metric single-valued neutrosophic distance measure (Shortly SMSVNDM).
Theorem 3
\(S\left(A, B\right)=1-d(A,B)\) is a similarity measure for every \(A, B\in SVNS\left(X\right)\).
Proof
We can easily verify definition 10 of similarity measure’s axioms [37].
Methodology
In this section, we propose a methodological approach to neutrosophic multi-attribute decision-making problems by using a sine metric single-valued neutrosophic distance measure. The following steps are the necessary steps for the proposed methodological approach to select the proper attributes and alternatives.
Step 1. Problem field Selection
Consider the multi-attribute decision-making problem with \(l\) conditioned attributes,\({R}_{1}, {R}_{2}, . . ., {R}_{l}\), \(n\) decision attributes \(\left\{{C}_{1}, {C}_{2}, . . ., {C}_{n}\right\}\) and the \(m\) alternatives are \({S}_{1}, {S}_{2}, . . ., {S}_{m}\) such that \(n\le m\) (Tables 1, 2).
Table 1
Conditioned attributes vs alternatives
\({S}_{1}\)
\({S}_{2}\)
\({S}_{m}\)
\({R}_{1}\)
\(({r}_{11})\)
\(({r}_{12})\)
\(({r}_{1m})\)
\({R}_{2}\)
\(({r}_{21})\)
\(({r}_{22})\)
\(({r}_{2m})\)
\({R}_{l}\)
\(({r}_{l1})\)
\(({r}_{l2})\)
\(({r}_{lm})\)
Table 2
Alternatives Vs Decision attributes
\({R}_{1}\)
\({R}_{2}\)
\({R}_{l}\)
\({C}_{1}\)
\(({c}_{11})\)
\(({c}_{12})\)
\(({c}_{1l})\)
\({C}_{2}\)
\(({c}_{21})\)
\(({c}_{22})\)
\(({c}_{2l})\)
\({C}_{n}\)
\(({c}_{n1})\)
\(({c}_{n2})\)
\(({c}_{nl})\)
Here all the attributes \({r}_{pj}\) and \({c}_{kp}(p=1, 2, \dots ,l; j=1, 2,\dots ,m;k=1, 2, \dots ,n)\) are all single valued neutrosophic sets.
Step 2. Distance measures for alternatives and attributes
Calculate the distance measure of alternative \({S}_{j}\) and decision attribute \({C}_{k}\) by using the following distance measure:
Tabulate all the calculated distance measures of alternatives and attributes put in the following Table (Tables 3).
Table 3
Distance measure table
\({C}_{k} {S}_{j}\)
\({C}_{1}\)
\({C}_{2}\)
\({C}_{n}\)
\({S}_{1}\)
\(d\left({S}_{1}, {C}_{1}\right)\)
\(d\left({S}_{1}, {C}_{2}\right)\)
\(d\left({S}_{1}, {C}_{n}\right)\)
\({S}_{2}\)
\(d\left({S}_{2}, {C}_{1}\right)\)
\(d\left({S}_{2}, {C}_{2}\right)\)
\(d\left({S}_{2}, {C}_{n}\right)\)
\({S}_{m}\)
\(d\left({S}_{m}, {C}_{1}\right)\)
\(d\left({S}_{m}, {C}_{2}\right)\)
\(d\left({S}_{m}, {C}_{n}\right)\)
Step 4. Problem decision
From the distance measure table, choose the attribute\({C}_{k}\), \(k=1, 2, . . .,n\) for the alternative\({S}_{j}\), \(j=1, 2, . . ., m\) by which the lowest distance measure value between the alternatives \({S}_{j}\) and the attributes\({C}_{k}\), and then conclude the attribute \({C}_{k}\) is the best attribute for alternative\({S}_{j}\).
Numerical example: application of sine metric single-valued neutrosophic distance measure
The course selection is one of the major multi-decision-making problems for students’ career development in the educational field. This plays a major role in defining their future. The student’s course selection starts from the schooling itself and it probably extends until getting a suitable job. In the stage of entering college, the students can considerably have more responsibility and curiosity in their course selection than school course selection. The students can choose their career by identifying the availability of courses, the infrastructure of the institution, the subject interest, and especially their economic background. But they can choose the course by their subject interesting only. The awareness and responsibility of student’s careers and course selection vary from student to student due to their subject interest, economic background, etc. An incorrect decision of a student’s course section in the field of education may ruin the academic as well as the professional career of the student. In this situation, the indeterminacy component may also arise and so here also neutrosophic set theory can apply to the multi-attribute decision-making problem for the student’s career determination.
