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

Zhang, Sumin; Ye, Jun

doi:https://doi.org/10.1155/2022/1021280

Journal of Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 1021280 | https://doi.org/10.1155/2022/1021280

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

Sumin Zhang¹and Jun Ye^1,2

Academic Editor: Ghous Ali

Received04 Jul 2022

Revised01 Aug 2022

Accepted05 Aug 2022

Published20 Sept 2022

Abstract

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 first 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 (SVNLNHEWAM_N) and SVNLNHE weighted geometric mean (SVNLNHEWGM_N) operators (forming SVNEs), and the SVNLNHEWAM_L and SVNLNHEWGM_L operators (forming LNEs). Next, MAGDM models are established based on the SVNLNHEWAM_N and SVNLNHEWGM_N operators or the SVNLNHEWAM_L and SVNLNHEWGM_L 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 the mixed information denotation, aggregation algorithms, and decision-making methods in handling MAGDM issues with the quantitative and qualitative attributes in the setting of SVNLNHSs.

1. Introduction

In general, there exist both quantitative attributes and qualitative attributes in multiple attribute (group) decision-making (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 difficult to represent qualitative arguments by numeric values, but they are easily represented by linguistic values. In inconsistent and indeterminate situations, a simplified neutrosophic set (SNS) [1], including an interval-valued 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 [6–14]. In the setting of SNSs, some researchers presented various aggregation operators and their MADM/MAGDM models to solve neutrosophic MADM/MAGDM problems [10, 15–20]. Then, 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]. Then, 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 single-valued 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. Therefore, 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 (SVNLNHEWAM_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)The proposed SVNLNHS/SVNLNHE solves the representation problem of single-valued neutrosophic and linguistic neutrosophic mixed information.(ii)The 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)The established MAGDM models can solve MAGDM issues with quantitative and qualitative attributes in a SVNLNHS circumstance.(iv)The 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.

The 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. The 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, SVNLNHEWAM_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.

2. Preliminaries of SVNEs and LNEs

This part reviews the basic notions and operations of SVNEs and LNEs.

2.1. Basic Notions and Operations of SVNEs

Set U = {u₁, u₂, …, u_m} as a universal set. Then, a SVNS ZN in U can be represented as [1, 3]where < u_i, x_ZN(u_i), y_ZN(u_i), z_ZN(u_i)> (i = 1, 2, …, m) is SVNE in ZN for u_j U and x_ZN(u_i), y_ZN(u_i), z_ZN(u_i) [0, 1], and then it is simply denoted as zn_i = < x_ZNi, y_ZNi, z_ZNi > .

For two SVNEs, zn₁ = < x_ZN₁, y_ZN₁, z_ZN₁>, ZN₂= < x_ZN2, y_ZN2, z_ZN₂>, and β > 0, and their relations are contained as follows [17]:(1)(2)(3)(4)

Suppose that there is a group of SVNEs zn_i = < x_{_ZNi}, y_{_ZNi}, z_{_ZNi}> (i = 1, 2, …, m) with their related weights β_i [0, 1] for . Then, the SVNE weighted arithmetic mean (SVNEWAM) operator and the SVNE weighted geometric mean (SVNEWGM) operator are introduced, respectively, as follows [17]:

To compare SVNEs, the score and accuracy functions of SVNEs and their ranking laws are introduced below [17].

Set zn_i = < x_{_ZNi}, y_{_ZNi}, z_{_ZNi} > as any SVNE. The score and accuracy functions of zn_i are presented, respectively, as follows:

Then, the sorting laws based on the score values of F(zn_i) and the accuracy values of G(zn_i) (i = 1, 2) are as follows [17]:(a)zn₁ > zn₂ for F(zn₁) > F(zn₂)(b)zn₁ > zn₂ for F(zn₁) = F(zn₂) and G(zn₁) > G(zn₂)(c)zn₁ = zn₂ for F(zn₁) = F(zn₂) and G(zn₁) = G(zn₂)

2.2. Basic Notions and Operations of LNEs

Let U = {u₁, u₂, …, u_m} be a universal set and be a linguistic term set (LTS) with an odd cardinality r + 1. Thus, a LNS LH is defined as follows [4]:where for u_iU is LNE in LH and are the truth, indeterminacy, and falsity linguistic variables, respectively. For convenience, LNE is simply denoted as .

