Group decision making method based on hybrid aggregation operator for intuitionistic uncertain linguistic variables

Abstract

For a multi-attribute group decision making (MAGDM) problem where the attribute values are intuitionistic uncertain linguistic variables (IULVs), we propose a novel decision making method based on hybrid aggregation operator of IULVs. Firstly, new operational rules of IULVs are proposed based on linguistic scale functions (LSFs) to overcome the existing shortcoming in which the operations on linguistic variables (LVs) directly based on the subscripts of linguistic terms (LTs) are not closed, then the expected value and accuracy function of IULVs are introduced. Further, some aggregation operators for IULVs are proposed, including the weighted geometric average operator for IULVs (IULWGA), ordered weighted geometric operator for IULVs (IULOWG), then the hybrid geometric operator for IULVs (IULHG) are developed, which could consider the weights of attributes and their ranking positions. However, because IULHG don’t meet some desirable properties, we proposed a new hybrid weighted geometric operator for IULVs (IULHWG) which can not only weight the importance of each argument and its ordered position but also maintain some desirable properties. Based on these operators, an approach to MAGDM with IULVs has been proposed. Finally, an illustrative example is provided to show the steps of the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Linguistic variables intuitionistic fuzzy set (IFS)MAGDM aggregation operator

1 Introduction

In day-to-day life, uncertainties play a crucial role in the decision making process, and intuitionistic fuzzy set (IFS), which was proposed by Atanassov [1, 2], is a very valid tool to depict these uncertainties by the parameters of membership degree (MD) and no-membership degree (NMD). Since the IFS considered the MD and NMD, obviously, it can describe and character the fuzzy essence of the objective world more exquisitely [1]. In addition, due to the increasing complexity and uncertainty of the MAGDM problem, it is difficult to measure the attribute values of the alternatives by the quantitative value. LTs originated from cognitive behavior are easy to describe the qualitative information [11 , 38]. For example, when we evaluation the speed of a car in high-way, we can easily use the LTs such as “slow”, “fast”, “very fast” and so on to give an assessment. However, sometimes, for some complicated decision makings, there is some hesitation for this given result by LTs. For example, when we performed a performance test on a smart phone, we gave a standard about LT “good”, however, some decision makers (DMs) think it doesn’t meet “good” standard, and it may be 80% meeting this standard, 10% not meeting this standard. At this point, it also reflects the certain defects in the practical application about the evaluation based on the LTs. To overcome this shortcoming of LTs, some new research achievements were proposed by combining IFSs with LTs [10 , 24]. For example, in order to accurately describe the MD and NMD of a certain LT, intuitionistic linguistic number [9 , 25] and interval-valued intuitionistic fuzzy linguistic number [36] have been defined, however, these research results still have some shortcomings when describing uncertain information. Further, by combining the concepts of intuitionistic fuzzy numbers (IFNs) and uncertain linguistic variables (ULVs) [33], Liu and Jin [12] proposed the concept of intuitionistic uncertain linguistic variables (IULVs).

The aggregation operators are important tool to solve the MAGDM problems, in which arithmetic operators and geometric operators as basic operators have attracted wide attention. Xu and Yager [39] introduced the weighted geometric (WG) operator, ordered weighted geometric (OWG) operator and hybrid geometric (HG) operator for IFNs. Zhang and Liu [19] proposed the weighted arithmetic averaging (WAA) operator for triangular IFNs in which the MD and the NMD were expressed by the triangular fuzzy numbers, and developed a MAGDM approach. Wang and Zhang [28] proposed WAA operator and WG operator for intuitionistic trapezoidal fuzzy numbers. Wei [29] proposed ordered weighted averaging (OWA) operator and hybrid aggregation operator for intuitionistic trapezoidal fuzzy numbers. Zeng et al. [41] extended the ordered weighted averaging (OWA) operator to the intuitionistic fuzzy induced ordered weighted averaging distance (IFIOWAD) operator. Merig__ al. [21] produced the belief structure— linguistic ordered weighted averaging (BS-LOWA), the BS— linguistic hybrid averaging (BS-LHA). Zeng et al. [42] proposed the Pythagorean fuzzy induced ordered weighted averaging-weighted average (PFIOWAWA) operator. Xu [33] also further proposed the operational laws of ULVs, and proposed uncertain linguistic weighted aggregation operator for ULVs. Then Xu [32 , 35] introduced the OWA operator for ULVs and the induced ordered weighted averaging (IOWA) operator for ULVs, and then applied them to MAGDM problems. Wei [30] developed some HG operators for ULVs.

For MAGDM problems where the attribute values take the form of IULVs, many scholars have proposed different types of decision-making methods. Based on arithmetic operators and geometric operators, Liu and Zhang [20] defined intuitionistic uncertain linguistic weighted arithmetic average (IULWAA) operator, intuitionistic uncertain linguistic ordered weighted averaging (IULOWA) operator and intuitionistic uncertain linguistic hybrid averaging (IULHA) operator. Based on the IULWAA and IULHA operators, an approach for solving MAGDM problems is developed. Liu and Jin [12] presented three intuitionistic uncertain linguistic aggregation operators, including intuitionistic uncertain linguistic weighted geometric average (IULWGA) operator, intuitionistic uncertain linguistic ordered weighted geometric (IULOWG) operator and intuitionistic uncertain linguistic hybrid geometric (IULHG) operator and provided two new MADGM methods.

Although these intuitionistic uncertain linguistic aggregation operators are widely used in various fields, they still have the following shortcomings:

(1)
The existing operational rules of IULVs are not closed. These operations defined in [12, 20] are directly based on the subscript of LTs. Although such operation rules are simple and easy to calculate, they have the following shortcomings: the computation results are easy to exceed the upper bound of the predefined linguistic term set (LTS), that is, the operations are not closed. For example, suppose S is a LTS with seven linguistic labels, ${\tilde{a}}_{1} = 〈 [s_{4}, s_{5}], (0.4, 0.6) 〉$ and ${\tilde{a}}_{2} = 〈 [s_{2}, s_{3}], (0.2, 0.3) 〉$ are two IULVs, according to the operations defined in [12, 20], we can obtain ${\tilde{a}}_{1} + {\tilde{a}}_{2} = 〈 [s_{6}, s_{8}], (0.52, 0.18) 〉$ . Obviously, the linguistic result exceeds the upper limit of S.

The IULHG operator can generalize both the IULWGA operator and IULOWG operator, and weigh the importance and the ordered position of the given arguments. However, there is a flaw that the IULHG operator do not satisfy some basic properties, such as idempotency and boundedness, which are desirable for aggregating a finite collection of IULVs.

The above analysis leads to the purpose of this study. Firstly, based on LSFs, we develop new operational laws of IULVs. New operational rules have good closure and can solve the cross-border phenomenon of existing operations. Then, based on new operations, new intuitionistic uncertain linguistic hybrid weighted geometric (IULHWG) operator is proposed. This operator can not only weigh the importance of each argument and its ordered position but also maintain some desirable properties, i.e., idempotency, boundedness and monotonicity. Furthermore, a novel method is developed to deal with MAGDM problems under intuitionistic uncertain linguistic environment based on the proposed IULHWG operator. At the same time, the novel decision approach proposed in this paper can be applied to the selection of suppliers in logistics, site selection, express evaluation and other decision-making problems.

The construct of this paper is arranged as follows. In Section 2, we briefly introduce some basic concepts and new operational laws of IULVs based on LSFs, and expected value and accuracy function of IULVs are introduced. In Section 3, we propose a WGA operator for IULVs (IULWGA), an OWG operator for IULVs (IULOWG), a hybrid geometric operator for IULVs (IULHG), and new hybrid weighted geometric operator for IULVs (IULHWG). In Section 4, based on these operators, we develop an approach to solve the MAGDM problems where the attribute information is expressed by IULVs. Section 5 gives a practical example to show the steps of the proposed approach, and explain its feasibility and validity by compared with other methods. Finally, some conclusions and future research directions are outlined in Section 6.
2 Preliminaries

2.1 Intuitionistic fuzzy set (IFS)

We first introduce some basic concepts and related operational laws for IFS.

Definition 1. [1] Let X be a given discourse. Then the IFS A in X is defined as: $A = {〈 x, u_{A} (x), v_{A} (x) 〉 | x \in X},$ (1) where u_A (x) ∈ [0, 1] and v_A (x) ∈ [0, 1], u_A (x) and v_A (x) represent membership degree (MD) and no-membership degree (NMD), with the condition: $0 \leq u_{A} (x) + v_{A} (x) \leq 1$ (2)

For convenience, Xu and Xia [37] called (u_a, v_a) an intuitionistic fuzzy number (IFN), where $u_{a} \in [0, 1], v_{a} \in [0, 1], u_{a} + v_{a} \leq 1 .$

Definition 2. [2, 13] Let a = (u_a, v_a) , b = (u_b, v_b) be two IFNs. The operational laws between two IFNs are shown as follows: $(1) a \oplus b = (u_{a} + u_{b} - u_{a} u_{b}, v_{a} v_{b})$ (3) $(2) a \otimes b = (u_{a} u_{b}, v_{a} + v_{b} - v_{a} v_{b})$ (4) $(3) na = (1 - {(1 - u_{a})}^{n}, {v_{a}}^{n}), n > 0$ (5) $(4) a^{n} = ({u_{a}}^{n}, 1 - {(1 - v_{a})}^{n}), n > 0$ (6)

2.2 The uncertain linguistic variables (ULVs)

Let S ={ s_i|i = 0, 1, 2, … 2t } be a LTS and t be odd number. If t = 3, then a LTS S can be defined as: S = {s₀ = very poor, s₁ = slight poor, s₂ = poor, s₃ = fair, s₄ = slight good, s₅ = good, s₆ = very good}. In general, the LTS S should satisfy the following characteristics [5].

If a > b, then s_a > s_b;

There is the negation operator, neg (s_a) = s_b when b = 2t - a.

If s_a ≥ s_b, then max(s_a, s_b) = s_a;

If s_a ≤ s_b, then min(s_a, s_b) = s_a.

In order to overcome the possible loss of information in the computation process of linguistic information, Xu [31] extended the discrete LTS S to a continuous LTS $\bar{S} = {{\bar{s}}_{i} | i \in [0, r]}$ , where r ≥ t.

Further, Xu [35] proposed the concept of ULVs.

Definition 3. [35] Suppose $\tilde{s} = [s_{a}, s_{b}]$ , $s_{a}, s_{b} \in \bar{S}$ and 0 < a ≤ b, s_a, s_b are the lower limit and upper limit of $\tilde{s}$ , respectively, then $\tilde{s}$ is called an uncertain linguistic variable (ULV).

Let $\tilde{S}$ be a set of all ULVs, for any ULVs ${\tilde{s}}_{1} = [s_{a_{1}}, s_{b_{1}}]$ and ${\tilde{s}}_{2} = [s_{a_{2}}, s_{b_{2}}]$ , the operation laws are defined as follows [22, 34]: $\begin{matrix} s_{1} \end{matrix} \oplus {\tilde{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] \oplus [s_{a_{2}}, s_{b_{2}}] = [s_{a_{1} + a_{2}}, s_{b_{1} + b_{2}}]$ (7) $\begin{matrix} s_{1} \end{matrix} \otimes {\tilde{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] \otimes [s_{a_{2}}, s_{b_{2}}] = [s_{a_{1} \times a_{2}}, s_{b_{1} \times b_{2}}]$ (8) $(3) λ {\tilde{s}}_{1} = λ [s_{a_{1}}, s_{b_{1}}] = [s_{λ \times a_{1}}, s_{λ \times b_{1}}]$ (9) $(4) {({\tilde{s}}_{1})}^{λ} = [s_{a_{1}}, s_{b_{1}}]^{λ} = [s_{{a_{1}}^{λ}}, s_{{b_{1}}^{λ}}]$ (10)Example 1. Let ${\tilde{s}}_{1} = [s_{1}, s_{2}]$ , ${\tilde{s}}_{2} = [s_{3}, s_{4}]$ , then according to (7)–(10), we have $(1) {\tilde{s}}_{1} + {\tilde{s}}_{2} = [s_{1}, s_{2}] + [s_{3}, s_{4}] = [s_{4}, s_{6}]$ (11) $(2) {\tilde{s}}_{1} \times {\tilde{s}}_{2} = [s_{1}, s_{2}] \times [s_{3}, s_{4}] = [s_{3}, s_{8}]$ (12) $(3) {\tilde{2s}}_{2} = 2 [s_{3}, s_{4}] = [s_{6}, s_{8}]$ (13) $(4) ({\tilde{s}}_{2})^{2} = [s_{3}, s_{4}]^{2} = [s_{6}, s_{8}]$ (14) Obviously, the operational results in the equations (12)–(14) have cross-border situation which can lead to information distortion. Therefore, new operational laws of ULVs need to be redefined.

