Multiple attribute group decision making approach based on extended VIKOR and linguistic neutrosophic Set

Abstract

In the intricate decision-making problems, the preference information of decision makers may be difficultly stated by numerical values due to the ambiguity of human thinking about the complex objective things in the real world, and sometimes, it may be easily expressed by LTs (linguistic terms). Thus, how to solve fuzzy MAGDM (multiple attribute group decision-making) problem in which the attributes are described by neutrosophic linguistic information and weights for the attributes and decision makers are fully unknown, has become an important research direction. To achieve this goal, this paper briefly reviews the basic concept of LTs, SVLNN (single-valued linguistic neutrosophic number), and the closeness degree, then we put forward the closeness degree between the LNNs (linguistic neutrosophic numbers) and the distance formula between LNNs, and they can be used to determine the weights of experts and attributes. Then, a MAGDM method based on the extended VIKOR (Visekriterijumska Optimizacija I Kom-promisno Resenje) method is established under the single-valued linguistic neutrosophic numbers (SVLNN) environment. Further, an illustrative example about selecting problems of fault handling point is presented to demonstrate the application of the developed approach, and the feasibility and effectiveness of the proposed method are proved by comparing this method with WAA (Weighted Arithmetic Averaging) and TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method.

Keywords

Multiple attribute group decision-making linguistic neutrosophic numbers closeness degree Visekriterijumska Optimizacija I Kom-promisno Resenje

1 Introduction

MAGDM is a hot research topic in modern decision theory and it has been widely used in various fields [1 –4], such as benefit assessment, supplier selection, personnel decision-making, etc., however, when the environment is complex, we cannot describe the attribute values of these problems by real numbers, and sometimes decision-makers (DMs) can easily describe them by linguistic terms (LTs). For example, the evaluation of Chinese mobile customer service, the customers can give the LTs like “very satisfied”, “satisfaction”, “General”, and “poor”, etc. Therefore, how to make MAGDM in the linguistic environment is becoming more and more concerned for researchers.

Due to the uncertainty of information and many limitations such as time pressure, lack of knowledge and information extraction difficulties, etc., it is difficult for DMs to express their preferences numerically in many complex realities [5 –10]. Then the theory of fuzzy set (FS) initiated by Zadeh [11] has attracted wide attention from different authors due to its effectiveness in dealing with uncertain situations in the real world. Further Atanassov [12] proposed the concept of intuitionistic FSs (IFSs), which is an extension of FS. IFSs consider membership (truth) function (TF) and non-membership (falsity) function (FF), and they can easier to describe complex information than FSs. By a variety of real-valued functions between IFSs, IFS has proven to be a powerful tool to quantify the similarity/dissimilarity of the two objects. However, it is difficult to express the indeterminate and inconsistent information. So Smarandache [13] proposed neutrosophic sets (NSs) by adding an independent indeterminacy function (IF) to overcome this shortcoming, which include a TF, an IF, and a FF, independently. Further, Wang et al. [14] proposed a single-valued NSs (SVNSs) and provided their various properties. Then Ye [15] put forward the information energy of SVNS, and then applied it to decision-making problems with SVNSs. The single-valued neutrosophic number (SVNN) is a basic element in a SVNS [16], which can only express a TF, an IF, and a FF independently, and can describe the incomplete, indeterminate, and inconsistent information; however, because the TF, IF, and FF of SVNNs can only be described by crisp numbers, and they cannot be expressed by linguistic information in linguistic decision-making problems, in order to overcome the defects, Ye [17] introduced the concept of the single-valued linguistic neutrosophic numbers (SVLNNs), and SVLNNs are very suitable for describing more complex linguistic information of human judgments under linguistic decision-making environment since LNNs contain the advantages of both SVNNs and linguistic variables (LVs), which imply the truth, falsity, and indeterminate linguistic information. However, none of the existing linguistic decision-making methods can handle decision problems with uncertain and inconsistent linguistic information. Therefore, how to carry out MAGDM in the linguistic environment is more and more concerned by the researchers [18, 19].

There are many specific methods for MAGDM, which have their own advantages in real applications. Commonly used multi-attribute decision-making methods are ELECTRE (elimination and choice translating reality) method, MABAC (multi-attribute border approximation area comparison) method, EDAS (estimation of distribution algorithms) method, Dempster-Shafer theory of evidence, TOPSIS method, VIKOR method, etc. Benayoun [20] proposed ELECTRE, and due to the ambiguity between schemes and the subjectivity of some thresholds in the method, this method has some unreasonable situations in the final sequencing of schemes. P. Ji, etc. proposed an MABAC–ELECTRE method to the selection of an outsourcing provider under the single-valued neutrosophic linguistic environments [21]. EDAS method was proposed in 1996, it is extended to resolve group decision-making problems in the linguistic neutrosophic environment [22 –24]. J.M. Merigó, etc. develop a new approach for decision making with Dempster-Shafer theory of evidence by using linguistic information, which realized the unification of mixed information such as exact numbers, interval numbers and linguistic variables, retained the uncertainty of information the comprehensive foreground value of the scheme reduces the loss of decision information. [25]. TOPSIS method was originally introduced by Hwang and Yoon [26] and is a widely used method for dealing with MAGDM problems. Its focuses on choosing the alternative with the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). The VIKOR method was first proposed by Opricovic as an efficient tool to handle conflicting criteria [27, 28], which ranks alternatives and determines a compromise solution that is the closest to the ideal. Its most prominent feature is to maximize the effectiveness of the community and minimize the individual regret, and the choice of compromise solution is accepted by DMs. Obviously, the VIKOR method has more advantages than the other methods in selecting the compromise solution, and then it has been developed rapidly since it was proposed. Opricovic [29, 30] proposed a fuzzy VIKOR method in which both the attribute values and weights could be triangular fuzzy numbers. Sayadi et al. [31] extended the VIKOR method to a decision-making problem with interval numbers. Wan et al. [32] developed an extended VIKOR method for MAGDM using triangular intuitionistic fuzzy numbers. However, now the existing VIKOR method cannot deal with the fuzzy MAGDM with SVNNs.

There is a great deal of MAGDM problems how to determine these weights was an important research topic. X.D. Peng and J.G. Dai [33] present the combined weights, which can show both the subjective information and the objective information, and the objective weights of various attributes are determined via gray system theory. A fuzzy best-worst method was used to obtain the weights of risk factors by Z.P. Tian, etc. [34], and an integrated structure based on fuzzy proximity and fuzzy similarity entropy weights was developed to obtain the weights of FMEA team members with respect to different risk factors. Regarding the determination of weights, many scholars have done a lot of relevant research and application [35 –39]. So far, there is no weight determination method for comprehensive application of closeness and distance measure. Because the SVLNNs are superior to other means in expressing complex uncertain information, and the VIKOR method has more advantages than the other methods in selecting the compromise solution, the objectives of this paper are to propose the extended VIKOR method for the MAGDM problem with SVLNNs. The main contributions of this paper are to propose an expert weight determination method based on closeness degree and the distance formula which is used to determine attribute weight; finally, we have verified the effectiveness and superiority of a MAGDM method based on extended VIKOR.

The rest of this paper is organized as follows: Section 2 briefly reviews the basic concept of LTS, SVNS, SVLNN and VIKOR method. Section 3, we put forward the closeness degree and extended the distance formula between LNNs, and, gave the steps of a MAGDM method based on the proposed VIKOR method under SVLNN environment. Section 4 gives an example and demonstrate the feasibility and validity of the method. Section 5 gives conclusions and future research directions.

2 Preliminaries

2.1 The LTS

There is a great deal of information in the real world which cannot be assessed by a quantitative form, especially for qualitative information, which is more easily evaluated by LTs. For the convenience of use, we have to choose the appropriate linguistic descriptors for the LT set (LTS) and their semantics, so we introduce some information about linguistic variables (LVs).