In this section, we demonstrate a numerical example of course selection for some particular students as a real-life application of our sine metric single-valued neutrosophic distance measure for the above-proposed methodology ineffective.
Step 1. Problem field selection.
Let \(S=\left\{{S}_{1}, {S}_{2}, {S}_{3}, {S}_{4}, {S}_{5}, {S}_{6}, {S}_{7}\right\}\) be the set of students, \(C=\{{C}_{1}\),\({C}_{2}\),\({C}_{3}\), \({C}_{4}\), \({C}_{5}\), \({C}_{6}\),\({C}_{7}\)} be the set of courses and \({R}_{C}=\{{R}_{1}\),\({R}_{2}\), \({R}_{3}\), \({R}_{4}\), \({R}_{5}\)} be the set of subject-related courses. Table 4 shows the information about the student’s subject interest, for example, the subject knowledge \({R}_{1}\) whose membership value for the student \({S}_{6}\) is \(0.9\), the non-membership value is \(0.5\) and the indeterminacy value is \(0.1\), and so denoted as \((0.9, 0.1, 0.5)\). Table 5 shows the information about subject knowledge and the courses, for example, the membership value of the course \({C}_{5}\) for the subject \({R}_{4}\) is \(0.1\), the non-membership value is \(0.6\) and the indeterminacy value is \(0\), it also denotes \((0.1, 0.6, 0)\).
Table 4
Students vs subjects
Student
subject
\({S}_{1}\)
\({S}_{2}\)
\({S}_{3}\)
\({S}_{4}\)
\({S}_{5}\)
\({S}_{6}\)
\({S}_{7}\)
\({R}_{1}\)
\((0.4, 0.6, 1)\)
\((0.5, 0.5, 0.9)\)
\((0.6, 0.4, 0.8)\)
\((0.7, 0.3, 0.7)\)
\((0.8, 0.2, 0.6)\)
\((0.9, 0.1, 0.5)\)
\((1, 0, 0.4)\)
\({R}_{2}\)
\((0.8, 0.4, 0.1)\)
\((0.5, 0.6, 0.2)\)
\((1, 0.7, 0.8)\)
\((0.3, 0.2, 0.9)\)
\((0.6, 0.7, 0.8)\)
\((0.9, 1, 0)\)
\((0, 0, 1)\)
\({R}_{3}\)
\((0.2, 0.1, 0)\)
\((0.3, 0.6, 0.7)\)
\((0.3, 0.2, 0.1)\)
\((0.7, 0.8, 1)\)
\((0, 0, 1)\)
\((0.6, 0.7, 0.3)\)
\((0.3, 0.2, 0.1)\)
\({R}_{4}\)
\((0.6, 0.5, 0.2)\)
\((0.7, 0, 0)\)
\((0.5, 0.5, 0.5)\)
\((0.6, 0.5, 0.1)\)
\((0.8, 0.8, 0.8)\)
\((0.3, 0.7, 0.6)\)
\((0.7, 0.8, 0.1)\)
\({R}_{5}\)
\((0.3, 0.3, 0.3)\)
\((0, 0, 1)\)
\((0.6, 0.5, 0.3)\)
\((0.4, 0.5, 0.6)\)
\((0.3, 0.2, 0.1)\)
\((0.2, 0.1, 0.1)\)
\((0.5, 0.6, 0.1)\)
Table 5
Subject vs courses
Subject course
\({R}_{1}\)
\({R}_{2}\)
\({R}_{3}\)
\({R}_{4}\)
\({R}_{5}\)
\({C}_{1}\)
\((0.4, 0.5, 1)\)
\((0.4, 0.5, 0.6)\)
\((0.8, 0.6, 0.1)\)
\((0.9, 0, 0)\)
\((0.1, 0.1, 0.1)\)
\({C}_{2}\)
\((0.8, 0.1, 0.1)\)
\(0.6, 0.5, 0.5)\)
\((0.3, 0.2, 0.1)\)
\((0.8, 0.9, 1)\)
\((0, 0, 1)\)
\({C}_{3}\)
\((1, 0, 0)\)
\((0.6, 0.4, 0.2)\)
\((0.5, 0.8, 0.9)\)
\((1, 0.3, 0.5)\)
\((0.3, 0.4, 0)\)
\({C}_{4}\)
\((0.5, 0.6, 1)\)
\((0, 0, 0.3)\)
\((0, 0.3, 0.3)\)
\((0.6, 0.3, 0.1)\)
\((0.7, 0.8, 0.1)\)
\({C}_{5}\)
\((0.6, 0.3, 0.1)\)
\((0.5, 0.3, 0.1)\)
\((0.7, 0.9, 0.3)\)
\((0.1, 0.6, 0)\)
\((0.5, 0.2, 0.1)\)
\({C}_{6}\)
(0, 0.7, 1)
\((0.5, 0.4, 0.2)\)
\((0.3, 0.2, 1)\)
\((0.5, 0.4, 0.1)\)
\((0.9, 0, 1)\)
\({C}_{7}\)
(0.8, 0.1, 0.3)
\((0.7, 0.5, 0.6)\)
\((0.8, 0.9, 0.1)\)
\((0.6, 0.7, 0.2)\)
\((0, 0, 1)\)
Step 2. Distance measure for the alternatives and attributes.
