Linguistic neutrosophic uncertain numbers and their multiple attribute group decision-making method

Abstract

To describe the linguistic neutrosophic information with uncertain variables in the complex decision-making problems, this paper originally defines the concept of a linguistic neutrosophic uncertain number (LNUN). The LNUN consists of three uncertain linguistic variables given by three neutrosophic linguistic numbers to describe the truth, falsity, and indeterminacy linguistic information independently. Then, both the basic operational laws of LNUNs and the score and accuracy functions of LNUNs are presented for the ranking of LNUNs. After that, two operators including a LNUN weighted arithmetic averaging (LNUNWAA) operator and a LNUN weighted geometric averaging (LNUNWGA) operator are developed to aggregate LNUN information. Based on the aggregation operators, a novel multiple attribute group decision-making (MAGDM) method is established under a LNUN environment. Finally, an example of investment decision is illustrated to demonstrate the application and the effectiveness of the developed method.

Keywords

Linguistic neutrosophic uncertain number linguistic neutrosophic uncertain number weighted arithmetic averaging operator linguistic neutrosophic uncertain number weighted geometric averaging operator score function accuracy function multiple attribute group decision-making neutrosophic linguistic number

1 Introduction

Due to the complexity of decision-making environment and the ambiguity and uncertainty of human cognition of objective things, linguistic variables are more effective than numerical values to describe the decision-making information. Thus, to improve the efficiency of decision-making, many researchers have studied the decision-making problems in a linguistic environment. And some achievements have been made in the last decades [4]. Zadeh [25] first defined the concept of a linguistic variable and applied it in fuzzy reasoning. Then, Herrera et al. [5, 7] proposed linguistic decision-making analysis, and then Xu developed linguistic aggregation operators [33, 35] and goal programming models [36]. Merigó et al. [15 , 20] presented more linguistic aggregation operators for the aggregation of linguistic variables in decision-making problems. To describe the uncertain decision-making information, Xu [34, 37] put forward uncertain linguistic variables given by interval values (interval linguistic variables). Then, Wei [11], Wei et al. [12], Jin et al. [13], Peng et al. [1], Zhang [14] developed various aggregation operators of uncertain linguistic variables for the MAGDM/MADM problems with uncertain linguistic information. After that, Peng et al. [2] proposed a cloud model with uncertain linguistic information for group decision-making.

To better cope with the incompleteness and uncertainty of the linguistic decision environment, some researchers extended the concept of linguistic variables by attaching reliability information. Based on intuitionistic fuzzy numbers and linguistic variables, Wang and Li [16] introduced the intuitionistic linguistic fuzzy number (ILFN), which is composed of a linguistic variable and an intuitionistic fuzzy number, to express both the decision-maker’s judgment and the reliability of the linguistic variable, and then they presented some aggregation operators of ILFNs for MAGDM problems. Next, Ye presented a single-valued neutrosophic linguistic number (SVNLN) [19], where the reliability of the linguistic variable is given by a single-valued neutrosophic number [6], and an interval neutrosophic linguistic number (INLN) [18], where the reliability is described by an interval-valued neutrosophic number. Then, Liu and Shi [26] proposed the concept of neutrosophic uncertain linguistic number (SVNULN), in which the linguistic variable is an interval value. Ye [22] further put forward interval neutrosophic uncertain linguistic variables (INULVs), where both the linguistic variable and the reliability of the linguistic variable are interval values. Later, Broumi et al. [31] presented an extended TOPSIS method for INULVs and applied it to multiple attribute decision making (MADM). However, all the above linguistic numbers cannot describe the truth, indeterminacy, and falsity linguistic information directly from a linguistic item set.

To describe the truth and falsity linguistic information directly from linguistic items, Chen et al. [38] firstly proposed the linguistic intuitionistic fuzzy number (LIFN), which consists of l_p and l_q for representing the truth/membership degree and the falsity/non-membership degree, respectively. And Liu et al. [27, 28] developed some aggregation operators of LIFNs for MADM. Then, Fang and Ye [39] further proposed linguistic neutrosophic number (LNN), where three independent linguistic variables are used to describe the truth, falsity, and indeterminacy linguistic information respectively, and then the LNN weighted arithmetic averaging (LNNWAA) and LNN weighted geometric averaging (LNNWGA) operators for MAGDM problems with LNN information. After that, some aggregation operators of LNNs, such as the LNN normalized weighted Bonferroni mean (LNNNWBM) and LNN normalized weighted geometric Bonferroni mean (LNNNWGBM) operators [3], the linguistic neutrosophic power weighted Heronian aggregation (LNPWHA) operator [29], the linguistic neutrosophic generalized weighted partitioned Bonferroni mean (LNGWPBM) operator [32], the linguistic neutrosophic Hamy mean (LNHM) and weighted linguistic neutrosophic Hamy mean (WLNHM) operators [30], were presented for MAGDM problems with LNN information. Moreover, to describe certain linguistic neutrosophic information and uncertain linguistic neutrosophic information simultaneously, Ye [23] proposed a linguistic neutrosophic cubic number and applied it in MADM problems. However, all aforementioned uncertain linguistic variables given by interval linguistic term values are not so flexible for uncertain linguistic expression and analysis.

Fortunately, to express the incomplete and indeterminate information, Samrandache defined a neutrosophic number (NN) [6 , 9], denoted by B = t + vI, where t is the determinate part and vI is the indeterminate part. To express uncertain linguistic information, Samrandache further proposed the neutrosophic linguistic number (NLN) [10], denoted by l_t+vI, where t + vI is a NN. Later, Ye [21] proposed aggregation operators of NLNs for MAGDM. Recently, Ye [24] presented hesitant neutrosophic linguistic numbers (HNLNs) and developed a MADM method under a HNLN environment. Obviously, the uncertain linguistic variables expressed by neutrosophic numbers are more flexible than those given by interval linguistic term values since NN is a changeable interval number.

Motivated by NLN and LNN, this paper proposes a new concept of LNUN, consisting of three uncertain linguistic variables given by neutrosophic linguistic numbers, to describe the truth, falsity, and indeterminacy linguistic information respectively. Then, we introduce the operational laws of LNUNs and the score and accuracy functions of LNUNs for the sorting of LNUNs. Moreover, we present a LNUN weighted arithmetic averaging (LNUNWAA) operator and a LNUN weighted geometric averaging (LNUNWGA) operator, and develop a novel MAGDM method based on the two operators under a LNUN environment.

In Section 2, the preliminaries of LNNs and NLNs are introduced, including the concept, the basic operational laws, and the ranking rules. In Section 3, the concept of LNUN, the operational laws of LNUNs, the expected value, and the score and accuracy functions of LNUNs are all presented for the ranking of LNUNs. In Section 4, the weighted aggregation operators of LNUNs including the LNUNWAA and LNUNWGA operators are put forward and proved. In section 5, a novel MAGDM method is developed based on the LNUNWAA or LNUNWGA operator under a LNUN environment. In section 6, an illustrative example on investment problems was provided to demonstrate the application of the proposed method. Finally, Section 7 presents comparative analysis, and then Section 8 gives conclusions and future research direction.

2 Preliminaries of LNNs and NLNs

2.1 Operational laws and score and accuracy functions of LNNs

In this subsection, the basic operational laws of LNNs and the score and accuracy functions of LNNs for ranking will be introduced according to the reference [39].

Definition 1. [39] Assume that L = {l₀, l₁, ⋯, l_s} with odd cardinality s + 1. A LNN can be defined as e = 〈l_{T
_a}, l_{U
_a}, l_{F
_a}〉 for l_{T
_a}, l_{U
_a}, l_{F
_a} ∈ L and T_a, U_a, F_a ∈ [0, s], where l_{T
_a}, l_{U
_a}, l_{F
_a} express the truth, indeterminacy, and falsity linguistic term values independently.

