Multiple attribute group decision-making method using correlation coefficients between linguistic neutrosophic numbers

Abstract

Linguistic neutrosophic numbers (LNNs) contain three linguistic variables which can independently represent the truth, indeterminacy, and falsity degrees in the real world. Correlation coefficient is a useful tool in decision-making problems. Hence, this paper proposes three new correlation coefficients between LNNs. Then, a multi-attribute group decision-making (MAGDM) method is developed based on the correlation coefficients under LNN environment. Finally, an example about a decision-making problem is presented to illustrate the application and effectiveness of the proposedmethod.

Keywords

Linguistic neutrosophic number correlation coefficient MAGDM method

1 Introduction

In complex decision-making problems, it is difficult to describe qualitative evaluation by real numbers instead of linguistic values due to the ambiguity of human thinking and the complexity of objective things. In this situation, Zadeh [43] proposed the concept of linguistic variable, which can describe qualitative information using words or sentences in natural language. For instance, an investment company wants to select an investment project. Then, a decision-maker can evaluate a project based on a linguistic term set as follows: $\tilde{L} = {{\tilde{l}}_{0} = extremely low, {\tilde{l}}_{1} = very low, {\tilde{l}}_{2} = low, {\tilde{l}}_{3} = slightly low, {\tilde{l}}_{4} = medium, {\tilde{l}}_{5} = slightly high, {\tilde{l}}_{6} = high, {\tilde{l}}_{7} = very high, {\tilde{l}}_{8} = extremely high}$ . Later, Herrera et al. applied linguistic information in solving a multi-criteria decision-making problem

[1, 2]. Next, many new aggregation operators were presented to solve decision-making problems with linguistic information [24–26 , 62–64]. Furthermore, many scholars extended linguistic variables to intuitionistic fuzzy variables and their intuitionistic fuzzy aggregation operators and applied them in decision-making problems [13,14 , 65]. Then, (interval) hesitant fuzzy linguistic term sets (HFLTSs) were introduced and applied to decision making problems [6 , 52]. Further, intuitionistic hesitant linguistic sets (IHLSs) were proposed for decision

making [16, 57]. However, the intuitionistic fuzzy linguistic set contains two linguistic variables, which can express the linguistic information of both the truth/membership and falsity/non-membership degrees, but they cannot describe indeterminate and intuitionistic fuzzy variable linguistic information. Neutrosophic set (NS) was firstly proposed by Smarandache [3, 5] and described by the truth, indeterminacy, and falsity membership functions independently, which can effectively express incomplete, indeterminate, and inconsistent information. A neutrosophic number (NN) is a basic element in NS, and it has been applied to solve indeterminate and inconsistent information for multi-attribute group decision-making (MAGDM) problems [32, 33]. Recently, some extensions of neutrosophic sets were widely studied for solving decision-making problems [17 , 66]. Hence, Fang and Ye extend NS concept to linguistic set to propose the concept of a linguistic neutrosophic number (LNN) [67], which was independently described by the truth, indeterminacy, and falsity degrees, for overcoming the insufficiency of the existing linguistic variables, and then Fang and Ye presented a LNN weighted geometric averaging (LNNWGA) operator and a LNN weighted arithmetic averaging (LNNWAA) operator for a MAGDM problems with LNNs [67]. Li et al. introduced the concept of linguistic neutrosophic sets (LNSs) and two aggregation operators for multicriteria decision-making (MCDM) problems under linguistic neutrosophic environments [61]. Shi and Ye proposed cosine similarity measures of LNNs and applied them in MAGDM problems [45]. Wang et al. presented some new decision-making methods using single valued interval neutrosophic linguistic numbers and interval neutrosophic linguistic numbers[34, 35].

Correlation coefficient is a useful mathematical tool for solving fuzzy problems [18 , 47]. Recently, some extensions of correlation coefficients of neutrosophic theory were widely studied, including neutrosophic sets (NSs) [22, 23], interval neutrosophic set (INS) [55], single valued neutrosophic sets (SVNSs) [37 –40], interval neutrosophic sets (INSs) [19], neutrosophic soft sets (NSSs) [7], simplified neutrosophic sets (SNSs) [46], single-valued neutrosophic refined soft sets [8] and normal neutrosophic sets [41], and so on. The above correlation coefficients have been applied in decision-making problems, but they cannot solve decision-making problems with LNN information. Hence, this paper extends correlation coefficients to LNNs and presents their application in MAGDM problems. To do so, the rest of the article is organized as follows. Section 2 briefly introduces some preliminaries related to the following analysis. Section 3 proposes three new correlation coefficients between LNNs. In Section 4, we establish a MAGDM method based on the correlation coefficients of LNNs. Section 5 gives an illustrative example about the decision-making problem to demonstrate the feasibility and effectiveness of the proposed method. Conclusions and future work are summarized inSection 6.

2 Preliminaries

Definition 1. [51] Let P and Q be two SVNSs in the universe of discourse $\hat{U} = {x_{1}, x_{2}, . . ., x_{n}},$ and $P = {x_{i}, T_{P} (x_{i}), I_{P} (x_{i}), F_{P} (x_{i}) | x_{i} \in \hat{U}}$ and $Q = {x_{i}, T_{Q} (x_{i}), I_{Q} (x_{i}), F_{Q} (x_{i}) | x_{i} \in \hat{U}}$ . Then, the similarity measure of two SVNSs P and Q was defined by Majumdar and Samanta as follows:

$ρ_{MS} (P, Q) = \frac{\sum_{i = 1}^{n} (\begin{matrix} \min (T_{P} (x_{i}), T_{Q} (x_{i})) \\ + \min (I_{P} (x_{i}), I_{Q} (x_{i})) \\ + \min (F_{P} (x_{i}), F_{Q} (x_{i})) \end{matrix})}{\sum_{i = 1}^{n} (\begin{matrix} \max (T_{P} (x_{i}), T_{Q} (x_{i})) \\ + \max (I_{P} (x_{i}), I_{Q} (x_{i})) \\ + \max (F_{P} (x_{i}), F_{Q} (x_{i})) \end{matrix})} .$ (1)