The distance measures between each student and each course with reference to the subjects is
From the above distance measure Table 6, we can observe that the smallest measure value for the student \({S}_{1}\) is 0.41165 to the course \({C}_{4}\). For the student \({S}_{2}\), 0.41161 is the smallest measure value to the course \({C}_{6}\) and for the student \({S}_{3},\) the smallest measure value is 0.49310 to the course \({C}_{4}\). The smallest measure value for the student \({S}_{4}\) is 0.48370 to the course \({C}_{5}\), for the student \({S}_{5},\) the smallest measure value is 0.47122 to the course \({C}_{2}\) and for the student \({S}_{6}\) the smallest measure value is 0.46446 to the course \({C}_{5}\). The smallest measure value for the student \({S}_{7}\) is 0.46002 to the course \({C}_{4}\). Thus the suitable course for the career development of students \({S}_{1}\), \({S}_{3}\) and \({S}_{7}\) is.
Table 6
Distance measure table
\({C}_{1}\)
\({C}_{2}\)
\({C}_{3}\)
\({C}_{4}\)
\({C}_{5}\)
\({C}_{6}\)
\({C}_{7}\)
\({S}_{1}\)
0.45196
0.56744
0.56159
0.41165
0.48999
0.44781
0.54979
\({S}_{2}\)
0.47293
0.49493
0.56483
0.52547
0.5661
0.41161
0.47658
\({S}_{3}\)
0.55025
0.52215
0.61279
0.49310
0.55340
0.57934
0.5515
\({S}_{4}\)
0.54791
0.66321
0.51833
0.56573
0.48370
0.54585
0.48526
\({S}_{5}\)
0.58725
0.47122
0.51831
0.64821
0.61773
0.6328
0.56088
\({S}_{6}\)
0.54135
0.55944
0.50384
0.68433
0.46446
0.71094
0.47421
\({S}_{7}\)
0.62231
0.51831
0.56469
0.46002
0.58827
0.68259
0.54925
\({C}_{4}\). The course \({C}_{6}\) is more suitable for the student \({S}_{2}\) and the course \({C}_{2}\) is suitable for student \({S}_{5}\). For the career development of students \({S}_{4}\) and \({S}_{6}\), \({C}_{5}\) is a suitable course.
Results and discussion
In this section, we discuss the decision of sine metric single-valued neutrosophic distance measure to other single-valued neutrosophic distance measures [27, 36], by comparing their results. The following tables: Tables 7, 8, 9, and 10 shows the calculated results of the distance measures \({d}_{1}\left(A,B\right), {d}_{2}\left(A,B\right), {D}_{1}(A,B)\) and \({D}_{2}\left(A,B\right)\) respectively for data of Tables 4 and 5.