2.3 Linguistic scale functions (LSFs)

Definition 4. [26, 39] Let S ={ s₀, s₁, …, s_2t } be a linguistic term set (LTS). For any real number θ_i (i = 0, 1, …, 2t), a LSF f conducting the mapping from s_i to θ_i (i = 0, 1, 2, …, 2t) can be represented as follows: $f : s_{i} \to θ_{i} (i = 0, 1, 2, \dots, 2 t)$ (15) where 0 ≤ θ₀ ≤ θ₁ ≤ … θ_2t ≤ 1. At the same time, the above function is expanded to $f^{*} : \bar{s} \to R^{+} (R^{+} = {r | r \geq 0, r \in R})$ , which satisfies f^* (s_i) = θ_i, and is a strictly monotonically increasing and continuous function. Therefore, the inverse function of f^* exists, denoted as f^*-1.

The following LSFs are provided by Wang [26, 27]. 1.

One simple LSF is defined as follows: $f_{1} (s_{i}) = θ_{i} = \frac{i}{2 t} (i = 0, 1, 2, \dots, 2 t)$ (16)

The second is defined as follows [3]: $\begin{matrix} f_{2} (s_{i}) = θ_{i} \\ = {\begin{matrix} \frac{ξ^{t} - ξ^{t - i}}{2 ξ^{t} - 2} (i = 0, 1, 2, \dots, t) \\ \frac{ξ^{t} + ξ^{i - t} - 2}{2 ξ^{t} - 2} (i = t + 1, t + 2, \dots, 2 t) \end{matrix} \end{matrix}$ (17) where the value range of parameter ζ is [1.36, 1, 4].

The third is defined based on the prospect theory as follows: $f_{3} (s_{i}) = θ_{i} = {\begin{matrix} \frac{t^{α} - {(t - i)}^{α}}{2 t^{α}} (i = 0, 1, 2, \dots, t) \\ \frac{t^{β} + {(i - t)}^{β}}{2 t^{β}} (i = t + 1, t + 2, \dots, 2 t) \end{matrix}$ (18) where α, β ∈ [0, 1]. When α = β = 1, this function reduces to $f_{1} (s_{i}) = θ_{i} = \frac{i}{2 t}$ .

Definition 5. Based on above definition, combining with the LTS S = {s₀ = very poor, s₁ = poor, s₂ = poor, s₃ = fair, s₄ = good, s₅ = good, s₆ = very good}, we can get $\begin{matrix} (1) If t = 3 then \\ f (s_{i}) = θ_{i} = \frac{i}{6} (i = 0, 1, 2, \dots, 6) \\ and f^{* - 1} (θ_{i}) = s_{6 θ_{i}} (θ_{i} \in [0, 1]) . \end{matrix}$ $\begin{matrix} (2) If t = 3 then \\ f (s_{i}) = θ_{i} = {\begin{matrix} \frac{ξ^{3} - ξ^{3 - i}}{2 ξ^{3} - 2} (i = 0, 1, 2, 3) \\ \frac{ξ^{3} + ξ^{i - 3} - 2}{2 ξ^{3} - 2} (i = 4, 5, 6) \end{matrix} and \\ f^{* - 1} (θ_{i}) = {\begin{matrix} s_{3 - {log}_{ξ} (ξ^{3} - (2 ξ^{3} - 2) θ_{i})}, θ_{i} \in [0, 0.5] \\ s_{3 + {log}_{ξ} ((2 ξ^{3} - 2) θ_{i} - ξ^{3} + 2)}, θ_{i} \in (0.5, 1] \end{matrix} . \end{matrix}$ $\begin{matrix} (3) If t = 3 then \\ f (s_{i}) = θ_{i} = {\begin{matrix} \frac{3^{α} - {(3 - i)}^{α}}{2 \times 3^{α}} (i = 0, 1, 2, 3) \\ \frac{3^{β} + {(i - 3)}^{β}}{2 \times 3^{β}} (i = 4, 5, 6) \end{matrix} and \\ f^{* - 1} (θ_{i}) = {\begin{matrix} s_{3 - {(3^{a} - 2 \times 3^{a} \times θ_{i})}^{1 / - a}}, θ_{i} \in [0, 0.5] \\ s_{3 + {(2 \times 3^{β} \times θ_{i} - 3^{β})}^{1 / - β}}, θ_{i} \in (0.5, 1] \end{matrix} . \end{matrix}$

Definition 6. Let $\tilde{S}$ be a set of all ULVs, f^* be LSFs, and λ ≥ 0, for any two ULVs ${\tilde{s}}_{1} = [s_{a_{1}}, s_{b_{1}}]$ and ${\tilde{s}}_{2} = [s_{a_{2}}, s_{b_{2}}]$ , the new operational laws are defined as follows: $\begin{matrix} (1) {\tilde{s}}_{1} \oplus {\tilde{s}}_{2} \\ = [f^{* - 1} (f^{*} (s_{a_{1}}) + f^{*} (s_{a_{2}}) - f^{*} (s_{a_{1}}) f^{*} (s_{a_{2}})), \\ f^{* - 1} (f^{*} (s_{b_{1}}) + f^{*} (s_{b_{2}}) - f^{*} (s_{b_{1}}) f^{*} (s_{b_{2}}))] \end{matrix}$ (19) $\begin{matrix} s_{1} \end{matrix} \otimes {\tilde{s}}_{2} = [f^{* - 1} ((f^{*} (s_{a_{1}})) \times (f^{*} (s_{a_{2}}))), f^{* - 1} ((f^{*} (s_{b_{1}})) \times (f^{*} (s_{b_{2}})))]$ (20) $\begin{matrix} (3) λ {\tilde{s}}_{1} = λ [s_{a_{1}}, s_{b_{1}}] = [f^{* - 1} (1 - {(1 - f^{*} (s_{a_{1}}))}^{λ}), \\ f^{* - 1} (1 - {(1 - f^{*} (s_{b_{1}}))}^{λ})] \end{matrix}$ (21) $(4) {\tilde{s}}_{1}^{λ} = [f^{* - 1} ({(f^{*} (s_{a_{1}}))}^{λ}), f^{* - 1} ({(f^{*} (s_{b_{1}}))}^{λ})]$ (22)Example 2. Let ${\tilde{s}}_{1} = [s_{1}, s_{2}]$ , ${\tilde{s}}_{2} = [s_{3}, s_{4}]$ , then according to the operational laws (19–22), we get (suppose $f (s_{i}) = θ_{i} = \frac{i}{6} (i = 0, 1, 2, \dots, 6)$ ) $\begin{matrix} (1) {\tilde{s}}_{1} \oplus {\tilde{s}}_{2} = [f^{* - 1} (\frac{1}{6} + \frac{3}{6} - \frac{1}{6} \times \frac{3}{6}), \\ f^{* - 1} (\frac{2}{6} + \frac{4}{6} - \frac{2}{6} \times \frac{4}{6})] = [s_{21 / 6, s 28 / 6}] \\ = [s_{3.5}, s_{4.7}] \end{matrix}$ (23) $\begin{matrix} (2) {\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = [f^{* - 1} (\frac{1}{6} \times \frac{3}{6}), f^{* - 1} (\frac{2}{6} \times \frac{4}{6})] \\ = [s_{3 / 6, s 8 / 6}] \\ = [s_{0.5}, s_{1.3}] \end{matrix}$ (24) $\begin{matrix} (3) 2 {\tilde{s}}_{2} = [f^{* - 1} (1 - {(1 - \frac{3}{6})}^{2}), \\ f^{* - 1} (1 - {(1 - \frac{4}{6})}^{2})] \end{matrix}$ (25) $\begin{matrix} (4) {\tilde{s}}_{2}^{2} = [f^{* - 1} ({(\frac{3}{6})}^{2}), f^{* - 1} ({(\frac{4}{6})}^{2})] \\ = [s_{9 / 6, s 16 / 6}] \\ = [s_{1.5}, s_{2.7}] \end{matrix}$ (26) Compared with the operational results in (11–14), the results in (23–26) are more reasonable because they can solve cross-border problems.

