# Multiple Attribute Group Decision-Making Method Based on Linguistic Neutrosophic Numbers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{p}, l

_{q}), where l

_{p}and l

_{q}stand for the linguistic variables of the truth/membership and falsity/non-membership degrees, respectively, and developed a MAGDM method with LIFNs. Then, Liu and Wang [11] presented some improved LIFN aggregation operators for MADM. It is obvious that the LIFN consists of two linguistic variables l

_{p}and l

_{q}and describes the linguistic information of both the truth/membership and falsity/non-membership degrees, which are expressed by linguistic values rather than exact values like IFNs. However, LIFNs cannot describe indeterminate and inconsistent linguistic information. Then, a single-valued neutrosophic number (SVNN), which is a basic element in a single-valued neutrosophic set (SVNS) [12,13], can only express the truth, indeterminacy, and falsity degrees independently, and describe the incomplete, indeterminate, and inconsistent information in SVNN rather than linguistic information; then, it cannot express linguistic information in linguistic decision-making problems, while linguistic variables can represent the qualitative information for attributes in complex MADM problems. Hence, Ye [13] proposed the single-valued neutrosophic linguistic number (SVNLN), which is composed of a linguistic variable and an SVNN, where the linguistic variable is represented as the decision-maker’s judgment to an evaluated object and the SVNN is expressed as the reliability of the given linguistic variable, and developed an extended TOPSIS method for MAGDM problems with SVNLNs. However, SVNLN cannot also describe the truth, indeterminacy, and falsity linguistic information according to a linguistic term set. Tian et al. [14] put forward a simplified neutrosophic linguistic MAGDM approach for green product development. Liu and Tang [15] presented an interval neutrosophic uncertain linguistic Choquet integral method for MAGDM. Liu and Shi [16] introduced some neutrosophic uncertain linguistic number Heronian mean operators for MAGDM. However, all existing linguistic decision-making methods cannot express and deal with decision-making problems with indeterminate and inconsistent linguistic information.

_{0}= extremely low, l

_{1}= very low, l

_{2}= low, l

_{3}= slightly low, l

_{4}= medium, l

_{5}= slightly high, l

_{6}= high, l

_{7}= very high, l

_{8}= extremely high}. If the evaluation of a supplier with respect to its service performance is given as l

_{6}for the truth/membership degree, l

_{2}for the indeterminacy degree, and l

_{3}for the falsity/non-membership degree, respectively, by the decision-maker corresponding to the linguistic term set L then, for the concept of an LNN, it can be expressed as the form of an LNN e = <l

_{6}, l

_{2}, l

_{3}>. Obviously, LIFN and SVNLN cannot express such kinds of linguistic evaluation values; while LNN can easily describe them in a linguistic setting by the extension of SVNN and LIFN to LNN. Therefore, it is necessary to introduce LNN for expressing indeterminate and inconsistent linguistic information corresponding to human fuzzy thinking about complex problems, especially for some qualitative evaluations for attributes, and solving linguistic decision-making problems with indeterminate and inconsistent linguistic information. However, LNNs 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, which imply the truth, falsity, and indeterminate linguistic information. To aggregate LNN information in MAGDM problems, we have to develop some weighted aggregation operators, including an LNN-weighted arithmetic averaging (LNNWAA) operator and an LNN-weighted geometric averaging (LNNWGA) operator, which are usually used for MADM/MAGDM problems, score, and accuracy functions for the comparison of LNNs, and their decision-making method. Thus, the purposes of this paper are (1) to propose LNNs and their basic operational laws; (2) to introduce the score and accuracy functions of the LNN for comparing LNNs; (3) to present the LNNWAA and LNNWGA operators, their properties, and special cases; (4) to develop a MAGDM method based on the LNNWAA or LNNWGA operator under an LNN environment; and (5) to explain the advantages of the proposed method.

## 2. Linguistic Intuitionistic Fuzzy Numbers

**Definition 1.**

_{0}, l

_{1}, …, l

_{t}} is a linguistic term set with odd cardinality t + 1, where l

_{j}(j = 0, 1, …, t) is a possible value for a linguistic variable. If there is s = (l

_{p}, l

_{q}) for l

_{p}, l

_{q}$\in $ L and p, q $\in $ [0, t], then s is called LIFN.

**Definition 2.**

**Definition 3.**

_{p}, l

_{q}) be a LIFN in L, then the score and accuracy functions are defined as follows:

**Definition 4.**

- (1)
- If S(s
_{1}) < S(s_{2}), then s_{1}$\prec $ s_{2}; - (2)
- If S(s
_{1}) > S(s_{2}), then s_{1}$\succ $ s_{2}; - (3)
- If S(s
_{1}) = S(s_{2}) and H(s_{1}) < H(s_{2}), then s_{1}$\prec $ s_{2}; - (4)
- If S(s
_{1}) = S(s_{2}) and H(s_{1}) > H(s_{2}), then s_{1}$\succ $ s_{2}; - (5)
- If S(s
_{1}) = S(s_{2}) and H(s_{1}) = H(s_{2}), then s_{1}= s_{2}.