For two LNEs, , , , and β > 0, and their operational relations are as follows [4]:(1)(2)(3)(4)

Suppose that there is a group of LNEs (i = 1, 2, …, m) with their related weights β_i ∈ [0, 1] for . Then, the LNE weighted arithmetic mean (LNEWAM) and LNE weighted geometric mean (LNEWGM) operators are introduced as follows [4]:

Set as any LNE. The score and accuracy functions of lh_i are defined, respectively, as follows [4]:

Then, the sorting laws based on the score values of P(lh_i) and the accuracy values of Q(lh_i) (i = 1, 2) are given as follows [4]:(a)lh₁ > lh₂ for P(lh₁) > P(lh₂)(b)lh₁ > lh₂ for P(lh₁) = P(lh₂) and Q(lh₁) > Q(lh₂)(c)lh₁ = lh₂ for P(lh₁) = P(lh₂) and Q(lh₁) = Q(lh₂)

3. SVNLNHSs and SVNLNHEs

This 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.

Definition 1. Let U = {u₁, u₂, …, u_m} be a universe set and S = {s_p|p = 0, 1, …, r} be LTS with an odd cardinality r + 1. Then, a SVNLNHS ML is defined bywhere 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 TL_ML(u_i), IL_ML(u_i), FL_ML(u_i) ∈ [0, 1] or the linguistic values for TL_ML(u_i) IL_ML(u_i), FL_ML(u_i) ∈ S and u_i ∈ U. Moreover, the SVNLNHS ML is composed of the q SVNEs for x_{_ZNi}, y_{_ZNi}, z_{_ZNi} ∈ [0, 1] (i = 1, 2, …, q) and the m − q LNEs for and a_i, b_i, c_i ∈ [0, r] (i = q + 1, q + 2, …, m).

Definition 2. Suppose that ML₁ and ML₂ are two SVNLNHSs, which contain q SVNEs zn_1i = < x_ZN1i, y_ZN1i, z_ZN1i> (i = 1, 2, …, q) and m − q LNEs for (i = q + 1, q + 2, …, m) and q SVNEs zn_2i = < x_ZN2i, y_ZN2i, z_ZN2i> (i = 1, 2, …, q) and m − q LNEs for (i = q + 1, q + 2, …, m). Thus, ML₁ and ML₂ imply the following relations:(1)ML₁ ⊆ ML₂ ⇔ zn_1i ⊆ zn_2i (i = 1, 2, …, q) and lh_1i ⊆ lh_2i (i = q + 1, q + 2, …, m), i.e., x_ZN1i ≤ x_ZN2i, y_ZN2i ≤ y_ZN1i, and z_ZN2i ≤ z_ZN1i for i = 1, 2, …, q and , , and for i = q + 1, q + 2, …, m;(2)ML₁ = ML₂ ⇔ zn_1i ⊆ zn_2i, zn_1i ⊇ zn_2i, lh₁ ⊆ lh₂, and lh₂ ⊆ lh₁, i.e., x_ZN1i = x_ZN2i, y_ZN2i = y_ZN1i, and z_ZN2i = z_ZN1i for i = 1, 2, …, q and , , and for i = q + 1, q + 2, …, m.

Definition 3. Set zn_i = < x_{_ZNi}, y_{_ZNi}, z_{_ZNi} > and as any SVNE and any LNE, respectively. Then, let a linguistic and neutrosophic conversion function be for a_i, b_i, c_i ∈ [0, r], and then its inverse conversion function is for x_{_ZNi}, y_{_ZNi}, z_{_ZNi} ∈ [0, 1]. Thus, some basic operations of SVNEs and LNEs are given as follows:(1)(2)(3)(4) for β > 0(5) for β > 0(6) for β > 0(7) for β > 0It 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.

4. Weighted Arithmetic and Geomatic Mean Operators of SVNLNHEs

This 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.