2.4 The intuitionistic uncertain linguistic set (IULS)

Definition 7. [12] Let $[s_{θ (x)}, s_{τ (x)}] \in \bar{S}$ , X is a given domain, then $A = {〈 x [[s_{θ (x)}, s_{τ (x)}], (u_{A} (x), v_{A} (x))] 〉 | x \in X}$ (27) is called an intuitionistic uncertain linguistic set (IULS). Where u_A (x): X → [0, 1] and v_A (x): X → [0, 1] represent the MD and NMD of the argument x to the ULV [s_θ(x), s_τ(x)], with the condition 0 ≤ u_A (x) + v_A (x) ≤1, 1pt ∀ x ∈ X. If π (x) =1 - u_A (x) - v_A (x) , ∀ x ∈ X, then π (x) is called the degree of indeterminacy of x to ULV [s_θ(x), s_τ(x)]. Then $〈 [s_{θ (x)}, s_{τ (x)}], (u_{A} (x), v_{A} (x)) 〉$ (28) is called an intuitionistic uncertain linguistic number (IULN).Definition 8. [12] Let ${\tilde{a}}_{1}$ = 〈 [s_θ(a₁) , s_τ(a₁)], (u (a₁) , v (a₁)) 〉 and ${\tilde{a}}_{2} = 〈 [s_{θ (a_{2})}, s_{τ (a_{2})}]$ , (u (a₂) , v (a₂)) 〉 be two IULNs and λ ≥ 0, then the operational rules for ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ are defined as follows: $\begin{matrix} (1) {\tilde{a}}_{1} + {\tilde{a}}_{2} = 〈 [s_{θ (a_{1}) + θ (a_{2})}, s_{τ (a_{1}) + τ (a_{2})}], \\ (u (a_{1}) + u (a_{2}) - u (a_{1}) u (a_{2}), v (a_{1}) v (a_{2})) 〉 \end{matrix}$ (29) $\begin{matrix} \begin{matrix} (2) {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = 〈 [s_{θ (a_{1}) \times θ (a_{2})}, s_{τ (a_{1}) \times τ (a_{2})}], \\ (u (a_{1}) u (a_{2}) v (a_{1}) + v (a_{2}) - v (a_{1}) v (a_{2})) 〉 \end{matrix} \end{matrix}$ (30) $\begin{matrix} (3) λ {\tilde{a}}_{1} = 〈 [s_{λ \times θ (a_{1})}, s_{λ \times τ (a_{1})}], \\ (1 - {(1 - u (a_{1}))}^{λ}, {(v (a_{1}))}^{λ}) 〉 \end{matrix}$ (31) $\begin{matrix} (4) {\tilde{a}}_{1}^{λ} = 〈 [s_{θ {(a_{1})}^{λ}}, s_{τ {(a_{1})}^{λ}}], \\ ({(u (a_{1}))}^{λ}, 1 - {(1 - v (a_{1}))}^{λ}) 〉 \end{matrix}$ (32)Example 3. Let ${\tilde{a}}_{1} = 〈 [s_{1}, s_{2}], (0.4, 0.6) 〉$ , ${\tilde{a}}_{2} = 〈 [s_{3}, s_{4}], (0.2, 0.3) 〉$ , then according above operational laws, we can get $(1) {\tilde{a}}_{1} + {\tilde{a}}_{2} = 〈 [s_{4}, s_{6}], (0.52, 0.18) 〉$ (33) $(2) {\tilde{a}}_{1} \times {\tilde{a}}_{2} = 〈 [s_{3}, s_{8}], (0.08, 0.72) 〉$ (34) $2 {\tilde{a}}_{2} = 〈 [s_{6}, s_{8}], (0.36, 0.09) 〉$ (35) $({\tilde{a}}_{2})^{2} = 〈 [s_{6}, s_{8}], (0.04, 0.51) 〉$ (36) Obviously, the operational results in (34–36) have the same cross-border problems. Therefore, based on LSFs, we redefine operational laws of IULNs.Definition 9. Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}]$ , (u (a₁) , v (a₁))〉 and ${\tilde{a}}_{2} = 〈 [s_{θ (a_{2})}, s_{τ (a_{2})}]$ , (u (a₂) , v (a₂)) 〉 be two IULNs and λ ≥ 0, and then based on LSFs, the new operation laws about ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ are defined as follows: $\begin{matrix} (1) {\tilde{a}}_{1} + {\tilde{a}}_{2} = 〈 [f^{* - 1} (f^{*} (s_{θ (a_{1})}) + f^{*} (s_{θ (a_{2})}) \\ - f^{*} (s_{θ (a_{1})}) f^{*} (s_{θ (a_{2})})), \\ f^{* - 1} (f^{*} (s_{τ (a_{1})}) + f^{*} (s_{τ (a_{2})}) \\ - f^{*} (s_{τ (a_{1})}) f^{*} (s_{τ (a_{2})}))], \\ (u (a_{1}) + u (a_{2}) - u (a_{1}) u (a_{2}), v (a_{1}) v (a_{2})) 〉 \end{matrix}$ (37) $\begin{matrix} (2) {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = 〈 [f^{* - 1} (f^{*} (s_{θ (a_{1})}) \times f^{*} (s_{θ (a_{2})})), \\ f^{* - 1} (f^{*} (s_{τ (a_{1})}) \times f^{*} (s_{τ (a_{2})}))], \\ (u (a_{1}) u (a_{2}), v (a_{1}) + v (a_{2}) - v (a_{1}) v (a_{2})) 〉 \end{matrix}$ (38) $\begin{matrix} (3) λ {\tilde{a}}_{1} = 〈 [f^{* - 1} (1 - {(1 - f^{*} (s_{θ (a_{1})}))}^{λ}), \\ f^{* - 1} (1 - {(1 - f^{*} (s_{τ (a_{1})}))}^{λ})] \\ (1 - {(1 - u (a_{1}))}^{λ}, {(v (a_{1}))}^{λ}) 〉 \end{matrix}$ (39) $\begin{matrix} (4) {\tilde{a}}_{1}^{λ} = 〈 [f^{* - 1} ({(f^{*} (s_{θ (a_{1})}))}^{λ}), \\ f^{* - 1} ({(f^{*} (s_{τ (a_{1})}))}^{λ})], \\ ({(u (a_{1}))}^{λ}, 1 - (1 - v (a_{1}))^{λ}) 〉 \end{matrix}$ (40)Example 4. Let ${\tilde{a}}_{1} = 〈 [s_{1}, s_{2}], (0.4, 0.6) 〉$ , ${\tilde{a}}_{2} = 〈 [s_{3}, s_{4}], (0.2, 0.3) 〉$ , then according above operational laws, we can get (suppose $f (s_{i}) = θ_{i} = \frac{i}{6} (i = 0, 1, 2, \dots, 6)$ ) $\begin{matrix} (1) {\tilde{a}}_{1} + {\tilde{a}}_{2} = 〈 [f^{* - 1} (\frac{1}{6} + \frac{3}{6} - \frac{3}{36}), \\ f^{* - 1} (\frac{2}{6} + \frac{4}{6} - \frac{8}{36})], (0.52, 0.18) 〉 \\ = 〈 [s_{3.5}, s_{4.7}], (0.52, 0.18) 〉 \end{matrix}$ (41) $\begin{matrix} (2) {\tilde{a}}_{1} \times {\tilde{a}}_{2} = 〈 [f^{* - 1} (\frac{1}{6} \times \frac{3}{6}), \\ f^{* - 1} (\frac{2}{6} \times \frac{4}{6})], (0.08, 0.72) 〉 \\ = 〈 [s_{0.5}, s_{1.3}], (0.08, 0.72) 〉 \end{matrix}$ (42) $\begin{matrix} (3) 2 {\tilde{a}}_{2} = 〈 [f^{* - 1} (1 - {(1 - \frac{3}{6})}^{2}), \\ f^{* - 1} (1 - {(1 - \frac{4}{6})}^{2})], (0.36, 0.09) 〉 \\ = 〈 [s_{4.5}, s_{5.3}], (0.36, 0.09) 〉 \end{matrix}$ (43) $\begin{matrix} (4) {({\tilde{a}}_{2})}^{2} = 〈 [f^{* - 1} ({(\frac{3}{6})}^{2}), f^{* - 1} ({(\frac{4}{6})}^{2})], \\ (0.04, 0.51) 〉 = 〈 [s_{1.5}, (3) s_{2.7}], (0.04, 0.51) 〉 \end{matrix}$ (44) Obviously, the results in (41–44) can eliminate cross-border problems, and they also prove the validity of the improved operational laws.Theorem 1. For any two IULNs ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ , the calculation rules have the properties shown as follows: $(1) {\tilde{a}}_{1} + {\tilde{a}}_{2} = {\tilde{a}}_{2} + {\tilde{a}}_{1}$ (45) $(2) {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = {\tilde{a}}_{2} \otimes {\tilde{a}}_{1}$ (46) $(3) λ ({\tilde{a}}_{1} + {\tilde{a}}_{2}) = λ {\tilde{a}}_{1} + λ {\tilde{a}}_{2}$ (47) $(4) {\tilde{a}}_{1}^{λ_{1}} \otimes {\tilde{a}}_{1}^{λ_{2}} = {({\tilde{a}}_{1})}^{λ_{1} + λ_{2}}, λ_{1}, λ_{2} \geq 0$ (48) $(5) λ_{1} {\tilde{a}}_{1} + λ_{2} {\tilde{a}}_{1} = (λ_{1} + λ_{2}) {\tilde{a}}_{1}, λ_{1}, λ_{2} \geq 0$ (49) $(6) {\tilde{a}}_{1}^{λ} \otimes {\tilde{a}}_{2}^{λ} = {({\tilde{a}}_{1} \otimes {\tilde{a}}_{2})}^{λ}, λ \geq 0$ (50)Proof: (1) According to formulas (37) and (38), the rules (1) and (2) are kept. (2) For rule (3), we have $\begin{matrix} (i) λ ({\tilde{a}}_{1} + {\tilde{a}}_{2}) \\ = λ 〈 [f^{* - 1} (f^{*} (s_{θ (a_{1})}) + f^{*} (s_{θ (a_{2})}) \\ - f^{*} (s_{θ (a_{1})}) f^{*} (s_{θ (a_{2})})), f^{* - 1} (f^{*} (s_{τ (a_{1})}) \\ + f^{*} (s_{τ (a_{2})}) - f^{*} (s_{τ (a_{1})}) f^{*} (s_{τ (a_{2})}))] 〉 \\ = (u (a_{1}) + u (a_{2}) - u (a_{1}) u (a_{2}), v (a_{1}) v (a_{2})) 〉 \end{matrix}$ $\begin{matrix} = 〈 [f^{* - 1} (1 - (1 - (f^{*} (s_{θ (a_{1})}) + f^{*} (s_{θ (a_{2})}) \\ {- f^{*} (s_{θ (a_{1})}) f^{*} (s_{θ (a_{2})})))}^{λ}), \\ f^{* - 1} (1 - (1 - (f^{*} (s_{τ (a_{1})}) + f^{*} (s_{τ (a_{2})}) \\ {- f^{*} (s_{τ (a_{1})}) f^{*} (s_{τ (a_{2})})))}^{λ})] \\ (1 - {(1 - u (a_{1}))}^{λ} {(1 - u (a_{2}))}^{λ}, {(v (a_{1}))}^{λ} {(v (a_{2}))}^{λ}) 〉 \end{matrix}$ and (ii) $\begin{matrix} λ {\tilde{a}}_{1} + λ {\tilde{a}}_{2} = 〈 [f^{* - 1} (1 - {(1 - f^{*} (s_{θ (a_{1})}))}^{λ}), f^{* - 1} (1 - {(1 - f^{*} (s_{τ (a_{1})}))}^{λ})], \\ (1 - {(1 - u (a_{1}))}^{λ}, {(v (a_{1}))}^{λ}) 〉 + 〈 [f^{* - 1} (1 - {(1 - f^{*} (s_{θ (a_{2})}))}^{λ}), f^{* - 1} (1 - {(1 - f^{*} (s_{τ (a_{2})}))}^{λ})], \\ (1 - {(1 - u (a_{2}))}^{λ}, {(v (a_{2}))}^{λ}) 〉 = 〈 [f^{* - 1} ((1 - {(1 - f^{*} (s_{θ (a_{1})}))}^{λ}) + (1 - {(1 - f^{*} (s_{θ (a_{2})}))}^{λ}) \\ - (1 - {(1 - f^{*} (s_{θ (a_{1})}))}^{λ}) (1 - {(1 - f^{*} (s_{θ (a_{2})}))}^{λ})), f^{* - 1} ((1 - {(1 - f^{*} (s_{τ (a_{1})}))}^{λ}) \\ + (1 - {(1 - f^{*} (s_{τ (a_{2})}))}^{λ}) - (1 - {(1 - f^{*} (s_{τ (a_{1})}))}^{λ}) (1 - {(1 - f^{*} (s_{τ (a_{2})}))}^{λ}))] \\ (1 - (1 - (1 - {(1 - u (a_{1}))}^{λ})) (1 - (1 - (1 - u {(a_{2})}^{λ}))), {(v (a_{1}))}^{λ} {(v (a_{2}))}^{λ}) 〉 \\ = 〈 [f^{* - 1} (1 - {(1 - f^{*} (s_{θ (a_{1})}))}^{λ} {(1 - f^{*} (s_{θ (a_{2})}))}^{λ}), f^{* - 1} (1 - {(1 - f^{*} (s_{τ (a_{1})}))}^{λ} {(1 - f^{*} (s_{τ (a_{2})}))}^{λ})] \\ (1 - {(1 - u (a_{1}))}^{λ} {(1 - u (a_{2}))}^{λ}, {(v (a_{1}))}^{λ} {(v (a_{2}))}^{λ}) 〉 = 〈 [f^{* - 1} (1 - (1 - (f^{*} (s_{θ (a_{1})}) + f^{*} (s_{θ (a_{2})}) \\ {- f^{*} (s_{θ (a_{1})}) f^{*} (s_{θ (a_{2})})))}^{λ}), f^{* - 1} (1 - {(1 - (f^{*} (s_{τ (a_{1})}) + f^{*} (s_{τ (a_{2})}) - f^{*} (s_{τ (a_{1})}) f^{*} (s_{τ (a_{2})})))}^{λ})] \\ = (1 - {(1 - u (a_{1}))}^{λ} {(1 - u (a_{2}))}^{λ}, {(v (a_{1}))}^{λ} {(v (a_{2}))}^{λ}) 〉 \end{matrix}$ By (i) and (ii), we get $λ ({\tilde{a}}_{1} + {\tilde{a}}_{2}) = λ {\tilde{a}}_{1} + λ {\tilde{a}}_{2}$ . Similarly, we can prove the ruled (4–6).

2.5 Comparison of two IULNs

Definition 10. Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}$ , s_τ(a₁)], (u (a₁) , v (a₁)) 〉 be an IULN, an expected value $E ({\tilde{a}}_{1})$ of ${\tilde{a}}_{1}$ can be represented as follows: $\begin{matrix} E ({\tilde{a}}_{1}) = (u (a_{1}) + 1 - v (a_{1})) \\ \times \frac{(f^{*} (s_{θ (a_{1})}) + f^{*} (s_{τ (a_{1})}))}{4} \end{matrix}$ (51)Definition 11. Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}$ , s_τ(a₁)], (u (a₁) , v (a₁)) 〉 be an IULN, an accuracy function $H ({\tilde{a}}_{1})$ of ${\tilde{a}}_{1}$ can be represented as follows $\begin{matrix} H ({\tilde{a}}_{1}) = (u (a_{1}) + v (a_{1})) \\ \times \frac{f^{*} (s_{θ (a_{1})}) + f^{*} (s_{τ (a_{1})})}{2} \end{matrix}$ (52)Definition 12. [20] Let ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ are any two IULNs, then,

If $E ({\tilde{a}}_{1}) > E ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

If $E ({\tilde{a}}_{1}) = E ({\tilde{a}}_{2})$ , then

If $H ({\tilde{a}}_{1}) > H ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

If $H ({\tilde{a}}_{1}) = H ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} = {\tilde{a}}_{2}$ .