## 3. Linguistic Neutrosophic Numbers

**Definition 5.**

_{0}, l

_{1}, …, l

_{t}} is a linguistic term set with odd cardinality t + 1. If e = <l

_{p}, l

_{q}, l

_{r}> is defined for l

_{p}, l

_{q}, l

_{r}$\in $ L and p, q, r $\in $ [0, t], where l

_{p}, l

_{q}, and l

_{r}express independently the truth degree, indeterminacy degree, and falsity degree by linguistic terms, respectively, then e is called an LNN.

**Definition 6.**

**Example 1.**

_{1}= <l

_{6}, l

_{2}, l

_{3}> and e

_{2}= <l

_{5}, l

_{1}, l

_{2}> be two LNNs in L and ρ = 0.5, then there are the following operational results:

- (1)
- $$\begin{array}{ll}{e}_{1}\oplus {e}_{2}& =\langle {l}_{{p}_{1}},{l}_{{q}_{1}},{l}_{{r}_{1}}\rangle \oplus \langle {l}_{{p}_{2}},{l}_{{q}_{2}},{l}_{{r}_{2}}\rangle =\langle {l}_{{p}_{1}+{p}_{2}-\frac{{p}_{1}{p}_{2}}{t}},{l}_{\frac{{q}_{1}{q}_{2}}{t}},{l}_{\frac{{r}_{1}{r}_{2}}{t}}\rangle \\ & =\langle {l}_{6+5-6\times 5/8},{l}_{2\times 1/8},{l}_{3\times 2/8}\rangle =\langle {l}_{7.25},{l}_{0.25},{l}_{0.75}\rangle ,\end{array}$$
- (2)
- $$\begin{array}{ll}{e}_{1}\otimes {e}_{2}& =\langle {l}_{{p}_{1}},{l}_{{q}_{1}},{l}_{{r}_{1}}\rangle \otimes \langle {l}_{{p}_{2}},{l}_{{q}_{2}},{l}_{{r}_{2}}\rangle =\langle {l}_{\frac{{p}_{1}{p}_{2}}{t}},{l}_{{q}_{1}+{q}_{2}-\frac{{q}_{1}{q}_{2}}{t}},{l}_{{r}_{1}+{r}_{2}-\frac{{r}_{1}{r}_{2}}{t}}\rangle \\ & =\langle {l}_{\frac{6\times 5}{8}},{l}_{2+1-\frac{2\times 1}{8}},{l}_{3+2-\frac{3\times 2}{8}}\rangle =\langle {l}_{3.75},{l}_{2.75},{l}_{4.25}\rangle ,\end{array}$$
- (3)
- $$\begin{array}{ll}\rho {e}_{1}& =\rho \langle {l}_{{p}_{1}},{l}_{{q}_{1}},{l}_{{r}_{1}}\rangle =\langle {l}_{t-t{\left(1-\frac{{p}_{1}}{t}\right)}^{\rho}},{l}_{t{\left(\frac{{q}_{1}}{t}\right)}^{\rho}},{l}_{t{\left(\frac{{r}_{1}}{t}\right)}^{\rho}}\rangle =\langle {l}_{8-8{\left(1-\frac{6}{8}\right)}^{0.5}},{l}_{8{\left(\frac{2}{8}\right)}^{0.5}},{l}_{8{\left(\frac{3}{8}\right)}^{0.5}}\rangle \\ & \hspace{1em}\text{\hspace{1em}}=\langle {l}_{4},{l}_{4},{l}_{4.899}\rangle ,\end{array}$$
- (4)
- $$\begin{array}{ll}{e}_{1}^{\rho}& ={\langle {l}_{{p}_{1}},{l}_{{q}_{1}},{l}_{{r}_{1}}\rangle}^{\rho}=\langle {l}_{t{\left(\frac{{p}_{1}}{t}\right)}^{\rho}},{l}_{t-t{\left(1-\frac{{q}_{1}}{t}\right)}^{\rho}},{l}_{t-t{\left(1-\frac{{r}_{1}}{t}\right)}^{\rho}}\rangle =\langle {l}_{8{\left(\frac{6}{8}\right)}^{0.5}},{l}_{8-8{\left(1-\frac{2}{8}\right)}^{0.5}},{l}_{8-8{\left(1-\frac{3}{8}\right)}^{0.5}}\rangle \\ & =\langle {l}_{6.9282},{l}_{1.0718},{l}_{1.6754}\rangle .\end{array}$$