Assume that the LTS N ={ n₀, n₁, n₂,. . . , n_r } consists of an odd number of elements that r + 1 is an odd number. In practice, r + 1 is generally 3, 5, 7, 9 and so on [40]. For example, when r = 8, it is expressed as N = {n₀ = extremely important, n₁ = very important, n₂ = important, n₃ = slightly important, n₄ = fair, n₅ = unimportant, n₆ = slightly unimportant, n₇ = very unimportant, n₈ = extremely unimportant}, where, n_i is called a LV.

For any LTS N, the following conditions are met [40].

If i > j, then n_i ≻ n_j;

Exist a negative operator neg (n_i) = n_j, so j = r - 1 - i;

If n_i ≥ n_j, then max(n_i, n_j) = n_i;

If n_i ≤ n_j, then min(n_i, n_j) = n_i.

For any LTS N = (n₀, n₁, n₂, . . , n_r), in order to minimize the loss of linguistic decision information in operations, the LTS N is extended to a continuous one N = (n_α|α ∈ R) [25]. Then operations of LVs are shown as follows [41]: $n_{i} \oplus n_{l} = n_{i + l}$ (1) $n_{i} \otimes n_{l} = n_{i \times l}$ (2) $β n_{i} = n_{β \times i}$ (3) ${(n_{i})}^{β} = n_{i^{β}}$ (4)

2.2 VIKOR method

The VIKOR method [27] is from an aggregation function developed by Lp-metric. The Lp-metric over the option O_i (i = 1, 2,. . . , m) for compromise programming is evaluated as: $\begin{matrix} L_{pi} = {(\sum_{j = 1}^{h} {(w_{j} \frac{O^{+} - O_{ij}}{O^{+} - O^{-}})}^{p})}^{\frac{1}{p}} \\ (p \in [0, \infty]; i = 1, 2, . . ., m; j = 1, 2, . . ., h) \end{matrix}$ (5) where O_ij is an evaluation value for criteria P_j (j = 1, 2,. . . , h) with respect to alternative K_i (i = 1, 2,. . . , m), w_j (j = 1, 2,. . . , h) are the criteria weights, O⁺ = max O_ij, O^- = min O_ij are the best and worst values, respectively. Here, the metric L_1i and L_∞i are mentioned to originate ‘slightly important’ for the majority of criteria and ‘important’ of each individual criterion, and L_pi represents the distance from a solution to an ideal solution. Its most prominent feature is to maximize the effectiveness of the community and minimize the individual regret, and the choice of compromise solution is accepted by DMs.

2.3 The SVLNN

Definition 1. [17] Assume that N = (n₀, n₁, n₂,. . . , n_r) is a LTS with odd number r + 1. If v = (n_a, n_b, n_c) is defined for n_a, n_b, n_c ∈ N, where n_a,n_b and n_c express the TF, IF, and FF by LT, respectively, then v is called an SVLNN.

Definition 2. [17] Let v = (n_a, n_b, n_c), v₁ = (n_{a
₁}, n_{b
₁}, n_{c
₁}) and v₂ = (n_{a
₂}, n_{b
₂}, n_{c
₂}) be three SVLNNs and, then there are the following operational laws.

$\begin{matrix} v_{1} \oplus v_{2} & = & (n_{a_{1}}, n_{b_{1}}, n_{c_{1}}) \oplus (n_{a_{2}}, n_{b_{2}}, n_{c_{2}}) \\ = & (n_{a_{1} + a_{2} - \frac{a_{1} a_{2}}{r}}, n_{\frac{b_{1} b_{2}}{r}}, n_{\frac{c_{1} c_{2}}{r}}); \end{matrix}$ (6)

$\begin{array}{l} v_{1} \otimes v_{2} & = & (n_{a_{1}}, n_{b_{1}}, n_{c_{1}}) \otimes (n_{a_{2}}, n_{b_{2}}, n_{c_{2}}) \\ = & (n_{\frac{a_{1} a_{2}}{r}}, n_{b_{1} + b_{2} - \frac{b_{1} b_{2}}{r}}, n_{c_{1} + c_{2} - \frac{c_{1} c_{2}}{r}}); \end{array}$ (7)

$ρ v = ρ (n_{a}, n_{b}, n_{c}) = (n_{r - r {(1 - \frac{a}{r})}^{ρ}}, n_{r {(\frac{b}{r})}^{ρ}}, n_{r {(\frac{c}{r})}^{ρ}});$ (8)

$\begin{matrix} v^{ρ} & = & {(n_{a}, n_{b}, n_{c})}^{ρ} \\ = & (n_{r {(\frac{a}{r})}^{ρ}}, n_{r - r {(r - \frac{b}{r})}^{ρ}}, n_{r - r {(1 - \frac{c}{r})}^{ρ}}); \end{matrix}$ (9)

Example 1. Assume that N = (n₀, n₁, n₂,. . . , n₈) is a LTS. Let v = {n₆, n₃, n₂}, v₁ = {n₆, n₁, n₁} and v₂ = {n₇, n₃, n₄} be three SVLNNs and ρ = 0.5 , then we can get

$\begin{matrix} (1) v_{1} \oplus v_{2} & = & (n_{a_{1} + a_{2} - \frac{a_{1} a_{2}}{r}}, n_{\frac{b_{1} b_{2}}{r}}, n_{\frac{c_{1} c_{2}}{r}}) \\ = & (n_{6 + 7 - \frac{6 \times 7}{8}}, n_{\frac{1 \times 3}{8}}, n_{\frac{1 \times 4}{8}}) \\ = & (n_{7.75} n_{0.375} n_{0.125}); \end{matrix}$

$\begin{matrix} (2) v_{1} \otimes v_{2} & = & (n_{\frac{a_{1} a_{2}}{r}}, n_{b_{1} + b_{2} - \frac{b_{1} b_{2}}{r}}, n_{c_{1} + c_{2} - \frac{c_{1} c_{2}}{r}}) \\ = & (n_{\frac{6 \times 7}{8}}, n_{1 + 3 - \frac{1 \times 3}{8}}, n_{1 + 4 - \frac{1 \times 4}{8}}) \\ = & (n_{5.25} n_{3.625} n_{4.5}); \end{matrix}$

$\begin{matrix} (3) ρ v & = & (n_{r - r {(1 - \frac{a}{r})}^{ρ}}, n_{r {(\frac{b}{r})}^{ρ}}, n_{r {(\frac{c}{r})}^{ρ}}) \\ = & (n_{8 - 8 {(1 - \frac{6}{8})}^{0.5}}, n_{8 {(\frac{3}{8})}^{0.5}}, n_{8 {(\frac{2}{8})}^{0.5}}) \\ = & (n_{4}, n_{4.90}, n_{4}); \end{matrix}$ $\begin{matrix} (4) v^{ρ} & = & (n_{r {(\frac{a}{r})}^{ρ}}, n_{r - r {(1 - \frac{b}{r})}^{ρ}}, n_{r - r {(1 - \frac{c}{r})}^{ρ}}) \\ = & (n_{8 {(\frac{6}{8})}^{0.5}}, n_{8 - 8 {(1 - \frac{3}{8})}^{0.5}}, n_{8 - 8 {(1 - \frac{2}{8})}^{0.5}}) \\ = & (n_{4}, n_{1.68}, n_{4}); \end{matrix}$

