Multiple Attribute Group Decision-Making Models Using Single-Valued Neutrosophic and Linguistic Neutrosophic Hybrid Element Aggregation Algorithms

Multiple attribute group decision-making (MAGDM) issues may involve quantitative and qualitative attributes. In inconsistent and indeterminate decision-making issues, current assessment information of quantitative and qualitative attributes with respect to alternatives only contains either numerical neutrosophic values or linguistic neutrosophic values as the single information expression. However, existing neutrosophic techniques cannot perform the mixed information denotation and aggregation operations of numerical neutrosophic values and linguistic neutrosophic values in neutrosophic decision-making issues. To solve the puzzles, this article presents the information denotation, aggregation operations, and MAGDM models of single-valued neutrosophic and linguistic neutrosophic hybrid sets/elements (SVNLNHSs/SVNLHEs) as new techniques to perform MAGDM issues with quantitative and qualitative attributes in the environment of SVNLNHEs. In this study, we rst propose a SVNLNHS/ SVNLNHE notion that consists of a single-valued neutrosophic element (SVNE) for the quantitative argument and a linguistic neutrosophic element (LNE) for the qualitative argument. According to a linguistic and neutrosophic conversion function and its inverse conversion function, we present some basic operations of single-valued neutrosophic elements and linguistic neutrosophic elements, the SVNLNHE weighted arithmetic mean (SVNLNHEWAMN) and SVNLNHE weighted geometric mean (SVNLNHEWGMN) operators (forming SVNEs), and the SVNLNHEWAML and SVNLNHEWGML operators (forming LNEs). Next, MAGDM models are established based on the SVNLNHEWAMN and SVNLNHEWGMN operators or the SVNLNHEWAML and SVNLNHEWGML operators to realize MAGDM issues with single-valued neutrosophic and linguistic neutrosophic hybrid information, and then their applicability and availability are indicated through an illustrative example in the SVNLNHE circumstance. By comparison with the existing techniques, our new techniques reveal obvious advantages in themixed information denotation, aggregation algorithms, and decision-making methods in handlingMAGDM issues with the quantitative and qualitative attributes in the setting of SVNLNHSs.