**Definition 7.**

_{p}, l

_{q}, l

_{r}> be an LNN in L. Then the score and accuracy functions of e are defined as follows:

**Definition 8.**

- (1)
- If Q(e
_{1}) < Q(e_{2}), then e_{1}$\prec $ e_{2}; - (2)
- If Q(e
_{1}) > Q(e_{2}), then e_{1}$\succ $ e_{2}; - (3)
- If Q(e
_{1}) = Q(e_{2}) and T(e_{1}) < T(e_{2}), then e_{1}$\prec $ e_{2}; - (4)
- If Q(e
_{1}) = Q(e_{2}) and T(e_{1}) > T(e_{2}), then e_{1}$\succ $ e_{2}; - (5)
- If Q(e
_{1}) = Q(e_{2}) and T(e_{1}) = T(e_{2}), then e_{1}= e_{2}.

**Example 2.**

_{1}= <l

_{6}, l

_{3}, l

_{4}>, e

_{2}= <l

_{5}, l

_{1}, l

_{3}>, and e

_{3}= <l

_{6}, l

_{4}, l

_{3}> be three LNNs in L, then the values of their score and accuracy functions are as follows:

- Q(e
_{1}) = (2 × 8 + 6 − 3 − 4)/24 = 0.625, Q(e_{2}) = (2 × 8 + 5 − 1 − 3)/24 = 0.7083, and Q(e_{3}) = (2 × 8 + 6 − 4 − 3)/24 = 0.625; - T(e
_{1}) = (6 − 4)/8= 0.25 and T(e_{3}) = (6 − 3)/8= 0.375. - According to Definition 8, their ranking order is e
_{2}$\succ $ e_{3}$\succ $ e_{1}.

## 4. Weighted Aggregation Operators of LNNs

#### 4.1. LNNWAA Operator

**Definition 9.**

_{j}$\in $ [0, 1] is the weight of e

_{j}(j = 1, 2, …, n), satisfying ${\sum}_{j=1}^{n}{w}_{j}}=1$.

**Theorem 1.**

_{j}$\in $ [0, 1] is the weight of e

_{j}(j = 1, 2, …, n), satisfying ${\sum}_{j=1}^{n}{w}_{j}}=1$.