Definition 2. [67]. Let $\tilde{L} = {{\tilde{l}}_{0}, {\tilde{l}}_{1}, . . ., {\tilde{l}}_{τ}}$ is a linguistic term set with odd cardinality τ+1. If $g = < {\tilde{l}}_{T}, {\tilde{l}}_{I}, {\tilde{l}}_{F} >$ is defined for ${\tilde{l}}_{T}, {\tilde{l}}_{I}, {\tilde{l}}_{F} \in L$ and T, I, F ∈ [0, τ] by linguistic terms, where ${\tilde{l}}_{T}$ represents the truth degree, ${\tilde{l}}_{I}$ represents indeterminacy degree, and ${\tilde{l}}_{F}$ represents falsity degree, then g is calleda LNN.

3 Correlation coefficients between LNNs

In this section, three new correlation coefficients between LNNs are proposed as extensions of Majumdar and Samanta’s similarity measure [51].

Definition 4. Given the linguistic term set $\tilde{L} = {{\tilde{l}}_{0}, {\tilde{l}}_{1}, . . ., {\tilde{l}}_{τ}}$ with odd cardinality τ+1. If P = {p₁, p₂,. . . , p_n} and Q = {q_1, q₂,. . . , q_n} are two sets of LNNs, where $p_{j} = 〈 {\tilde{l}}_{T_{p_{j}}}, {\tilde{l}}_{I_{p_{j}}}, {\tilde{l}}_{F_{p_{j}}} 〉 and q_{j} = 〈 {\tilde{l}}_{T_{q_{j}}}, {\tilde{l}}_{I_{q_{j}}}, {\tilde{l}}_{F_{q_{j}}} 〉$ are two LNNs with ${\tilde{l}}_{T_{p_{j}}}, {\tilde{l}}_{I_{p_{j}}}, {\tilde{l}}_{F_{p_{j}}}, {\tilde{l}}_{T_{q_{j}}}, {\tilde{l}}_{I_{q_{j}}}, {\tilde{l}}_{F_{q_{j}}} \in \tilde{L}$ and $f ({\tilde{i}}_{j}) = j$ is a linguistic scale function for $T_{p_{j}}, I_{p_{j}}, F_{p_{j}}, T_{q_{j}}, I_{q_{j}}, F_{q_{j}} \in [0, τ]$ and j = 1, 2,. . . , n . Then, three correlation coefficients between P and Q are proposed as follows:

$\begin{matrix} ρ_{1} (P, Q) \\ = \frac{1}{n} \sum_{j = 1}^{n} \frac{(\begin{matrix} min (f ({\tilde{l}}_{T_{p_{j}}}), f ({\tilde{l}}_{T_{q_{j}}})) + min (f ({\tilde{l}}_{I_{p_{j}}}), \\ f ({\tilde{l}}_{I_{q_{j}}})) + min (f ({\tilde{l}}_{F_{p_{j}}}), f ({\tilde{l}}_{F_{q_{j}}})) \end{matrix})}{(\begin{matrix} \sqrt{f ({\tilde{l}}_{T_{p_{j}}}) f ({\tilde{l}}_{T_{q_{j}}})} + \sqrt{f ({\tilde{l}}_{I_{p_{j}}}) f ({\tilde{l}}_{I_{q_{j}}})} \\ + \sqrt{f ({\tilde{l}}_{F_{p_{j}}}) f ({\tilde{l}}_{F_{p_{j}}})} \end{matrix})} \\ = \frac{1}{n} \sum_{j = 1}^{n} \frac{(\begin{matrix} min (T_{p_{j}}, T_{q_{j}}) + min (I_{p_{j}}, I_{q_{j}}) \\ + min (F_{p_{j}}, F_{q_{j}}) \end{matrix})}{(\begin{matrix} \sqrt{T_{p_{j}} T_{q_{j}}} + \sqrt{I_{p_{j}} I_{q_{j}}} \\ + \sqrt{F_{p_{j}} F_{q_{j}}} \end{matrix})} \end{matrix}$ (2)

$ρ_{2} (P, Q) = \frac{1}{n} \sum_{j = 1}^{n} \frac{(f ({\tilde{l}}_{T p_{j}}) f ({\tilde{l}}_{T q_{j}}) + f ({\tilde{l}}_{I p_{j}}) f ({\tilde{l}}_{I q_{j}}))}{({(\max (f ({\tilde{l}}_{T p_{j}}), f ({\tilde{l}}_{T q_{j}})))}^{2} + {(\max (f ({\tilde{l}}_{I p_{j}}), f ({\tilde{l}}_{I p_{j}})))}^{2} + {(\max (f ({\tilde{l}}_{F p_{j}}), f ({\tilde{l}}_{F p_{j}})))}^{2})} = \frac{1}{n} \sum_{j = 1}^{n} \frac{T p_{j} T q_{j} + I p_{j} I q_{j} + F p_{j} F q_{j}}{({(\max (T p_{j}, T q_{j}))}^{2} + {(\max (I p_{j}, I q_{j}))}^{2} + {(\max (F p_{j}, F q_{j}))}^{2})}$ (3)

$ρ_{3} (P, Q) = \frac{1}{n} \sum_{j = 1}^{n} \frac{\min (f ({\tilde{l}}_{T p_{j}}), f ({\tilde{l}}_{T q_{j}})) + \min (f ({\tilde{l}}_{I p_{j}}), f ({\tilde{l}}_{I q_{j}})) + \min (f ({\tilde{l}}_{F p_{j}}), f ({\tilde{l}}_{F q_{j}}))}{\max (f ({\tilde{l}}_{T p_{j}}), f ({\tilde{l}}_{T q_{j}})) + \max (f ({\tilde{l}}_{I p_{j}}), f ({\tilde{l}}_{I q_{j}})) + \max (f ({\tilde{l}}_{F p_{j}}), f ({\tilde{l}}_{F q_{j}}))} = \frac{1}{n} \sum_{j = 1}^{n} \frac{(\min (T p_{j}, T q_{j}) + \min (I p_{j}, I q_{j}) + \min (F p_{j}, F q_{j}))}{(\max (T p_{j}, T q_{j}) + \max (I p_{j}, I q_{j}) + \max (F p_{j}, F q_{j}))}$ (4)

where, the symbols “min” and “max” represent the minimum and maximum operations, respectively.