Definition 2. [39] Let e = 〈l_{T
_a}, l_{U
_a}, l_{F
_a}〉, e₁ = 〈l_{T
_{a
₁}}, l_{U
_{a
₁}}, l_{F
_{a
₁}}〉, and e₂ = 〈l_{T
_{a
₂}}, l_{U
_{a
₂}}, l_{F
_{a
₂}}〉 be three LNNs in L and ρ > 0, there exist the following operational laws:

$\begin{matrix} e_{1} \oplus e_{2} = & 〈 l_{T_{a_{1}}}, l_{U_{a_{1}}}, l_{F_{a_{1}}} 〉 \oplus 〈 l_{T_{a_{2}}}, l_{U_{a_{2}}}, l_{F_{a_{2}}} 〉 \\ = & 〈 l_{T_{a_{1}} + T_{a_{2}} - \frac{T_{a_{1}} T_{a_{2}}}{s}}, l_{\frac{U_{a_{1}} U_{a_{2}}}{s}}, l_{\frac{F_{a_{1}} F_{a_{2}}}{s}} 〉 \end{matrix}$ ;

$\begin{matrix} e_{1} \otimes e_{2} = & 〈 l_{T_{a_{1}}}, l_{U_{a_{1}}}, l_{F_{a_{1}}} 〉 \otimes 〈 l_{T_{a_{2}}}, l_{U_{a_{2}}}, l_{F_{a_{2}}} 〉 \\ = & 〈 l_{\frac{T_{a_{1}} T_{a_{2}}}{s}}, l_{U_{a_{1}} + U_{a_{2}} - \frac{U_{a_{1}} U_{a_{2}}}{s}}, l_{F_{a_{1}} + F_{a_{2}} - \frac{F_{a_{1}} F_{a_{2}}}{s}} 〉 \end{matrix}$ ;

$ρ e = ρ 〈 l_{T_{a}}, l_{U_{a}}, l_{F_{a}} 〉 = 〈 l_{s - s {(1 - \frac{T_{a}}{s})}^{ρ}}, l_{s {(\frac{U_{a}}{s})}^{ρ}}, l_{s {(\frac{F_{a}}{s})}^{ρ}} 〉$ ;

$e^{ρ} = {〈 l_{T_{a}}, l_{U_{a}}, l_{F_{a}} 〉}^{ρ} = 〈 l_{s (\frac{T_{a}}{s})^{ρ}}, l_{s - s (1 - \frac{U_{a}}{s})^{ρ}}, l_{s - s (1 - \frac{F_{a}}{s})^{ρ}} 〉$ .

It is clear that the above operational results of the LNNs are still LNNs. To compare the LNNs, Fang and Ye [39] defined the score and accuracy functions of LNNs.

Definition 3. [39] Let e =〈 l_{T
_a}, l_{U
_a}, l_{F
_a} 〉 be a LNN in L. Then the score and accuracy functions of e are $Q (e) = \frac{2 s + T_{a} - U_{a} - F_{a}}{3 s} for Q (e) \in [0, 1],$ (1) $T (e) = \frac{T_{a} - F_{a}}{s} for T (e) \in [- 1, 1] .$ (2)

Definition 4. [39] Let e₁ =〈 l_{T
_{a
₁}}, l_{U
_{a
₁}}, l_{F
_{a
₁}} 〉 and e₂ =〈 l_{T
_{a
₂}}, l_{U
_{a
₂}}, l_{F
_{a
₂}} 〉 be two LNNs, then they have the following comparative relations:

if Q (e₁) > Q (e₂), then e₁ > e₂;

if Q (e₁) < Q (e₂), then e₁ < e₂;

if Q (e₁) = Q (e₂), and T (e₁) > T (e₂), then e₁ > e₂;

if Q (e₁) = Q (e₂), and T (e₁) < T (e₂), then e₁ < e₂;

if Q (e₁) = Q (e₂), and T (e₁) = T (e₂), then e₁ = e₂.

2.2 Operational laws and expected values of NLNs

In this subsection, both the operational laws and the expected values of the NLNs will be introduced according to the reference [21].

Definition 5. [21] Suppose that l_t₁+v₁I and l_t₂+v₂I are two NLNs. Then they have the following operational laws:

l_t₁+v₁I + l_t₂+v₂I = l_{t₁+t₂+(v₁+v₂)I};

l_t₁+v₁I - l_t₂+v₂I = l_{t₁-t₂+(v₁-v₂)I};

l_t₁+v₁I × l_t₂+v₂I = l_{t₁t₂+(t₁v₂+t₂v₁+v₁v₂)I};

$\frac{l_{t_{1} + v_{1} I}}{l_{t_{2} + v_{2} I}} = l_{\frac{t_{1}}{t_{2}} + \frac{t_{2} v_{1} - t_{1} v_{2}}{t_{2} (t_{2} + v_{2})} I}$ for t₂ ≠ 0 and t₂ ≠ - v₂;

ρl_t₁+v₁I = l_{ρt₁+ρv₁I} for ρ ≥ 0;

$(l_{t_{1} + v_{1} I})^{ρ} = l_{t_{1}^{ρ} + [(t_{1} + v_{1})^{ρ} - t_{1}^{ρ}] I}$ for ρ ≥ 0.

It is clear that the above operational results are still NLNs. Then, Ye [21] put forward the expected value of a NLN and the ranking rules based on the expected values.

Definition 6. [21] Assume that L = {l₀, l₁, ⋯, l_s} is a finite linguistic term set with cardinality s + 1, where l_i = l_{t_i+v_iI} is a NLN with I ∈ [inf I, sup I]. The expected value of l_i can be written as $E (l_{i}) = \frac{(t_{i} + v_{i} inf I) + (t_{i} + v_{i} sup I)}{2 s} .$ (3)

Definition 7. [21] Let l_i = l_{t_i+v_iI} and l_j = l_{t_j+v_jI} be two NLNs. There exist the following relations:

If E (l_i) > E (l_j), then l_i > l_j;

If E (l_i) = E (l_j), then l_i = l_j.

3 Linguistic neutrosophic uncertain numbers

A LNN is used to express the truth, indeterminacy, and falsity linguistic term values independently by three certain linguistic variables l_{T
_a}, l_{U
_a}, and l_{I
_a}. However, the truth, indeterminacy, and falsity degrees need to be given by uncertain linguistic information especially in some complex decision situations. NNs seem simpler and more flexible to express the uncertain information than the traditional interval values since NN can be considered as a changeable interval number. Thus, in this section, we propose the LNUN, which consists of the truth, indeterminacy, and falsity neutrosophic linguistic variables/numbers.

Definition 8. Assume that L = {l₀, l₁, ⋯, l_s} is a linguistic term set with odd cardinality s + 1. A LNUN in L is constructed as h =〈 l_{T_a+T_bI}, l_{U_a+U_bI}, l_{F_a+F_bI} 〉 with three uncertain linguistic variables l_{T_a+T_bI}, l_{U_a+U_bI}, and l_{F_a+F_bI} representing the truth, indeterminacy, and falsity NLNs independently, where T_a + T_bI, U_a + U_bI, F_a + F_bI ∈ [0, s] and I ∈ [inf I, sup I].

Definition 9. Let h =〈 l_{T_a+T_bI}, l_{U_a+U_bI}, l_{F_a+F_bI} 〉, h₁ =〈 l_{T_{a
₁}+T_{b
₁}I}, l_{U_{a
₁}+U_{b
₁}I}, l_{F_{a
₁}+F_{b
₁}I} 〉, and h₂ =〈 l_{T_{a
₂}+T_{b
₂}I}, l_{U_{a
₂}+U_{b
₂}I}, l_{F_{a
₂}+F_{b
₂}I} 〉 be three LNUNs in L and ρ > 0, the operational laws of the LNUNs are given by $\begin{matrix} h_{1} \oplus h_{2} & = & 〈 l_{T_{a_{1}} + T_{b_{1}} I}, l_{U_{a_{1}} + U_{b_{1}} I}, l_{F_{a_{1}} + F_{b_{1}} I} 〉 \\ \oplus & 〈 l_{T_{a_{2}} + T_{b_{2}} I}, l_{U_{a_{2}} + U_{b_{2}} I}, l_{F_{a_{2}} + F_{b_{2}} I} 〉 \\ = & 〈 l_{T_{a_{1}} + T_{b_{1}} I + T_{a_{2}} + T_{b_{2}} I - \frac{(T_{a_{1}} + T_{b_{1}} I) (T_{a_{2}} + T_{b_{2}} I)}{s}}, \\ l_{\frac{(U_{a_{1}} + U_{b_{1}} I) (U_{a_{2}} + U_{b_{2}} I)}{s}}, l_{\frac{(F_{a_{1}} + F_{b_{1}} I) (F_{a_{2}} + F_{b_{2}} I)}{s}} 〉, \end{matrix}$ (4) $\begin{matrix} h_{1} \otimes h_{2} & = & 〈 l_{T_{a_{1}} + T_{b_{1}} I}, l_{U_{a_{1}} + U_{b_{1}} I}, l_{F_{a_{1}} + F_{b_{1}} I} 〉 \\ \oplus & 〈 l_{T_{a_{2}} + T_{b_{2}} I}, l_{U_{a_{2}} + U_{b_{2}} I}, l_{F_{a_{2}} + F_{b_{2}} I} 〉 \\ = & 〈 l_{\frac{(T_{a_{1}} + T_{b_{1}} I) (T_{a_{2}} + T_{b_{2}} I)}{s}}, \\ l_{U_{a_{1}} + U_{b_{1}} I + U_{a_{2}} + U_{b_{2}} I - \frac{(U_{a_{1}} + U_{b_{1}} I) (U_{a_{2}} + U_{b_{2}} I)}{s}}, \\ l_{F_{a_{1}} + F_{b_{1}} I + F_{a_{2}} + F_{b_{2}} I - \frac{(F_{a_{1}} + F_{b_{1}} I) (F_{a_{2}} + F_{b_{2}} I)}{s}} 〉, \end{matrix}$ (5) $\begin{matrix} ρ h = ρ 〈 l_{T_{a} + T_{b} I}, l_{U_{a} + U_{b} I}, l_{F_{a} + F_{b} I} 〉 \\ = 〈 l_{s - s (1 - \frac{T_{a} + T_{b} I}{s})^{ρ}}, l_{s (\frac{U_{a} + U_{b} I}{s})^{ρ}}, l_{s (\frac{F_{a} + F_{b} I}{s})^{ρ}} 〉, \end{matrix}$ (6) $\begin{matrix} h^{ρ} & = & {〈 l_{T_{a} + T_{b} I}, l_{U_{a} + U_{b} I}, l_{F_{a} + F_{b} I} 〉}^{ρ} \\ = & 〈 l_{s (\frac{T_{a} + T_{b} I}{s})^{ρ}}, l_{s - s (1 - \frac{U_{a} + U_{b} I}{s})^{ρ}}, l_{s - s (1 - \frac{F_{a} + F_{b} I}{s})^{ρ}} 〉 . \end{matrix}$ (7)