**Proof.**

- (1)
- When n = 2, by Equation (9), we obtain:$${w}_{1}{e}_{1}=\langle {l}_{t-t{(1-\frac{{p}_{1}}{t})}^{{w}_{1}}},{l}_{t{(\frac{{q}_{1}}{t})}^{{w}_{1}}},{l}_{t{(\frac{{r}_{1}}{t})}^{{w}_{1}}}\rangle ,$$$${w}_{2}{e}_{2}=\langle {l}_{t-t{(1-\frac{{p}_{2}}{t})}^{{w}_{2}}},{l}_{t{(\frac{{q}_{2}}{t})}^{{w}_{2}}},{l}_{t{(\frac{{r}_{2}}{t})}^{{w}_{2}}}\rangle .$$By Equation (7), there is the following result:$$\begin{array}{l}LNNWAA\left({e}_{1},{e}_{2}\right)={w}_{1}{e}_{1}\oplus {w}_{2}{e}_{2}=\text{\hspace{0.17em}}\langle {l}_{t-t{(1-\frac{{p}_{1}}{t})}^{{w}_{1}}+t-t{(1-\frac{{p}_{2}}{t})}^{{w}_{2}}-\frac{(t-t{(1-\frac{{p}_{1}}{t})}^{{w}_{1}})(t-t{(1-\frac{{p}_{2}}{t})}^{{w}_{2}})}{t}},{l}_{t{(\frac{{q}_{1}}{t})}^{{w}_{1}}{(\frac{{q}_{2}}{t})}^{{w}_{2}}},{l}_{t{(\frac{{r}_{1}}{t})}^{{w}_{1}}{(\frac{{r}_{2}}{t})}^{{w}_{2}}}\rangle \\ \text{\hspace{1em}}=\langle {l}_{t-t{(1-\frac{{p}_{1}}{t})}^{{w}_{1}}+t-t{(1-\frac{{p}_{2}}{t})}^{{w}_{2}}-(t-t{(1-\frac{{p}_{1}}{t})}^{{w}_{1}}-t{(1-\frac{{p}_{2}}{t})}^{{w}_{2}}+t{(1-\frac{{p}_{1}}{t})}^{{w}_{1}}{(1-\frac{{p}_{2}}{t})}^{{w}_{2}})},{l}_{t{(\frac{{q}_{1}}{t})}^{{w}_{1}}{(\frac{{q}_{2}}{t})}^{{w}_{2}}},{l}_{t{(\frac{{r}_{1}}{t})}^{{w}_{1}}{(\frac{{r}_{2}}{t})}^{{w}_{2}}}\rangle \\ \text{\hspace{1em}}=\langle {l}_{t-t{(1-\frac{{p}_{1}}{t})}^{{w}_{1}}{(1-\frac{{p}_{2}}{t})}^{{w}_{2}}},{l}_{t{(\frac{{q}_{1}}{t})}^{{w}_{1}}{(\frac{{q}_{2}}{t})}^{{w}_{2}}},{l}_{t{(\frac{{r}_{1}}{t})}^{{w}_{1}}{(\frac{{r}_{2}}{t})}^{{w}_{2}}}\rangle =\langle {l}_{t-t{\displaystyle \prod _{j=1}^{2}{(1-\frac{{p}_{j}}{t})}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{2}{(\frac{{q}_{j}}{t})}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{2}{(\frac{{r}_{j}}{t})}^{{w}_{j}}}}\rangle .\end{array}$$
- (2)
- When n = k, by applying Equation (14), we obtain:$$LNNWAA({e}_{1},{e}_{2},\mathrm{...},{e}_{k})={\displaystyle \sum _{j=1}^{k}{w}_{j}{e}_{j}}=\langle {l}_{t-t{\displaystyle \prod _{j=1}^{k}{\left(1-\frac{{p}_{j}}{t}\right)}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{k}{\left(\frac{{q}_{j}}{t}\right)}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{k}{\left(\frac{{r}_{j}}{t}\right)}^{{w}_{j}}}}\rangle ,$$
- (3)
- When n = k + 1, by applying Equations (15) and (16), which yields:$$\begin{array}{l}LNNWAA\left({e}_{1},{e}_{2},\mathrm{...},{e}_{k+1}\right)={\displaystyle \sum _{j=1}^{k+1}{w}_{j}{e}_{j}}\\ \text{\hspace{1em}}=\text{\hspace{0.17em}}\langle {l}_{t-t{\displaystyle \prod _{j=1}^{k}{(1-\frac{{p}_{j}}{t})}^{{w}_{j}}}+t-t{(1-\frac{{p}_{k+1}}{t})}^{{w}_{k+1}}-\frac{(t-t{\displaystyle \prod _{j=1}^{k}{(1-\frac{{p}_{j}}{t})}^{wj}})(t-t{(1-\frac{{p}_{k+1}}{t})}^{{w}_{k+1}})}{t}},{l}_{t{\displaystyle \prod _{j=1}^{k}{(\frac{{q}_{j}}{t})}^{{w}_{j}}}{(\frac{{q}_{k+1}}{t})}^{{w}_{k+1}}},{l}_{t{\displaystyle \prod _{j=1}^{k}{(\frac{{r}_{j}}{t})}^{{w}_{j}}}{(\frac{{r}_{k+1}}{t})}^{{w}_{k+1}}}\rangle \\ \text{\hspace{1em}}=\langle {l}_{t-t{\displaystyle \prod _{j=1}^{k}{(1-\frac{{p}_{j}}{t})}^{{w}_{j}}}+t-t{(1-\frac{{p}_{k+1}}{t})}^{{w}_{k+1}}-(t-t{\displaystyle \prod _{j=1}^{k}{(1-\frac{{p}_{j}}{t})}^{{w}_{j}}}-t{(1-\frac{{p}_{k+1}}{t})}^{{w}_{k+1}}+t{\displaystyle \prod _{j=1}^{k}{(1-\frac{{p}_{j}}{t})}^{{w}_{j}}}{(1-\frac{{p}_{k+1}}{t})}^{{w}_{k+1}})},{l}_{t{\displaystyle \prod _{j=1}^{k}{(\frac{{q}_{j}}{t})}^{{w}_{j}}}{(\frac{{q}_{k+1}}{t})}^{{w}_{k+1}}},{l}_{t{\displaystyle \prod _{j=1}^{k}{(\frac{{r}_{j}}{t})}^{{w}_{j}}}{(\frac{{r}_{k+1}}{t})}^{{w}_{k+1}}}\rangle \\ \text{\hspace{1em}}=\langle {l}_{t-t{\displaystyle \prod _{j=1}^{k}{(1-\frac{{p}_{j}}{t})}^{{w}_{j}}}{(1-\frac{{p}_{k+1}}{t})}^{{w}_{k+1}}},{l}_{t{\displaystyle \prod _{j=1}^{k}{(\frac{{q}_{j}}{t})}^{{w}_{j}}}{(\frac{{q}_{k+1}}{t})}^{{w}_{k+1}}},{l}_{t{\displaystyle \prod _{j=1}^{k}{(\frac{{r}_{j}}{t})}^{{w}_{j}}}{(\frac{{r}_{k+1}}{t})}^{{w}_{k+1}}}\rangle =\langle {l}_{t-t{\displaystyle \prod _{j=1}^{k+1}{(1-\frac{{p}_{j}}{t})}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{k+1}{(\frac{{q}_{j}}{t})}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{k+1}{(\frac{{r}_{j}}{t})}^{{w}_{j}}}}\rangle .\end{array}$$