Example 5. Let ${\tilde{a}}_{1} = 〈 [s_{1}, s_{2}], (0.4, 0.6) 〉$ , ${\tilde{a}}_{2}$ = 〈 [s₃, s₄], (0.2, 0.3) 〉, if f (s_i) = θ_i = $\frac{i}{6} (i = 0, 1, 2$ , …, 6), then according to the Definitions 10,11 and 12, then $E ({\tilde{a}}_{1}) = \frac{(0.4 + 1 - 0.6) \times (\frac{1}{6} + \frac{2}{6})}{4} = 0.1,$ $E ({\tilde{a}}_{2}) = \frac{(0.2 + 1 - 0.3) \times (\frac{3}{6} + \frac{4}{6})}{4} = 0.2625 .$ Thus, we have ${\tilde{a}}_{1} < {\tilde{a}}_{2}$ .

3 The intuitionistic uncertain linguistic aggregation operators

Definition 13. [12] Let ${\tilde{a}}_{i}$ = 〈 [s_{θ(a_i)} , s_{τ(a_i)}], (u (a_i) , v (a_i)) 〉 (i = 1, 2, …, n) be a group of the IULNs, and IULWGA: Ωⁿ → Ω, if

$IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{i = 1}^{n} {({\tilde{a}}_{i})}^{ω_{i}}$ (53) where Ω is the set of all IULNs, and ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector of ${\tilde{a}}_{i} (i = 1, 2, \dots, n)$ , $ω_{i} \in [0, 1], \sum_{i = 1}^{n} ω_{i} = 1$ , then IULWGA is called intuitionistic uncertain linguistic weighted geometric average operator.

Theorem 2. Let ${\tilde{a}}_{i}$ = 〈 [s_{θ(a_i)} , s_{τ(a_i)}], (u (a_i) , v (a_i)) 〉 (i = 1, 2, …, n) be a group of the IULNs, the aggregated result from definition 13 is still an IULN, then

$\begin{matrix} IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \prod_{i = 1}^{n} {({\tilde{a}}_{i})}^{ω_{i}} = 〈 [f^{* - 1} (\prod_{i = 1}^{n} {(f^{*} (s_{θ (a_{i})}))}^{ω_{i}}), \\ f^{* - 1} (\prod_{i = 1}^{n} {(f^{*} (s_{τ (a_{i})}))}^{ω_{i}})], \\ (\prod_{i = 1}^{n} {(u (a_{i}))}^{ω_{i}}, 1 - \prod_{i = 1}^{n} {(1 - v (a_{i}))}^{ω_{i}}) 〉 \end{matrix}$ (54)

Theorem 2 can be proved by mathematical induction method, the steps are shown as follows:

Proof:

(1) When n = 1, obviously, it is right. (2) When n = 2,

${({\tilde{a}}_{1})}^{ω_{1}} = 〈 [f^{* - 1} ({(f^{*} (s_{θ (a_{1})}))}^{ω_{1}}), f^{* - 1} ({(f^{*} (s_{τ (a_{1})}))}^{ω_{1}})], ({(u (a_{1}))}^{ω_{1}}, 1 - {(1 - v (a_{1}))}^{ω_{1}}) 〉 {({\tilde{a}}_{2})}^{ω_{2}} = 〈 [f^{* - 1} ({(f^{*} (s_{θ (a_{2})}))}^{ω_{2}}), f^{* - 1} ({(f^{*} (s_{τ (a_{2})}))}^{ω_{2}})], ({(u (a_{2}))}^{ω_{2}}, 1 - {(1 - v (a_{2}))}^{ω_{2}})] IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}) = {({\tilde{a}}_{1})}^{ω_{1}} \times {({\tilde{a}}_{2})}^{ω_{2}} = 〈 [f^{* - 1} ({(f^{*} (s_{θ (a_{1})}))}^{ω_{1}}), f^{* - 1} ({(f^{*} (s_{τ (a_{1})}))}^{ω_{1}})], ({(u (a_{1}))}^{ω_{1}}, 1 - {(1 - v (a_{1}))}^{ω_{1}}) 〉 \times 〈 [f^{* - 1} ({(f^{*} (s_{θ (a_{2})}))}^{ω_{2}}), f^{* - 1} ({(f^{*} (s_{τ (a_{2})}))}^{ω_{2}})], ({(u (a_{2}))}^{ω_{2}}, 1 - {(1 - v (a_{2}))}^{ω_{2}}) 〉 = 〈 [f^{* - 1} ({(f^{*} (s_{θ (a_{1})}))}^{ω_{1}} \times {(f^{*} (s_{θ (a_{2})}))}^{ω_{2}}), f^{* - 1} ({(f^{*} (s_{θ (a_{1})}))}^{ω_{1}} \times {(f^{*} (s_{τ (a_{2})}))}^{ω_{2}})], ({(u (a_{1}))}^{ω_{1}} \times {(u (a_{2}))}^{ω_{2}}, 1 - {(1 - v (a_{1}))}^{ω_{1}} {(1 - v (a_{2}))}^{ω_{2}}) 〉 = 〈 [f^{* - 1} (\prod_{i = 1}^{2} {(f^{*} (s_{θ (a_{i})}))}^{ω_{i}}), f^{* - 1} (\prod_{i = 1}^{2} {(f^{*} (s_{τ (a_{i})}))}^{ω_{i}})], (\prod_{i = 1}^{2} {(u (a_{i}))}^{ω_{i}}, 1 - \prod_{i = 1}^{2} {(1 - v (a_{i}))}^{ω_{i}}) 〉$ So, when n = 2, formula (54) is right, too. (3) Suppose when n = k, formula (54) is right, i.e., $\begin{matrix} IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{k}) \\ = 〈 [f^{* - 1} (\prod_{i = 1}^{k} {(f^{*} (s_{θ (a_{i})}))}^{ω_{i}}), f^{* - 1} (\prod_{i = 1}^{k} {(f^{*} (s_{τ (a_{i})}))}^{ω_{i}})], (\prod_{i = 1}^{k} {(u (a_{i}))}^{ω_{i}}, 1 - \prod_{i = 1}^{k} {(1 - v (a_{i}))}^{ω_{i}}) 〉 . \end{matrix}$

Then, when n = k + 1, $\begin{matrix} IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{k}, {\tilde{a}}_{k + 1}) = IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{k}) \times {({\tilde{a}}_{k + 1})}^{ω_{k + 1}} \\ = 〈 [f^{* - 1} (\prod_{i = 1}^{k} {(f^{*} (s_{θ (a_{i})}))}^{ω_{i}}), f^{* - 1} (\prod_{i = 1}^{k} {(f^{*} (s_{τ (a_{i})}))}^{ω_{i}})], (\prod_{i = 1}^{k} {(u (a_{i}))}^{ω_{i}}, 1 - \prod_{i = 1}^{k} {(1 - v (a_{i}))}^{ω_{i}}) 〉 \\ \times 〈 [f^{* - 1} ({(f^{*} (s_{θ (a_{k + 1})}))}^{ω_{k + 1}}), f^{* - 1} ({(f^{*} (s_{τ (a_{k + 1})}))}^{ω_{k + 1}})], ({(u (a_{k + 1}))}^{ω_{k + 1}}, 1 - {(1 - v (a_{k + 1}))}^{ω_{k + 1}}) 〉 \\ 〈 [f^{* - 1} (\prod_{i = 1}^{k} {(f^{*} (s_{θ (a_{i})}))}^{ω_{i}} \times {(f^{*} (s_{θ (a_{k + 1})}))}^{ω_{k + 1}}), f^{* - 1} (\prod_{i = 1}^{k} {(f^{*} (s_{τ (a_{i})}))}^{ω_{i}} \times {(f^{*} (s_{θ (a_{k + 1})}))}^{ω_{k + 1}})] \\ (\prod_{i = 1}^{k} {(u (a_{i}))}^{ω_{i}} \times {(u (a_{k + 1}))}^{ω_{k + 1}}, 1 - \prod_{i = 1}^{k} {(1 - v (a_{i}))}^{ω_{i}} \times {(1 - v (a_{k + 1}))}^{ω_{k + 1}}) 〉 \\ 〈 [f^{* - 1} (\prod_{i = 1}^{k + 1} {(f^{*} (s_{θ (a_{i})}))}^{ω_{i}}), f^{* - 1} (\prod_{i = 1}^{k + 1} {(f^{*} (s_{τ (a_{i})}))}^{ω_{i}})], (\prod_{i = 1}^{k + 1} {(u (a_{i}))}^{ω_{i}}, 1 - \prod_{i = 1}^{k + 1} {(1 - v (a_{i}))}^{ω_{i}}) 〉 \end{matrix}$ So, when n = k + i, formula (54) is right, too. According to steps (1), (2) and (3), we can conclude that formula (54) is right for all n. The IULWGA operator has the following properties:Property 1. (Idempotency) Let ${\tilde{a}}_{i} = \tilde{a}$ for all i = 1, 2, …, n, then $IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ .Proof: Since ${\tilde{a}}_{i} = \tilde{a}$ , for all i = 1, 2, …, n, we have $\begin{matrix} {IULWGA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{i = 1}^{n} ({\tilde{a}}_{i})^{ω_{i}} \\ = \prod_{i = 1}^{n} (\tilde{a})^{ω_{i}} = 〈 [f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{θ (a)}))^{ω_{i}}), \\ f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{τ (a)}))^{ω_{i}})], \\ (\prod_{i = 1}^{n} (u (a))^{ω_{i}}, 1 - \prod_{i = 1}^{n} (1 - v (a))^{ω_{i}}) 〉 \\ = 〈 [f^{* - 1} ((f^{*} (s_{θ (a)}))^{\sum_{i = 1}^{n} ω_{i}}), \\ f^{* - 1} ((f^{*} (s_{τ (a)}))^{\sum_{i = 1}^{n} ω_{i}})], \\ ((u (a))^{\sum_{i = 1}^{n} ω_{i}}, 1 - (1 - v (a))^{\sum_{i = 1}^{n} ω_{i}}) 〉 \\ = 〈 [s_{θ (a),} s_{τ (a)}], (u (a), v (a)) 〉 = \tilde{a} . \end{matrix}$ Property 2. (Boundedness) Let ${\tilde{a}}^{-} = 〈 [s_{min θ (a_{i})}, s_{min τ (a_{i})}], (min_{i} {u (a_{i})}, max_{i} {v (a_{i})}) 〉$ , ${\tilde{a}}^{+} = 〈 [s_{max θ (a_{i})}, s_{max τ (a_{i})}], (max_{i} {u (a_{i})}$ , $min_{i} {v (a_{i})}) 〉$ , then ${\tilde{a}}^{-}$ ≤ IULWGA_ω $({\tilde{a}}_{1}$ , ${\tilde{a}}_{2}$ , …, ${\tilde{a}}_{n})$ ≤ ${\tilde{a}}^{+}$ .Proof: For ∀i, it follows $min_{i} {u (a_{i})} \leq u (a_{i}) \leq max_{i} {u (a_{i})}$ , and $min_{i} {v (a_{i})} \leq v (a_{i}) \leq max_{i} {v (a_{i})}$ . Since y = x^a (0 < a < 1) is a monotonic increasing function when x > 0, it holds, which is equivalent to $min {u (a_{i})}^{\sum_{i = 1}^{n} ω_{i}} \leq \prod_{i = 1}^{n} (u (a_{i}))^{ω_{i}} \leq max {u (a_{i})}^{\sum_{i = 1}^{n} ω_{i}},$ i.e., $min {u (a_{i})} \leq \prod_{i = 1}^{n} (u (a_{i}))^{ω_{i}} \leq max {u (a_{i})}$ . Analogously, we have $min {v (a_{i})} \leq 1 - \prod_{i = 1}^{n} (1 - v (a_{i}))^{ω_{i}} \leq max {v (a_{i})} .$ Let IULWGA_ω $({\tilde{a}}_{1}$ , ${\tilde{a}}_{2}$ , …, ${\tilde{a}}_{n})$ = $\tilde{a}$ = 〈 [s_θ(a) , s_τ(a)], (u (a) , v (a)) 〉, by combining above formulas, it follows that $\begin{matrix} E ({\tilde{a}}^{-}) = (min {u (a_{i})} + 1 - max {v (a_{i})}) \\ \times \frac{f^{*} (s_{min θ (a_{i})}) + f^{*} (s_{min τ (a_{i})})}{4}, \\ E ({\tilde{a}}^{+}) = (max {u (a_{i})} + 1 - min {v (a_{i})}) \\ \times \frac{f^{*} (s_{max θ (a_{i})}) + f^{*} (s_{max τ (a_{i})})}{4}, \\ E (\tilde{a}) = (u (a) + 1 - v (a)) \times \frac{f^{*} (s_{θ (a)}) + f^{*} (s_{τ (a)})}{4}, \end{matrix}$ then $E ({\tilde{a}}^{-}) \leq E (\tilde{a}) \leq E ({\tilde{a}}^{+}) .$ There may be three different cases:

If $E ({\tilde{a}}^{-}) < E (\tilde{a}) < E ({\tilde{a}}^{+})$ , it follows from the comparison law that ${\tilde{a}}^{-} < \tilde{a} < {\tilde{a}}^{+}$ .