Obviously, the above operational results are still LNUNs. To rank LNUNs, the expected values of LNUNs and the score and accuracy functions of the LNUNs should be further defined.

Definition 10. Let a LNUN be h =〈 l_{T_a+T_bI}, l_{U_a+U_bI}, l_{F_a+F_bI} 〉 in L. The expected value of h is $\begin{matrix} E (h) & = & 〈 l_{E (T_{a} + T_{b} I)}, l_{E (U_{a} + U_{b} I)}, l_{E (F_{a} + F_{b} I)} 〉 \\ = & 〈 l_{\frac{(T_{a} + T_{b} inf I) + (T_{a} + T_{b} sup I)}{2}}, l_{\frac{(U_{a} + U_{b} inf I) + (U_{a} + U_{b} sup I)}{2}} 〉, \\ l_{\frac{(F_{a} + F_{b} inf I) + (F_{a} + F_{b} sup I)}{2}} 〉 . \end{matrix}$ (8)

Thus, the expected value of a LNUN can be turned into a LNN. According to Equations (1 and 2), the score and accuracy functions of LNUNs can be further defined below.

Definition 11. Let a LNUN be h =〈 l_{T_a+T_bI}, l_{U_a+U_bI}, l_{F_a+F_bI} 〉 in L. The score and accuracy functions of h are $\begin{matrix} S (h) & = & \frac{[\begin{matrix} 4 s + T_{a} + T_{b} inf I + T_{a} + T_{b} sup I \\ - (U_{a} + U_{b} inf I + U_{a} + U_{b} sup I) \\ - (F_{a} + F_{b} inf I + F_{a} + F_{b} sup I) \end{matrix}]}{6 s} \\ for S (h) \in [0, 1], \end{matrix}$ (9) $\begin{matrix} H (h) = \frac{[\begin{array}{l} T_{a} + T_{b} \inf I + T_{a} + T_{b} \sup I \\ - (F_{a} + F_{b} \inf I + F_{a} + F_{b} \sup I) \end{array}]}{2 s} \\ f o r H (h) \in [- 1, 1] . \end{matrix}$ (10)

Definition 12. Let h₁ = 〈l_{T_{a
₁}+T_{b
₁}I}, l_{U_{a
₁}+U_{b
₁}I}, l_{F_{a
₁}+F_{b
₁}I}〉 and h₂ = 〈l_{T_{a
₂}+T_{b
₂}I}, l_{U_{a
₂}+U_{b
₂}I}, l_{F_{a
₂}+F_{b
₂}I}〉 be two LNUNs, then they have the following comparative relations:

If S (h₁) > S (h₂), then h₁ > h₂;

If S (h₁) < S (h₂), then h₁ < h₂;

If S (h₁) = S (h₂) and H (h₁) > H (h₂), then h₁ > h₂;

If S (h₁) = S (h₂) and H (h₁) < H (h₂), then h₁ < h₂;

If S (h₁) = S (h₂) and H (h₁) = H (h₂), then h₁ = h₂.

4 Weighted aggregation operators of LNUNs

4.1 LNUNWAA operator

Definition 13. Provided that h_i = 〈l_{T_{a
_i}+T_{b
_i}I}, l_{U_{a
_i}+U_{b
_i}I}, l_{F_{a
_i}+F_{b
_i}I}〉 (i = 1, 2, ⋯, m) is a collection of LNUNs in L. The corresponding LNUNWAA operator is written as $LNUNWAA (h_{1}, h_{2}, \dots, h_{m}) = \sum_{i = 1}^{m} ω_{i} h_{i},$ (11) where ω_i ∈ [0, 1] is the weight of h_i, under the condition of $\sum_{i = 1}^{m} ω_{i} = 1$ .

According to Definition 9 and Definition 13, we have the following theorem.

Theorem 1. If h_i =〈 l_{T_{a
_i}+T_{b
_i}I}, l_{U_{a
_i}+U_{b
_i}I}, l_{F_{a
_i}+F_{b
_i}I} 〉 (i = 1, 2, ⋯, m) is a collection of LNUNs in L, the aggregation result of Equation (11) is still a LNUN, which is calculated by the following aggregation formula: $\begin{array}{l} L N U N W A A (h_{1}, h_{2}, \dots, h_{m}) = \sum_{i = 1}^{m} ω_{i} h_{i} \\ =< l_{s - s \prod_{i = 1}^{m} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{m} {(\frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{m} {(\frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}} >, \end{array}$ (12) where ω_i ∈ [0, 1] is the weight of h_i, satisfying the condition of $\sum_{i = 1}^{m} ω_{i} = 1$ .

In the following, Theorem 1 can be proved by using the mathematical induction.

Proof of Theorem 1:

It is obvious that when m = 1, the theorem is valid.

When m = 2, by Equation (6) we can obtain

ω_{1} h_{1} = < l_{s - s {(1 - \frac{T_{a_{1}} + T_{b_{1}} I}{s})}^{ω_{1}}}, l_{s {(\frac{U_{a_{1}} + U_{b_{1}} I}{s})}^{ω_{1}}}, l_{s {(\frac{F_{a_{1}} + T_{b_{1}} I}{s})}^{ω_{1}}} >

(13)

ω_{2} h_{2} = < l_{s - s {(1 - \frac{T_{a_{2}} + T_{b_{2}} I}{s})}^{ω_{2}}}, l_{s {(\frac{U_{a_{2}} + U_{b_{2}} I}{s})}^{ω_{2}}}, l_{s {(\frac{F_{a_{2}} + T_{b_{2}} I}{s})}^{ω_{2}}} >

(14)