Introduction
In general, there exist both quantitative attributes and qualitative attributes in multiple attribute (group) decisionmaking (MADM/MAGDM) issues. In the assessment process, the assessment information of quantitative attributes is usually represented by numerical values because numerical values are more suitable to the denotation form of quantitative arguments, while the assessment information of qualitative attributes is usually assigned by linguistic term values because the linguistic value is more suitable to human judgment and thinking/expression habits. Generally speaking, it is di cult to represent qualitative arguments by numeric values, but they are easily represented by linguistic values. In inconsistent and indeterminate situations, a simpli ed neutrosophic set (SNS) [1], including an intervalvalued neutrosophic set/element (IVNS/IVNE) [2] and a single-valued neutrosophic set/element (SVNS/SVNE) [3], is depicted by the truth, falsity, and indeterminacy membership degrees, while a linguistic neutrosophic set/element (LNS/LNE) [4] is depicted by the truth, falsity, and indeterminacy linguistic values. Since the neutrosophic set theories [5], including SNS, SVNS, IVNS, and LNS, are vital mathematical tools to denote and handle indeterminate and inconsistent issues in the real world, they have been widely applied in decision-making issues [6][7][8][9][10][11][12][13][14]. In the setting of SNSs, some researchers presented various aggregation operators and their MADM/MAGDM models to solve neutrosophic MADM/MAGDM problems [10,[15][16][17][18][19][20].
en, other researchers introduced various extended versions of SNSs, including single-valued neutrosophic rough sets [21], normal neutrosophic sets [22], bipolar neutrosophic sets [23], simplified neutrosophic indeterminate sets [24], and neutrosophic Z-numbers [25], and used them in MADM/ MAGDM issues. In the setting of LNEs, some researchers proposed several aggregation operators of LNEs and their MAGDM models to carry out linguistic neutrosophic MAGDM problems [26,27]. en, some extended linguistic sets, such as linguistic neutrosophic uncertain sets and linguistic neutrosophic cubic sets, were also presented to perform some linguistic neutrosophic MAGDM problems [28]. Unfortunately, the existing neutrosophic theories and MADM models [28,29] cannot yet resolve the denotation, operations, and MADM issues of the mixed information of SVNEs and LNEs. However, the existing assessment information of the quantitative or qualitative attributes with respect to alternatives only gives either numerical neutrosophic information or linguistic neutrosophic information as a single information expression. In the case of singlevalued neutrosophic and linguistic neutrosophic mixed information, existing neutrosophic technologies cannot represent the mixed information of SVNE and LNE nor can they perform mixed operations of the two. erefore, the mixed information representation and aggregation operations and decision-making problems pose challenges in this study, which motivates our research to address them. To solve these problems, the aims of this article are as follows: (1) to propose a single-valued neutrosophic and linguistic neutrosophic hybrid set/element (SVNLNHS/SVNLNHE) for the mixed information representation of both SVNE and LNE, (2) to present basic operations of SVNEs and LNEs according to a linguistic and neutrosophic conversion function and its inverse conversion function, (3) to propose the single-valued neutrosophic and linguistic neutrosophic hybrid element weighted arithmetic mean (SVNLNHE-WAM N ) and single-valued neutrosophic and linguistic neutrosophic hybrid element weighted geometric mean (SVNLNHEWGM N ) operators for the aggregated SVNEs and the SVNLNHEWAM L and SVNLNHEWGM L operators for the aggregated LNEs, (4) to establish MAGDM models based on the SVNLNHEWAM N and SVNLNHEWGM N operators or the SVNLNHEWAM L and SVNLNHEWGM L operators in the setting of SVNLNHSs, and (5) to apply the established MAGDM models to an illustrative example on the selection problem of industrial robots that contain both quantitative and qualitative attributes in a SVNLNHS circumstance.
Generally, the main contributions of this article are summarized as follows: (i) e proposed SVNLNHS/SVNLNHE solves the representation problem of single-valued neutrosophic and linguistic neutrosophic mixed information. (ii) e proposed weighted aggregation operators of SVNLNHEs based on the linguistic and neutrosophic conversion function and its inverse conversion function provide the effective aggregation algorithms of SVNLNHEs. (iii) e established MAGDM models can solve MAGDM issues with quantitative and qualitative attributes in a SVNLNHS circumstance. (iv) e established MAGDM models can solve the selection problem of industrial robots that contain both quantitative and qualitative attributes and show the availability and rationality of the new techniques in a SVNLNHS circumstance.
e remaining structure of this article consists of the following sections. Section 2 reviews the basic concepts and operations of SVNEs and LNEs as the preliminaries of this study. e notions of SVNLNHS and SVNLNHE and some basic operations of SVNEs and LNEs based on the linguistic and neutrosophic conversion function and its inverse conversion function are proposed in Section 3. In Section 4, the SVNLNHEWAM N , SVNLNHEWGM N , SVNLNHE-WAM L , and SVNLNHEWGM L operators are presented in terms of the basic operations of SVNEs and LNEs. In Section 5, two new MAGDM models are established by the SVNLNHEWAM N and SVNLNHEWGM N operators or the SVNLNHEWAM L and SVNLNHEWGM L operators. Section 6 presents an illustrative example on the selection problem of industrial robots that contains both quantitative and qualitative attributes and then gives a comparative analysis with the existing techniques to show the availability and rationality of the new techniques. Finally, conclusions and future research are summarized in Section 7.

Preliminaries of SVNEs and LNEs
is part reviews the basic notions and operations of SVNEs and LNEs.

Basic Notions and Operations of LNEs.
Let U = {u 1 , u 2 , . . ., u m } be a universal set and S � s p |p � 0, 1, . . . , r be a linguistic term set (LTS) with an odd cardinality r + 1. us, a LNS LH is defined as follows [4]: For two LNEs, lh 1 � 〈s a 1 , s b 1 , s c 1 〉, lh 2 � 〈s a 2 , s b 2 , s c 2 〉, , and β > 0, and their operational relations are as follows [4]: en, the LNE weighted arithmetic mean (LNEWAM) and LNE weighted geometric mean (LNEWGM) operators are introduced as follows [4]: Set lh i � 〈s a i , s b i , s c i 〉 as any LNE. e score and accuracy functions of lh i are defined, respectively, as follows [4]:

SVNLNHSs and SVNLNHEs
is section proposes SVNLNHS/SVNLNHE for the mixed information representation of both SVNE and LNE and then presents some basic operations of SVNEs and LNEs according to a linguistic and neutrosophic conversion function and its inverse conversion function.
en, a SVNLNHS ML is defined by where TL ML (u i ), IL ML (u i ), and FL ML (u i ) are the truth, indeterminacy, and falsity membership functions, and their values are either the fuzzy values for . ., m). us, ML 1 and ML 2 imply the following relations: us, some basic operations of SVNEs and LNEs are given as follows: It is obvious that the operational results of (2), (4), (5), and (8) are LNEs and the operational results of (3) and (7), and (9) are SVNEs.