If $E (\tilde{a}) = E ({\tilde{a}}^{+})$ , that is to say,

$\begin{matrix} (u (a) + 1 - v (a)) \times \frac{f^{*} (s_{θ (a)}) + f^{*} (s_{τ (a)})}{4} \\ = (max {u (a_{i})} + 1 - min {v (a_{i})}) \\ \times \frac{f^{*} (s_{max θ (a_{i})}) + f^{*} (s_{max τ (a_{i})})}{4} . \end{matrix}$

Since we have proven that u (a)≤ max { u (a_i) }, v (a) ≥ min { v (a_i) }, it holds only where u (a) = max{ u (a_i) }, v (a) = min { v (a_i) }, $\frac{f^{*} (s_{θ (a)}) + f^{*} (s_{τ (a)})}{4} = \frac{f^{*} (s_{max θ (a_{i})}) + f^{*} (s_{max τ (a_{i})})}{4}$ , which implies $\tilde{a} = {\tilde{a}}^{+}$ .

If $E (\tilde{a}) = E ({\tilde{a}}^{-})$ , similarly, we can derive $\tilde{a} = {\tilde{a}}^{-}$ as well.

Combining these three cases, it holds ${\tilde{a}}^{-} \leq IULWG A_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ .Property 3. (Monotonicity) Let ${\tilde{a}}_{i}^{1}$ = 〈 [s_{θ(a_i¹)} , s_{τ(a_i¹)}], (u (a_i¹) , v (a_i¹))〉, ${\tilde{a}}_{i}^{2}$ = 〈 [s_{θ(a_i²)} , s_{τ(a_i²)}], (u (a_i²) , v (a_i²)) 〉 be two groups of IULNs, if u (a_i¹) ≤ u (a_i²) , v (a_i¹) ≥ v (a_i²) , s_{θ(a_i¹)} ≤ s_{θ(a_i²)}, s_{τ(a_i¹)} ≤ s_{τ(a_i²)} for i = 1, 2, …, n, then $\begin{matrix} IULWG A_{ω} ({\tilde{a}}_{1}^{1}, {\tilde{a}}_{2}^{1}, \dots, {\tilde{a}}_{n}^{1}) \\ \leq IULWG A_{ω} ({\tilde{a}}_{1}^{2}, {\tilde{a}}_{2}^{2}, \dots, {\tilde{a}}_{n}^{2}) . \end{matrix}$ Proof: Considering ∀i, u (a_i¹) ≤ u (a_i²) , v (a_i¹) ≥ v (a_i²), s_{θ(a_i¹)} ≤ s_{θ(a_i²)}, s_{τ(a_i¹)} ≤ s_{τ(a_i²)}, we have $\begin{matrix} \prod_{i = 1}^{n} (u (a_{i}^{1}))^{ω_{i}} \leq \prod_{i = 1}^{n} (u (a_{i}^{2}))^{ω_{i}}, \\ 1 - \prod_{i = 1}^{n} (1 - v (a_{i}^{1}))^{ω_{i}} \leq 1 - \prod_{i = 1}^{n} (1 - v (a_{i}^{2}))^{ω_{i}}, \\ f^{*} (s_{θ (a_{i}^{1})}) \leq f^{*} (s_{θ (a_{i}^{2})}), \\ f^{*} (s_{τ (a_{i}^{1})}) \leq f^{*} (s_{τ (a_{i}^{2})}) . \end{matrix}$ Let IULWGA_ω $({\tilde{a}}_{1}^{1}$ , ${\tilde{a}}_{2}^{1}$ , …, ${\tilde{a}}_{n}^{1})$ = ${\tilde{a}}^{1}$ = 〈 [s_θ(a¹) , s_τ(a¹)], (u (a¹) , v (a¹)) 〉, IULWGA_ω $({\tilde{a}}_{1}^{2}$ , ${\tilde{a}}_{2}^{2}$ , …, ${\tilde{a}}_{n}^{2})$ = ${\tilde{a}}^{2}$ = 〈 [s_θ(a²) , s_τ(a²)], (u (a²) , v (a²))〉, then u (a¹) ≤ u (a²), v (a¹) ≥ v (a²), and f^* (s_θ(a¹)) ≤ f^* (s_θ(a²)), f^* (s_τ(a¹)) ≤ f^* (s_τ(a²)). Thus, $\begin{matrix} E ({\tilde{a}}^{1}) = (u (a^{1}) + 1 - v (a^{1})) \\ \times \frac{f^{*} (s_{θ (a^{1})}) + f^{*} (s_{τ (a^{1})})}{4} \\ \leq E ({\tilde{a}}^{2}) = (u (a^{2}) + 1 - v (a^{2})) \\ \times \frac{f^{*} (s_{θ (a^{2})}) + f^{*} (s_{τ (a^{2})})}{4} . \end{matrix}$ Similar to the proof of (2), we can derive ${\tilde{a}}^{1} \leq {\tilde{a}}^{2}$ , i.e., $IULWG A_{ω} ({\tilde{a}}_{1}^{1}, {\tilde{a}}_{2}^{1}, \dots, {\tilde{a}}_{n}^{1}) \leq IULWG A_{ω} ({\tilde{a}}_{1}^{2}, {\tilde{a}}_{2}^{2}, \dots, {\tilde{a}}_{n}^{2})$ .Definition 14. [8] Let ${\tilde{a}}_{i}$ = 〈 [s_{θ(a_i)} , s_{τ(a_i)}], (u (a_i) , v (a_i)) 〉 (i = 1, 2, …, n) be a group of the IULNs, and IULOWG: Ωⁿ → Ω, if $IULOW G_{W} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{i = 1}^{n} {({\tilde{a}}_{i})}^{w_{ɛ (i)}}$ (55) where Ω is the set of all IULNs and W = (w₁, w₂, …, w_n) ^T is an associated weight vector with IULOWG operator, $w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1$ ; where ${\tilde{a}}_{i}$ is the ɛ (i) th largest argument of ${\tilde{a}}_{i} (i = 1, 2, \dots, n)$ , then IULOWG is called the intuitionistic uncertain linguistic ordered weighted geometric operator.w_i is only decided by the ɛ (i) th position in the aggregation process. w_i can be determined in the following ways.Definition 15. The position weighted vector W can be determined by using a regular increasing monotone quantifier Q, which is defined as follows [7]: $w_{i} = Q (\frac{i}{n}) - Q (\frac{i - 1}{n}), i = 1, 2, \dots, n$ (56) where Q can be given by (1)

Q (r) = r^α, α ≥ 0 (basic RIM quantifier)

Q (r) = 1 - (1 - r) ^α, α ≥ 0 (polynomial quantifier)

$Q (r) = {(\frac{e^{r} - 1}{e - 1})}^{α}$ (exponential quantifier)

Q (r) = arcsin(rα) (trigonometric quantifier)

Theorem 3. Let

{\tilde{a}}_{i}

= 〈 [s_{θ(a_i)} , s_{τ(a_i)}], (u (a_i) , v (a_i)) 〉, (i = 1, 2, …, n) be a group of the IULNs, then the result aggregated from Definition 14 is still an IULN, even

\begin{matrix} IULOW G_{W} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = 〈 [f^{* - 1} (\prod_{i = 1}^{n} {(f^{*} (s_{θ (a_{i})}))}^{w_{ɛ (i)}}), \\ f^{* - 1} (\prod_{i = 1}^{n} {(f^{*} (s_{τ (a_{i})}))}^{w_{ɛ (i)}})], \\ (\prod_{i = 1}^{n} {(u (a_{i}))}^{w_{ɛ (i)}}, 1 - \prod_{i = 1}^{n} {(1 - v (a_{i}))}^{w_{ɛ (i)}}) 〉 . \end{matrix}

(57) Similar to Theorem 2, Theorem 3 can be proved by mathematical induction method, and proof steps are omitted here. The IULOWG operator has the following properties:Property 4. (Idempotency) Let

{\tilde{a}}_{i} = \tilde{a}

, for all i = 1, 2, …, n, then

IULOW G_{W} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}

.Proof: Similar with proof of Property 1, it is omitted here.Property 5. (Boundedness)Let

{\tilde{a}}^{-}

= 〈 [s_{minθ(a_i)} , s_{minτ(a_i)}],

(min_{i} {u (a_{i})}

max_{i} {v (a_{i})}) 〉

{\tilde{a}}^{+}

= 〈 [s_{maxθ(a_i)} , s_{maxτ(a_i)}],

(max_{i} {u (a_{i})}

min_{i} {v (a_{i})}) 〉

, then

{\tilde{a}}^{-}

≤ IULOWG_W

({\tilde{a}}_{1}

{\tilde{a}}_{2}

, …,

{\tilde{a}}_{n})

≤

{\tilde{a}}^{+}

.Proof: Similar with proof of Property 2, it is omitted here.Property 6. (Monotonicity)Let

{\tilde{a}}_{i}^{1} = 〈 [s_{θ ({a_{i}}^{1})}, s_{τ ({a_{i}}^{1})}], (u ({a_{i}}^{1}), v ({a_{i}}^{1})) 〉

{\tilde{a}}_{i}^{2} = 〈 [s_{θ ({a_{i}}^{2})}, s_{τ ({a_{i}}^{2})}], (u ({a_{i}}^{2}), v ({a_{i}}^{2})) 〉

be two groups of IULNs, if u (a_i¹) ≤ u (a_i²) , v (a_i¹) ≥ v (a_i²) , s_{θ(a_i¹)} ≤ s_{θ(a_i²)}, s_{τ(a_i¹)} ≤ s_{τ(a_i²)} for i = 1, 2, …, n, then

IULOW G_{W} ({\tilde{a}}_{1}^{1}, {\tilde{a}}_{2}^{1}, \dots, {\tilde{a}}_{n}^{1}) \leq IULOW G_{W} ({\tilde{a}}_{1}^{2}, {\tilde{a}}_{2}^{2}, \dots, {\tilde{a}}_{n}^{2})

.Proof: Similar with proof of Property 3, it is omitted here. In order to consider the important degree of input arguments and their ordered positions, the IULHG operator is defined as follows.Definition 16. [14] Let

{\tilde{a}}_{i}

= 〈 [s_{θ(a_i)}, s_{τ(a_i)}] , (u (a_i) , v (a_i)) 〉 (i = 1, 2, …, n) be a group of the IULNs, and IULHG: Ωⁿ → Ω, if