According to the above definition, the three correlation coefficients ρi (P, Q) (i = 1, 2, 3) for LNNs satisfy the following properties (c₁)– (c₃):

0≤ρ _i (P, Q)≤1;

ρ _i (P, Q) =ρ _i (Q, P);

If P = Q, then ρ _i (P, Q) = 1.

Proof. Firstly, we prove the properties (c₁)– (c₃) of ρ₁(P, Q).

(c₁) Since T_{p_j}, I_{q_j}, F_{p_j}, T_{q_j}, I_{p_j}, F_{q_j} ∈ [0, τ] there exist the following inequalities: min (T_{p
_j}, ≤ $\sqrt{T_{p_{j}} T_{q_{j}}}$ , min (I_{p
_j}, I_{q
_j}) ≤ $\sqrt{I_{p_{j}} I_{q_{j}}}$ , $min (F_{p_{j}}, F_{q_{j}}) \leq \sqrt{F_{p_{j}} F_{q_{j}}}, and min (T_{p_{j}}, T_{q_{j}}) + \min (I_{p_{j}}, I_{q_{j}}) + \min (F_{p_{j}}, F_{q_{j}}) \leq \sqrt{T_{p_{j}} T_{q_{j}}} + \sqrt{I_{p_{j}} I_{q_{j}}} + \sqrt{F_{p_{j}} F_{q_{j}}}$ . Hence 0 ≤ ρ₁ (E, G) ≤ 1.

(c₂) It is obvious that the property is true.

(c₃) For any two sets of LNNs P and Q, if P = Q, there exist P ⊇ QandP ⊆ Q, then ${\tilde{l}}_{T_{p_{j}}} = {\tilde{l}}_{T_{q_{j}}}$ , ${\tilde{l}}_{I_{p_{j}}} = {\tilde{l}}_{I_{q_{j}}}$ , ${\tilde{l}}_{F_{p_{j}}} = {\tilde{l}}_{F_{q_{j}}},$ for j = 1, 2,. . . , n . Hence, we have $f ({\tilde{l}}_{T p_{j}}) = f ({\tilde{l}}_{T q_{j}}), f ({\tilde{l}}_{I p_{j}}) = f ({\tilde{l}}_{I q_{j}}), and f ({\tilde{l}}_{F p_{j}}) = f ({\tilde{l}}_{F q_{j}}),$ , this implies T_{p_j} = T_{q_j}, I_{p_j} = I_{q_j} and F_{p_j} = F_{q_j} for j = 1, 2,. . . , n . Obviously, there exist $\min (T p_{j}, T q_{j}) = \sqrt{T p_{j} T q_{j}}, \min (I p_{j}, I q_{j}) = \sqrt{I p_{j} I q_{j}}$ , and $\min (F p_{j}, F q_{j}) = \sqrt{F p_{j} F q_{j}}$ . Hence, ρ₁ (P, Q) =1 holds.

Secondly, we prove the properties (c₁)– (c₃) of ρ₂(P, Q).

(c₁) Since T_{p
_j}, I_{p
_j}, F_{p
_j}, T_{q
_j}, I_{q
_j}, F_{q
_j} ∈ [0, τ], there are the following inequalities: 0 ≤ T_{p
_j} T_{q
_j} ≤ (max(T_{p
_j}, T_{q
_j}))², $0 \leq I_{p_{j}} I_{q_{j}} \leq {(\max (I_{p_{j}}, I_{q_{j}}))}^{2}, 0 \leq F_{p_{j}} F_{q_{j}} {(\max (F_{p_{j}}, F_{q_{j}}))}^{2}, and T_{p_{j}} T_{q_{j}} + I_{p_{j}} I_{q_{j}} + F_{p_{j}} F_{q_{j}} \leq {(\max (T_{p_{j}} T_{q_{j}}))}^{2} + {(\max (I_{p_{j}}, I_{q_{j}}))}^{2} + {(\max (F_{p_{j}}, F_{q_{j}}))}^{2}$ , Hence 0 ≤ ρ₂ (P, Q) ≤1 .

(c₂) It is obvious that the property is true.

(c₃) For any two sets of LNNs P and Q, if P = Q, there exist P ⊇ QandP ⊆ Q, then ${\tilde{l}}_{T_{p_{j}}} = {\tilde{l}}_{T_{q_{j}}}, {\tilde{l}}_{I_{p_{j}}} = {\tilde{l}}_{I_{q_{j}}},$ ${\tilde{l}}_{F_{p_{j}}} = {\tilde{l}}_{F_{q_{j}}}$ for j = 1, 2,. . . , n . This implies T_{p
_j} = T_{q
_j}, I_{p
_j} I_{q
_j}, and F_{p
_j} = F_{q
_j} for j = 1, 2,. . . , n . Obviously, there exist T_{p
_j}T_{q
_j} = (max(T_{p
_j}, T_{q
_j})) ², I_{p
_j} I_{q
_j} and $F_{p_{j}} F_{q_{j}} = {(\max (F_{p_{j}} F_{q_{j}}))}^{2}$ . Hence, ρ₂(P, Q) = 1 holds.