According to the above operational rules, we can get the following properties: $\begin{matrix} (1) v_{1} \oplus v_{2} = v_{2} \oplus v_{1}; \end{matrix}$ (10) $\begin{matrix} (2) v_{1} \otimes v_{2} = v_{2} \otimes v_{1}; \end{matrix}$ (11) $\begin{matrix} (3) ρ v_{1} \oplus ρ v_{2} = ρ (v_{1} \oplus v_{2}); \end{matrix}$ (12) $\begin{matrix} (4) ρ_{1} v \oplus ρ_{2} v = (ρ_{1} + ρ_{2}) v; \end{matrix}$ (13) $\begin{matrix} (5) v_{1}^{ρ} \otimes v_{2}^{ρ} = {(v_{1} \otimes v_{2})}^{ρ}; \end{matrix}$ (14) $\begin{matrix} (6) v^{ρ_{1}} \otimes v^{ρ_{2}} = v^{(ρ_{1} + ρ_{2})}; \end{matrix}$ (15) $\begin{matrix} (7) (v_{1} \oplus v_{2}) \oplus v_{3} = v_{1} \oplus (v_{2} \oplus v_{3}); \end{matrix}$ (16) $\begin{matrix} (8) (v_{1} \otimes v_{2}) \otimes v_{3} = v_{1} \otimes (v_{2} \otimes v_{3}); \end{matrix}$ (17)

Definition 3. [17] Assume that N = { n₀, n₁, n₂, . . . , n_r} is a LTS with odd number r + 1. Let v = (n_a, n_b, n_c) be an SVLNN. Then the score and accuracy functions of v are defined as follows: $(1) Y (v) = \frac{2 r + a - b - c}{3 r}$ (18) Obviously, we have Y (v) ∈ [0, 1]. $(2) V (v) = \frac{a - c}{r}$ (19)

Obviously, we have V (v) ∈ [- 1, 1].

Definition 4. [17] Let v₁ = (n_{a
₁}, n_{b
₁}, n_{c
₁}) and v₂ = (n_{a
₂}, n_{b
₂}, n_{c
₂}) be two SVLNNs, then their comparative relations are defined as follows:

If Y (v₁) < Y (v₂), then v₁ ≺ v₂; item If Y (v₁) = Y (v₂), and (a) V (v₁) < V (v₂), then v₁ ≺ v₂; (b) V (v₁) = V (v₂), then v₁ = v₂.

Example 2. Assume that N = { n₀, n₁, n₂, n₃, n₄, n₅, n₆, n₇, n₈} is a LTS. Let v₁ = (n₅, n₂, n₃), v₂ = (n₇, n₁, n₃) and v₃ = (n₅, n₃, n₂) be three SVLNNs, then their score and accuracy functions are shown as follows: $Y (v_{1}) = \frac{2 \times 8 + 5 - 2 - 3}{24} = 0.667;$ $Y (v_{2}) = \frac{2 \times 8 + 7 - 1 - 3}{24} = 0.792;$ $and Y (v_{3}) = \frac{2 \times 8 + 5 - 3 - 2}{24} = 0.667 .$ $V (v_{1}) = \frac{5 - 3}{8} = 0.25; And$ $V (v_{3}) = \frac{5 - 2}{8} = 0.375 .$

According to Definition 7, their ranking order is v₂ ≻ v₃ ≻ v₁.

2.4 SVLNNWAA/SVLNNWGA operator

Definition 5. [17] Let v_i = (n_{a
_i}, n_{b
_i}, n_{c
_i}) (i = 1, 2,. . . , m) be a collection of SVLNNs, then we can define SVLNNWAA operators as follows: $\begin{matrix} SVLNNWAA (v_{1}, v_{2}, . . ., v_{r}) = \sum_{i = 1}^{r} w_{i} v_{i} \\ = (n_{r - r \prod_{i = 1}^{r} {(1 - \frac{a_{i}}{r})}^{w_{i}}}, n_{r \prod_{i = 1}^{r} {(\frac{b_{i}}{r})}^{w_{i}}}, n_{r \prod_{i = 1}^{r} {(\frac{c_{i}}{r})}^{w_{i}}}) \end{matrix}$ (20) Where w_i ∈ [0, 1] is the weight of v_i (i = 1, 2,. . . , m), and satisfying $\sum_{i = 1}^{m} w_{i} = 1$ .

Definition 6. [17] Let v_i = (n_{a
_i}, n_{b
_i}, n_{c
_i}) (i = 1, 2,. . . , m) be a collection of SVLNNs, then we can define SVLNNWGA operators as follows: $\begin{matrix} SVLNNWGA (v_{1}, v_{2}, . . ., v_{r}) = \prod_{i = 1}^{r} v_{i}^{w_{i}} \\ = (n_{r \prod_{i = 1}^{r} {(\frac{a_{i}}{r})}^{w_{i}}}, n_{r - r \prod_{i = 1}^{r} {(1 - \frac{b_{i}}{r})}^{w_{i}}}, n_{r - r \prod_{i = 1}^{r} {(1 - \frac{c_{i}}{r})}^{w_{i}}}) \end{matrix}$ (21) Where w_i ∈ [0, 1] is the weight of v_i (i = 1, 2,. . . , m), and satisfying $\sum_{i = 1}^{m} w_{i} = 1$ .

3 A new MAGDM method for SVLNN based on the extended VIKOR

In this section, we propose the extended VIKOR to solve MAGDM problems with SVLNNs.

3.1 A description of the decision problem

For a MAGDM problem, let K = (K₁, K₂,. . . , K_m) be a set of alternatives, P = (P₁, P₂,. . . , P_h) be a set of attributes and Z = (Z₁, Z₂,. . . , Z_t) be a set of DMs. w = (w₁, w₂,. . . , w_h) is the attribute weight vector, and $w_{j} \in [0, 1] (j = 1, 2, 3, . . ., h), \sum_{j = 1}^{h} w_{j} = 1$ . ω = (ω₁, ω₂,. . . , ω_t) is a weight vector of DMs, and ω_e ∈ [0, 1] (e = 1, 2,. . . , t), $\sum_{e = 1}^{t} ω_{e} = 1$ . The evaluation value of the alternative K_i under the attribute P_j by the DM Z_e is expressed as $O_{ij}^{e} = (n_{ij}^{a^{e}}, n_{ij}^{b^{e}}, n_{ij}^{c^{e}})$ in SVLNN, then a SVLNN decision matrix $A^{e} = {⌊ O_{ij}^{e} ⌋}_{m \times h}$ (e = 1, 2,. . . , t ; i = 1, 2,. . . , m ; j = 1, 2,. . . , h) is constructed, and the goal is to rank the alternatives.

3.2 Determining the expert weight

There are a great deal of MAGDM problems in which the expert weights are unknown, how to determine these weights is an important research topic. According to the traditional idea of maximizing the closeness between each alternative and the ideal solution [41], an optimal model is established to obtain the expert weights.

Definition 7. [42] Suppose the domain is U, the mapping is cl : F (U). A, B and C are any three fuzzy set, A ∈ U, B ∈ U and C ∈ U, A = {x_a1, x_a2,. . . , x_aj}, B = {x_b1, x_b2,. . . , x_bj} and C = {x_c1, x_c2,. . . , x_cj}. F (U) → [0, 1] meets the conditions:

cl (A, A) = 1, cl (∅ , U) = 0;

cl (A, B) = cl (B, A);

A ⊆ B ⊆ C ⇒ cl (A, C) ≤ cl (A, B) ∩ cl (B, C).

The traditional closeness formula is defined as follows [42]: $cl (A, B) = \frac{\sum_{j = 1}^{h} x_{aj} \land x_{bj}}{\sum_{j = 1}^{h} x_{aj} \lor x_{bj}}$ (22)

Then cl (A, B) is called closeness between A and B.

Where cl (A, B) ∈ [0, 1], the closer to 0, the greater the distance between the two sets A and B, the thinner the relationship, otherwise, the closer the relationship between the two sets A and B.

Definition 8. For any three SVLNNs, v₁ = (n_{a
₁}, n_{b
₁}, n_{c
₁}), v₂ = (n_{a
₂}, n_{b
₂}, n_{c
₂}), v₃ = (n_{a
₃}, n_{b
₃}, n_{c
₃}), we can define the closeness between v₁ and v₂ as follows.