- (1)
- Idempotency: Let e
_{j}(j = 1, 2, …, n) be a collection of LNNs in L. If e_{j}(j = 1, 2, …, n) is equal, i.e., e_{j}= e for j = 1, 2, …, n, then $LNNWAA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)=e$. - (2)
- Boundedness: Let e
_{j}(j = 1, 2, …, n) be a collection of LNNs in L and let ${e}^{-}=\langle \underset{j}{\mathrm{min}}({l}_{{p}_{j}}),\underset{j}{\mathrm{max}}({l}_{{q}_{j}}),\underset{j}{\mathrm{max}}({l}_{{r}_{j}})\rangle $ and ${e}^{+}=\langle \underset{j}{\mathrm{max}}({l}_{{p}_{j}}),\underset{j}{\mathrm{min}}({l}_{{q}_{j}}),\underset{j}{\mathrm{min}}({l}_{{r}_{j}})\rangle $. Then ${e}^{-}\le LNNWAA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)\le {e}^{+}$. - (3)
- Monotonicity: Let e
_{j}(j = 1, 2, …, n) be a collection of LNNs in L. If e_{j}≤ ${e}_{j}^{*}$ for j = 1, 2, …, n, then $LNNWAA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)\le LNNWAA\left({e}_{1}^{*},{e}_{2}^{*},\cdots ,{e}_{n}^{*}\right)$.

**Proof.**

- (1)
- Since e
_{j}= e, i.e., p_{j}= p; q_{j}= q; t_{j}= r for j = 1, 2, …, n, we have:$$\begin{array}{l}LNNWAA({e}_{1},{e}_{2},\mathrm{...},{e}_{n})={\displaystyle \sum _{j=1}^{n}{w}_{j}{e}_{j}}=\langle {l}_{t-t{\displaystyle \prod _{j=1}^{n}{\left(1-\frac{{p}_{j}}{t}\right)}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{n}{\left(\frac{{q}_{j}}{t}\right)}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{n}{\left(\frac{{r}_{j}}{t}\right)}^{{w}_{j}}}}\rangle \\ \text{\hspace{1em}}=\langle {l}_{t-t{\left(1-\frac{p}{t}\right)}^{{\displaystyle \sum _{j=1}^{n}{w}_{j}}}},{l}_{t{\left(\frac{q}{t}\right)}^{{\displaystyle \sum _{j=1}^{n}{w}_{j}}}},{l}_{t{\left(\frac{r}{t}\right)}^{{\displaystyle \sum _{j=1}^{n}{w}_{j}}}}\rangle =\langle {l}_{t-t\left(1-\frac{p}{t}\right)},{l}_{t\left(\frac{q}{t}\right)},{l}_{t\left(\frac{r}{t}\right)}\rangle \\ \text{\hspace{1em}}=\langle {l}_{p},{l}_{q},{l}_{r}\rangle =e.\end{array}$$ - (2)
- Since the minimum LNN is e
^{−}and the maximum LNN is e^{+}, e^{−}≤ e_{j}≤ e^{+}. Thus, $\sum _{j=1}^{n}{w}_{j}{e}^{-}}\le {\displaystyle \sum _{j=1}^{n}{w}_{j}{e}_{j}}\le {\displaystyle \sum _{j=1}^{n}{w}_{j}{e}^{+}$. According to the above property (1), ${e}^{-}\le {\displaystyle \sum _{j=1}^{n}{w}_{j}{e}_{j}}\le {e}^{+}$, i.e., ${e}^{-}\le LNNWAA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)\le {e}^{+}$. - (3)
- Since ${e}_{j}\le {e}_{j}^{*}$ for j = 1, 2, …, n, $\sum _{j=1}^{n}{w}_{j}{e}_{j}}\le {\displaystyle \sum _{j=1}^{n}{w}_{j}{e}_{j}^{*}$, i.e., $LNNWAA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)\le LNNWAA\left({e}_{1}^{*},{e}_{2}^{*},\cdots ,{e}_{n}^{*}\right)$.

_{j}= 1/n for j = 1, 2, …, n, the LNNWAA operator is reduced to the LNN arithmetic averaging operator.

#### 4.2. LNNWGA Operator

**Definition 10.**

_{j}$\in $ [0, 1] is the weight of e

_{j}(j = 1, 2, …, n), satisfying ${\sum}_{j=1}^{n}{w}_{j}}=1$.