Thirdly, we prove the properties (c₁)– (c₃) of ρ₃(P, Q).

(c₁) The inequality 0≤ρ₃(P, Q)≤1is obvious.

(c₂) It is obvious that the property is true.

(c₃) For any two sets of LNNs P and Q, if P = Q, this implies T_{p
_j} = F_{q
_j}, I_{p
_j} = I_{q
_j} and F_{p
_j} = _{F_{q
_j}} for j = 1, 2,. . . , n . Then min (T_{p
_j}, T_{q
_j}) = max(T_{p
_j}, T_{q
_j}), min (I_{p
_j}, I_{q
_j}) = max(I_{p_j,I_{q
_j}}), and min (F_{p
_j}, F_{q
_j}) = max (I_{p
_j}, I_{q
_j}). Hence, ρ₃(P, Q) = 1 holds.

Thus, we have finished the proofs of these properties.

If we consider the weights of the elements p _j and q _j (j = 1, 2,. . . . , n) , the three weighted correlation coefficients between P and Q are proposed, respectively, as follows: $ρ_{1 ω} (P, Q) = \sum_{j = 1}^{n} ω_{j} \frac{(\min (T_{p_{j}}, T_{q_{j}}) + \min (I_{p_{j}}, I_{q_{j}}) + \min (F_{p_{j}}, F_{q_{j}}))}{\sqrt{(T_{p_{j}}, T_{q_{j}})} + \sqrt{(I_{p_{j}}, I_{q_{j}})} + \sqrt{(F_{p_{j}}, F_{q_{j}})}},$ (5)

$ρ_{2 ω} (P, Q) = \sum_{j = 1}^{n} ω_{j} \frac{T_{p_{j}} T_{q_{j}} + I_{p_{j}} I_{q_{j}} + F_{p_{j}} F_{q_{j}}}{({(\min (T_{p_{j}}, T_{q_{j}}))}^{2} + {(\min (I_{p_{j}}, I_{q_{j}}))}^{2} + {(\min (F_{p_{j}}, F_{q_{j}}))}^{2})},$ (6) $ρ_{3 ω} (P, Q) = \sum_{j = 1}^{n} ω_{j} \frac{(\min (T_{p_{j}}, T_{q_{j}}) + \min (I_{p_{j}}, I_{q_{j}}) + \min (F_{p_{j}}, F_{q_{j}}))}{(\min (T_{p_{j}}, T_{q_{j}}) + \min (I_{p_{j}}, I_{q_{j}}) + \min (F_{p_{j}}, F_{q_{j}}))},$ (7)

where ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ for j = 1, 2,. . . , n .

It is obvious that the three weighted correlation coefficients ρ_iω (P, Q) (i = 1, 2, 3) also satisfy the following properties (c₁)– (c₃):

0≤ρ _i _ω(P, Q)≤1;

ρ _i _ω(P, Q) =ρ _i _ω(Q, P);

If P = Q, then ρ _i _ω(P, Q) = 1_.

Especially when ω _j = 1/n for j = 1, 2,. . . , n, Equations (5)– (7) reduce to Equations (2)– (4), respectively.

We can easily prove the properties (c₁)– (c₃) for ρ_iω (P, Q) (i = 1, 2, 3) by the similar proof way. Hence, these proofs are omitted here.

4 MAGDM method based on correlation coefficients of LNNs

For a MAGDM problem, assume that y decision makers D = {D₁, D₂,. . . , Dy} evaluate a set of m alternatives S = {s₁, s₂,. . . , s_m} under a set of n attributes A = {a₁, a₂,. . . , a_n} by LNN information based on the linguistic term set $\tilde{L} = {{\tilde{l}}_{0}, {\tilde{l}}_{1}, . . ., {\tilde{l}}_{τ}}$ with odd cardinality τ+1. The corresponding weight vector of the decision makers is $ω^{d} = (ω_{1}^{d}, ω_{2}^{d}, \dots, ω_{y}^{d})^{T},$ satisfying $ω_{k}^{d} \in [0, 1]$ and $\sum_{k = 1}^{y} ω_{k}^{d} = 1$ (k = 1, 2,. . . , y) . The corresponding weight vector of the attributes is $ω^{a} = (ω_{1}^{a}, ω_{2}^{a}, \dots, ω_{n}^{a})^{T},$ satisfying $ω_{j}^{a} \in [0, 1]$ and $\sum_{j = 1}^{n} ω_{j}^{a} = 1$ (k = 1, 2,. . . , n) . Each decision maker D_k (k = 1, 2,. . . , y) . gives his/her evaluation values and establishes the corresponding LNN decision matrix as follows:

$\begin{matrix} M^{(k)} = (m_{ij}^{(k)})_{m \times n} = {[M_{1}^{(k)} M_{2}^{(k)} \dots M_{m}^{(k)}]}^{T} \\ = [\begin{matrix} < {\tilde{l}}_{T_{11}}^{(k)}, {\tilde{l}}_{I_{11}}^{(k)}, {\tilde{l}}_{F_{11}}^{(k)} > . . . < {\tilde{l}}_{T_{1 n}}^{(k)}, {\tilde{l}}_{I_{1 n}}^{(k)}, {\tilde{l}}_{F_{1 n}}^{(k)} > \\ < {\tilde{l}}_{T_{21}}^{(k)}, {\tilde{l}}_{I_{21}}^{(k)}, {\tilde{l}}_{F_{21}}^{(k)} > . . . < {\tilde{l}}_{T_{2 n}}^{(k)}, {\tilde{l}}_{I_{2 n}}^{(k)}, {\tilde{l}}_{F_{2 n}}^{(k)} > \\ ⋮ . . . ⋮ \\ < {\tilde{l}}_{T_{m 1}}^{(k)}, {\tilde{l}}_{I_{m 1}}^{(k)}, {\tilde{l}}_{F_{m 1}}^{(k)} > . . . < {\tilde{l}}_{T_{mn}}^{(k)}, {\tilde{l}}_{I_{mn}}^{(k)}, {\tilde{l}}_{F_{mn}}^{(k)} > \end{matrix}] \\ (k = 1, 2, . . ., y), \end{matrix}$