From Equation (4), the weighted aggregation result can be calculated by $\begin{matrix} L N U N W A A (h_{1}, h_{2}) = ω_{1} h_{1} \oplus ω_{2} h_{2} \\ = < l_{\begin{matrix} s - s {(1 - \frac{T_{a_{1}} + T_{b_{1}} I}{s})}^{ω_{1}} + s - s {(1 - \frac{T_{a_{2}} + T_{b_{2}} I}{s})}^{ω_{2}} \\ - \frac{[s - s {(1 - \frac{T_{a_{1}} + T_{b_{1}} I}{s})}^{ω_{1}}] [s - s {(1 - \frac{T_{a_{2}} + T_{b_{2}} I}{s})}^{ω_{2}}]}{s} \end{matrix}}, \\ l_{\frac{[s {(\frac{U_{a_{1}} + U_{b_{1}} I}{s})}^{ω_{1}}] [s {(\frac{U_{a_{2}} + U_{b_{2}} I}{s})}^{ω_{2}}]}{s}}, l_{\frac{[s {(\frac{F_{a_{1}} + F_{b_{1}} I}{s})}^{ω_{1}}] [s {(\frac{F_{a_{2}} + F_{b_{2}} I}{s})}^{ω_{2}}]}{s}} > \\ = < l_{s - s {(1 - \frac{T_{a_{1}} + T_{b_{1}} I}{s})}^{ω_{1}} {(1 - \frac{T_{a_{2}} + T_{b_{2}} I}{s})}^{ω_{2}}}, l_{s {(\frac{U_{a_{1}} + U_{b_{1}} I}{s})}^{ω_{1}} {(\frac{U_{a_{2}} + U_{b_{2}} I}{s})}^{ω_{2}}}, \\ l_{s {(\frac{F_{a_{1}} + F_{b_{1}} I}{s})}^{ω_{1}} {(\frac{F_{a_{2}} + F_{b_{2}} I}{s})}^{ω_{2}}} > \\ = < l_{s - s \prod_{i = 1}^{2} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{2} {(\frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{2} {(\frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}} > . \end{matrix}$ (15)

(3) Let m = k, the aggregation result of LNUNs can be expressed as $\begin{matrix} L N U N W A A (h_{1}, h_{2}, \dots, h_{k}) = \sum_{i = 1}^{k} ω_{i} h_{i} \\ =< l_{s - s \prod_{i = 1}^{k} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{k} {(\frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{k} {(\frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}} > . \end{matrix}$ (16)

(4) Let m = k + 1, from Equations (15 and 16), the aggregation result of LNUNs with k + 1 items is calculated by $\begin{matrix} L N U N W A A (h_{1}, h_{2}, \dots, h_{k + 1}) = \sum_{i = 1}^{k} ω_{i} h_{i} \oplus (ω_{k + 1} h_{k + 1}) \\ = 〈 l_{\begin{matrix} s - s \prod_{i = 1}^{k} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}} + s - s {(1 - \frac{T_{a_{k + 1}} + T_{b_{k + 1}} I}{s})}^{ω_{k + 1}} \\ - \frac{[s - s \prod_{i = 1}^{k} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}] [s - s {(1 - \frac{T_{a_{k + 1}} + T_{b_{k + 1}} I}{s})}^{ω_{k + 1}}]}{s} \end{matrix}}, \\ l_{\frac{[s \prod_{i = 1}^{k} {(\frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}] [s {(\frac{U_{a_{k + 1}} + U_{b_{k + 1}} I}{s})}^{ω_{k + 1}}]}{s}}, l_{\frac{[s \prod_{i = 1}^{k} {(\frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}] [s {(\frac{F_{a_{k + 1}} + F_{b_{k + 1}} I}{s})}^{ω_{k + 1}}]}{s}} 〉 \\ =〈 l_{s - [s \prod_{i = 1}^{k} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}] {(1 - \frac{T_{a_{k + 1}} + T_{b_{k + 1}} I}{s})}^{ω_{k + 1}}}, \\ l_{[s \prod_{i = 1}^{k} {(\frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}] {(\frac{U_{a_{k + 1}} + U_{b_{k + 1}} I}{s})}^{ω_{k + 1}}}, l_{[s \prod_{i = 1}^{k} {(\frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}] {(\frac{F_{a_{k + 1}} + F_{b_{k + 1}} I}{s})}^{ω_{k + 1}}} 〉 \\ =〈 l_{s - s \prod_{i = 1}^{k + 1} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{k + 1} {(\frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{k + 1} {(\frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}} 〉 . \end{matrix}$ (17)

The above operational results prove that Equation(12) is valid for any m.

Moreover, assume that h_i (i = 1, 2, ⋯, m) is a group of LNUNs. Then, the LNUNWAA operator has the following properties:

Idempotency: If h_i = h for i = 1, 2, ⋯, m, then LNUNWAA (h₁, h₂, ⋯, h_m) = h.

Boundedness: Provided that the minimum LNUN is $h^{-} = 〈 min_{i} (l_{T_{a_{i}} + T_{b_{i}} I}), max_{i}$ $(l_{U_{a_{i}} + U_{b_{i}} I}), max_{i} (l_{F_{a_{i}} + F_{b_{i}} I}) 〉$ and the maximum LNUN is $h^{+} = 〈 max_{i} (l_{T_{a_{i}} + T_{b_{i}} I}), min_{i}$ $(l_{U_{a_{i}} + U_{b_{i}} I}), min_{i} (l_{F_{a_{i}} + F_{b_{i}} I}) 〉$ for i = 1, 2, ⋯, m, then h⁻ ≤ LNUNWAA (h₁, h₂, ⋯ h_m) ≤h⁺.

Monotonicity: If $h_{i} \leq h_{i}^{*}$ for i = 1, 2, ⋯, m, then $LNUNWAA (h_{1}, h_{2}, \dots, h_{m}) \leq LNUNWAA (h_{1}^{*}, h_{2}^{*}, \dots, h_{m}^{*})$ .

Proof.

(1) Assume that h =〈 l_{T_a+T_bI}, l_{U_a+U_bI}, l_{F_a+F_bI} 〉 is a LNUN. Since h_i = h for i = 1, 2, ⋯, m, the corresponding LNUNWAA result can be calculated by $\begin{matrix} L N U N W A A (h_{1}, h_{2}, \dots, h_{m}) = \sum_{i = 1}^{m} ω_{i} h_{i} \\ =〈 l_{s - s \prod_{i = 1}^{m} {(1 - \frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{m} {(\frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}}, l_{s \prod_{i = 1}^{m} {(\frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}} > \\ = 〈 l_{s - s {(1 - \frac{T_{a} + T_{b} I}{s})}^{\sum_{i = 1}^{m} ω_{i}}}, l_{s {(\frac{U_{a} + U_{b} I}{s})}^{\sum_{i = 1}^{m} ω_{i}}}, l_{s {(\frac{F_{a} + F_{b} I}{s})}^{\sum_{i = 1}^{m} ω_{i}}} > \\ = 〈 l_{s - s (1 - \frac{T_{a} + T_{b} I}{s})}, l_{s (\frac{U_{a} + U_{b} I}{s})}, l_{s (\frac{F_{a} + F_{b} I}{s})} > . \end{matrix}$ (18)

(2) Since e⁻ is the minimum LNUN and e⁺ is the maximum LNUN, there is e⁻ ≤ e_i ≤ e⁺. Thus, $\sum_{i = 1}^{m} ω_{i} h^{-} \leq \sum_{i = 1}^{m} ω_{i} h_{i} \leq \sum_{i = 1}^{m} ω_{i} h^{+}$ . From the property (1), there are $\sum_{i = 1}^{m} ω_{i} h^{-} = h^{-}$ and $\sum_{i = 1}^{m} ω_{i} h^{+} = h^{+}$ . Therefore, h⁻ ≤ LNUNWAA (h₁, h₂, ⋯, h_m) ≤ h⁺ holds.

(3) Since $h_{i} \leq h_{i}^{*}$ for i = 1, 2, ⋯, m, there exists $\sum_{i = 1}^{m} ω_{i} h_{i} \leq \sum_{i = 1}^{m} ω_{i} h_{i}^{*}$ . Therefore, $LNUNWAA (h_{1}, h_{2}, \dots, h_{m}) \leq LNUNWAA (h_{1}^{*}, h_{2}^{*}, \dots, h_{m}^{*})$ holds.

Thus, the proofs of these properties of the LNUNWAA operator are completed. Especially when ω_i = 1/m for i = 1, 2, ⋯, m, the LNUNWAA operator is reduced to the LNUN arithmetic averaging operator.

4.2 LNUNWGA operator

Definition 14. Let h_i = 〈l_{T_{a
_i}+T_{b
_i}I}, l_{U_{a
_i}+U_{b
_i}I}, l_{F_{a
_i}+F_{b
_i}I}〉 (i = 1, 2, ⋯, m) be a collection of LNUNs in L, then the LNUNWGA operator can be defined as $LNUNWGA (h_{1}, h_{2}, \dots, h_{m}) = \prod_{i = 1}^{m} h_{i}^{ω_{i}},$ (19) where ω_i ∈ [0, 1] is the weight of h_i (i = 1, 2, ⋯, m) under the condition of $\sum_{i = 1}^{m} ω_{i} = 1$ .

From the Definition 9 and Definition 14, we can give the following theorem.