$\begin{matrix} S (v_{1}, v_{2}) \\ = \frac{\sum_{j = 1}^{h} (n_{a_{1}} \land n_{a_{2}})}{3 \sum_{j = 1}^{h} (n_{a_{1}} \lor n_{a_{2}})} & + & \frac{\sum_{j = 1}^{h} (n_{b_{1}} \land n_{b_{2}})}{3 \sum_{j = 1}^{h} (n_{b_{1}} \lor n_{b_{2}})} \\ + & \frac{\sum_{j = 1}^{h} (n_{c_{1}} \land n_{c_{2}})}{3 \sum_{j = 1}^{h} (n_{c_{1}} \lor n_{c_{2}})} \end{matrix}$ (23)

Where, ∧ represents the operation of selecting the minimum number. ∨ represents the operation of selecting the maximum number.

Obviously, (23) can meet the properties (i)–(iii) in definition 10.

Then we propose the method of determining the expert weights as follows.

Step 1. The decision matrix A^e provided by expertis converted into the following matrix: $D^{j} = [\begin{matrix} X_{11}^{j} & X_{12}^{j} & \cdot \cdot \cdot & X_{1 t}^{j} \\ X_{21}^{j} & X_{22}^{j} & \cdot \cdot \cdot & X_{2 t}^{j} \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ X_{m 1}^{j} & X_{m 2}^{j} & \cdot \cdot \cdot & X_{mt}^{j} \end{matrix}]$ (24)

Where $X_{ie}^{j} = O_{ij}^{e} = (n_{i, a^{e}}^{j}, n_{i, b^{e}}^{j}, n_{i, c^{e}}^{j})$ (e = 1, 2,. . . , t ; i = 1, 2,. . . , m ; j = 1, 2,. . . , h).

Step 2. Calculate the average value of expert, we have ${\bar{v}}_{i}^{j} = (\bar{n_{ai}^{j}}, \bar{n_{bi}^{j}}, \bar{n_{ci}^{j}})$ (25)

Where, ${\bar{n}}_{ai}^{j} = \frac{1}{t} \sum_{e = 1}^{t} (n_{i, a^{e}}^{j})$ , ${\bar{n}}_{bi}^{j} = \frac{1}{t} \sum_{e = 1}^{t} (n_{i, b^{e}}^{j})$ , ${\bar{n}}_{ci}^{j} = \frac{1}{t} \sum_{e = 1}^{t} (n_{i, c^{e}}^{j})$ .

Step 3. Calculate the closeness degree matrix which can be constructed by using the formula (23) and (25) for the attribute P_j, shown as follows: $S^{j} = [\begin{matrix} S (X_{11}^{j}, {\bar{v}}_{1}^{j}) & S (X_{12}^{j}, {\bar{v}}_{1}^{j}) & \cdot \cdot \cdot & S (X_{1 t}^{j}, {\bar{v}}_{1}^{j}) \\ S (X_{21}^{j}, {\bar{v}}_{2}^{j}) & S (X_{22}^{j}, {\bar{v}}_{2}^{j}) & \cdot \cdot \cdot & S (X_{2 t}^{j}, {\bar{v}}_{2}^{j}) \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ S (X_{m 1}^{j}, {\bar{v}}_{m}^{j}) & S (X_{m 2}^{j}, {\bar{v}}_{m}^{j}) & \cdot \cdot \cdot & S (X_{mt}^{j}, {\bar{v}}_{m}^{j}) \end{matrix}]$ (26)

Where, S is defined by (23).

Step 4. The weight of the expert Z_e can be calculated as follows: $ω_{e} = \frac{\sum_{j = 1}^{h} \sum_{i = 1}^{m} S (X_{ie}^{j}, {\bar{v}}_{i}^{j})}{\sum_{e = 1}^{t} \sum_{j = 1}^{h} \sum_{i = 1}^{m} S (X_{ie}^{j}, {\bar{v}}_{i}^{j})}$ (27)

3.3 Determining the attribute weights

Based on the Euclidean distance formula, we use the maximum deviation method to determine attribute weights. When the difference between the maximum attribute value and the minimum attribute value of the same attribute is used to represent the fluctuation amplitude of the attribute, this difference is called the maximum deviation [43]. According to the maximum deviation value of each attribute, the weight of the attribute is determined, which is called the maximum deviation fixed weight principle.

Definition 9. Assume that N ={ n₀, n₁, n₂,. . . , n_r } is a LTS. Let v₁ = (n_{a
₁}, n_{b
₁}, n_{c
₁}) and v₂ = (n_{a
₂}, n_{b
₂}, n_{c
₂}) be two SVLNNs, then its standardized Euclidean distance can be defined as follows:

$\begin{matrix} H (v_{1}, v_{2}) = \\ | \sqrt{((a_{1} - a_{2})^{2} + (b_{1} - b_{2})^{2} + (c_{1} - c_{2})^{2})} | \end{matrix}$ (28)

The method of determining the attribute weights is proposed as follows.

Step 1. In a SVLNN decision matrix A^e = ${⌊ O_{ij}^{e} ⌋}_{m \times h}$ (e = 1, 2,. . . , t ; i = 1, 2,. . . , m ; j = 1, 2,. . . , h) (where $O_{ij}^{e} = (n_{ij}^{a^{e}}, n_{ij}^{b^{e}}, n_{ij}^{c^{e}})$ ), the maximum decision value is chosen as the fixed value. The calculation method is as follows: $O_{i}^{e +} = (\overset{h}{max_{j = 1}} (O_{ij}^{ae}), \overset{h}{min_{j = 1}} (O_{ij}^{be}), \overset{h}{min_{j = 1}} (O_{ij}^{ce}))$ (29)

Step 2. The Euclidean distance value of the information is evaluated by using formulas (28) and (29), the Euclidean distance matrix of SVLNN is established as follows: $G = [\begin{matrix} H (O_{11}^{e}, O_{1}^{e +}) & H (O_{12}^{e}, O_{1}^{e +}) & \cdot \cdot \cdot & H (O_{1 h}^{e}, O_{1}^{e +}) \\ H (O_{21}^{e}, O_{2}^{e +}) & H (O_{22}, O_{2}^{e +}) & \cdot \cdot \cdot & H (O_{2 h}, O_{2}^{e +}) \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ \cdot & \cdot & \cdot \cdot \cdot & \cdot \\ H (O_{m 1}^{e}, O_{m}^{e +}) & H (O_{m 2}^{e}, O_{m}^{e +}) & \cdot \cdot \cdot & H (O_{mh}^{e}, O_{m}^{e +}) \end{matrix}]$ (30)

Step 3. According to the maximum deviation fixed weight principle, calculate the maximum deviation of the attribute on the distance value [44]:

$\begin{matrix} M_{j} & = {max}_{i = 1}^{m} (\sum_{e = 1}^{t} H (O_{ij}^{e}, O_{i}^{e +})) \\ - {min}_{i = 1}^{m} (\sum_{e = 1}^{t} H (O_{ij}^{e}, O_{i}^{e +})) \end{matrix}$ (31)

Step 4. After normalizing the upper coefficients, the weights obtained are as follows: $w_{j} = M_{j} / - \sum_{j = 1}^{h} M_{j}$ (32)

3.4 Decision steps of extended VIKOR

The specific steps of the extended VIKOR method in the SVLNN environment are shown as follows:

Step 1. Normalization for attribute values.

According to attribute types, if one attribute is cost index, it should be converted to benefit type by $O_{ij}^{e} = (n_{ij}^{c^{e}}, 1 - n_{ij}^{b^{e}}, n_{ij}^{a^{e}})$ .

Step 2. Determining the expert weights.

According to section 3.2, we can get the expert weight vector ω = (ω₁, ω₂, ⋯ , ω_t).