where $m_{ij}^{(k)} = < {\tilde{l}}_{T_{ij}}^{(k)}, {\tilde{l}}_{T_{ij}}^{(k)}, {\tilde{l}}_{F_{ij}}^{(k)} >$ , for ${\tilde{l}}_{T_{ij}}^{(k)}, {\tilde{l}}_{I_{ij}}^{(k)},$ ${\tilde{l}}_{F_{ij}}^{(k)} \in L$ represents the evaluation value of the i-th alternative s_i under the j-th attribute a_j given by the k-th decision maker D_k by the form of LNN.

Then, we apply the correlation coefficients of LNNs to MAGDM problems, and then decision steps are described as follows:

Step 1. Construct a LNN matrix of the ideal alternatives as follows: $\begin{matrix} M^{*} & = (m_{ij}^{*})_{m \times n} = [M_{1}^{*}, M_{2}^{*}, \dots, M_{m}^{*}]^{T} with \\ m_{ij}^{*} & = < \max_{i} ({\tilde{l}}_{T_{ij}}), \min_{i} ({\tilde{l}}_{I_{ij}}), \min_{i} ({\tilde{l}}_{F_{ij}}) \\ > (i = 1, 2, . . ., m; j = 1, 2, . . ., n) . \end{matrix}$

Step 2. Corresponding to the weight vector of the attributes $ω^{a} = (ω_{1}^{a}, ω_{2}^{a}, \dots, ω_{n}^{a})^{T}$ , calculate the weighted correlation coefficient values between M_i⁽ ^k ⁾ and M_i^* by Equations (5) or (6) or Equation (7), and obtain the values of ρ_1ω (Mi^(k), Mi^*) or ρ_2ω (Mi^(k), Mi^*) or ρ_3ω (Mi^(k), Mi^*) (i = 1, 2,. . . , m) .

Step 3. Corresponding to the weight vector of the decision makers $ω^{d} = (ω_{1}^{d}, ω_{2}^{d}, \dots, ω_{y}^{d})^{T}$ , calculate the overall weighted correlation coefficient values by one of Equations (8)– (10): $\begin{matrix} ρ_{1 ω} (s_{i}, M^{*}) = \sum_{k = 1}^{y} ω_{k}^{d} \cdot ρ_{1 ω} (M_{i}^{(k)}, M_{i}^{*}) \\ (i = 1, 2, \dots, m), \end{matrix}$ (8) $\begin{matrix} ρ_{2 ω} (s_{i}, M^{*}) = \sum_{k = 1}^{y} ω_{k}^{d} \cdot ρ_{2 ω} (M_{i}^{(k)}, M_{i}^{*}) \\ (i = 1, 2, \dots, m), \end{matrix}$ (9) $\begin{matrix} ρ_{3 ω} (s_{i}, M^{*}) = \sum_{k = 1}^{y} ω_{k}^{d} \cdot ρ_{3 ω} (M_{i}^{(k)}, M_{i}^{*}) \\ 1 pt (i = 1, 2, \dots, m), \end{matrix}$ (10)

where $ω_{k}^{d} \in [0, 1]$ and $\sum_{k = 1}^{y} ω_{k}^{d} = 1 (k = 1, 2, . . ., y) .$

Step 4. Rank the alternatives according to the correlation coefficient values of ρ_1ω (s_i, M^*) or ρ_2ω (s_i, M^*) or ρ_3ω (s_i, M^*) (i = 1, 2,. . . , m) , and the best alternative q_i_* is derived by $\begin{matrix} i^{*} = arg \max_{1 \leq i \leq m} {ρ_{j ω} (s_{i}, M^{*})} \\ (i = 1, 2, . . ., m; j = 1, 2, 3) \end{matrix}$ (11)

Step 5. End.

5 Illustrative example and comparison analysis

To demonstrate the applications of the presented approach, this section provides an illustrative example about the MAGDM problem of investment alternatives adapted from [67].

5.1 Illustrative example

An investment company needs to take the best alternative of investing a sum of money. There is a set of four possible investment alternatives S = {s₁: a car company, s₂: a food company, s₃: a computer company, s₄: an arms company}. The alternatives must be evaluated based on a set of three attributes A = {a₁: the risk factor, a₂: the growth factor, a₃: the environmental factor} with LNN information from the linguistic term set $\tilde{L}$ , where $\tilde{L}$ = { ${\tilde{l}}_{0}$ = extremely low, ${\tilde{l}}_{1}$ = very low, ${\tilde{l}}_{2}$ = low, ${\tilde{l}}_{3}$ = slightly low, ${\tilde{l}}_{4}$ = medium, ${\tilde{l}}_{5}$ = slightly high, ${\tilde{l}}_{6}$ = high, ${\tilde{l}}_{7}$ = very high, ${\tilde{l}}_{8}$ = extremely high}. The importance weight vector of the three attributes is ω^a = (0.35, 0.25, 0.4) ^T . A set of three decision-makers D = {D₁, D₂, D₃} is invited to evaluate the four alternatives, and their corresponding weight vector is ω^d = (0.37, 0.33, 0.3) ^T.