Theorem 2. Let h_i = 〈l_{T_{a
_i}+T_{b
_i}I}, l_{U_{a
_i}+U_{b
_i}I}, l_{F_{a
_i}+F_{b
_i}I}〉 (i = 1, 2, ⋯, m) be a collection of LNUNs in L. Then the aggregation result of Equation (19) is still a LNUN, which is calculated by the following aggregation formula: $\begin{matrix} LNUNWGA (h_{1}, h_{2}, \dots, h_{m}) = \sum_{i = 1}^{m} h_{i}^{ω_{i}} \\ = 〈 l_{s \prod_{i = 1}^{m} {(\frac{T_{a_{i}} + T_{b_{i}} I}{s})}^{ω_{i}}}, l_{s - s \prod_{i = 1}^{m} {(1 - \frac{U_{a_{i}} + U_{b_{i}} I}{s})}^{ω_{i}}}, l_{s - s \prod_{i = 1}^{m} {(1 - \frac{F_{a_{i}} + F_{b_{i}} I}{s})}^{ω_{i}}} 〉, \end{matrix}$ (20) where ω_i ∈ [0, 1] is the weight of h_i (i = 1, 2, ⋯, m) satisfying the condition of $\sum_{i = 1}^{m} ω_{i} = 1$ . Especially when ω_i = 1/m for i = 1, 2, ⋯, m, the LNUNWGA operator is reduced to the LNUN geometric averaging operator.

The proof for Theorem 2 is not presented here due to the similarity of the proof manner to Theorem 1.

It is obvious that the LNUNWGA operator contains the following properties:

Idempotency: If h_i = h for i = 1, 2, ⋯, m, then LNUNWGA (h₁, h₂, ⋯, h_m) = h.

Boundedness: If the minimum LNUN is $h^{-} = 〈 min_{i} (l_{T_{a_{i}} + T_{b_{i}} I}), max_{i} (l_{U_{a_{i}} + U_{b_{i}} I}), max_{i}$ (l_{F_{a
_i}+F_{b
_i}I}) 〉 and the maximum LNUN is $h^{+} = 〈 max_{i} (l_{T_{a_{i}} + T_{b_{i}} I}), min_{i} (l_{U_{a_{i}} + U_{b_{i}} I}), min_{i}$ (l_{F_{a
_i}+F_{b
_i}I}) 〉 for i = 1, 2, ⋯, m, then h⁻ ≤ LNUNWGA (h₁, h₂, ⋯, h_m) ≤ h⁺.

Monotonicity: If $h_{i} \leq h_{i}^{*}$ for i = 1, 2, ⋯, m, then LNUNWAA (h₁, h₂, $\dots, h_{m}) \leq LNUNWAA (h_{1}^{*}, h_{2}^{*}, \dots, h_{m}^{*})$ .

The proof is similar to that of the LNUNWAA operator, so it is omitted here.

5 MAGDM method based on the LNUNWAA or LNUNWGA operator

In this section, a novel decision-making method based on the proposed LNUNWAA or LNUNWGA operator is presented to deal with the MAGDM problems with LNUN information.

In a MAGDM problem, d decision-makers (denoted by M = (M₁, M₂, ⋯, M_d)) are assigned to evaluate m alternatives (denoted by X = (X₁, X₂, ⋯, X_m)) on n attributes (denoted by P = (P₁, P₂, ⋯, P_n)) by LNUNs from the linguistic term set L = {l₀, l₁, ⋯, l_s} with the odd cardinality s + 1. The weight vector of the decision-makers is W = (ω₁, ω₂, ⋯, ω_d) ^T, whilst that of the attributes is V = (ω₁, ω₂, ⋯, ω_n) ^T. Each linguistic term value of the attribute evaluation is given by a LNUN. Thus, the LNUN decision matrix $D_{m \times n}^{k}$ can be constructed as $D_{m \times n}^{k} = [\begin{matrix} h_{11}^{k} & h_{12}^{k} & h_{13}^{k} & \dots & h_{1 n}^{k} \\ h_{21}^{k} & h_{22}^{k} & h_{23}^{k} & \dots & h_{2 n}^{k} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ h_{m 1}^{k} & h_{m 2}^{k} & h_{m 3}^{k} & \dots & h_{mn}^{k} \end{matrix}],$ where $h_{ij}^{k} = 〈 l_{T_{a_{ij}}^{k} + T_{b_{ij}}^{k} I}, l_{U_{a_{ij}}^{k} + U_{b_{ij}}^{k} I}, l_{F_{a_{ij}}^{k} + F_{b_{ij}}^{k} I} 〉$ (i = 1, 2, ⋯, m; j = 1, 2, ⋯, n; k = 1, 2, ⋯, d) is a LNUN.

To solve this MAGDM problem, the LNUNWAA or LNUNWGA operator of LNUNs and the score and accuracy functions of LNUNs can be used. The decision-making steps are introduced in details as follows:

Step 1. Integrate the decision matrices $D_{m \times n}^{k} = (h_{ij}^{k}) (k = 1, 2, \dots, d)$ given from all the decision-makers into one decision matrix D_m×n = (h_ij) _m×n by applying the LNUNWAA operator, where h_ij =〈 l_{T_{a
_ij}+T_{b
_ij}I}, l_{U_{a
_ij}+U_{b
_ij}I}, l_{F_{a
_ij}+F_{b
_ij}I} 〉 (i = 1, 2, …, m; j = 1, 2, …, n) is an integrated LNUN obtained by the following formula. $\begin{matrix} h_{ij} = LNUNWAA (h_{ij}^{1}, h_{ij}^{2}, \dots, h_{ij}^{d}) = \sum_{k = 1}^{d} ω_{k} h_{ij}^{k} \\ = 〈 l_{s - s \prod_{k = 1}^{d} {(1 - \frac{T_{a_{ij}}^{k} + T_{b_{ij}}^{k} I}{s})}^{ω_{k}}}, l_{s \prod_{k = 1}^{d} {(\frac{U_{a_{ij}}^{k} + U_{b_{ij}}^{k} I}{s})}^{ω_{k}}}, \\ l_{s \prod_{k = 1}^{d} {(\frac{F_{a_{ij}}^{k} + F_{b_{ij}}^{k} I}{s})}^{ω_{k}}} 〉, \end{matrix}$ (21) where ω_k is the weight of the kth decision maker M_k.

Step 2. For each alternative X_i (i = 1, 2, ⋯, m), calculate the LNUNs h_ij (j = 1, 2, ⋯, n) of all attributes P_j (i = 1, 2, ⋯, n) as one LNUN h_i by the following LNUNWAA or LNUNWGA operator: $\begin{matrix} h_{i} = LNUNWAA (h_{i 1}, h_{i 2}, \dots, h_{in}) = \sum_{j = 1}^{n} ω_{j} h_{ij} \\ = 〈 l_{s - s \prod_{j = 1}^{n} {(1 - \frac{T_{a_{ij}} + T_{b_{ij}} I}{s})}^{ω_{j}}}, l_{s \prod_{j = 1}^{n} {(\frac{U_{a_{ij}} + U_{b_{ij}} I}{s})}^{ω_{j}}}, l_{s \prod_{j = 1}^{n} {(\frac{F_{a_{ij}} + F_{b_{ij}} I}{s})}^{ω_{j}}} 〉 \end{matrix}$ or $\begin{matrix} h_{i} = L N U N W A A (h_{i 1}, h_{i 2}, \dots, h_{i n}) = \sum_{j = 1}^{n} ω_{j} h_{i j} \\ = 〈 l_{s - s \prod_{j = 1}^{n} {(1 - \frac{T_{a_{i j}} + T_{b_{i j}} I}{s})}^{ω_{j}}}, l_{s \prod_{j = 1}^{n} {(\frac{U_{a_{i j}} + U_{b_{i j}} I}{s})}^{ω_{j}}}, l_{s \prod_{j = 1}^{n} {(\frac{F_{a_{i j}} + F_{b_{i j}} I}{s})}^{ω_{j}}} > \end{matrix}$ (22) where ω_j is the weight of an attribute P_j.