Weighted Arithmetic and Geomatic Mean
Operators of SVNLNHEs is section proposes some weighted aggregation operators of SVNLNHEs corresponding to the linguistic and neutrosophic conversion function and its inverse conversion function, and then indicates their properties.

Aggregation Operators of SVNLNHEs Corresponding to the Linguistic and Neutrosophic Conversion Function. Let
. ., m) be q SVNEs and m − q LNEs, respectively. en, based on Definition 3 and the SVNEWAM and SVNEWGM operators of Eqs. (2) and (3) [17], the weighted arithmetic and geomatic mean operators of SVNLNHEs corresponding to the linguistic and neutrosophic conversion function are proposed by the SVNLNHEWAM N and SVNLNHEWGM N operators, where β i ∈ [0, 1] is the weight of zn i (i � 1, 2, . . ., q) and lh i (i � q + 1, q + 2, . . ., m) with m i�1 β i � 1. en, the aggregated results of the SVNLNHEWAM N and SVNLNHEWGM N operators are SVNEs.
Based on the properties of the SVNEWAM and SVNEWGM operators [17], it is obvious that the SVNLNHEWAM N and SVNLNHEWGM N operators also contain the following properties:

MAGDM Models in the Environment of SVNLNHSs
In this section, novel MAGDM models are developed in terms of the SVNLNHEWAM N and SVNLNHEWGM N operators and the SVNLNHEWAM L and SVNLNHEWGM L operators to perform MAGDM issues with quantitative and qualitative attributes in the mixed information environment of SVNEs and LNEs. Regarding a mixed information MAGDM issue in the circumstance of SVNLNHSs, there exist t alternatives, denoted by a set of them E � E 1 , E 2 , . . . E t , and then they are satisfactorily assessed over m attributes, denoted by a set of them V = {v 1 , v 2 , . . ., v q , v q + 1 , v q + 2 , . . ., v m }, which contains q quantitative attributes and m − q qualitative attributes.
Step 4: the alternatives are sorted in descending order based on the sorting laws of SVNEs, and the first one is the best choice.

Model 2.
A MAGDM model using the SVNLNHEWAM L and SVNLNHEWGM L operators is developed to perform the MAGDM issue with SVNLNHEs. Its detailed steps are presented as follows: Step 1': the same as Step 1.
Step 4': the alternatives are sorted in descending order based on the sorting laws of LNEs, and then the first one is the best choice.

Illustrative Example on the Selection Problem of Industrial Robots Containing Both Quantitative and Qualitative Attributes
is section applies the proposed MAGDM models to an illustrative example on the selection problem of industrial robots that contains both quantitative and qualitative attributes in the circumstance of SVNLNHSs to prove their usefulness, and then gives a comparison with existing techniques to show the availability and rationality of the new techniques.