Thus, we can construct the LNN decision matrix M_i⁽ ^k ⁾ according to each decision-maker D_k (k = 1, 2, 3) as follows: $\begin{matrix} M^{(1)} = (m_{ij}^{(1)})_{4 \times 3} = {[M_{1}^{(1)} M_{2}^{(1)} M_{3}^{(1)} M_{4}^{(1)}]}^{T} \\ = [\begin{matrix} < {\tilde{l}}_{6}, {\tilde{l}}_{1}, {\tilde{l}}_{2} > < {\tilde{l}}_{7}, {\tilde{l}}_{2}, {\tilde{l}}_{1} > < {\tilde{l}}_{6}, {\tilde{l}}_{2}, {\tilde{l}}_{2} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{3}, {\tilde{l}}_{2} > < {\tilde{l}}_{7}, {\tilde{l}}_{2}, {\tilde{l}}_{1} > \\ < {\tilde{l}}_{6}, {\tilde{l}}_{2}, {\tilde{l}}_{2} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{6}, {\tilde{l}}_{2}, {\tilde{l}}_{2} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{2} > < {\tilde{l}}_{7}, {\tilde{l}}_{2}, s_{3} > < {\tilde{l}}_{7}, {\tilde{l}}_{2}, {\tilde{l}}_{1} > \end{matrix}], \end{matrix}$ $\begin{matrix} M^{(2)} = (m_{ij}^{(2)})_{4 \times 3} = {[M_{1}^{(2)} M_{2}^{(2)} M_{3}^{(2)} M_{4}^{(2)}]}^{T} \\ = [\begin{matrix} < {\tilde{l}}_{6}, {\tilde{l}}_{1}, {\tilde{l}}_{2} > < {\tilde{l}}_{6}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{4}, {\tilde{l}}_{2}, {\tilde{l}}_{3} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{2}, {\tilde{l}}_{3} > < {\tilde{l}}_{6}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{4}, {\tilde{l}}_{2}, {\tilde{l}}_{3} > \\ < {\tilde{l}}_{5}, {\tilde{l}}_{1}, {\tilde{l}}_{2} > < {\tilde{l}}_{5}, {\tilde{l}}_{1}, {\tilde{l}}_{2} > < {\tilde{l}}_{5}, {\tilde{l}}_{4}, {\tilde{l}}_{2} > \\ < {\tilde{l}}_{6}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{5}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{5}, {\tilde{l}}_{2}, {\tilde{l}}_{3} > \end{matrix}], \end{matrix}$ $\begin{matrix} M^{(3)} = (m_{ij}^{(3)})_{4 \times 3} = {[M_{1}^{(3)} M_{2}^{(3)} M_{3}^{(3)} M_{4}^{(3)}]}^{T} \\ = [\begin{matrix} < {\tilde{l}}_{7}, {\tilde{l}}_{3}, {\tilde{l}}_{4} > < {\tilde{l}}_{7}, {\tilde{l}}_{3}, {\tilde{l}}_{3} > < {\tilde{l}}_{5}, {\tilde{l}}_{2}, {\tilde{l}}_{5} > \\ < {\tilde{l}}_{6}, {\tilde{l}}_{3}, {\tilde{l}}_{4} > < {\tilde{l}}_{5}, {\tilde{l}}_{1}, {\tilde{l}}_{2} > < {\tilde{l}}_{6}, {\tilde{l}}_{2}, {\tilde{l}}_{3} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{2}, {\tilde{l}}_{4} > < {\tilde{l}}_{6}, {\tilde{l}}_{1}, {\tilde{l}}_{2} > < {\tilde{l}}_{7}, {\tilde{l}}_{2}, {\tilde{l}}_{4} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{2}, {\tilde{l}}_{3} > < {\tilde{l}}_{5}, {\tilde{l}}_{2}, {\tilde{l}}_{1} > < {\tilde{l}}_{6}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > \end{matrix}] . \end{matrix}$

Then, we can apply the developed MAGDM method to this decision-making problem by the following steps:

Step 1. Establish the LNN matrix $M^{*} = (m_{ij}^{*})_{4 \times 3}$ of the ideal alternatives (ideal solutions) $M_{i}^{*} (i = 1, 2, 3, 4)$ as follows: $\begin{matrix} M^{*} = (m_{ij}^{*})_{4 \times 3} = {[M_{1}^{*} M_{2}^{*} M_{3}^{*} M_{4}^{*}]}^{T} \\ = [\begin{matrix} < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > \\ < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > \end{matrix}], \end{matrix}$

where $m_{ij}^{*} = < \max ({\tilde{l}}_{T_{ij}}), \min ({\tilde{l}}_{I_{ij}}), \min ({\tilde{l}}_{F_{ij}})$ $> = < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} >$ (i = 1, 2, 3, 4 ; j = 1, 2, 3)and $M_{i}^{*} = {< {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} > < {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} >$ $< {\tilde{l}}_{7}, {\tilde{l}}_{1}, {\tilde{l}}_{1} >} (i = 1, 2, 3, 4)$ represents a set of the ideal evaluation values of the i-th alternative s_i (i = 1, 2, 3, 4) with respect to the attribute a_j (j = 1, 2, 3) by LNNs.