Step 3. Obtain the matrix W_ij by the matrix ${(O_{ij}^{e})}_{m \times h} (e = 1, 2, . . ., t; i = 1, 2, . . ., m; j = 1, 2,$ . . . , h), where $O_{ij}^{e} = (n_{ij}^{a^{e}}, n_{ij}^{b^{e}}, n_{ij}^{c^{e}})$ is an integrated SVLNN, by using the follow SVLNNWAA operator or SVLNNWGA operator [17]: $\begin{matrix} W_{ij} & = & SVLNNWAA (O_{ij}^{1} O_{ij}^{2}, . . ., O_{ij}^{t}) \\ = & \sum_{e = 1}^{t} ω_{e} O_{ij}^{e} = (n_{t - t \prod_{e = 1}^{t} {(1 - \frac{a_{ij}^{e}}{t})}^{ω_{e}}}, \\ n_{t \prod_{e = 1}^{t} {(\frac{b_{ij}^{e}}{t})}^{ω_{e}}}, n_{t \prod_{e = 1}^{t} {(\frac{c_{ij}^{e}}{t})}^{ω_{e}}}) \end{matrix}$ (33) or

$\begin{array}{l} W_{i j} & = & S V L N N W G A (O_{i j}^{1} O_{i j}^{2}, ..., O_{i j}^{t}) \\ = & \sum_{e = 1}^{t} {(O_{i j}^{e})}^{w^{e}} = (n_{t {\prod_{e = 1}^{t} (\frac{a_{i j}^{e}}{t})}^{w e}}, n_{t - t {\prod_{e = 1}^{t} (1 - \frac{b_{i j}^{e}}{t})}^{w_{e}}}, n_{t - t {\prod_{e = 1}^{t} (1 - \frac{c_{i j}^{e}}{t})}^{w_{e}}}) \end{array}$ (34)

Step 4. Determining the attribute weights.

According to section 3.3, we can get the attribute weight vector w = (w₁, w₂, ⋯ , w_h).

Step 5. Determine the positive and negative ideal solution. $W^{+} = (W_{a}^{+}, W_{b}^{+}, W_{c}^{+})$ (35) where $W_{a}^{+} = max (W_{ij}^{a})$ , $W_{b}^{+} = min (W_{ij}^{b}),$ $W_{c}^{+} = min (W_{ij}^{c});$ $W^{-} = (W_{a}^{-}, W_{b}^{-}, W_{c}^{-})$ (36) Where $W_{a}^{-} = min (W_{ij}^{a}), W_{b}^{-} = max (W_{ij}^{b}),$ $W_{c}^{-} = max (W_{ij}^{c});$

Step 6. Calculate group utility values S_i and individual regrets values R_i. $S_{i} = (S_{i}^{a}, S_{i}^{b}, S_{i}^{c})$ (37) $S_{i}^{a} = \sum_{j = 1}^{h} w_{j} \frac{W_{a}^{+} - W_{ij}^{a}}{W_{a}^{+} - W_{a}^{-}}$ (38) $S_{i}^{b} = \sum_{j = 1}^{h} w_{j} \frac{W_{b}^{+} - W_{ij}^{b}}{W_{b}^{+} - W_{b}^{-}}$ (39) $S_{i}^{c} = \sum_{j = 1}^{h} w_{j} \frac{W_{c}^{+} - W_{ij}^{c}}{W_{c}^{+} - W_{c}^{-}}$ (40) $R_{i} = (R_{i}^{a}, R_{i}^{b}, R_{i}^{c})$ (41) $R_{i}^{a} = max (w_{j} \frac{W_{a}^{+} - W_{ij}^{a}}{W_{a}^{+} - W_{a}^{-}})$ (42) $R_{i}^{b} = max (w_{j} \frac{W_{b}^{+} - W_{ij}^{b}}{W_{b}^{+} - W_{b}^{-}})$ (43) $R_{i}^{c} = max (w_{j} \frac{W_{c}^{+} - W_{ij}^{c}}{W_{c}^{+} - W_{c}^{-}})$ (44)

Step 7. Calculate the evaluation value Q_i. $Q_{i} = (Q_{i}^{a}, Q_{i}^{b}, Q_{i}^{c})$ (45) $Q_{i}^{a} = δ \frac{S_{i}^{a} - S_{i}^{a -}}{S_{i}^{a +} - S_{i}^{a -}} + (1 - δ) \frac{R_{i}^{a} - R_{i}^{a -}}{R_{i}^{a +} - R_{i}^{a -}}$ (46) $Q_{i}^{b} = δ \frac{S_{i}^{b} - S_{i}^{b -}}{S_{i}^{b +} - S_{i}^{b -}} + (1 - δ) \frac{R_{i}^{b} - R_{i}^{b -}}{R_{i}^{b +} - R_{i}^{b -}}$ (47) $Q_{i}^{c} = δ \frac{S_{i}^{c} - S_{i}^{c -}}{S_{i}^{c +} - S_{i}^{c -}} + (1 - δ) \frac{R_{i}^{c} - R_{i}^{c -}}{R_{i}^{c +} - R_{i}^{c -}}$ (48) Where $S_{i}^{a +} = max (S_{i}^{a})$ , $S_{i}^{a -} = min (S_{i}^{a})$ ; $S_{i}^{b +} = max (S_{i}^{b})$ , $S_{i}^{b -} = min (S_{i}^{b})$ ; $S_{i}^{c +} = max (S_{i}^{c})$ , $S_{i}^{c -} = min (S_{i}^{c})$ . δ is the group utility weight, and δ ∈ [0, 1]. In general, δ = 0.5.

Step 8. Calculate the values $\bar{S_{i}}$ , $\bar{R_{i}}$ and $\bar{Q_{i}}$ , using the following equation. $\bar{S_{i}} = \frac{S_{i}^{a} + S_{i}^{b} + S_{i}^{c}}{3}$ (49) $\bar{R_{i}} = \frac{R_{i}^{a} + R_{i}^{b} + R_{i}^{c}}{3}$ (50) $\bar{Q_{i}} = \frac{Q_{i}^{a} + Q_{i}^{b} + Q_{i}^{c}}{3}$ (51)

Step 9. Definite compromise solution and sort by size.

Step 10. Determine a compromise solution. The alternative denoted K₍₁₎ is the best one using the measure $\bar{Q}$ (minimum), so it is considered a promising solution if the following two conditions are satisfied:

Cond1: $\bar{Q} (K_{(2)}) - \bar{Q} (K_{(1)}) \geq \frac{1}{r - 1}$ (If r ≤ 4, then $\frac{1}{r - 1} = \frac{1}{4}$ [32]), K₍₂₎ is the solution of the second smallest value in $\bar{Q_{i}} (i = 1, 2, . . ., m)$ , $\bar{Q} (K_{(1)})$ and $\bar{Q} (K_{(2)})$ are the compromise evaluation values of scheme K₍₁₎ and K₍₂₎, respectively;

Cond2: In the sorting of $\bar{S_{i}}$ and $\bar{R_{i}}$ , K₍₁₎ has at least one sort of the nearest one. If there is a condition not satisfied, then get a compromise set: (1) If only condition 1 is satisfied, the compromise set is {K₍₁₎, K₍₂₎}; (2) If only condition 2 is satisfied, At this point, a maximum M can be determined in the i value satisfying inequality $\bar{Q} (K_{(i)}) - \bar{Q} (K_{(1)}) < \frac{1}{r - 1}$ , the compromise solution set is { K₍₁₎, K₍₂₎,. . . , K_(r) } .

4 Application

4.1 An illustrative example

In this section, we use an example (adapted from [17]) to show the application of the VIKOR method in SVLNN environment.