4.1. Aggregation Operators of SVNLNHEs Corresponding to the Linguistic and Neutrosophic Conversion Function

Let zn_i = < x_{_ZNi}, y_{_ZNi}, z_{_ZNi}> (i = 1, 2, …, q) and (i = q + 1, q + 2, …, m) be q SVNEs and m − q LNEs, respectively. Then, 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 . Then, the aggregated results of the SVNLNHEWAM_N and SVNLNHEWGM_N operators are SVNEs.

Especially, when q = m (without LNEs), the SVNLNHEWAM_N and SVNLNHEWGM_N operators are reduced to the SVNEWAM and SVNEWGM operators [17], i.e., Eq. (2) and Eq. (3).

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:(1)Idempotency: if zn_i = f(lh_i) = zn for i = 1, 2, …, q, q + 1, q + 2, …, m, there are and .(2)Boundedness: let and be the minimum and maximum SVNEs for i = 1, 2, …, m, and then there are the inequalities and .(3)Monotonicity: if the condition of (i = 1, 2, …, q) and (i = q + 1, q + 2, …, m) exists, and also exist.

4.2. Aggregation Operators of SVNLNHEs According to the Inverse Conversion Function

Let zn_i = < x_{_ZNi}, y_{_ZNi}, z_{_ZNi}> (i = 1, 2, …, q) and (i = q + 1, q + 2, …, m) be q SVNEs and m − q LNEs, respectively. Then, based on Definition 3 and the LNEWAM and LNEWGM operators of Eqs. (7) and (8) [4], the weighted arithmetic and geomatic mean operators of SVNLNHEs corresponding to the inverse conversion function are proposed by the SVNLNHEWAM_L and SVNLNHEWGM_L 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 . Then, the aggregated results of the SVNLNHEWAM_L and SVNLNHEWGM_L operators are LNEs.

Especially, when q = 0 (without SVNEs), the SVNLNHEWAM_L and SVNLNHEWGM_L operators are reduced to the LNEWAM and LNEWGM operators [4], i.e., Eq. (7) and Eq. (8).

Based on the characteristics of the LNEWAM and LNEWGM operators [4], it is obvious that the SVNLNHEWAM_L and SVNLNHEWGM_L operators also contain the following characteristics:(1)Idempotency: if f^− 1(zn_i) = lh_i = lh for i = 1, 2, …, q, q + 1, q + 2, …, m, there are and .(2)Boundedness: let and be the minimum and maximum LNEs for i = 1, 2, …, m, and then there are the inequalities and .(3)Monotonicity: if the condition of for i = 1, 2, …, q and for i = q + 1, q + 2, …, m exists, and also exist.

5. 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 alternatives, denoted by a set of them , and then they are satisfactorily assessed over m attributes, denoted by a set of them V = {₁, ₂, …, _q, _q ₊ ₁, _q ₊ ₂, …, _m}, which contains q quantitative attributes and m − q qualitative attributes. Then, there is a group of decision makers with their weight vector for α_k [0, 1] and . The assessment values of each alternative over the q quantitative attributes are given by the decision makers g_k (k = 1, 2, …, e) and represented by SVNEs for (k = 1, 2, …, e; j = 1, 2, …, t; i = 1, 2, …, q), and then the assessment values of each alternative over the m − q qualitative attributes are represented by LNEs for (k = 1, 2, …, e; j = 1, 2, …, t; i = q + 1, q + 2, …, m) from the LTS S = {s_p|p = 0, 1, 2, …, r}. Thus, all assessed values can be constructed as the e decision matrices of SVNLNHEs M_k = ()_t× m (k = 1, 2, …, e). Then, a weight vector is specified to consider the weights β_i of attributes v_i (i = 1, 2, …, m) with β_i [0, 1] and .

Thus, two MAGDM models are developed in terms of the SVNLNHEWAM_N and SVNLNHEWGM_N operators or the SVNLNHEWAM_L and SVNLNHEWGM_L operators to perform MAGDM issues with the mixed evaluation information of SVNEs and LNEs.