Step 3. Calculate the score function S (h_i) and accuracy function H (h_i) of h_i for each alternative X_i by Equations (9 and 10). $S (h_{i}) = \frac{[\begin{matrix} 4 s + T_{a_{i}} + T_{b_{i}} inf I + T_{a_{i}} + T_{b_{i}} sup I \\ - (U_{a_{i}} + U_{b_{i}} inf I + U_{a_{i}} + U_{b_{i}} sup I) \\ - (F_{a_{i}} + F_{b_{i}} inf I + F_{a_{i}} + F_{b_{i}} sup I) \end{matrix}]}{6 s},$ (23) $H (h_{i}) = \frac{[\begin{matrix} T_{a_{i}} + T_{b_{i}} inf I + T_{a_{i}} + T_{b_{i}} sup I \\ - (F_{a_{i}} + F_{b_{i}} inf I + F_{a_{i}} + F_{b_{i}} sup I) \end{matrix}]}{2 s} .$ (24)

Step 4. According to Definition 12, rank the alternatives by the obtained score values (accuracy values if necessary) and select the best one.

Step 5. End.

6 Illustrative example

By considering an example adapted from Reference [39], a project decision-making problem in an investment company is illustrated to demonstrate the application of the proposed MAGDM method in this section.

The investment company needs to choose an appropriate industry for investment. A set of four alternatives denoted by X = (X₁, X₂, X₃, X₄) is provided where X₁ is a food company, X₂ is a computer company, X₃ is a car company, and X₄ is an electrical appliance company. Three attributes of each alternative, including the risk (denoted by P₁), the growth (denoted by P₂) and the environmental impact (denoted by P₃), should be considered by the different importance given by a weight vector V = (0.3, 0.45, 0.25) ^T. Three decision-makers (denoted by M = (M₁, M₂, M₃)) are invited with the different importance expressed by a weight vector W = (0.3, 0.35, 0.35) ^T. Now the three decision-makers are required to give the suitability evaluation of the four possible alternatives X_i (i = 1, 2, 3, 4) with respect to the three attributes P_j (j = 1, 2, 3) by using the expression of LNUNs from the linguistic term set L = {l₀ = extremely low, l₁ = very low, l₂ = low, l₃ = slightly low, l₄ = medium, l₅ = slightly high, l₆ = high, l₇ = very high, l₈ = extremely high} with the odd cardinality s + 1 =9. Thus, the linguistic evaluation information given by each decision-maker M_k (k = 1, 2, 3) can be constructed as $D^{1} = [\begin{matrix} 〈 l_{6 + I}, l_{1}, l_{2 + I} 〉 & 〈 l_{7 + I}, l_{2}, l_{1 + I} 〉 & 〈 l_{6 + I}, l_{2 + I}, l_{2} 〉 \\ 〈 l_{7 + I}, l_{1}, l_{1} 〉 & 〈 l_{7 + I}, l_{3 + I}, l_{2} 〉 & 〈 l_{7}, l_{2 + I}, l_{1 + I} 〉 \\ 〈 l_{6}, l_{2 + I}, l_{2 + I} 〉 & 〈 l_{7 + I}, l_{1}, l_{1} 〉 & 〈 l_{6 + I}, l_{2}, l_{2} 〉 \\ 〈 l_{7 + I}, l_{1 + I}, l_{2} 〉 & 〈 l_{7}, l_{2 + I}, l_{3 + I} 〉 & 〈 l_{7 + I}, l_{2 + I}, l_{1} 〉 \end{matrix}],$ $D^{2} = [\begin{matrix} 〈 l_{6 + I}, l_{1}, l_{2 + I} 〉 & 〈 l_{6 + I}, l_{1}, l_{1 + I} 〉 & 〈 l_{4 + I}, l_{2 + I}, l_{3} 〉 \\ 〈 l_{7 + I}, l_{2}, l_{3} 〉 & 〈 l_{6 + I}, l_{1 + I}, l_{1} 〉 & 〈 l_{4}, l_{2 + I}, l_{3 + I} 〉 \\ 〈 l_{5}, l_{1 + I}, l_{2 + I} 〉 & 〈 l_{5 + I}, l_{1}, l_{2} 〉 & 〈 l_{5 + I}, l_{4}, l_{2} 〉 \\ 〈 l_{6 + I}, l_{1 + I}, l_{1} 〉 & 〈 l_{5}, l_{1 + I}, l_{1 + I} 〉 & 〈 l_{5 + I}, l_{2 + I}, l_{3} 〉 \end{matrix}],$ $D^{3} = [\begin{matrix} 〈 l_{7 + I}, l_{3}, l_{4 + I} 〉 & 〈 l_{7 + I}, l_{3}, l_{3 + I} 〉 & 〈 l_{5 + I}, l_{2 + I}, l_{5} 〉 \\ 〈 l_{6 + I}, l_{3}, l_{4} 〉 & 〈 l_{5 + I}, l_{1 + I}, l_{2} 〉 & 〈 l_{6}, l_{2 + I}, l_{3 + I} 〉 \\ 〈 l_{7}, l_{2 + I}, l_{4 + I} 〉 & 〈 l_{6 + I}, l_{1}, l_{2} 〉 & 〈 l_{7 + I}, l_{2}, l_{4} 〉 \\ 〈 l_{7 + I}, l_{2 + I}, l_{3} 〉 & 〈 l_{5}, l_{2 + I}, l_{1 + I} 〉 & 〈 l_{6 + I}, l_{1 + I}, l_{1} 〉 \end{matrix}] .$

Obviously, the decision-making can be carried out by using the MAGDM method based on the LNUNWAA operator with the following steps:

Step 1. Get the following integrated matrix D = (h_ij) _m×n by Equation (21): $\begin{matrix} D = [\begin{matrix} 〈 l_{6 . 431 + 1 . 569 I}, l_{1.469}, l_{2.549 + 1.038 I} 〉 \\ 〈 l_{6 . 725 + 1 . 275 I}, l_{1.872}, l_{2.386} 〉 \\ 〈 l_{6.192}, l_{1.569 + 1.034 I}, l_{2.549 + 1.038 I} 〉 \\ 〈 l_{6 . 725 + 1 . 275 I}, l_{1.275 + 1.03 I}, l_{1.808} 〉 \end{matrix} \\ \begin{matrix} 〈 l_{6 . 725 + 1 . 275 I}, l_{1.808}, l_{1.469 + 1.08 I} 〉 \\ 〈 l_{6 . 128 + 1 . 872 I}, l_{1.39 + 1.072 I}, l_{1.569} 〉 \\ 〈 l_{6 . 128 + 1 . 872 I}, l_{1}, l_{1.625} 〉 \\ 〈 l_{5.842}, l_{1.569 + 1.034 I}, l_{1.39 + 1.072 I} 〉 \end{matrix} \\ \begin{matrix} 〈 l_{5 . 062 + 1 . 066 I}, l_{2 + I}, l_{3.176} 〉 \\ 〈 l_{5.929}, l_{2 + I}, l_{2.158 + 1.091 I} 〉 \\ 〈 l_{6 . 192 + 1 . 808 I}, l_{2.549}, l_{2.549} 〉 \\ 〈 l_{6 . 128 + 1 . 872 I}, l_{1 . 569 + 1 . 034 I}, l_{1.469} 〉 \end{matrix}] . \end{matrix}$

Step 2. From Equation (22), aggregating all the LNUNs h_ij (j = 1, 2, 3) for each alternative X_i (i = 1, 2, 3, 4), the collective overall LNUNs h_i (i = 1, 2, 3, 4) are as follows: $\begin{matrix} h_{1} = 〈 l_{6.328 + 1.672 I}, l_{1.742 + 0.186 I}, l_{2.102 + 0.882 I} 〉, \\ h_{2} = 〈 l_{6.289 + 1.711 I}, l_{1.665 + 0.718 I}, l_{1.927 + 0.208 I} 〉, \\ h_{3} = 〈 l_{6.163 + 1.837 I}, l_{1.446 + 0.237 I}, l_{2.081 + 0.225 I} 〉, \\ h_{4} = 〈 l_{6.222 + 1.778 I}, l_{1.474 + 1.036 I}, l_{1.525 + 0.447 I} 〉 . \end{matrix}$

Step 3. By Equation (24), the score values of h_i (i = 1, 2, 3, 4) for I ∈ [0, 1] are as follows: $\begin{matrix} S (h_{1}) = 0.7828, S (h_{2}) = 0.7954, \\ S (h_{3}) = 0.8051, S (h_{4}) = 0.8071 . \end{matrix}$

Step 4. According to the score values, the ranking order of the four alternatives is X₄ > X₃ > X₂ > X₁. Hence, X₄ is the best choice.

Following the above 4 steps, the ranking results for different ranges of the indeterminate degree I can be obtained, as shown in Table 1.