Step 2. Calculate the weighted correlation coefficient values between $M_{i}^{(k)}$ and $M_{i}^{*} (i = 1, 2, 3, 4)$ by Equations (5) or (6) or (7), and obtain the values of $ρ_{1 ω} (M_{i}^{(k)}$ , $M_{i}^{*}) or ρ_{2 ω} (M_{i}^{(k)}$ , $M_{i}^{*})$ or $ρ_{3 ω} (M_{i}^{(k)}$ , $M_{i}^{*})$ (i = 1, 2, 3, 4) as following correlation coefficients matrices: $\begin{matrix} ρ_{1 ω} (M, M^{*}) \\ = [\begin{matrix} ρ_{1 ω} (M^{(1)}, M^{*}) \\ ρ_{1 ω} (M^{(2)}, M^{*}) \\ ρ_{1 ω} (M^{(3)}, M^{*}) \end{matrix}] = [\begin{matrix} ρ_{1 ω} (M_{1}^{(1)}, M_{1}^{*}) ρ_{1 ω} (M_{2}^{(1)}, M_{2}^{*}) \\ ρ_{1 ω} (M_{1}^{(2)}, M_{1}^{*}) ρ_{1 ω} (M_{2}^{(2)}, M_{2}^{*}) \\ ρ_{1 ω} (M_{1}^{(3)}, M_{1}^{*}) ρ_{1 ω} (M_{2}^{(3)}, M_{2}^{*}) \end{matrix} \\ \begin{matrix} ρ_{1 ω} (M_{3}^{(1)}, M_{3}^{*}) ρ_{1 ω} (M_{4}^{(1)}, M_{4}^{*}) \\ ρ_{1 ω} (M_{3}^{(2)}, M_{3}^{*}) ρ_{1 ω} (M_{4}^{(2)}, M_{4}^{*}) \\ ρ_{1 ω} (M_{3}^{(3)}, M_{3}^{*}) ρ_{1 ω} (M_{4}^{(3)}, M_{4}^{*}) \end{matrix}] \\ = [\begin{matrix} 0.8975 0.9542 0.8945 0.9388 \\ 0.8350 0.8307 0.8043 0.8602 \\ 0.8012 0.8166 0.8730 0.8979 \end{matrix}], \end{matrix}$

$\begin{matrix} ρ_{2 ω} (M, M^{*}) \\ = [\begin{matrix} ρ_{2 ω} (M^{(1)}, M^{*}) \\ ρ_{2 ω} (M^{(2)}, M^{*}) \\ ρ_{2 ω} (M^{(3)}, M^{*}) \end{matrix}] = [\begin{matrix} ρ_{2 ω} (M_{1}^{(1)}, M_{1}^{*}) ρ_{2 ω} (M_{2}^{(1)}, M_{2}^{*}) \\ ρ_{2 ω} (M_{1}^{(2)}, M_{1}^{*}) ρ_{2 ω} (M_{2}^{(2)}, M_{2}^{*}) \\ ρ_{2 ω} (M_{1}^{(3)}, M_{1}^{*}) ρ_{2 ω} (M_{2}^{(3)}, M_{2}^{*}) \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} ρ_{2 ω} (M_{3}^{(1)}, M_{3}^{*}) ρ_{2 ω} (M_{4}^{(1)}, M_{4}^{*}) \\ ρ_{2 ω} (M_{3}^{(2)}, M_{3}^{*}) ρ_{2 ω} (M_{4}^{(2)}, M_{4}^{*}) \\ ρ_{2 ω} (M_{3}^{(3)}, M_{3}^{*}) ρ_{2 ω} (M_{4}^{(3)}, M_{4}^{*}) \end{matrix}] \\ = [\begin{matrix} 0.5694 0.6098 0.5781 0.6097 \\ 0.4344 0.4346 0.4185 0.4454 \\ 0.4257 0.4836 0.5331 0.5271 \end{matrix}], \end{matrix}$ $\begin{matrix} ρ_{3 ω} (M, M^{*}) \\ = [\begin{matrix} ρ_{3 ω} (M^{(1)}, M^{*}) \\ ρ_{3 ω} (M^{(2)}, M^{*}) \\ ρ_{3 ω} (M^{(3)}, M^{*}) \end{matrix}] = [\begin{matrix} ρ_{3 ω} (M_{1}^{(1)}, M_{1}^{*}) ρ_{3 ω} (M_{2}^{(1)}, M_{2}^{*}) \\ ρ_{3 ω} (M_{1}^{(2)}, M_{1}^{*}) ρ_{3 ω} (M_{2}^{(2)}, M_{2}^{*}) \\ ρ_{3 ω} (M_{1}^{(3)}, M_{1}^{*}) ρ_{3 ω} (M_{2}^{(3)}, M_{2}^{*}) \end{matrix} \\ \begin{matrix} ρ_{3 ω} (M_{3}^{(1)}, M_{3}^{*}) ρ_{3 ω} (M_{4}^{(1)}, M_{4}^{*}) \\ ρ_{3 ω} (M_{3}^{(2)}, M_{3}^{*}) ρ_{3 ω} (M_{4}^{(2)}, M_{4}^{*}) \\ ρ_{3 ω} (M_{3}^{(3)}, M_{3}^{*}) ρ_{3 ω} (M_{4}^{(3)}, M_{4}^{*}) \end{matrix}] \\ = [\begin{matrix} 0.7959 0.8975 0.7955 0.8625 \\ 0.7022 0.6847 0.6354 0.7389 \\ 0.5981 0.6417 0.7192 0.7931 \end{matrix}], \end{matrix}$

where ρ_1ω (M^(k), M^*) , ρ_2ω (M^(k), M^*) , and ρ_1ω (M^(k), M^*) represent the correlation coefficients between the ideal matrix and the evaluation matrix of the k-th decision-maker for the four alternatives by Equations (5)– (7).

Step 3. Corresponding to the weight vector of the decision-makers ω^d = (0.37, 0.33, 0.3) ^T, calculate the overall weighted correlation coefficients values ρ_1ω (s_i, M^*) (i = 1, 2,. . . , m) by Equation (8) as follows: $\begin{matrix} ρ_{1 ω} (s_{1}, M^{*}) & = 0.37 \times 0.8975 + 0.33 \times 0.8350 \\ + 0.3 \times 0.8012 = 0.8480; \end{matrix}$ $\begin{matrix} ρ_{1 ω} (s_{2}, M^{*}) & = 0.37 \times 0.9542 + 0.33 \times 0.6847 \\ + 0.3 \times 0.8166 = 0.8722; \end{matrix}$ $\begin{matrix} ρ_{1 ω} (s_{3}, M^{*}) & = 0.37 \times 0.8945 + 0.33 \times 0.8043 \\ + 0.3 \times 0.8730 = 0.8583; \end{matrix}$ $\begin{matrix} ρ_{1 ω} (s_{4}, M^{*}) & = 0.37 \times 0.9388 + 0.33 \times 0.8602 \\ + 0.3 \times 0.8979 = 0.9006 . \end{matrix}$