**Theorem 2.**

_{j}$\in $ [0, 1] is the weight of e

_{j}(j =1, 2, …, n), satisfying ${\sum}_{j=1}^{n}{w}_{j}}=1$. Especially when w

_{j}= 1/n for j = 1, 2, …, n, the LNNWGA operator is reduced to the LNN geometric averaging operator.

- (1)
- Idempotency: Let e
_{j}(j = 1, 2, …, n) be a collection of LNNs in L. If e_{j}(j = 1, 2, …, n) is equal, i.e., e_{j}= e for j = 1, 2, …, n, then $LNNWGA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)=e$. - (2)
- Boundedness: Let e
_{j}(j = 1, 2, …, n) be a collection of LNNs in L and let ${e}^{-}=\langle \underset{j}{\mathrm{min}}({l}_{{p}_{j}}),\underset{j}{\mathrm{max}}({l}_{{q}_{j}}),\underset{j}{\mathrm{max}}({l}_{{r}_{j}})\rangle $ and ${e}^{+}=\langle \underset{j}{\mathrm{max}}({l}_{{p}_{j}}),\underset{j}{\mathrm{min}}({l}_{{q}_{j}}),\underset{j}{\mathrm{min}}({l}_{{r}_{j}})\rangle $. Then ${e}^{-}\le LNNWGA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)\le {e}^{+}$. - (3)
- Monotonicity: Let e
_{j}(j = 1, 2, …, n) be a collection of LNNs in L. If e_{j}≤ ${e}_{j}^{*}$ for j = 1, 2, …, n, then $LNNWGA\left({e}_{1},{e}_{2},\cdots ,{e}_{n}\right)\le LNNWGA\left({e}_{1}^{*},{e}_{2}^{*},\cdots ,{e}_{n}^{*}\right)$.

## 5. MAGDM Method Based on the LNNWAA or LNNWGA Operator

_{1}, Y

_{2}, …, Y

_{m}} be a set of alternatives and Z = {Z

_{1}, Z

_{2}, …, Z

_{n}} be a set of attributes. The weigh vector of the attributes Z

_{j}(j = 1, 2, …, n) is W = (w

_{1}, w

_{2}, .…, w

_{n})

^{T}. Then, a group of decision-makers D = {D

_{1}, D

_{2}, …, D

_{d}} can be assigned with a corresponding weight vector

**ω**= (ω

_{1}, ω

_{2}, …, ω

_{d})

^{T}to evaluate the alternatives Y

_{i}(i = 1, 2, …, m) on the attributes Z

_{j}(j = 1, 2, …, n) by LNNs from the linguistic term set L = {l

_{0}= extremely low, l

_{1}= very low, l

_{2}= low, l

_{3}= slightly low, l

_{4}= medium, l

_{5}= slightly high, l

_{6}= high, l

_{7}= very high, l

_{8}= extremely high}. In the evaluation process, the decision-makers can assign the three linguistic values of the truth, falsity, and indeterminacy degrees, composed of an LNN, to each attribute Z

_{j}on an alternative Y

_{i}according to the linguistic terms. Thus, the LNN evaluation information of the attributes Z

_{j}(j = 1, 2, …, n) on the alternatives Y

_{i}(i = 1, 2, …, m) provided by each decision maker D

_{k}(k = 1, 2, …, d) can be established as an LNN decision matrix M

^{k}= (${e}_{ij}^{k}$)

_{m×n}, where ${e}_{ij}^{k}=\langle {l}_{{p}_{ij}^{k}}^{},{l}_{{q}_{ij}^{k}}^{},{l}_{{r}_{ij}^{k}}^{}\rangle $ (k = 1, 2, …, d; i = 1,2, …, m; j = 1, 2, …, n) is an LNN.