Table 1
Decision results based on the LNUNWAA operator by choosing different indeterminate ranges for I in LNUNs

I	LNUNWAA	Ranking
I ∈ [−1, 0]	S (h₁) =0.7576, S (h₂) =0.7627, S (h₃) =0.7478, S (h₄) =0.7948	X₄ > X₂ > X₁ > X₃
I ∈ [−0.7, 0]	S (h₁) =0.7614, S (h₂) =0.7676, S (h₃) =0.7564, S (h₄) =0.7966	X₄ > X₂ > X₁ > X₃
I ∈ [−0.5, 0]	S (h₁) =0.7639, S (h₂) =0.7709, S (h₃) =0.7622, S (h₄) =0.7978	X₄ > X₂ > X₁ > X₃
I ∈ [−0.3, 0]	S (h₁) =0.7664, S (h₂) =0.7742, S (h₃) =0.7679, S (h₄) =0.7991	X₄ > X₂ > X₃ > X₁
I ∈ [−0. 1, 0]	S (h₁) =0.7689, S (h₂) =0.7774, S (h₃) =0.7736, S (h₄) =0.8003	X₄ > X₂ > X₃ > X₁
I = 0	S (h₁) =0.7702, S (h₂) =0.7791, S (h₃) =0.7765, S (h₄) =0.8009	X₄ > X₂ > X₃ > X₁
I ∈ [0, 0. 1]	S (h₁) =0.7714, S (h₂) =0.7807, S (h₃) =0.7793, S (h₄) =0.8015	X₄ > X₂ > X₃ > X₁
I ∈ [0, 0. 3]	S (h₁) =0.7740, S (h₂) =0.7840, S (h₃) =0.7851, S (h₄) =0.8028	X₄ > X₃ > X₂ > X₁
I ∈ [0, 0. 5]	S (h₁) =0.7765, S (h₂) =0.7872, S (h₃) =0.7908, S (h₄) =0.8040	X₄ > X₃ > X₂ > X₁
I ∈ [0, 0. 7]	S (h₁) =0.7790, S (h₂) =0.7905, S (h₃) =0.7965, S (h₄) =0.8052	X₄ > X₃ > X₂ > X₁
I ∈ [0, 1]	S (h₁) =0.7828, S (h₂) =0.7954, S (h₃) =0.8051, S (h₄) =0.8071	X₄ > X₃ > X₂ > X₁

On the other hand, the MAGDM method based on the LNUNWGA operator can be also used for the decision-making problem by the following procedure.

Step 1’. Get the same result as Step 1.

Step 2’. Accumulate all LNUNs h_ij (j = 1, 2, 3) by applying Equation (23), the collective overall LNUNs of h_i (i = 1, 2, 3, 4) for X_i (i = 1, 2, 3, 4) are as follows: $\begin{matrix} h_{1} = 〈 l_{6.181 + 1.303 I}, l_{1.758 + 0.278 I}, l_{2.265 + 0.773 I} 〉, \\ h_{2} = 〈 l_{6.250 + 1.173 I}, l_{1.693 + 0.742 I}, l_{1.972 + 0.304 I} 〉, \\ h_{3} = 〈 l_{6.163 + 1.245 I}, l_{1.59 + 0.328 I}, l_{2.151 + 0.359 I} 〉, \\ h_{4} = 〈 l_{6.167 + 0.777 I}, l_{1.482 + 01.033 I}, l_{1.538 + 0.495 I} 〉 . \end{matrix}$

Step 3’. By Equation (24), the score values of h_i (i = 1, 2, 3, 4) for I ∈ [0, 1] are as follows: $\begin{matrix} S (h_{1}) = 0.7618, S (h_{2}) = 0.7770, \\ S (h_{3}) = 0.7792, S (h_{4}) = 0.7822 . \end{matrix}$

Step 4’. Ranking all the alternatives according to the score values as X₄ > X₃ > X₂ > X₁. Thus, X₄ is the best choice.

Similarly, following the above 4 steps, the ranking results for different ranges of the indeterminate degree I can be obtained, as shown in Table 2.

Table 2

Decision results based on the LNUNWGA operator by choosing different indeterminate ranges for I in LNUNs

I	LNUNWGA	Ranking
I ∈ [−1, 0]	S (h₁) =0.7513, S (h₂) =0.7717, S (h₃) =0.756, S (h₄) =0.8134	X₄ > X₂ > X₃ > X₁
I ∈ [−0.7, 0]	S (h₁) =0.7529, S (h₂) =0.7725, S (h₃) =0.7595, S (h₄) =0.8087	X₄ > X₂ > X₃ > X₁
I ∈ [−0.5, 0]	S (h₁) =0.7539, S (h₂) =0.773, S (h₃) =0.7618, S (h₄) =0.8056	X₄ > X₂ > X₃ > X₁
I ∈ [−0.3, 0]	S (h₁) =0.755, S (h₂) =0.7735, S (h₃) =0.7641, S (h₄) =0.8025	X₄ > X₂ > X₃ > X₁
I ∈ [−0. 1, 0]	S (h₁) =0.7561, S (h₂) =0.7741, S (h₃) =0.7664, S (h₄) =0.7994	X₄ > X₂ > X₃ > X₁
I = 0	S (h₁) =0.7566, S (h₂) =0.7743, S (h₃) =0.7676, S (h₄) =0.7978	X₄ > X₂ > X₃ > X₁
I ∈ [0, 0. 1]	S (h₁) =0.7571, S (h₂) =0. 7746, S (h₃) =0. 7688, S (h₄) =0. 7962	X₄ > X₂ > X₃ > X₁
I ∈ [0, 0. 3]	S (h₁) =0. 7582, S (h₂) =0. 7751, S (h₃) =0. 7711, S (h₄) =0. 7931	X₄ > X₂ > X₃ > X₁
I ∈ [0, 0. 5]	S (h₁) =0. 7592, S (h₂) =0. 7757, S (h₃) =0. 7734, S (h₄) =0. 79	X₄ > X₂ > X₃ > X₁
I ∈ [0, 0. 7]	S (h₁) =0. 7603, S (h₂) =0. 7762, S (h₃) =0. 7757, S (h₄) =0. 7869	X₄ > X₃ > X₂ > X₁
I ∈ [0, 1]	S (h₁) =0. 7618, S (h₂) =0. 7770, S (h₃) =0. 7792, S (h₄) =0. 7822	X₄ > X₃ > X₂ > X₁

7 Comparative analysis

To illustrate the advantages of the proposed LNUN and MAGDM method, we compare the proposed LNUN MAGDM method using the LNUNWAA and LNUNWGA operators with existing related MAGDM methods [1 , 39] in LNN setting.

Firstly, from Table 1 the ranking orders obtained based on the LNUNWAA operator change three times in the range of I from [−1,0] to [0,1]. Herewith, there are X₄ > X₂ > X₁ > X₃ from I ∈ [-1, 0] to I ∈ [-0.5, 0], X₄ > X₂ > X₃ > X₁ from I ∈ [-0.3, 0] to I ∈ [0, 0.1], and X₄ > X₃ > X₂ > X₁ from I ∈ [0, 0.3] to I ∈ [0, 1]. And the best alternative is X₄. Similarly, from Table 2, the ranking orders obtained based on the LNUNWGA operator are X₄ > X₂ > X₃ > X₁ from I ∈ [-1, 0] to I ∈ [0, 0.5] and X₄ > X₃ > X₂ > X₁ from I ∈ [0, 0.7] to I ∈ [0, 1]. Then X₄ is the best one. The illustrative example shows that different ranges of the indeterminate degrees for I in LNUNs can result in different ranking orders of alternatives. Thus, based on the MAGDM method with LNUN information proposed in this paper, the decision-makers will have flexibility in real decision-making problems by selecting different ranges of indeterminate degrees according to their preference or real requirements. It should be noted that if the indeterminacy I is not considered (i.e., I = 0), the decision-making information represented by LNUNs is reduced to the LNN information, and then the MAGDM method with the LNN information in [39] is the same as the MAGDM method proposed in this study and they give the same ranking orders. Obviously, existing LNNWAA and LNNWGA operators and their MAGDM method [39] are special cases of the proposed LNUNWAA and LNUNWGA operators and their MAGDM method in this study for I = 0. Hence, the new MAGDM method is superior to existing MAGDM method [39].