IULH G_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{i = 1}^{n} {({\tilde{b}}_{σ_{i}})}^{w_{i}}

(58) where Ω is the set of all IULNs, and W = (w₁, w₂, …, w_n) ^T is an associated weight vector with IULHG,

w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1

;

{\tilde{b}}_{σ_{i}}

is ith the largest of weighted argument

{\tilde{b}}_{k} (k = 1, 2, \dots, n)

{\tilde{b}}_{k} = {({\tilde{a}}_{k})}^{n ω_{k}} (k = 1, 2, \dots, n)

, ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector of

{\tilde{a}}_{i} (i = 1, 2, \dots, n)

ω_{i} \in [0, 1], \sum_{i = 1}^{n} ω_{i} = 1

, and n is the balancing coefficient, then IULHG is called the intuitionistic uncertain linguistic hybrid geometric operator.Theorem 4. Let

{\tilde{a}}_{i}

= 〈 [s_{θ(a_i)} , s_{τ(a_i)}], (u (a_i) , v (a_i)) 〉 (i = 1, 2, …, n) be a group of the IULNs, then the result aggregated from formula (58) is still an IULN, and even

\begin{matrix} IULH G_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = 〈 [f^{* - 1} (\prod_{i = 1}^{n} {(f^{*} (s_{θ (b_{σ_{i}})}))}^{w_{i}}), \\ f^{* - 1} (\prod_{i = 1}^{n} {(f^{*} (s_{τ (b_{σ_{i}})}))}^{w_{i}})], \\ (\prod_{i = 1}^{n} {(u (b_{σ_{i}}))}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - v (b_{σ_{i}}))}^{w_{i}}) 〉 . \end{matrix}

(59) From Theorems 2, 3 and 4, we know that the IULHG operator generalizes both the IULWGA and IULOWG operators by weighting the given importance and the ordered position of the arguments. However, there is a flaw that the proposed IULHG operator does not satisfy some desirable properties, such as boundedness and idempotency.Example 6. Assume

{\tilde{a}}_{1} = 〈 [s_{2}, s_{4}], (0.4, 0.3) 〉

, (0.4,0.3), (0.4,0.3) are three IULNs, whose weighting vector is ω = (1, 0, 0) ^{^T}, and the associated vector is also W = (1, 0, 0) ^T. Then,

\begin{matrix} {\tilde{b}}_{1} = {({\tilde{a}}_{1})}^{3 ω_{1}} = 〈 [s_{2,} s_{4}], (0.064, 0.657) 〉, \\ {\tilde{b}}_{2} = {({\tilde{a}}_{2})}^{3 ω_{2}} = 〈 [s_{2,} s_{4}], (1, 0) 〉, \\ {\tilde{b}}_{3} = {({\tilde{a}}_{3})}^{3 ω_{3}} = 〈 [s_{2,} s_{4}], (1, 0) 〉 . \end{matrix}

If t = 3 and

f (s_{i}) = θ_{i} = \frac{i}{6} (i = 0, 1, 2, \dots, 6)

, according to the formula (51) and Definitions 10 and 12,

E ({\tilde{b}}_{1}) = 0.10175

E ({\tilde{b}}_{2}) = 0.5

E ({\tilde{b}}_{3}) = 0.5

. Thus,

E ({\tilde{b}}_{1}) < E ({\tilde{b}}_{2}) = E ({\tilde{b}}_{3})

. By formula (59), we can get

IULH G_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, {\tilde{a}}_{3}) = 〈 [s_{2,} s_{4}], (1, 0) 〉 > 〈 [s_{2,} s_{4}], (0.4, 0.3) 〉 .

As we can see from the above example, IULHG operator does not satisfy some desirable properties. However, boundedness and idempotency are the most important properties for every aggregation operator, which are desirable for aggregating a finite collection of arguments. In the IULHG operator, we need to calculate

{\tilde{b}}_{k} = {({\tilde{a}}_{k})}^{n ω_{k}} (k = 1, 2, \dots, n)

, then rank

{\tilde{b}}_{k}

. Obviously, this process is also relatively complex, and it is also the reason which leads to these shortcomings. Thus, it is natural to develop a new HG operator that can not only weigh the importance of each argument and its ordered position but also keep these desirable properties. Therefore, a new hybrid weighted geometric operator for IULNs (IULHWG) can be generated.Definition 17. Let

{\tilde{a}}_{i}

= 〈 [s_{θ(a_i)} , s_{τ(a_i)}], (u (a_i) , v (a_i)) 〉 (i = 1, 2, …, n) be a group of the IULNs, and IULHWG: Ωⁿ → Ω, if

IULHW G_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{i = 1}^{n} {({\tilde{a}}_{i})}^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}

(60) where Ω is the set of all IULNs, and W = (w₁, w₂, …, w_n) ^T is an associated weight vector with IULHWG,

w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1

; ɛ:{ 1, 2, …, n } → { 1, 2, …, n } is the permutation such that

{\tilde{a}}_{i}

is the ɛ (i) th largest element of

{\tilde{a}}_{i} (i = 1, 2, \dots, n)

, and ω = (ω₁, ω₂, …, ω_n) ^T is the weighting vector of the arguments

{\tilde{a}}_{i} (i = 1, 2, \dots, n)

, with

ω_{i} \in [0, 1], \sum_{i = 1}^{n} ω_{i} = 1

.Theorem 5. Let

{\tilde{a}}_{i} = 〈 [s_{θ (a_{i})}, s_{τ (a_{i})}], (u (a_{i}),

v (a_i)) 〉 (i = 1, 2, …, n) be a group of the IULNs, then, the result aggregated from formula (60) is still an IULN, and even

\begin{matrix} {IULHWG}_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = 〈 [f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{θ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}), \\ f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{τ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}})], \\ (\prod_{i = 1}^{n} (u (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}, \\ 1 - \prod_{i = 1}^{n} (1 - v (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}) 〉 . \end{matrix}

(61)

Proof: According to the Definition 17, it is obvious that the aggregated value with IULHWG operator is also an IULN. According to the formula (40), we can get

\begin{matrix} ({\tilde{a}}_{i})^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}} \\ = 〈 [f^{* - 1} ((f^{*} (s_{θ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}), \\ f^{* - 1} ((f^{*} (s_{τ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}})], \\ ((u (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}, 1 - (1 - v (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}) 〉 . \end{matrix}

Then,

\begin{matrix} {IULHWG}_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \prod_{i = 1}^{n} ({\tilde{a}}_{i})^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}} \\ = \prod_{i = 1}^{n} 〈 [f^{* - 1} ((f^{*} (s_{θ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}), \\ f^{* - 1} ((f^{*} (s_{τ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}})], \\ ((u (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}, 1 - (1 - v (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}) 〉 \\ = 〈 [f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{θ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}), \\ f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{τ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}})], \\ (\prod_{i = 1}^{n} (u (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}, \\ 1 - \prod_{i = 1}^{n} (1 - (1 - (1 - v (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}))) 〉 \\ = 〈 [f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{θ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}), \\ f^{* - 1} (\prod_{i = 1}^{n} (f^{*} (s_{τ (a_{i})}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}})], \\ (\prod_{i = 1}^{n} (u (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}, \\ 1 - \prod_{i = 1}^{n} (1 - v (a_{i}))^{\frac{w_{ɛ (i)} ω_{i}}{\sum_{i = 1}^{n} w_{ɛ (i)} ω_{i}}}) 〉 . \end{matrix}

Example 7. Let

{\tilde{a}}_{1} = 〈 [s_{2}, s_{4}], (0.2, 0.4) 〉

{\tilde{a}}_{2} = 〈 [s_{2}, s_{4}], (0.2, 0.6) 〉

and

{\tilde{a}}_{3} = 〈 [s_{2}, s_{4}], (0.7, 0.1) 〉

be three IULNs, whose weighting vector is ω = (0.15, 0.3, 0.55) ^{^T}, and the aggregation associated vector is W = (0.3, 0.4, 0.3) ^T. If

f (s_{i}) = θ_{i} = \frac{i}{6} (i = 0, 1, 2, \dots, 6)

, according to the formula (51) and Definitions 10 and 12, we obtain

E ({\tilde{a}}_{1}) = 0.2, E ({\tilde{a}}_{2}) = 0.15, E ({\tilde{a}}_{3}) = 0.4,

and

E ({\tilde{a}}_{3}) > E ({\tilde{a}}_{1}) > E ({\tilde{a}}_{2})

, which implies

{\tilde{a}}_{3} > {\tilde{a}}_{1} > {\tilde{a}}_{2}

. Hence, ɛ (1) = 2, ɛ (2) = 3, ɛ (3) = 1. Then,

\begin{matrix} w_{ɛ (1)} ω_{1} / - \sum_{i = 1}^{3} w_{ɛ (j)} ω_{j} = 0.19, \\ w_{ɛ (2)} ω_{2} / - \sum_{i = 1}^{3} w_{ɛ (j)} ω_{j} = 0.286, \\ w_{ɛ (3)} ω_{3} / - \sum_{i = 1}^{3} w_{ɛ (j)} ω_{j} = 0.524 . \end{matrix} .

According to the formula (61), we can obtain

\begin{matrix} IULHW G_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, {\tilde{a}}_{3}) \\ = 〈 [s_{2}, s_{4}], (0.3856, 0.3392) 〉 . \end{matrix}

Obviously, if the associated weighting vector W = (1/ - n, 1/ - n, …, 1/ - n) ^T, then the IULHWG operator reduces to the IULWGA operator; if ω = (1/ - n, 1/ - n, …, 1/ - n) ^T, then the IULHWG operator reduces to the IULOWG operator. So the IULHWG operator is a generalization of the IULWGA operator and the IULOWG operator. The IULHWG operator satisfies the following properties:Property 7. (Idempotency). Let

{\tilde{a}}_{i} = \tilde{a}

for all i = 1, 2, …, n, then

{IULHWG}_{W, ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}

.Proof: Similar with proof of Property 1, it is omitted here.Property 8. (Boundedness)Let

{\tilde{a}}^{-} = 〈 [s_{min θ (a_{i})}, s_{min τ (a_{i})}], (min_{i} {u (a_{i})}

max_{i} {v (a_{i})}) 〉

{\tilde{a}}^{+} = 〈 [s_{max θ (a_{i})}, s_{max τ (a_{i})}], (max_{i}

{u (a_{i})}, min_{i} {v (a_{i})}) 〉,

then

{\tilde{a}}^{-} \leq IULHW G_{W, ω}

({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}

.Proof: Similar with proof of Property 2, it is omitted here.Property 9. (Monotonicity)Let

{\tilde{a}}_{i}^{1} = 〈 [s_{θ ({a_{i}}^{1})}, s_{τ ({a_{i}}^{1})}], (u ({a_{i}}^{1}), v ({a_{i}}^{1})) 〉

{\tilde{a}}_{i}^{2} = 〈 [s_{θ ({a_{i}}^{2})}, s_{τ ({a_{i}}^{2})}], (u ({a_{i}}^{2}), v ({a_{i}}^{2})) 〉

be two groups of IULNs, if u (a_i¹) ≤ u (a_i²) , v (a_i¹) ≥ v (a_i²) , s_{θ(a_i¹)} ≤ s_{θ(a_i²)}, s_{τ(a_i¹)} ≤ s_{τ(a_i²)} for i = 1, 2, …, n, then

\begin{matrix} IULHW G_{W, ω} ({\tilde{a}}_{1}^{1}, {\tilde{a}}_{2}^{1}, \dots, {\tilde{a}}_{n}^{1}) \\ \leq IULHW G_{W, ω} ({\tilde{a}}_{1}^{2}, {\tilde{a}}_{2}^{2}, \dots, {\tilde{a}}_{n}^{2}) . \end{matrix}

Proof: Similar with proof of Property 3, it is omitted here. Since the IULHWG operator can not only weigh both the given arguments and their ordered positions simultaneously but also keeps those ideal properties, such as idempotency, boundedness and monotonicity, it is more powerful and efficient in fusing IULNs. Thus, the proposed IULHWG operator has more wide applications in the practical decision-making process.