In the process industry, the occurrence of failures is inevitable, and the decision of fault handling is also worth studying. In order to solve the failure to determine the best processing point of the decision-making program, take an industrial explosion as an example, K₁, K₂, K₃, K₄ correspond to three fault handling points, and they are evaluated by three attributes (denoted by SVLNN): (1) P₁ expresses the staff action; (2) P₂ represents the storage capacity of the device;(3) P₃ represents the personnel dispatching force, which weight vector is w = (w₁, w₂,. . . , w_h). Z¹, Z², Z³ represent three experts to form decision groups, Then, three DMs are invited to evaluate K₁, K₂, K₃, K₄ based on three attributes, and the importance of the three DMs is given as a weight vector ω = (ω₁, ω₂,. . . , ω_e). The goal is to be ranking K₁, K₂, K₃, K₄.

The evaluation information of each decision maker is expressed as the following SVLNN decision matrix [17]:

$Z^{1} = [\begin{matrix} (n_{6}, n_{1}, n_{2}) (n_{7}, n_{2}, n_{1}) (n_{6}, n_{2}, n_{2}) \\ (n_{7}, n_{1}, n_{1}) (n_{7}, n_{3}, n_{2}) (n_{7}, n_{2}, n_{1}) \\ (n_{6}, n_{2}, n_{2}) (n_{7}, n_{1}, n_{1}) (n_{6}, n_{2}, n_{2}) \\ (n_{7}, n_{1}, n_{2}) (n_{7}, n_{2}, n_{3}) (n_{7}, n_{2}, n_{1}) \end{matrix}],$ $Z^{2} = [\begin{matrix} (n_{6}, n_{1}, n_{2}) (n_{6}, n_{1}, n_{1}) (n_{4}, n_{2}, n_{3}) \\ (n_{7}, n_{2}, n_{3}) (n_{6}, n_{1}, n_{1}) (n_{4}, n_{2}, n_{3}) \\ (n_{5}, n_{1}, n_{2}) (n_{5}, n_{1}, n_{2}) (n_{5}, n_{4}, n_{2}) \\ (n_{6}, n_{1}, n_{1}) (n_{5}, n_{1}, n_{1}) (n_{5}, n_{2}, n_{1}) \end{matrix}],$ $Z^{3} = [\begin{matrix} (n_{7}, n_{3}, n_{4}) (n_{7}, n_{3}, n_{3}) (n_{5}, n_{2}, n_{5}) \\ (n_{6}, n_{3}, n_{4}) (n_{5}, n_{1}, n_{2}) (n_{6}, n_{2}, n_{3}) \\ (n_{7}, n_{2}, n_{4}) (n_{6}, n_{1}, n_{2}) (n_{7}, n_{2}, n_{4}) \\ (n_{7}, n_{2}, n_{3}) (n_{5}, n_{2}, n_{1}) (n_{6}, n_{1}, n_{1}) \end{matrix}],$

The following steps are given to solve this MAGDM problem.

Step 1. Because all attribute values are benefit type, there is no need to do normalization.

Step 2. Determine the expert weight ω by using the Equations (24)–(27). $ω_{1} = 0.3340, ω_{2} = 0.3355, ω_{3} = 0.3305 .$ Step 3. Calculate integrated matrix (W_ij) _m×h (i = 1, 2,. . . , m ; j = 1, 2,. . . , h) by using Equation (33): $\begin{matrix} {(W_{ij})}_{m \times h} = \\ [\begin{matrix} (n_{6 . . 4095}, n_{1.4378}, n_{2.5149}) \\ (n_{6.7426}, n_{1.8142}, n_{2.2859}) \\ (n_{6.1777}, n_{1.5850}, n_{2.5149}) \\ (n_{6.7382}, n_{1.2574}, n_{1.8123}) \end{matrix} \begin{matrix} (n_{6.7382}, n_{1.8123}, n_{1.4378}) \\ (n_{6.1858}, n_{1.4433}, n_{1.5850}) \\ (n_{6.1821}, n_{1.0000}, n_{1.5867}) \\ (n_{5.9214}, n_{1.5850}, n_{1.4433}) \end{matrix} \begin{matrix} (n_{5.1145}, n_{2.0000}, n_{3.1019}) \\ (n_{5.9979}, n_{2.0000}, n_{2.0786}) \\ (n_{6.1777}, n_{2.5236}, n_{2.5149}) \\ (n_{6.1821}, n_{1.5905}, n_{1.4457}) \end{matrix}] \end{matrix}$

Step 4. Determine the attribute weight w by using the Equations (29)–(32). $w_{1} = 0.255, w_{2} = 0.3285, w_{3} = 0.4165 .$

Step 5. Determine the positive and negative ideal solution by using Equations (35) and (36). $\begin{matrix} (1) W^{+} = (W_{a}^{+}, W_{b}^{+}, W_{c}^{+}) = (n_{6.7426}, n_{1.0000}, n_{1.4378}); \end{matrix}$ $\begin{matrix} (2) W^{-} = (W_{a}^{-}, W_{b}^{-}, W_{c}^{-}) = (n_{5.1145}, n_{2.5236}, n_{3.1019}) . \end{matrix}$

Step 6. Calculate group utility values S_i and individual regrets values R_i by using Equations (37)–(44). $S_{1} = (0.4694, 0.5218, 0.5816); S_{2} = (0.3028, 0.5053, 0.3195);$ $S_{3} = (0.3461, 0.5144, 0.4641); S_{4} = (0.3098, 0.3296, 0.0605) .$ $R_{1} = (0.4165, 0.2734, 0.4165); R_{2} = (0.1905, 0.2734, 0.1604);$ $R_{3} = (0.1445, 0.4165, 0.2696); R_{4} = (0.1657, 0.1614, 0.0574) .$

Step 7. Calculate the evaluation value Q_i by using Equations (45)–(48). $Q_{1} = (1.0000, 0.7195, 1.0000), Q_{2} = (0.0846, 0.6605, 0.3919)$ $Q_{3} = (0.1300, 0.9639, 0.6827), Q_{4} = (0.0600, 0.0000, 0.0000)$

Step 8. Calculate the values $\bar{S_{i}}$ , $\bar{R_{i}}$ and $\bar{Q_{i}}$ by using Equations (49)–(51). 1) $\bar{S_{1}} = 0.5243, \bar{S_{2}} = 0.3759, \bar{S_{3}} = 0.4415, \bar{S_{4}} = 0.2333$ ; 2) $\bar{R_{1}} = 0.3688, \bar{R_{2}} = 0.2081, \bar{R_{3}} = 0.2769, \bar{R_{4}} = 0.1282$ ; 3) $\bar{Q_{1}} = 0.9065, \bar{Q_{2}} = 0.3790, \bar{Q_{3}} = 0.5922,$ $\bar{Q_{4}} = 0.0200$ .

Step 9. Definite compromise solution which is shown in Table 1.

Table 1
Ranking of alternatives by $\bar{S_{i}}$ , $\bar{R_{i}}$ and $\bar{Q_{i}}$ in ascending order

$\bar{S_{i}}$ $\bar{R_{i}}$ $\bar{Q_{i}}$

K ₁ 0.5243 0.3688 0.9065

K ₂ 0.3759 0.2081 0.3790

K ₃ 0.4415 0.2769 0.5922

K ₄ 0.2333 0.1282 0.0200

The ranking order K₄ ≻ K₂ ≻ K₃ ≻ K₁ K₄ ≻ K₂ ≻ K₃ ≻ K₁ K₄ ≻ K₂ ≻ K₃ ≻ K₁

	$\bar{S_{i}}$	$\bar{R_{i}}$	$\bar{Q_{i}}$
K ₁	0.5243	0.3688	0.9065
K ₂	0.3759	0.2081	0.3790
K ₃	0.4415	0.2769	0.5922
K ₄	0.2333	0.1282	0.0200
The ranking order	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K₄ ≻ K₂ ≻ K₃ ≻ K₁

Step 10. Determine a compromise solution. $\begin{matrix} \bar{Q} (K_{2}) - \bar{Q} (K_{4}) & = 0.3790 - 0.0200 \\ = 0.3590 \geq \frac{1}{8 - 1} = 0.1429, \end{matrix}$ K₄ is the best ranked using the measure $\bar{Q_{i}} (i = 1, 2, . . ., m)$ , K₂ is the solution of the second smallest value in $\bar{Q_{i}} (i = 1, 2, . . ., m)$ , so the condition 1 is satisfied. At the same time, in the sorting of $\bar{S_{i}}$ and $\bar{R_{i}}$ , K₄ has at least one sort of the nearest one, so the condition 2 is satisfied. Therefore, scenario K₄ is the best compromise option for alternative low carbon suppliers.