Secondly, the proposed LNUNWAA and LNUNWGA operators can aggregate not only the LNUN information but also the LNN information because the LNN is only a special case of the LNUN with the indeterminacy degree of I being zero (i.e., I = 0); while all existing aggregation operators of LNNs, such as LNNWAA and LNNWGA [39], LNNNWBM and LNNNWGBM [1], LNPWHA [29], LNGWPBM [32], LNHM and WLNHM [30] operators, cannot aggregate LNUN information for the reason that the LNUN is an extension of LNN and contains much more information than LNN. Furthermore, existing MAGDM methods introduced in [1 , 39] cannot also handle such a MAGDM problem with LNUN information.

In general, the LNUN is a new generalization of LNN and NLN, which inherits the advantages of both LNN and NLN, and expresses the decision-making information with three uncertain linguistic variables including the truth, indeterminacy and falsity uncertain linguistic degrees given by flexible neutrosophic numbers. Compared with LNN, the LNUN with neutrosophic linguistic variables is more suitable for the expression of the real decision-making information with both partial linguistic certain and partial linguistic uncertain evaluations. The proposed MAGDM method with LNUN information can provide a more general and flexible selecting way for decision-makers by changing the indeterminate range of I desired by the decision-makers. The LNUNs enrich the neutrosophic theory and decision-making method under linguistic neutrosophic environments.

8 Conclusions

This paper firstly defined the concept of LNUNs by the combination of LNN and NLN, the operational laws, and the score and accuracy function of LNUNs. Then, two aggregation operators, including the LNUNWAA and LNUNWGA operators, were developed to aggregate LNUNs. Based on them, a novel MAGDM method was proposed in a LNUN environment. Finally, a MAGDM example was illustrated to demonstrate the application of the developed method. LNUNs inherit the advantages of both NLN and LNN and are more suitable for the practical science and engineering applications in linguistic uncertain environments. The proposed MAGDM method also enriches linguistic decision-making theory and provides a new way for decision-makers under LNUN environment. In the future research, we shall further develop new aggregation operators of LNUNs and apply them to decision-making, pattern recognition, medical diagnosis, resource allocation problems and so on.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Nos. 71471172, 61703280).

References

Peng

, Ye

and Zeng

, Uncertain pure linguistic hybrid harmonic averaging operator and generalized interval aggregation operator based approach to group decision making, Knowledge-Based Systems 36 (2012), 175–181.

Peng

, Zhou

and Peng

, Cloud model based approach to group decision making with uncertain pure linguistic information, Journal of Intelligent & Fuzzy Systems 32(3) (2017), 1959–1968.

Fan

C.X.

, Ye

, Hu

K.L.

and Fan

, Bonferroni mean operators of linguistic neutrosophic numbers and their multiple attribute group decision-making methods, Information 8(3) (2017), 107.

, Li

D.F.

, Merigó

J.M.

and Fang

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

Herrera

, Herrera-Viedma

and Verdegay

J.L.

, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets & Systems 78 (1996), 73–87.

Smarandache

Neutrosophy: Neutrosophic probability, set, and logic, American Research Press, Rehoboth, 1998.

Herrera

and Herrera-Viedma

, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets & Systems 115 (2000), 67–82.

Smarandache

Introduction to neutrosophic measure, neutrosophic integral, and neutrosophic probability, Sitech & Education Publisher, Craiova, Columbus 2013.

Smarandache

Introduction to neutrosophic statistics, Sitech & Education Publisher, Craiova, Columbus, 2014.

10.

Smarandache

Symbolic neutrosophic theory, EuropaNova asbl, Bruxelles, 2015.

11.

Wei

G.W.

, Uncertain linguistic hybrid geometric mean operator and its application to group decision making under uncertain linguistic environment, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17 (2009), 251–267.

12.

Wei

G.W.

, Zhao

X.F.

, Lin

and Wang

H.J.

, Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making, Applied Mathematical Modelling 37 (2013), 5277–5285.

13.

Jin

H.P.

, Min

G.G.

and Kwun

Y.C.

, Uncertain linguistic harmonic mean operators and their applications to multiple attribute group decision making, Computing 93 (2011), 47–64.

14.

Zhang

H.M.

, Uncertain linguistic power geometric operators and their use in multi-attribute group decision making, Mathematical Problems in Engineering 2015 (2015), 1–10.

15.

Merigó

J.M.

, Casanovas

and Martínez

, 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.

16.

Wang

J.Q.

and Li

H.B.

, Multi-criteria decision making based on aggregation operators for intuitionistic linguistic fuzzy numbers, Control and Decision 25 (2010), 1571–1574.

17.

Merigó

J.M.

and Gil-Lafuente

A.M.

, Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making, Information Sciences 236 (2013), 1–16.

18.

, Some aggregation operators of interval neutrosophic linguistic numbers for multiple attribute decision making, Journal of Intelligent & Fuzzy Systems 27 (2014), 2231–2241.

19.

, An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers, Journal of Intelligent & Fuzzy Systems 28 (2015), 247–255.

20.

Merigó

J.M.

, Palacios-Marqués

and Zeng

S.Z.

, Subjective and objective information in linguistic multi-criteria group decision making, European Journal of Operational Research 248 (2016), 522–531.

21.

, Aggregation operators of neutrosophic linguistic numbers for multiple attribute group decision making, Springerplus 5 (2016), 1691.

22.

, Multiple attribute group decision making based on interval neutrosophic uncertain linguistic variables, International Journal of Machine Learning & Cybernetics 8 (2017), 837–848.

23.

, Linguistic neutrosophic cubic numbers and their multiple attribute decision-making method, Information 8(3) (2017), 110.

24.

, Multiple attribute decision-making methods based on the expected value and the similarity measure of hesitant neutrosophic linguistic numbers, Cognitive Computation 10 (2018), 454–463.

25.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning, Learning Systems and Intelligent Robots 8 (1974), 199–249.

26.

Liu

P.D.

and Shi

L.L.

, Some neutrosophic uncertain linguistic number Heronian mean operators and their application to multi-attribute group decision making, Neural Computing and Application 28(5) (2017), 1079–1093.

27.

Liu

P.D.

and Wang

, Some improved linguistic intuitionistic fuzzy aggregation operators and their applications to multiple-attribute decision making, International Journal of Information Technology & Decision Making 16(3) (2017), 817–850.

28.

Liu

P.D.

, Liu

J.L.

and Merigó

J.M.

, Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making, Applied Soft Computing 62 (2018), 395–422.

29.

Liu

P.D.

, Mahmood

and Khan

, group decision making based on power Heronian aggregation operators under linguistic neutrosophic environment, International Journal of Fuzzy Systems 20(3) (2018), 970–985.

30.

Liu

P.D.

and You

X.L.

, Some linguistic neutrosophic Hamy mean operators and their application to multi-attribute group decision making, PLoS One 13(3) (2018), e0193027.

31.

Broumi

, Ye

and Smarandache

, An extended TOPSIS method for multiple attribute decision making based on interval neutrosophic uncertain linguistic variables, Neutrosophic Sets & Systems 8 (2015), 22–31.

32.

Wang

Y.M.

and Liu

P.D.

, linguistic neutrosophic generalized partitioned Bonferroni mean operators and their application to multi-attribute group decision making, Symmetry 10 (2018), 160.

33.

Z.S.

, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Information Sciences 166 (2004), 19–30.

34.

Z.S.

, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Information Sciences 168 (2004), 171–184.

35.

Z.S.

, A Note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information, Group Decision & Negotiation 15 (2006), 593–604.

36.

Z.S.

, Goal programming models for multiple attribute decision making under linguistic setting, Group Decision & Negotiation 9(2) (2006), 9–17.

37.

Z.S.

, Induced uncertain linguistic OWA operators applied to group decision making, Information Sciences 7 (2006), 231–238.

38.

Chen

Z.C.

, Liu

P.H.

and Pei

, An approach to multiple attribute group decision making based on linguistic intuitionistic fuzzy numbers, International Journal of Computational Intelligence Systems 8 (2015), 747–760.

39.

Fang

Z.B.

and Ye

, Multiple attribute group decision-making method based on linguistic neutrosophic numbers, Symmetry 9(7) (2017), 111.

Linguistic neutrosophic uncertain numbers and their multiple attribute group decision-making method

Abstract

Keywords

1 Introduction

2 Preliminaries of LNNs and NLNs

2.1 Operational laws and score and accuracy functions of LNNs

4.1 LNUNWAA operator

Table 1 Decision results based on the LNUNWAA operator by choosing different indeterminate ranges for I in LNUNs

8 Conclusions

Footnotes

Acknowledgments

References

Table 1
Decision results based on the LNUNWAA operator by choosing different indeterminate ranges for I in LNUNs