**Step 1:**Obtain the integrated matrix R = (e_{ij})_{m}_{×n}, where ${e}_{ij}^{}=\langle {l}_{{p}_{ij}}^{},{l}_{{q}_{ij}}^{},{l}_{{r}_{ij}}^{}\rangle $ (i = 1, 2, …, m; j = 1, 2, …, n) is an integrated LNN, by using the following LNNWAA operator:$${e}_{ij}=LNNWAA({e}_{ij}^{1},{e}_{ij}^{2},\mathrm{...},{e}_{ij}^{d})={\displaystyle \sum _{k=1}^{d}{\omega}_{k}{e}_{ij}^{k}}=\langle {l}_{t-t{\displaystyle \prod _{k=1}^{d}{\left(1-\frac{{p}_{ij}^{k}}{t}\right)}^{{\omega}_{k}}}},{l}_{t{\displaystyle \prod _{k=1}^{d}{\left(\frac{{q}_{ij}^{k}}{t}\right)}^{{\omega}_{k}}}},{l}_{t{\displaystyle \prod _{k=1}^{d}{\left(\frac{{r}_{ij}^{k}}{t}\right)}^{{\omega}_{k}}}}\rangle .$$**Step 2:**Obtain the collective overall LNN e_{i}for Y_{i}(i = 1, 2, …, m) by using the following LNNWAA operator or LNNWGA operator:$${e}_{i}=LNNWAA({e}_{i1},{e}_{i2},\mathrm{...},{e}_{in})={\displaystyle \sum _{j=1}^{n}{w}_{j}{e}_{ij}}=\langle {l}_{t-t{\displaystyle \prod _{j=1}^{n}{\left(1-\frac{{p}_{ij}}{t}\right)}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{n}{\left(\frac{{q}_{ij}}{t}\right)}^{{w}_{j}}}},{l}_{t{\displaystyle \prod _{j=1}^{n}{\left(\frac{{r}_{ij}}{t}\right)}^{{w}_{j}}}}\rangle ,$$$${e}_{i}=LNNWGA({e}_{i1},{e}_{i2},\mathrm{...},{e}_{in})={\displaystyle \prod _{j=1}^{n}{e}_{ij}^{{w}_{j}}}=\langle {l}_{t{\displaystyle \prod _{j=1}^{n}{\left(\frac{{p}_{ij}}{t}\right)}^{{w}_{j}}}},{l}_{t-t{\displaystyle \prod _{j=1}^{n}{\left(1-\frac{{q}_{ij}}{t}\right)}^{{w}_{j}}}},{l}_{t-t{\displaystyle \prod _{j=1}^{n}{\left(1-\frac{{r}_{ij}}{t}\right)}^{{w}_{j}}}}\rangle .$$**Step 3:**Calculate the score function Q(e_{i}) (accuracy function T(e_{i}) if necessary) (i = 1, 2, …, m) of the collective overall LNN e_{i}(i = 1, 2, …, m) by Equation (11) (Equation (12) if necessary).**Step 4:**Rank the alternatives corresponding to the score (accuracy if necessary) values, and then select the best one.**Step 5:**End.

## 6. An Illustrative Example

_{1}, Y

_{2}, Y

_{3}, Y

_{4}}, where Y

_{1}is a car company; Y

_{2}is a food company; Y

_{3}is a computer company; Y

_{4}is an arms company. The evaluation of the four alternatives must satisfy the requirements of three attributes: (1) Z

_{1}is the risk; (2) Z

_{2}is the growth; (3) Z

_{3}is the environmental impact. The importance of the three attributes is provided by the weigh vector

**W**= (0.35, 0.25, 0.4)

^{T}. Then, three decision-makers are invited and denoted as a set of the decision-makers D = {D

_{1}, D

_{2}, D

_{3}} and the importance of the three decision-makers is given as a weight vector

**ω**= (0.37, 0.33, 0.3)

^{T}. The three decision-makers are required to give the suitability evaluation of the four possible alternatives Y

_{i}(i = 1, 2, 3, 4) with respect to the three attributes Z

_{j}(j = 1, 2, 3) by the expression of the linguistic values of LNNs from the linguistic term set L = {l

_{0}= extremely low, l

_{1}= very low, l

_{2}= low, l

_{3}= slightly low, l

_{4}= medium, l

_{5}= slightly high, l

_{6}= high, l

_{7}= very high, l

_{8}= extremely high} with the odd cardinality t + 1 = 9. Thus, the linguistic evaluation information given by each decision-maker D

_{k}(k = 1, 2, 3) can be established as the following the LNN decision matrix M

^{k}:

**Step 1:**Get the following integrated matrix R = (e_{ij})_{m}_{×n}by using Equation (19):$$R=\left[\begin{array}{ccc}\langle {l}_{6.3755},\text{}{l}_{1.3904},\text{}{l}_{2.4623}\rangle & \langle {l}_{6.7430},\text{}{l}_{1.7969},\text{}{l}_{1.3904}\rangle & \langle {l}_{5.1608},\text{}{l}_{2.0000},\text{}{l}_{3.0097}\rangle \\ \langle \text{}{l}_{6.7689},\text{}{l}_{1.7477},\text{}{l}_{2.1781}\rangle & \langle {l}_{6.2523},\text{}{l}_{1.5015},\text{}{l}_{1.5911}\rangle & \langle {l}_{6.0547},\text{}{l}_{2.0000},\text{}{l}_{1.9980}\rangle \\ \langle \text{}{l}_{6.1429},\text{}{l}_{1.5911},\text{}{l}_{2.4623}\rangle & \langle {l}_{6.2309},\text{}{l}_{1.0000},\text{}{l}_{1.5476}\rangle & \langle {l}_{6.1429},\text{}{l}_{2.5140},\text{}{l}_{2.4623}\rangle \\ \langle {l}_{6.7430},\text{}{l}_{1.2311},\text{}{l}_{1.7969}\rangle & \langle {l}_{6.0020},\text{}{l}_{1.5911},\text{}{l}_{1.5015}\rangle & \langle {l}_{6.2309},\text{}{l}_{1.6245},\text{}{l}_{1.4370}\rangle \end{array}\right].$$**Step 2:**By using Equation (20), the collective overall LNNs of e_{i}for Y_{i}(i = 1, 2, 3, 4) can be obtained as follows:$${e}_{1}=\langle {l}_{6.0951},{l}_{1.7145},{l}_{2.3129}\rangle ,\text{\hspace{0.17em}}{e}_{2}=\langle {l}_{6.3863},{l}_{1.7759},{l}_{1.9453}\rangle ,\text{\hspace{0.17em}}{e}_{3}=\langle {l}_{6.1653},{l}_{1.7011},{l}_{2.1924}\rangle ,\mathrm{and}\text{\hspace{0.17em}}{e}_{4}=\langle {l}_{6.3818},{l}_{1.4666},{l}_{1.5711}\rangle .$$**Step 3:**Calculate the score values of Q(e_{i}) (i = 1, 2, 3, 4) of the collective overall LNNs of e_{i}(i = 1, 2, 3, 4) by Equation (11):Q(e_{1}) = 0.7528, Q(e_{2}) = 0.7777, Q(e_{3}) = 0.7613, and Q(e_{4}) = 0.8060.**Step 4:**Ranking order of the four alternatives is Y_{4}$\succ $ Y_{2}$\succ $ Y_{3}$\succ $ Y_{1}corresponding to the score values. Thus, the alternative Y_{4}is the best choice among the four alternatives.

**Step 1’:**The same as Step 1.**Step 2’:**By using Equation (21), the collective overall LNNs of e_{i}for Y_{i}(i = 1, 2, 3, 4) are obtained as follows:$${e}_{1}=\langle {l}_{5.9413},{l}_{1.7414},{l}_{2.4479}\rangle ,\text{\hspace{0.17em}}{e}_{2}=\langle {l}_{6.3464},{l}_{1.7902},{l}_{1.9634}\rangle ,{e}_{3}=\langle {l}_{6.1648},{l}_{1.8433},{l}_{2.2465}\rangle ,\text{}\mathrm{and}\text{\hspace{0.17em}}{e}_{4}=\langle {l}_{6.3459},{l}_{1.4810},{l}_{1.5811}\rangle .$$**Step 3’:**By using Equation (11), we calculate the score values of Q(e_{i}) (i = 1, 2, 3, 4) of the collective overall LNNs of e_{i}(i = 1, 2, 3, 4) as follows:Q(e_{1}) = 0.7397, Q(e_{2}) = 0.7747, Q(e_{3}) = 0.7531, and Q(e_{4}) = 0.8035.**Step 4’:**The ranking order of the four alternatives is Y_{4}$\succ $ Y_{2}$\succ $ Y_{3}$\succ $ Y_{1}. Thus, the alternative Y_{4}is still the best choice among the four alternatives.

- (1)
- The developed new method is more suitable for expressing and handling indeterminate and inconsistent linguistic information in linguistic decision-making problems to overcome the insufficiency of various linguistic decision-making methods in the existing literature.
- (2)
- The developed new method contains much more information (the three linguistic variables of truth, indeterminate, and falsity degrees contained in an LNN) than the existing method in [10,11] (the two linguistic variables of truth and falsity degrees contained in a LIFN) and can better describe people’s linguistic expression to objective things evaluated in detail.
- (3)
- The developed new method enriches the neutrosophic theory and decision-making method under a linguistic environment and provides a new way for solving linguistic MAGDM problems with indeterminate and inconsistent linguistic information.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Fang, Z.; Ye, J.
Multiple Attribute Group Decision-Making Method Based on Linguistic Neutrosophic Numbers. *Symmetry* **2017**, *9*, 111.
https://doi.org/10.3390/sym9070111

**AMA Style**

Fang Z, Ye J.
Multiple Attribute Group Decision-Making Method Based on Linguistic Neutrosophic Numbers. *Symmetry*. 2017; 9(7):111.
https://doi.org/10.3390/sym9070111

**Chicago/Turabian Style**

Fang, Zebo, and Jun Ye.
2017. "Multiple Attribute Group Decision-Making Method Based on Linguistic Neutrosophic Numbers" *Symmetry* 9, no. 7: 111.
https://doi.org/10.3390/sym9070111