Table 7
Distance Measure table for \({d}_{1}\left(A,B\right)\)
\({C}_{1}\)
\({C}_{2}\)
\({C}_{3}\)
\({C}_{4}\)
\({C}_{5}\)
\({C}_{6}\)
\({C}_{7}\)
\({S}_{1}\)
0.26
0.36667
0.38
0.24
0.30333
0.2667
0.35333
\({S}_{2}\)
0.2133
0.3333
0.3667
0.3533
0.3467
0.22
0.2933
\({S}_{3}\)
0.3267
0.34667
0.3933
0.3067
0.34
0.36
0.3533
\({S}_{4}\)
0.32
0.44
0.30667
0.34
0.2667
0.34
0.2933
\({S}_{5}\)
0.4
0.28
0.3067
0.4067
0.4267
0.42
0.3867
\({S}_{6}\)
0.367
0.34
0.2867
0.467
0.26
0.4933
0.2733
\({S}_{7}\)
0.3933
0.3533
0.38
0.2667
0.3667
0.4933
0.36
Table 8
Distance measure table for \({d}_{2}\left(A,B\right)\)
\({C}_{1}\)
\({C}_{2}\)
\({C}_{3}\)
\({C}_{4}\)
\({C}_{5}\)
\({C}_{6}\)
\({C}_{7}\)
\({S}_{1}\)
0.32146
0.44497
0.48647
0.31827
0.39749
0.38816
0.43895
\({S}_{2}\)
0.33467
0.4719
0.47188
0.44944
0.42425
0.32762
0.37058
\({S}_{3}\)
0.36968
0.41793
0.4539
0.39327
0.41713
0.4472
0.41069
\({S}_{4}\)
0.3759
0.49396
0.37505
0.40082
0.36693
0.40906
0.36514
\({S}_{5}\)
0.49529
0.39327
0.38471
0.46402
0.51639
0.48921
0.50332
\({S}_{6}\)
0.43436
0.40415
0.35681
0.53665
0.32146
0.54771
0.35869
\({S}_{7}\)
0.45093
0.46544
0.45822
0.35926
0.43127
0.59098
0.45018
Table 9
Distance measure table for \({D}_{1}\left(A,B\right)\)
\({C}_{1}\)
\({C}_{2}\)
\({C}_{3}\)
\({C}_{4}\)
\({C}_{5}\)
\({C}_{6}\)
\({C}_{7}\)
\({S}_{1}\)
0.198
0.3553
0.3553
0.16
0.26467
0.25467
0.3387
\({S}_{2}\)
0.21067
0.308
0.332
0.27067
0.31067
0.158
0.27733
\({S}_{3}\)
0.3073
0.3457
0.39233
0.20267
0.31
0.3113
0.35067
\({S}_{4}\)
0.3067
0.44
0.2733
0.26533
0.18933
0.3393
0.2923
\({S}_{5}\)
0.3733
0.252
0.28
0.414
0.388
0.41
0.266
\({S}_{6}\)
0.36467
0.32867
0.2865
0.428
0.2393
0.49132
0.27133
\({S}_{7}\)
0.36133
0.31933
0.32467
0.2653
0.30867
0.488
0.3367
Table 10
Distance measure table for \({D}_{2}\left(A,B\right)\)
\({C}_{1}\)
\({C}_{2}\)
\({C}_{3}\)
\({C}_{4}\)
\({C}_{5}\)
\({C}_{6}\)
\({C}_{7}\)
\({S}_{1}\)
0.05
0.05
0.0847
0.0547
0.02933
0.05067
0.094
\({S}_{2}\)
0.11733
0.03867
0.052
0.067
0.133
0.01867
0.11467
\({S}_{3}\)
0.086
0.05933
0.1733
0.0347
0.03733
0.116
0.07677
\({S}_{4}\)
0.123
0.06733
0.1087
0.07467
0.138
0.01
0.1833
\({S}_{5}\)
0.1107
0.052
0.17867
0.0567
0.1267
0.0246
0.188
\({S}_{6}\)
0.01067
0.034
0.07133
0.03867
0.10467
0.07067
0.022
\({S}_{7}\)
0.0467
0.03467
0.01467
0.0587
0.08467
0.14
0.06733
Simulation and comparison study
This section deals some simulation and comparison study of the Tables 6, 7, 8, 9 and 10.
In Fig. 1, the red line indicates the decision of student \({S}_{2}\) and the lowest value is at sixth red box. Therefore the suitable course for the student \({S}_{2}\) is the course \({C}_{6}\).
Fig. 1
Graphical representation of the distance measure Table 6
×
The following Table 11 shows the decision of \({d}_{1}(A,B)\),\({d}_{2}(A,B)\),\({D}_{1}\left(A, B\right)\),\({D}_{2}\left(A, B\right)\) and SMSVNDM \(d(A,B)\) by the smallest measure value of the distance measures from Tables 6, 7, 8, 9, 10 as well as Figs. 1, 2, 3, 4, 5.