Model 1. A MAGDM model using the SVNLNHEWAM_N and SVNLNHEWGM_N operators is developed to perform the MAGDM issue with SVNLNHEs. Its detailed steps are presented as follows: Step 1: using the SVNEWAM operator of Eq. (2) and the LNEWAM operator of Eq. (7), the decision matrices (k = 1, 2, …, e) are aggregated into the overall decision matrix . Step 2: using the SVNLNHEWAM_N operator of Eq. (12) or the SVNLNHEWGM_N operator of Eq. (13), the aggregated result for E_j (j = 1, 2, ⋯, t) is obtained by the following equation: or Step 3: the score values of F(zn_j) (j = 1, 2, …, t) are given by Eq. (4) and the accuracy values of G(zn_j) (j = 1, 2, …, t) are given by Eq. (5) if necessary. Step 4: the alternatives are sorted in descending order based on the sorting laws of SVNEs, and the first one is the best choice. Step 5: end.

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 2’: using the SVNLNHEWAM_L operator of Eq. (14) or the SVNLNHEWGM_L operator of Eq. (15), the aggregated result for is given by the following equation: or Step 3’: the score values of P(lh_j) (j = 1, 2, …, t) are given by Eq. (9) and the accuracy values of Q(lh_j) (j = 1, 2, …, t) are given by Eq. (10) if necessary. 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. Step 5’: end.

6. Illustrative Example on the Selection Problem of Industrial Robots Containing Both Quantitative and Qualitative Attributes

This 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.

6.1. Illustrative Example

This 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. The technical department preliminarily provides four types of industrial robots/alternatives, denoted as their set E = {E₁, E₂, E₃, E₄}. Then, they must satisfy four requirements/attributes: operating accuracy (₁), carrying capacity (₂), control performance (₃), and operating space and dexterity (₄). The weight vector of the four attributes is given by β = (0.25, 0.3, 0.25, 0.2). Thus, 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 for (k = 1, 2, 3; i = 1, 2; j = 1, 2, 3, 4) regarding the quantitative attributes ₁ and ₂ and the LNEs for (k = 1, 2, 3; i = 3, 4; j = 1, 2, 3, 4) regarding the qualitative attributes ₃ and ₄ from the LTS S = {very unsatisfactory, unsatisfactory, slight unsatisfactory, medium, slight satisfactory, satisfactory, very satisfactory} = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} with r = 6. The weight vector of the three decision makers is given by α = (0.4, 0.35, 0.25). Thus, the three decision matrices are constructed as follows:

Thus, the two MAGDM models developed can be utilized in the example to perform the MAGDM issue with SVNLNHEs.

Model 1. The MAGDM model using the SVNLNHEWAM_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), we give the aggregated values: zn₁ = < 0.7795, 0.1958, 0.1804>, zn₂ = < 0.7921, 0.1884, 0.2392>, zn₃ = < 0.7751, 0.2049, 0.2231>, and zn₄ = < 0.8290, 0.1761, 0.2178 > . Or zn₁ = < 0.7789, 0.2324, 0.1885>, zn₂ = < 0.7876, 0.1934, 0.2464>, zn₃ = < 0.7749, 0.2276, 0.2754>, and zn₄ = < 0.8265, 0.1850, 0.2504 > . Step 3: by Eq. (4), the score values of F(zn_j) for E_j (j = 1, 2, 3, 4) are given below: F(zn₁) = 0.8011, F(zn₂) = 0.7882, F(zn₃) = 0.7824, and F(zn₄) = 0.8117. Or F(zn₁) = 0.7860, F(zn₂) = 0.7826, F(zn₃) = 0.7573, and F(zn₄) = 0.7970. Step 4: the sorting order of the four alternatives is E₄ > E₁ > E₂ > E₃.

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. The MAGDM model using the SVNLNHEWAM_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 2’: by Eq. (18) or Eq. (19), we give the aggregated values: lh₁ = < s_4.6773, s_1.1747, s_1.0823>, lh₂ = < s_4.7528, s_1.1302, s_1.4353>, lh₃ = < s_4.6508, s_1.2297, s_1.3384>, and lh₄ = < s_4.9739, s_1.0564, s_1.3070 > , or lh₁ = < s_4.6736, s_1.3944, s_1.1311>, lh₂ = < s_4.7255, s_1.1603, s_1.4783>, lh₃ = < s_4.6494, s_1.3657, s_1.6526>, and lh₄ = < s_4.9591, s_1.1100, s_1.5027 > . Step 3’: by (9), the score values of P(lh_j) for E_j (j = 1, 2, 3, 4) are given as follows: P(lh₁) = 0.8011, P(lh₂) = 0.7882, P(lh₃) = 0.7824, and P(lh₄) = 0.8117, or P(lh₁) = 0.7860, P(lh₂) = 0.7826, P(lh₃) = 0.7573, and P(lh₄) = 0.7970. Step 4’: the sorting order of the four alternatives is E₄ > E₁ > E₂ > E₃.