Illustrative Example.
is subsection applies the proposed MAGDM models to the selection problem of industrial robots containing both quantitative and qualitative attributes to illustrate their application and availability in the circumstance of SVNLNHSs.
Some industrial company wants to buy a type of industrial robots for a manufacturing system. e technical department preliminarily provides four types of industrial robots/alternatives, denoted as their set E � {E 1 , E 2 , E 3 , E 4 }. en, they must satisfy four requirements/attributes: operating accuracy (v 1 ), carrying capacity (v 2 ), control performance (v 3 ), and operating space and dexterity (v 4 ). e weight vector of the four attributes is given by β � (0.25, 0.3, 0.25, 0.2). us, three experts/decision makers are invited to satisfactorily assess each alternative over the four attributes by their truth, falsity, and indeterminacy options/judgments, where the assessment values can be specified in the mixed forms of both the SVNEs zn k ji � 〈x k ZNji , y k ZNji , z k ZNji 〉 for x k ZNji , y k ZNji , z k ZNji ∈ [0, 1] (k � 1, 2, 3; i � 1, 2; j � 1, 2, 3, 4) regarding the quantitative attributes v 1 and v 2 and the LNEs lh k ji � 〈s a k ji , s b k ji , s c k ji 〉 for s a k ji , s b k ji , s c k ji ∈ S (k � 1, 2, 3; i � 3, 4; j � 1, 2, 3, 4) regarding the qualitative attributes v 3 and v 4 from the LTS S � {very unsatisfactory, unsatisfactory, slight unsatisfactory, medium, slight satisfactory, satisfactory, very satisfactory} � {s 0 , s 1 , s 2 , s 3 , s 4 , s 5 , s 6 } with r � 6. e weight vector of the three decision makers is given by α � (0.4, 0.35, 0.25). us, the three decision matrices are constructed as follows: Journal of Mathematics us, the two MAGDM models developed can be utilized in the example to perform the MAGDM issue with SVNLNHEs.
Model 1. e MAGDM model using the SVNLNHE-WAM N and SVNLNHEWGM N operators can be applied in the example, and then its detailed steps are depicted as follows: Step 1: using the SVNEWAM operator of Eq. (2) and the LNEWAM operator of Eq. (7), the above three decision matrices are aggregated into the following overall decision matrix: Step 2: by Eq. (16) or Eq. (17) Step 4: the sorting order of the four alternatives is Clearly, the sorting orders obtained by the SVNLNHEWAM N operator of Eq. (16) and the SVNLNHEWGM N operator of Eq. (17) are identical in this example.
Model 2. e MAGDM model using the SVNLNHE-WAM L and SVNLNHEWGM L operators can also be applied in the example, and then its detailed steps are depicted as follows: Step 1': the same as Step 1.
Step 4': the sorting order of the four alternatives is Hence, the sorting orders obtained by the SVNLNHE-WAM L operator of Eq. (18) [17] are only the special cases of the SVNLNHEWAM N and SVNLNHEWGM N operators, and then the existing LNEWAM and LNEWGM operators [4] are only the special cases of the SVNLNHEWAM L and SVNLNHEWGM L operators. Furthermore, the various existing aggregation operators cannot aggregate SVNLNHEs. (3) Since the existing MAGDM models with the single evaluation information of SVNEs or LNEs [4,17] are the special cases of our new MAGDM models, our new MAGDM models are broader and more versatile than the existing MAGDM models [4,17]. Furthermore, the various existing MAGDM models cannot carry out MAGDM problems with SVNLNHE information.
Generally, the new techniques solve the SVNLNHE denotation, aggregation operations, and MAGDM issues in the mixed information situation of SVNEs and LNEs. It is clear that our new techniques are very suitable for such decision-making issues with quantitative and qualitative attributes and overcome the defects of the existing decisionmaking techniques subject to the single evaluation information of SVNEs or LNEs. erefore, our new techniques reveal obvious superiorities over the existing techniques in the neutrosophic information denotation, aggregation operations, and decision-making methods.

Conclusion
Due to the lack of the SVNLNHE denotation, operations, and decision-making models in existing neutrosophic theory and applications, the proposed notion of SVNLNHS/ SVNLNHE and the defined linguistic and neutrosophic conversion function solved the hybrid neutrosophic information denotation and operational problems of SVNEs and LNEs. en, the proposed SVNLNHEWAM N , SVNLNHEWGM N , SVNLNHEWAM L , and SVNLNHEWGM L operators provided necessary aggregation algorithms for handling MAGDM issues with SVNLNHEs. e established MAGDM models solved such decision-making issues with quantitative and qualitative attributes in the SVNLNHE circumstance. Since the evaluation values of quantitative and qualitative attributes in the decision-making process are easily represented in SVNEs and LNEs that are given in view of decision makers' preferences/thinking habits, the managerial implications of this original research will be reinforced in neutrosophic decision-making methods and applications. Finally, an illustrative example was given and compared with the existing techniques to show the availability and rationality of the new techniques. Moreover, our new techniques not only overcome the insufficiencies of the existing techniques but also are broader and more versatile than the existing techniques when dealing with MAGDM issues in the setting of SVNLNHEs. However, in this study, the new techniques of the SVNLNHE denotation, aggregation algorithms, and MAGDM models reflected their superiority over existing techniques.
Regarding future research, these new techniques will be further extended to other areas, such as medical diagnosis, slope risk/instability evaluation, default diagnosis, and mechanical concept design, in the mixed information situation of SVNEs and LNEs. en, we shall also develop more aggregation algorithms, such as Hamacher, Dombi, and Bonferroni aggregation operators, and their applications in clustering analysis, information fusion, image processing, and mine risk/safety evaluation in the mixed information situation of both SVNE and LNE or both IVNE and uncertain LNE.