Table 1

Decision matrix ${\tilde{R}}^{1}$

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s₅, s₅] , (0.2, 0.7)〉	〈 [s₂, s₃] , (0.4, 0.6)〉	〈 [s₅, s₆] , (0.5, 0.5)〉	〈 [s₃, s₄] , (0.2, 0.6)〉
X ₂	〈 [s₄, s₅] , (0.4, 0.6)〉	〈 [s₅, s₅] , (0.4, 0.5)〉	〈 [s₃, s₄] , (0.1, 0.8)〉	〈 [s₄, s₄] , (0.5, 0.5)〉
X ₃	〈 [s₃, s₄] , (0.2, 0.7)〉	〈 [s₄, s₄] , (0.2, 0.7)〉	〈 [s₄, s₅] , (0.3, 0.7)〉	〈 [s₄, s₅] , (0.2, 0.7)〉
X ₄	〈 [s₆, s₆] , (0.5, 0.4)〉	〈 [s₂, s₃] , (0.2, 0.8)〉	〈 [s₃, s₄] , (0.2, 0.6)〉	〈 [s₃, s₃] , (0.3, 0.6)〉

Table 2

Decision matrix ${\tilde{R}}^{2}$

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s₄, s₄] , (0.1, 0.7)〉	〈 [s₃, s₄] , (0.2, 0.7)〉	〈 [s₃, s₄] , (0.2, 0.8)〉	〈 [s₆, s₆] , (0.4, 0.5)〉
X ₂	〈 [s₅, s₆] , (0.4, 0.5)〉	〈 [s₃, s₄] , (0.3, 0.6)〉	〈 [s₄, s₅] , (0.2, 0.6)〉	〈 [s₃, s₄] , (0.2, 0.7)〉
X ₃	〈 [s₄, s₅] , (0.2, 0.6)〉	〈 [s₄, s₄] , (0.2, 0.7)〉	〈 [s₂, s₃] , (0.4, 0.6)〉	〈 [s₃, s₄] , (0.3, 0.7)〉
X ₄	〈 [s₅, s₅] , (0.3, 0.6)〉	〈 [s₄, s₅] , (0.4, 0.5)〉	〈 [s₂, s₃] , (0.3, 0.6)〉	〈 [s₄, s₄] , (0.2, 0.6)〉

Table 3

Decision matrix ${\tilde{R}}^{3}$

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s₅, s₅] , (0.2, 0.6)〉	〈 [s₃, s₄] , (0.3, 0.7)〉	〈 [s₄, s₅] , (0.4, 0.5)〉	〈 [s₄, s₄] , (0.2, 0.7)〉
X ₂	〈 [s₄, s₅] , (0.3, 0.7)〉	〈 [s₅, s₅] , (0.3, 0.6)〉	〈 [s₂, s₃] , (0.1, 0.8)〉	〈 [s₃, s₄] , (0.4, 0.6)〉
X ₃	〈 [s₄, s₄] , (0.2, 0.7)〉	〈 [s₅, s₅] , (0.3, 0.6)〉	〈 [s₁, s₃] , (0.1, 0.8)〉	〈 [s₄, s₄] , (0.2, 0.7)〉
X ₄	〈 [s₃, s₄] , (0.2, 0.7)〉	〈 [s₃, s₄] , (0.1, 0.7)〉	〈 [s₄, s₅] , (0.3, 0.6)〉	〈 [s₅, s₅] , (0.4, 0.5)〉

4 An approach to group decision making based on the IULNs

Consider a MAGDM with the information expressed by IULNs: there are m alternatives X = {X₁, X₂, …, X_m } and n attributes C = {C₁, C₂, …, C_n }, ω = (ω₁, ω₂, …, ω_n) ^T is the weighting vector of the attribute C_j (j = 1, 2, …, n), where ω_j ≥ 0, j = 1, 2, …, n, $\sum_{j = 1}^{n}$ ω_j = 1. Let D = {D₁, D₂, …, D_p } be the set of DMs, and μ = (μ₁, μ₂, …, μ_p) ^T be their weight vector, where μ_k ≥ 0, k = 1, 2, …, p, $\sum_{k = 1}^{p}$ μ_k = 1. Suppose that ${\tilde{R}}^{k}$ = ${[{\tilde{r}}_{ij}^{k}]}_{m \times n}$ is the decision matrix, where ${\tilde{R}}_{ij}^{k}$ = $〈 [a_{ijk}^{L}$ , $a_{ijk}^{U}]$ , (u_ijk, v_ijk) 〉 takes the form of the IULNs, given by the DM D_k for alternative X_i with respect to attribute C_j, and, $a_{ijk}^{L}$ ≤ $a_{ijk}^{U}$ , $a_{ijk}^{L}$ , $a_{ijk}^{U}$ ∈ S. Then, the ranking of alternatives is required [24].

In the following, we apply the IULHWG operator to solve this MAGDM problem. The method involves the following steps:

Step 1. Utilize the IULHWG operator $\begin{matrix} {\tilde{r}}_{ij} = IULHW G_{W, μ} ({\tilde{r}}_{ij}^{1}, {\tilde{r}}_{ij}^{2}, \dots, {\tilde{r}}_{ij}^{p}) \\ = \prod_{k = 1}^{p} {({\tilde{r}}_{ij}^{k})}^{\frac{w_{ɛ (k)} μ_{k}}{\sum_{k = 1}^{p} w_{ɛ (k)} μ_{k}}}, \\ i = 1, 2, \dots, m; j = 1, 2, \dots, n . \end{matrix}$

to aggregate all the decision matrices and get a group decision matrix $\tilde{R} = {[{\tilde{r}}_{ij}]}_{m \times n}$ for alternative X_i with respect to criteria C_j.

Step 2. Utilize the IULHWG operator $\begin{matrix} {\tilde{r}}_{i} = IULHW G_{W, ω} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = \prod_{j = 1}^{n} {({\tilde{r}}_{ij})}^{\frac{w_{ɛ (j)} ω_{j}}{\sum_{j = 1}^{n} w_{ɛ (j)} ω_{j}}}, \\ i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ to get the comprehensive group attribute value ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternatives.

Step 3. Utilize expectation function and accuracy function of the IULNs to calculate the expected values $E ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ and the accuracy degrees $H ({\tilde{r}}_{i})$ of the comprehensive group attribute value ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ .

Step 4. According to the Definition 12 to rank all the alternatives X_i (i = 1, 2, …, m) and select the best one(s).

Step 5. End.

Table 4
The collective decision matrix $\tilde{R}$

C ₁ C ₂ C ₃ C ₄

X ₁ 〈 [s_4.80, s_4.80] , (0.176, 0.652)〉〈 [s_2.65, s_3.67] , (0.304, 0.673)〉〈 [s_4.41, s_5.43] , (0.414, 0.569)〉〈 [s_4.62, s_5.02] , (0.295, 0.566)〉

X ₂ 〈 [s_4.53, s_5.54] , (0.383, 0.566)〉〈 [s_4.60, s_4.82] , (0.362, 0.538)〉〈 [s_3.31, s_4.34] , (0.147, 0.705)〉〈 [s_3.62, s_4.28] , (0.381, 0.584)〉

X ₃ 〈 [s_3.75, s_4.54] , (0.20, 0.646)〉〈 [s_4.49, s_4.49] , (0.246, 0.652)〉〈 [s_2.84, s_4.17] , (0.274, 0.699)〉〈 [s_3.76, s_4.62] , (0.218, 0.700)〉

X ₄ 〈 [s_5.24, s_5.45] , (0.394, 0.501)〉〈 [s_3.23, s_4.26] , (0.258, 0.633)〉〈 [s_3.23, s_4.26] , (0.265, 0.600)〉〈 [s_4.20, s_4.20] , (0.315, 0.551)〉

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s_4.80, s_4.80] , (0.176, 0.652)〉	〈 [s_2.65, s_3.67] , (0.304, 0.673)〉	〈 [s_4.41, s_5.43] , (0.414, 0.569)〉	〈 [s_4.62, s_5.02] , (0.295, 0.566)〉
X ₂	〈 [s_4.53, s_5.54] , (0.383, 0.566)〉	〈 [s_4.60, s_4.82] , (0.362, 0.538)〉	〈 [s_3.31, s_4.34] , (0.147, 0.705)〉	〈 [s_3.62, s_4.28] , (0.381, 0.584)〉
X ₃	〈 [s_3.75, s_4.54] , (0.20, 0.646)〉	〈 [s_4.49, s_4.49] , (0.246, 0.652)〉	〈 [s_2.84, s_4.17] , (0.274, 0.699)〉	〈 [s_3.76, s_4.62] , (0.218, 0.700)〉
X ₄	〈 [s_5.24, s_5.45] , (0.394, 0.501)〉	〈 [s_3.23, s_4.26] , (0.258, 0.633)〉	〈 [s_3.23, s_4.26] , (0.265, 0.600)〉	〈 [s_4.20, s_4.20] , (0.315, 0.551)〉

5 Example

Let us suppose a university wants to expand a campus in a certain area [40]. Now there are four areas X₁, X₂, X₃ and X₄ as alternatives, respectively. The university evaluates these areas from the following four attributes (suppose that the weight vector of four attributes is ω = (0.32, 0.26, 0.18, 0.24) ^T): (1) C₁ is the traffic network layout; (2) C₂ is the local government support; (3) C₃ is the surrounding educational environment and resources; (4) C₄ is the economic development. The four possible alternatives {X₁, X₂, X₃, X₄} are evaluated using the LTS S = (s₀, s₁, s₂, s₃, s₄, s₅, s₆) by three DMs (whose weight vector μ = (0.4, 0.32, 0.28) ^T) under the above four attributes, and the decision matrices ${\tilde{R}}^{k} = {[{\tilde{r}}_{ij}^{k}]}_{4 \times 4} (k = 1, 2, 3)$ are built and listed in Tables 1–3.

Based on the method proposed in Section 4, let f^* (s_i) = i/ - 6 (i = 1, 2, …, 6), the steps are shown as follows:

Step 1. According to the decision information in matrix ${\tilde{R}}^{k} (k = 1, 2, 3)$ and the IULHWG operator $\begin{matrix} {\tilde{r}}_{ij} = IULHW G_{W, μ} ({\tilde{r}}_{i 1}^{1}, {\tilde{r}}_{i 2}^{2}, {\tilde{r}}_{in}^{3}) \\ = \prod_{k = 1}^{3} {({\tilde{r}}_{ij}^{k})}^{\frac{w_{ɛ (k)} μ_{k}}{\sum_{k = 1}^{3} w_{ɛ (k)} μ_{k}}}, \\ i = 1, 2, 3, 4; j = 1, 2, 3, 4 \end{matrix}$ to obtain a collective decision matrix $\tilde{R} = {[{\tilde{r}}_{ij}]}_{4 \times 4}$ , in which the weighted vector W can be determined by using formula (56), and get W = (0.58, 0.24, 0.18) ^T.

The results are shown in Table 4.

Step 2. According to the weight vector of attributes ω = (0.32, 0.26, 0.18, 0.24) ^T and the IULHWG operator $\begin{matrix} {\tilde{r}}_{i} & = & IULHW G_{W, ω} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = & \prod_{j = 1}^{n} {({\tilde{r}}_{ij})}^{\frac{w_{ɛ (j)} ω_{j}}{\sum_{j = 1}^{n} w_{ɛ (j)} ω_{j}}} \end{matrix}$ to obtain the comprehensive group attribute value ${\tilde{r}}_{i} (i = 1, 2, 3, 4)$ of the alternative X_i, in which the weighted vector W can be determined by using formula (56), and get W = (0.5, 0.21, 0.16, 0.13) ^T.