Hence, the sorting orders obtained by the SVNLNHEWAM_L operator of Eq. (18) and the SVNLNHEWGM_L operator of Eq. (19) are identical in this example.

Obviously, the score values and sorting orders between Model 1 and Model 2 reflect the same results. Moreover, we see that whether SVNEs are converted to LNEs or LNEs to SVNEs in the aggregation operations, their final decision results are actually identical. Thus, decision makers can choose Model 1 or Model 2 in MAGDM applications. Therefore, it is obvious that our new techniques are valid and reasonable.

6.2. Comparative Analysis with the Existing Neutrosophic MAGDM Models

Since the assessed values of SVNLNHEs are given in this illustrative example, the existing neutrosophic MAGDM models [4, 17] cannot deal with this illustrative example in the situation of SVNLNHEs. Then, our new techniques can handle neutrosophic MAGDM issues with SVNEs and/or LNEs and show the following highlights and advantages:(1)The proposed SVNLNHEs can conveniently denote the mixed information of SVNEs and LNEs regarding the assessment objects of quantitative and qualitative attributes, which is suitable for human judgment and thinking/expression habits, while existing neutrosophic expressions cannot represent SVNLNHE information.(2)The proposed SVNLNHEWAM_N and SVNLNHEWGM_N operators or the proposed SVNLNHEWAM_L and SVNLNHEWGM_L operators provide the necessary aggregation tools for handling MAGDM issues in the SVNLNHE circumstance, while the existing SVNEWAM and SVNEWGM operators [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 decision-making techniques subject to the single evaluation information of SVNEs or LNEs. Therefore, our new techniques reveal obvious superiorities over the existing techniques in the neutrosophic information denotation, aggregation operations, and decision-making methods.

7. 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. Then, the proposed SVNLNHEWAM_N, SVNLNHEWGM_N, SVNLNHEWAM_L, and SVNLNHEWGM_L operators provided necessary aggregation algorithms for handling MAGDM issues with SVNLNHEs. The 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. Then, 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.

Abbreviations

IVNS:	Interval-valued neutrosophic set
IVNE:	Interval-valued neutrosophic element
LTS:	Linguistic term set
LNS:	Linguistic neutrosophic set
LNE:	Linguistic neutrosophic element
LNEWAM:	Linguistic neutrosophic element weighted arithmetic mean
LNEWGM:	Linguistic neutrosophic element weighted geometric mean
MADM:	Multiple attribute decision-making
MAGDM:	Multiple attribute group decision-making
SNS:	Simplified neutrosophic set
SVNS:	Single-valued neutrosophic set
SVNE:	Single-valued neutrosophic element
SVNEWAM:	Single-valued neutrosophic element weighted arithmetic mean
SVNEWGM:	Single-valued neutrosophic element weighted geometric mean
SVNLNHS:	Single-valued neutrosophic and linguistic neutrosophic hybrid set
SVNLNHE:	Single-valued neutrosophic and linguistic neutrosophic hybrid element
SVNLNHEWAMN:	Single-valued neutrosophic and linguistic neutrosophic hybrid element weighted arithmetic mean (for the aggregated SVNE)
SVNLNHEWGMN:	Single-valued neutrosophic and linguistic neutrosophic hybrid element weighted geometric mean (for the aggregated SVNE)
SVNLNHEWAML:	Single-valued neutrosophic and linguistic neutrosophic hybrid element weighted arithmetic mean (for the aggregated LNE)
SVNLNHEWGML:	Single-valued neutrosophic and linguistic neutrosophic hybrid element weighted geometric mean (for the aggregated LNE).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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Copyright © 2022 Sumin Zhang and Jun Ye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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