Graphical representation of the distance measure Table 7
Fig. 3
Graphical representation of the distance measure Table 8
Fig. 4
Graphical representation of the distance measure Table 9
Fig. 5
Graphical representation of the distance measure Table 10
×
×
×
×
From the above comparison Table 11 and Figs. 1, 2, 3, 4, 5, we can observe that for student \({S}_{1}\), decisions of \({d}_{1}\left(A,B\right)\),\({d}_{2}\left(A,B\right)\) and \({D}_{1}\left(A, B\right)\) are the same, that is \({C}_{4}\) but the decision of \({D}_{2}\left(A, B\right)\) is \({C}_{5}\). The majority decision is \({C}_{4}\) which is coinciding with the decision of the proposed distance measure \(d\left(A,B\right)\). For student \({S}_{2}\), the decision of \({d}_{2}\left(A,B\right)\),\({D}_{1}\left(A, B\right)\) and \({D}_{2}\left(A, B\right)\) is \({C}_{6}\) which coincides with the decision of the proposed distance measure \(d\left(A,B\right)\), but the decision of \({d}_{1}\left(A,B\right)\) is \({C}_{1}\), which is the one not same to other decision. The decision of \({ d}_{1}\left(A,B\right)\),\({D}_{1}\left(A, B\right)\) and \({D}_{2}\left(A, B\right)\) for student \({S}_{3}\) is \({C}_{4}\) which is also coinciding to the decision of \(d\left(A,B\right)\), but not coinciding to the decision of \({d}_{2}\left(A,B\right)\), whose decision is \({C}_{1}\). The course ‘\({C}_{5}\)’ is the common decision of \({d}_{1}\left(A,B\right)\), and \({D}_{1}\left(A, B\right)\) for the student \({S}_{4}\) but the decision of \({d}_{2}\left(A, B\right)\) for the student \({S}_{4}\) is the course ‘\({C}_{7}\)’ and the decision of \({D}_{2}\left(A, B\right)\) for this same student is the course ‘\({C}_{6}\)’ which are not same. In this situation, for the student \({S}_{4}\) the majority decision is ‘\({C}_{5}\)’ which is the same as the decision of distance measure \(d(A,B)\). Similarly for the student \({S}_{5}\), ‘\({C}_{2}\)’ course is the common decision of \({d}_{1}(A,B)\), and \({D}_{1}\left(A, B\right)\) but the decision of \({d}_{2}\left(A, B\right),{D}_{2}\left(A, B\right)\) is not same. So for the student \({S}_{5}\), the majority decision is ‘\({C}_{2}\)’which is same as the decision of the distance measure \(d(A,B)\). The decision of distance measures \({d}_{1}(A,B)\),\({d}_{2}\left(A,B\right),\) and \({D}_{1}\left(A, B\right)\) for the student \({S}_{6}\) are \({C}_{5}\) but the decision of distance measure \({D}_{2}\left(A, B\right)\) is \({C}_{1}\) which is not the same to other. In this situation for the student \({S}_{6}\), the majority decision is ‘\({C}_{5}\)’ which is also coinciding with the decision of our proposed distance measure. For the student \({S}_{7}\), the decision of \({d}_{1}(A,B)\),\({d}_{2}(A,B)\), and \({D}_{1}\left(A, B\right)\) is \({C}_{4}\) which also coincides with \(d(A,B)\), even though the decision of \({D}_{2}\left(A, B\right)\) is \({C}_{3}\) which is not same. Thus final decision for each student is the majority decision, which is the suitable course for the career development of students \({S}_{1}\), \({S}_{3}\) and \({S}_{7}\) is \({C}_{4}\). The course ‘\({C}_{6}\)’is more suitable for the student \({S}_{2}\) and \({C}_{2}\) is suitable for the student \({S}_{5}\). For the career development of students \({S}_{4}\) and \({S}_{6}\),\({C}_{5}\) is a suitable course.