4.2 Sensitivity analysis

In actual decision-making, experts may have different decision-making attitude, and then take a different compromise coefficient, that is, δ can choose the number between [0, 1]. In the VIKOR method, the compromise evaluation value of each alternative may be affected by the compromise coefficient δ. To consider the influence of different value δ on the decision-making results, sensitivity analysis is performed by setting different δ values; the ranking results are shown in Table 2 and Fig. 1.

Fig. 1

The change of the compromise value $\bar{Q}$ with the compromise coefficient δ.

Table 2

The compromise value $\bar{Q}$ obtained with the change of δ

δ	$\bar{Q} (K_{1})$	$\bar{Q} (K_{2})$	$\bar{Q} (K_{3})$	$\bar{Q} (K_{4})$	Rank order	Eclectic program
0.0	0.8130	0.2983	0.5303	0.0260	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K₄, K₂
0.1	0.8317	0.3144	0.5427	0.0248	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K₄, K₂
0.3	0.8691	0.3467	0.5674	0.0224	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K ₄
0.5	0.9065	0.3790	0.5922	0.0200	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K ₄
0.7	0.9439	0.4113	0.6169	0.0176	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K₄, K₂
0.9	0.9813	0.4435	0.6417	0.0152	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K₄, K₂
1.0	1.0000	0.4597	0.6540	0.0140	K₄ ≻ K₂ ≻ K₃ ≻ K₁	K₄, K₂

From Fig. 1, it can be seen that under different value δ, the ranking results are stable; the optimal scheme remains unchanged, and δ has a limited impact on the ranking results. To sum up, the VIKOR decision method used in this paper is relatively insensitive to the coefficient δ, and the decision results have good stability.

4.3 Comparative analysis

To verify the effectiveness of the proposed method, we compare this method with the literature [2] (TOPSIS method) and [17] (WAA method), and we obtain the same ranking result, i.e., K₄ ≻ K₂ ≻ K₃ ≻ K₁ (Table 3). From this, the decision method proposed in this paper is effective.

In addition, from Fig. 2, it can be seen that the VIKOR method fluctuates greatly, and the ranking results can be clearly seen. The WAA and TOPSIS methods tend to be stable and the sorting results are not obvious.

Table 3
Comparison of the different methods

VIKOR TOPSIS WAA

Ranking Ranking Ranking

K ₁ 0.9065 4 0.9279 4 0.7528 4

K ₂ 0.3790 2 0.8866 2 0.7777 2

K ₃ 0.5922 3 0.9113 3 0.7613 3

K ₄ 0.0200 1 0.8654 1 0.8060 1

	VIKOR	TOPSIS	WAA
K ₁	0.9065	4	0.9279	4	0.7528	4
K ₂	0.3790	2	0.8866	2	0.7777	2
K ₃	0.5922	3	0.9113	3	0.7613	3
K ₄	0.0200	1	0.8654	1	0.8060	1

Fig. 2

Comparison of value fluctuations about WAA, TOPSIS and VIKOR.

In a word, our proposed fuzzy MAGDM method in this paper is the generalization of the existing decision-making method and can deal with the MAGDM with SVLNN information in which the expert weight and attribute weight are all unknown.

5 Conclusions

Because the SVLNN can clearly express the linguistic information, VIKOR algorithm has more advantages than the other methods in selecting the compromise solution, in this paper, we briefly reviews the basic concept of LTs, neutrosophic sets (NSs), single-valued linguistic neutrosophic number (SVLNN), and the closeness degree, then we put forward the closeness degree between the linguistic neutrosophic numbers (LNNs) and the distance formula between LNNs, and then we use them to determine the weights of experts and attributes. Based on the traditional VIKOR, a MAGDM method based on the extended VIKOR method is proposed under SVLNN environment. An illustrative example was given to demonstrate the steps and effectiveness of the proposed method by the comparison with existing methods. In the future, it is necessary to apply the proposed method to solve the real decision-making problems, such as evaluations on population resources and environment [45 –48] or Chinese culture [44]. And take VIKOR into PFS or IVFSS [49, 50] in the linguistic neutrosophic environment.

Footnotes

Acknowledgment

This work was supported by Key R & D project of Shandong Province, China (2017XCGC0605).

References

Chen

, LIU

and PEI

, An Approach to Multiple Attribute Group Decision Making Based on Linguistic Intuitionistic Fuzzy Numbers, International Journal of Computational Intelligence Systems 8(4) (2015), 747–760.

, An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers, J. Intell. Fuzzy Syst 28 (2015), 247–255.

Wan

S.P.

and Li

D.F.

, Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees, Omega-International Journal of Management Science 41 (2013), 925–940.

Kahraman

and Cebi

, A New Multiple Attribute Decision Making Method: Hierarchical Fuzzy Axiomatic Design, Expert Systems with Applications 36 (2009), 4848–4861.

Vinodh

and Vimal

K.E.K.

, Thirty criteria based leanness assessment using fuzzy logic approach,–, International Journal of Advanced Manufacturing Technology 60(9–12) (2012), 1185–1195.

Liu

P.D.

and Guan

Z.L.

, A Hybrid Multiple Attribute Decision Making Method Based on Binary Semantics, Control and Decision Making 24(7) (2009), 1074–1078.

Smarandache

, Neutrosophic Masses & Indeterminate Models Applications to Information Fusion, Advanced Mechatronic Systems (2009), 18–21.

Zhu

W.D.

and Zhou

G.Z.

, Group Decision-making Method Based on Two-dimensional Linguistic Evaluation Information, Systems Engineering 27(2) (2009), 113–118.

Aggarwal

, Biswas

and Ansari

A.Q.

, Neutrosophic Modeling and Control, IEEE Computer & Communication Technology (2010), 718–723.

10.

Liu

P.D.

and Zhang

X.A.

, Multi-attribute Decision-making Method Based on Interval Gray Linguistic Variables Mixed Aggregation Weighted Integration Operator, Control and Decision-Making 26(5) (2011), 743–747.

11.

Zadeh

L.A.

, Fuzzy sets, Inf Control 8(3) (1965), 338–353.

12.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets Syst 20(1986), 87–96.

13.

Ali

, Smarandache

and Vladareanu

, Neutrosophic set and Logic, mss, 1995, http://fs.allup.unm.edu/neutrosophy.htm.

14.

Wang

, Smarandache

and Zhang

Y.Q.

, R, Sunderraman Single valued neutrosophic sets, Multispace and Multistructure 4 (2010), 410–413.

15.

, Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment, International Journal of General Systems 42(4) (2013), 386–394.

16.

Smarandache

, Neutrosophy, Neutrosophic probability, set, and logic. Analytic synthesis and synthetic analysis, Philosophy, 1998.

17.

Fang

Z.B.

and Ye

, Multiple Attribute Group Decision Making Based on Linguistic Neutrosophic Numbers, Symmetry 9(111) (2017), 1–12.

18.

Merigó

J.M.

and Gil-Lafuente

A.M.

, Generalization of the linguistic aggregation operator and its application in decision making, Journal of Systems Engineering and Electronics 22(4) (2011), 593–603.

19.

Y.Y.