Obviously, the ranking order is: $\begin{matrix} ρ_{1 ω} (s_{4}, M^{*}) & ≻ ρ_{1 ω} (s_{2}, M^{*}) ≻ ρ_{1 ω} (s_{3}, M^{*}) \\ ≻ ρ_{1 ω} (s_{1}, M^{*}) . \end{matrix}$

Similarly, we can calculate the overall weighted correlation coefficients values by Equations (9) or (10), respectively, and get ρ_2ω (s_i, M^*) or ρ_3ω (s_i, M^*) as follows: $ρ_{2 ω} (s_{1}, M^{*}) = 0.4817, ρ_{2 ω} (s_{2}, M^{*}) = 0.5141,$ $ρ_{2 ω} (s_{3}, M^{*}) = 0.5120, ρ_{2 ω} (s_{4}, M^{*}) = 0.5307;$ $ρ_{3 ω} (s_{1}, M^{*}) = 0.7056, ρ_{3 ω} (s_{2}, M^{*}) = 0.7505,$ $ρ_{3 ω} (s_{3}, M^{*}) = 0.7198, ρ_{3 ω} (s_{4}, M^{*}) = 0.8009 .$

There are the ranking orders: $\begin{matrix} ρ_{2 ω} (s_{4}, M^{*}) & ≻ ρ_{2 ω} (s_{2}, M *) ≻ ρ_{2 ω} (s_{3}, M^{*}) \\ ≻ ρ_{2 ω} (s_{1}, M^{*}); \end{matrix}$ $\begin{matrix} ρ_{3 ω} (s_{4}, M^{*}) & ≻ ρ_{3 ω} (s_{2}, M^{*}) ≻ ρ_{3 ω} (s_{3}, M^{*}) \\ ≻ ρ_{3 ω} (s_{1}, M^{*}) . \end{matrix}$

Step 4. According to the above ranking orders of the correlation coefficient values, there is the same ranking order s₄ ≻ s₂ ≻ s₃ ≻ s₁. Thus, based on Equation (11) the alternative s₄ is the best choice among the four alternatives.

5.2 Related comparison

For convenient comparison, Table 1 lists all the MAGDM results based on the LNNWGA and LNNWAA operators in the relevant paper [67] and the correlation coefficients of LNNs presented in this paper, respectively.

Table 1
MAGDM results based on LNNs

MAGDM Method Correlation coefficients values (Score value [67]) Ranking order The best Alternative

ρ _1ω 0.8480, 0.8722, 0.8583, 0.9006 s₄ ≻ s₂ ≻ s₃ ≻ s₁

ρ _2ω 0.4817, 0.5141, 0.5120, 0.5307 s₄ ≻ s₂ ≻ s₃ ≻ s₁

ρ _3ω 0.7056, 0.7505, 0.7198, 0.8009 s₄ ≻ s₂ ≻ s₃ ≻ s₁

LNNWGA Operator [67] 0.7397, 0.7747, 0.7513, 0.8035 s₄ ≻ s₂ ≻ s₃ ≻ s₁

LNNWAA Operator [67] 0.7528, 0.7770, 0.7613, 0.8060 s₄ ≻ s₂ ≻ s₃ ≻ s₁

MAGDM Method	Correlation coefficients values (Score value [67])	Ranking order	The best Alternative
ρ _1ω	0.8480, 0.8722, 0.8583, 0.9006	s₄ ≻ s₂ ≻ s₃ ≻ s₁
ρ _2ω	0.4817, 0.5141, 0.5120, 0.5307	s₄ ≻ s₂ ≻ s₃ ≻ s₁
ρ _3ω	0.7056, 0.7505, 0.7198, 0.8009	s₄ ≻ s₂ ≻ s₃ ≻ s₁
LNNWGA Operator [67]	0.7397, 0.7747, 0.7513, 0.8035	s₄ ≻ s₂ ≻ s₃ ≻ s₁
LNNWAA Operator [67]	0.7528, 0.7770, 0.7613, 0.8060	s₄ ≻ s₂ ≻ s₃ ≻ s₁

From the result of Table 1, we can find that the ranking orders based on the three new correlation coefficients are in accordance with the results provided by Fang and Ye [67]. Firstly, compared with the existing related approaches based on the LNNWGA and LNNWAA operators [67], the calculation processes of the proposed methods are relatively simple. Secondly, the LNNWGA and LNNWAA operators contain some unreasonable results, and then the aggregated result of the LNNWGA operator tends to the maximum weight value, while the aggregated result of the LNNWAA operator tends to the maximum value in some cases [42]. Obviously, the proposed correlation coefficients can avoid the aforementioned insufficiencies and show their advantages. Furthermore, the method proposed in this paper can solve decision-making problems with indeterminate and inconsistent linguistic information; while the method provided in the relevant papers [ 6 , 37–41] cannot deal with LNNinformation.

6 Conclusion

Under linguistic environment, three new correlation coefficients between LNNs were presented in this paper. Then, we established a MAGDM method based on the proposed correlationcoefficients of LNNs. Finally, an illustrative example on the MAGDM problems with LNN information is provided to demonstrate the application and effectiveness of the presented method. The decision-making results show that the MAGDM method proposed in this paper can effectively deal with LNN information. In the future work, we shall extend the presented method to fault diagnosis and the three correlation coefficients to Pythagorean Fuzzy sets [9 –11] and picture fuzzy sets [12].

Footnotes

Acknowledgments

This research was supported by the Public Technology Research Project of Zhejiang Province (LGG18F030008).

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