Advantages of the work
From the comparative studies between [27, 36], we can state one of the advantages of the proposed method by our sine distance measure is that the final decision of the proposed method is the most common decision in multi-attribute decision-making problems for the single-valued neutrosophic distance measure. From Tables 6, 7, 8, 9, and 10, all distance measures values of \(d\left(A,B\right)\),\({d}_{1}\left(A,B\right)\),\({d}_{2}\left(A,B\right)\),\({D}_{1}\left(A, B\right)\) and \({D}_{2}\left(A, B\right)\) satisfy the inequality
This is another advantage. Hence, we can consider this method as one of the common methods and is the best one to find alternatives with suitable attributes. Our proposed distance measure \(d\left(A,B\right)\) is entirely different from other distance measures \({d}_{1}\left(A,B\right)\),\({d}_{2}\left(A,B\right)\),\({D}_{1}\left(A, B\right)\) and \({D}_{2}\left(A, B\right)\). The sine single-valued neutrosophic distance measure \(d\left(A,B\right)\) in multi-attribute decision-making problems will be more effective over the other single-valued neutrosophic distance measures \({d}_{1}\left(A,B\right)\),\({d}_{2}\left(A,B\right)\),\({D}_{1}\left(A, B\right)\) and \({D}_{2}\left(A, B\right)\), because the majority decisions of \({d}_{1}(A,B)\),\({d}_{2}(A,B)\),\({D}_{1}\left(A, B\right)\), \({D}_{2}\left(A, B\right)\) is coinciding to the decision of \(d(A,B)\) until not find another single-valued neutrosophic distance measure which satisfies the above said hypotheses. This method is simple, reliable, and dependable and can be used distance measures as well as similarity measures in multi-attribute decision-making problems as well as in multi-criterion decision-making problems. One more advantage of this method is that it can take not only the single-valued neutrosophic sets but also can take neutrosophic sets. We can also model the real-life application by the neutrosophic score functions, the neutrosophic matrix, and neutrosophic operators for this measure.
Limitations
One of the limitations of the present work is the number of attributes should not exceed the number of alternatives. The other limitation is that this method does not use any neutrosophic score functions, neutrosophic matrix, and neutrosophic operators. The collected data should be in single-valued neutrosophic form. All the distance measure values should be in [0, 1].
Assumptions in the proposed method
We first assumed the collected data should be in neutrosophic form and also assumed the values of membership functions, non-membership functions and indeterminacy functions should lie in the standard unit interval [0, 1]. Using sine metric distance measures, we need to find the distance between any two single-valued neutrosophic sets. Next, we assumed all sine metric distance measure values belong in the interval [0, 1]. The least distance measure value is assumed for the decision of selected problems. The least measured value means the assumed alternatives are much closer to the assumed attributes. The alternatives and the attributes are assumed to be single-valued neutrosophic sets.
Conclusions and future study
In the real world, many indeterminate elements such as ambiguous, incomplete, inexact, vague, unclear and uncertain elements occur. Therefore, we need more neutrosophic procedures than classical procedures. Neutrosophy is a branch of philosophy, which deals with the nature, origin, and scope of neutralities, introduced by Florentin Smarandache [24]. A single-valued neutrosophic set is a special neutrosophic set that is also a mathematical tool to handle imprecise, incomplete, and inconsistent information in multi-decision-making problems.
The distance measure as well as the similarity measure is a measure between two neutrosophic sets which helps to calculate the relation between information. The cross-entropy helps to measure the discrimination information between objects and now the cross-entropy extends to quadri-partitioned single-valued neutrosophic sets, interval-valued neutrosophic sets and interval quadri-partitioned neutrosophic sets. In multi-decision-making problems, distance measures and similarity measures on single-valued neutrosophic sets play an important role in making decisions regarding alternatives. There are many distance measures and similarity measures [18‐30, 36, 37, 47, 48] introduced and applied in medical diagnosis, data analysis, cross-entropy, and pattern recognition to deal with multi-decision-making problems. Majumdar and Samanta [27] defined the axioms of distance measure and introduced two different distance measures respectively called normalized hamming distance measure and normalized Euclidian distance measure. The cosine similarity measure [28], cotangent similarity measure [29] and tangent similarity measures [30] are neutrosophic similarity measures, but not sine functions without trigonometric identities under a neutrosophic environment. Chai et al. [36] further introduced some distance measures.
This paper defined a sine distance measure on single-valued neutrosophic sets and also showed that the proposed distance measure is a metric by satisfying metric axioms. Further, we have formulated a method using the proposed measure and demonstrated a real-life application by solving multi-attribute decision-making problems. Moreover, we have compared the decisions of various single-valued neutrosophic distance measures [27, 36] with the decision of the proposed distance measure. Finally, we showed that the decision of the proposed method coincided with the most common decision and stated that the decision of the proposed method is the best decision of the problem. For future studies, the proposed sine metric single-valued distance measure and the proposed methodology could be applied to many other research areas such as artificial intelligence, medical diagnosis, pattern recognition, data analysis etc. This research could be extended to the TOPSIS method, VIKOR method, TODIM method, ELECTRIC method, cross-entropy, interval-valued neutrosophic set theory, rough set theory, etc.
Acknowledgements
The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality and presentation of the paper.
Declarations
Conflict of interest
No conflict of interest regarding the paper.
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