, Zhang

H. Y.

and Wang

J. Q.

, Linguistic neutrosophic sets and its application to multi-criteria decision-making problems, 7(2) (2017).

20.

Benayoun

, Roy

and Sussman

, ELECTRE: Une méthode pour guider le choix en présence de points de vue multiples, Rev.franaise Informat.recherche Opérationnelle 3 (1966), 31–56.

21.

, Zhang

H.Y.

and Wang

J. Q.

, Selecting an outsourcing provider based on the combined MABAC-ELECTRE method using single-valued neutrosophic linguistic sets, Computers & Industrial Engineering (2018), 120.

22.

Peng

and Liu

, Algorithms for Neutrosophic Soft Decision Making Based on EDAS and New Similarity Measure, Journal of Intelligent & Fuzzy Systems 32(1) (2017), 955–968.

23.

Peng

X.D.

and Dai

J.G.

, Algorithms for interval neutrosophic multiple attribute decision making based on MABAC, Similarity Measure and EDAS 7(5) (2018).

24.

Y.Y.

, Wang

J. Q.

and Wang

T. L.

, A Linguistic Neutrosophic Multi-criteria Group Decision-Making Approach with EDAS Method, Arabian Journal for Science & Engineering, 1–13, doi: 10.1007/s13369-018-3487-5.

25.

Merigó

J.M.

, Casanovas

and Martìnez

, Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of evidence, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 18(3) (2010), 287–304.

26.

Hwang

C.L.

, Yoon

, Multiple Attribute Decision Making, Springer Berlin Heidelberg, 1981.

27.

Opricovic

, Multi-Criteria Optimization of Civil Engineering Systems, Faculty of Civil Engineering, Belgrade, 1998.

28.

Opricovic

and Tzeng

, Multicriteria Planning of Post-Earthquake Sustainable Reconstruction, Computer-Aided Civil and Infrastructure Engineering 17(3) (2010), 211–220.

29.

Opricovic

and Tzeng

G.H.

, Extended VIKOR method in comparison with outranking methods, Eur. J. Oper. Res 178 (2007), 514–529.

30.

Opricovic

, Fuzzy VIKOR with an application to water resources planning, Expert Systems with Application 38 (2011), 12983–12990.

31.

Sayadi

M.K.

, Heydari

and Shahanaghi

, Extension of VIKOR method for decision making problem with interval numbers, Applied Mathematical Modelling 33 (2009), 2257–2262.

32.

Wan

S.P.

, Wang

Q. Y.

and Dong

J. Y.

, The extended VIKOR method for multi-attribute group decision making with triangular intuitionistic fuzzy numbers, Elsevier Science Publishers B. V., 2013.

33.

Peng

X.D.

and Dai

J.G.

, Approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function, Neural Computing and Applications 29(10) (2018), 939–954.

34.

Tian

Z.P.

, Wang

J. Q.

and Zhang

H. Y.

, An integrated approach for failure mode and effects analysis based on fuzzy best-worst, relative entropy, and VIKOR methods, Applied Soft Computing, 2018.

35.

Shen

K.W.

, Wang

J. Q.

. Z-VIKOR method based on a new weighted comprehensive distance measure of Z-number and its application, IEEE Transactions on Fuzzy Systems 99 (2018), 1–1.

36.

Dey

, Broumi

, Le

H.S.

, Bakali

, Talea

and Smarandache

, A new algorithm for finding minimum spanning trees with undirected neutrosophic graphs, Granular Computing 3 (2018), 1–7.

37.

Broumi

, Smarandache

, Talea

and Bakali

, Single valued neutrosophic graphs: Degree, order and size// IEEE International Conference on Fuzzy Systems, IEEE, 2016.

38.

, Li

D.F.

, Merigó

J. M.

, Fang

. Mapping development of linguistic decision making studies, Journal of Intelligent & Fuzzy Systems 30(5) (2016), 2727–2736.

39.

Liu

P.D.

, Research on the Method and Application of Enterprise Informatization Level Evaluation Based on Fuzzy Multiple Attribute Decision, Beijing: Beijing Jiaotong University, 2009, 65–69.

40.

Wang

J.Q

, Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators, Neural Computer Applic (2016), 1–19.

41.

Wei

G.W.

, Luo

Y.J.

and Yao

H. S.

, PLWC-EOWA Operator and Its Application to Decision Making under uncertain linguistic setting, The Fifth Wuhan International Conference on E-Business Innovation Management Track 5 (2006), 2369–2373.

42.

Smarandache

, A Unifying field in logics: Neutrosophy: Neutrosophic Logic, Multiple-Valued Logic: An International Journal 8(3) (2002), 385–438.

43.

Liu

X.M.

and Zhao

K.Q.

, Triangular Fuzzy Number Multi-attribute Decision-making with the Attribute Weight Unknown Based on Connection Number, Fuzzy Systems and Mathematics 31(2) (2017), 95–106.

44.

Hou

Y.S

, Reflection on the Inheritance and Development of Wang PI opera in Shandong, Dong Yue Tribune 37 (2016), 142–146.

45.

Zha

and Kavuri

A.S.

, Effects of technical and allocative inefficiencies on energy and nonenergy elasticities: an analysis of energy-intensive industries in China, Chinese Journal of Population Resources and Environment 14 (2016), 292–297.

46.

Zhang

, He

and Pan

, A study on the design of a hybrid policy for carbon abatement, Chinese Journal of Population, Resources and Environment 15(1) (2017), 50–57.

47.

Zhang

and Shi

, Analysis of the relationship between environmental policies and air quality during major social events, Chinese Journal of Population Resources and Environment 14 (2016), 167–173.

48.

Zhu

, The 2030 Agenda for Sustainable Development and China’ simplementation, Chinese Journal of Population Resources and Environment 15 (2017), 142–146.

49.

Peng

X.D.

and Yang

, Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight, Applied Soft Computing 54 (2017), 415–430.

50.

Peng

and Yang

, Some Results for Pythagorean Fuzzy Sets, International Journal of Intelligent Systems 30(11) (2015), 1133–1160.

	VIKOR		TOPSIS		WAA
	Ranking		Ranking		Ranking
K ₁	0.9065	4	0.9279	4	0.7528	4
K ₂	0.3790	2	0.8866	2	0.7777	2
K ₃	0.5922	3	0.9113	3	0.7613	3
K ₄	0.0200	1	0.8654	1	0.8060	1

Multiple attribute group decision making approach based on extended VIKOR and linguistic neutrosophic Set

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 The LTS

3.1 A description of the decision problem

3.2 Determining the expert weight

4.1 An illustrative example

Table 1 Ranking of alternatives by S i ¯ , R i ¯ and Q i ¯ in ascending order S i ¯ R i ¯ Q i ¯ K 1 0.5243 0.3688 0.9065 K 2 0.3759 0.2081 0.3790 K 3 0.4415 0.2769 0.5922 K 4 0.2333 0.1282 0.0200 The ranking order K4 ≻ K2 ≻ K3 ≻ K1 K4 ≻ K2 ≻ K3 ≻ K1 K4 ≻ K2 ≻ K3 ≻ K1

Table 3 Comparison of the different methods VIKOR TOPSIS WAA Ranking Ranking Ranking K 1 0.9065 4 0.9279 4 0.7528 4 K 2 0.3790 2 0.8866 2 0.7777 2 K 3 0.5922 3 0.9113 3 0.7613 3 K 4 0.0200 1 0.8654 1 0.8060 1

Footnotes

Acknowledgment

References

Table 3
Comparison of the different methods

VIKOR TOPSIS WAA

Ranking Ranking Ranking

K ₁ 0.9065 4 0.9279 4 0.7528 4

K ₂ 0.3790 2 0.8866 2 0.7777 2

K ₃ 0.5922 3 0.9113 3 0.7613 3

K ₄ 0.0200 1 0.8654 1 0.8060 1