By the formula (61), we can get the following results. $\begin{matrix} {\tilde{r}}_{1} = 〈 [s_{4.2104}, s_{4.8908}], (0.3016, 0.3990) 〉, \\ {\tilde{r}}_{2} = 〈 [s_{4.2903}, s_{5.0914}], (0.3490, 0.4266) 〉, \\ {\tilde{r}}_{3} = 〈 [s_{4.0037}, s_{4.4912}], (0.2314, 0.3383) 〉, \\ {\tilde{r}}_{4} = 〈 [s_{4.5013}, s_{4.9107}], (0.3442, 0.4653) 〉 . \end{matrix}$

Step 3. Calculate the expected values $E ({\tilde{r}}_{i}) (i = 1, 2, 3, 4)$ of the collective overall IULNs ${\tilde{r}}_{i} (i = 1, 2, 3, 4)$ as follows: $\begin{matrix} E ({\tilde{r}}_{1}) = 0.3423, \\ E ({\tilde{r}}_{2}) = 0.3606, \\ E ({\tilde{r}}_{3}) = 0.3161, \\ E ({\tilde{r}}_{4}) = 0.3446 . \end{matrix}$

Step 4. Rank all the alternatives X_i (i = 1, 2, 3, 4) in accordance with $E ({\tilde{r}}_{i}) (i = 1, 2, 3, 4)$ of the collective overall IULNs ${\tilde{r}}_{i} (i = 1, 2, 3, 4)$ , and get X₂ ≻ X₄ ≻ X₁ ≻ X₃ and thus the most desirable alternative is X₂.

Table 5
Comparison with different methods

Method Whether the Ranking order

operations

are closed

The proposed method yes X₂ ≻ X₄ ≻ X₁ ≻ X₃

in this paper

The method in [20] no X₂ ≻ X₄ ≻ X₁ ≻ X₃

The method In [12] no X₂ ≻ X₄ ≻ X₁ ≻ X₃

Method	Whether the	Ranking order
The proposed method	yes	X₂ ≻ X₄ ≻ X₁ ≻ X₃
in this paper
The method in [20]	no	X₂ ≻ X₄ ≻ X₁ ≻ X₃
The method In [12]	no	X₂ ≻ X₄ ≻ X₁ ≻ X₃

To further prove the effectiveness of the developed method in this paper, we solve the same illustrative example by using the two existing MAGDM methods including IULHA operator proposed by Liu and Zhang [20] and IULHG operator proposed by Liu and Jin [12]. The ranking results are shown in Table 5.

From the Table 5, we can see that all methods in [12, 20] and in this paper produced the same ranking results. Obviously, this verifies the validity of the proposed method in this paper.

Next, we prove the advantages of the proposed method in this paper by comparing the following cases with three different methods. Suppose three the decision matrices ${\tilde{R}}^{k} = {[{\tilde{r}}_{ij}^{k}]}_{4 \times 4} (k = 1, 2, 3)$ are shown in Tables 6–8. The final ranking results are shown in the Table 12.

Table 6

New decision matrix ${\tilde{R}}^{1}$

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s₅, s₅] , (0.2, 0.7)〉	〈 [s₂, s₃] , (0.4, 0.6)〉	〈 [s₅, s₆] , (0.5, 0.5)〉	〈 [s₃, s₄] , (0.2, 0.6)〉
X ₂	〈 [s₄, s₅] , (0.4, 0.6)〉	〈 [s₅, s₅] , (0.4, 0.5)〉	〈 [s₃, s₄] , (0.1, 0.8)〉	〈 [s₄, s₄] , (0.5, 0.5)〉
X ₃	〈 [s₃, s₄] , (0.2, 0.7)〉	〈 [s₄, s₄] , (0.2, 0.7)〉	〈 [s₄, s₅] , (0.3, 0.7)〉	〈 [s₄, s₅] , (0.2, 0.7)〉
X ₄	〈 [s₆, s₆] , (0.5, 0.4)〉	〈 [s₂, s₃] , (0.2, 0.8)〉	〈 [s₃, s₄] , (0.2, 0.6)〉	〈 [s₃, s₃] , (0.3, 0.6)〉

Table 7

New decision matrix ${\tilde{R}}^{2}$

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s₅, s₅] , (0.2, 0.7)〉	〈 [s₂, s₃] , (0.4, 0.6)〉	〈 [s₅, s₆] , (0.5, 0.5)〉	〈 [s₃, s₄] , (0.2, 0.6)〉
X ₂	〈 [s₅, s₆] , (0.4, 0.5)〉	〈 [s₃, s₄] , (0.3, 0.6)〉	〈 [s₄, s₅] , (0.2, 0.6)〉	〈 [s₃, s₄] , (0.2, 0.7)〉
X ₃	〈 [s₄, s₅] , (0.2, 0.6)〉	〈 [s₄, s₄] , (0.2, 0.7)〉	〈 [s₂, s₃] , (0.4, 0.6)〉	〈 [s₃, s₄] , (0.3, 0.7)〉
X ₄	〈 [s₅, s₅] , (0.3, 0.6)〉	〈 [s₄, s₅] , (0.4, 0.5)〉	〈 [s₂, s₃] , (0.3, 0.6)〉	〈 [s₄, s₄] , (0.2, 0.6)〉

Table 8

New decision matrix ${\tilde{R}}^{3}$

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s₅, s₅] , (0.2, 0.7)〉	〈 [s₂, s₃] , (0.4, 0.6)〉	〈 [s₅, s₆] , (0.5, 0.5)〉	〈 [s₃, s₄] , (0.2, 0.6)〉
X ₂	〈 [s₄, s₅] , (0.3, 0.7)〉	〈 [s₅, s₅] , (0.3, 0.6)〉	〈 [s₂, s₃] , (0.1, 0.8)〉	〈 [s₃, s₄] , (0.4, 0.6)〉
X ₃	〈 [s₄, s₄] , (0.2, 0.7)〉	〈 [s₅, s₅] , (0.3, 0.6)〉	〈 [s₁, s₃] , (0.1, 0.8)〉	〈 [s₄, s₄] , (0.2, 0.7)〉
X ₄	〈 [s₃, s₄] , (0.2, 0.7)〉	〈 [s₃, s₄] , (0.1, 0.7)〉	〈 [s₄, s₅] , (0.3, 0.6)〉	〈 [s₅, s₅] , (0.4, 0.5)〉

Table 9

The comprehensive attribute value ${\tilde{r}}_{i}^{k}$

DM1	DM2	DM3
X ₁	〈 [s_3.74, s_4.42] , (0.32, 0.61)〉	〈 [s_3.74, s_4.42] , (0.32, 0.61)〉	〈 [s_3.74, s_4.42] , (0.32, 0.61)〉
X ₂	〈 [s_4.08, s_4.58] , (0.38, 0.58)〉	〈 [s_3.82, s_4.82] , (0.30, 0.59)〉	〈 [s_3.66, s_4.40] , (0.29, 0.66)〉
X ₃	〈 [s_3.68, s_4.42] , (0.22, 0.70)〉	〈 [s_3.40, s_4.14] , (0.26, 0.65)〉	〈 [s_3.72, s_4.08] , (0.21, 0.69)〉
X ₄	〈 [s_3.70, s_4.14] , (0.33, 0.57)〉	〈 [s_3.96, s_4..40] , (0.31, 0.57)〉	〈 [s_3.66, s_4.16] , (0.25, 0.63)〉

Table 10

The comprehensive attribute value ${\tilde{r}}_{i}^{k}$

DM1	DM2	DM3
X ₁	〈 [s_3.49, s_4.29] , (0.28, 0.62)〉	〈 [s_3.49, s_4.29] , (0.28, 0.62)〉	〈 [s_3.49, s_4.29] , (0.28, 0.62)〉
X ₂	〈 [s_4.03, s_4.55] , (0.33, 0.61)〉	〈 [s_3.72, s_4.74] , (0.28, 0.60)〉	〈 [s_3.49, s_4.32] , (0.26, 0.68)〉
X ₃	〈 [s_3.85, s_4.39] , (0.22, 0.70)〉	〈 [s_3.30, s_4.08] , (0.25, 0.65)〉	〈 [s_3.30, s_4.03] , (0.20, 0.70)〉
X ₄	〈 [s_3.37, s_3.94] , (0.30, 0.62)〉	〈 [s_3.79, s_4..32] , (0.29, 0.58)〉	〈 [s_3.57, s_4.08] , (0.21, 0.64)〉

(1) Ranking the alternatives by the method in [20] based on the IULWAA and IULHA operators.

By the first step in [20], we can get the following result in the Table 9 (suppose the attribute weight is ω = (0.32, 0.26, 0.18, 0.24) ^T).

Then, utilize the IULHA operator to get the collective overall attribute values ${\tilde{r}}_{i}$ of alternative X_i (suppose the position weight and DMs weight are W = (0.25, 0.50, 0.25) ^T and μ = (0.4, 0.32, 0.28) ^T, respectively). The results are shown as follows: $\begin{matrix} {\tilde{r}}_{1} = 〈 [s_{3.70}, s_{4.38}], (0.32, 0.61) 〉, \\ {\tilde{r}}_{2} = 〈 [s_{3.83}, s_{4.61}], (0.32, 0.60) 〉, \\ {\tilde{r}}_{3} = 〈 [s_{3.52}, s_{4.17}], (0.24, 0.67) 〉, \\ {\tilde{r}}_{4} = 〈 [s_{3.78}, s_{4.23}], (0.30, 0.59) 〉 . \end{matrix}$

Finally, calculate the expected value of ${\tilde{r}}_{i}$ , we can get the following results.

$\begin{matrix} E ({\tilde{r}}_{1}) = s_{1.4342}, \\ E ({\tilde{r}}_{2}) = s_{1.5192}, \\ E ({\tilde{r}}_{3}) = s_{1.0958}, \\ E ({\tilde{r}}_{4}) = s_{1.4218} . \end{matrix}$

Then, we can get X₂ ≻ X₁ ≻ X₄ ≻ X₃.

Table 11

The collective decision matrix $\tilde{R}$

C ₁	C ₂	C ₃	C ₄
X ₁	〈 [s₅, s₅] , (0.2, 0.7)〉	〈 [s₂, s_3.] , (0.4, 0.6)〉	〈 [s₅, s₆] , (0.5, 0.5)〉	〈 [s₃, s₄] , (0.2, 0.6)〉
X ₂	〈 [s_4.53, s_5.54] , (0.383, 0.566)〉	〈 [s_4.60, s_4.82] , (0.362, 0.538)〉	〈 [s_3.31, s_4.34] , (0.147, 0.705)〉	〈 [s_3.6171, s_4.2758] , (0.3812, 0.5843)〉
X ₃	〈 [s_3.75, s_4.54] , (0.200, 0.646)〉	〈 [s_4.49, s_4.49] , (0.246, 0.652)〉	〈 [s_2.84, s_4.17] , (0.274, 0.699)〉	〈 [s_3.7614, s_4.6201] , (0.2181, 0.70)〉
X ₄	〈 [s_5.24, s_5.45] , (0.394, 0.501)〉	〈 [s_3.23, s_4.26] , (0.258, 0.633)〉	〈 [s_3.23, s_4.26] , (0.265, 0.60)〉	〈 [s_4.2045, s_4.2045] , (0.3154, 0.5506)〉

(2) Ranking the alternatives by the method in [12] based on the IULWGA and IULHG operators.

By the first step in [12], we can obtain the aggregation results shown in Table 10 (suppose the attribute weight is ω = (0.32, 0.26, 0.18, 0.24) ^T).

Next, utilize the IULHG operator to get the collective comprehensive attribute values ${\tilde{r}}_{i}$ of alternative X_i (suppose the position weight and DMs weight are W = (0.25, 0.50, 0.25) ^T and μ = (0.4, 0.32, 0.28) ^T, respectively). The results are shown as follows: $\begin{matrix} {\tilde{r}}_{1} = 〈 [s_{3.38}, s_{4.15}], (0.36, 0.55) 〉, \\ {\tilde{r}}_{2} = 〈 [s_{3.71}, s_{4.52}], (0.29, 0.62) 〉, \\ {\tilde{r}}_{3} = 〈 [s_{3.84}, s_{4.56}], (0.20, 0.71) 〉, \\ {\tilde{r}}_{4} = 〈 [s_{3.73}, s_{4.35}], (0.26, 0.63) 〉 . \end{matrix}$

Then, calculate the expected value of ${\tilde{r}}_{i}$ , we can get the following results. $\begin{matrix} E ({\tilde{r}}_{1}) = s_{1.5248}, \\ E ({\tilde{r}}_{2}) = s_{1.3785}, \\ E ({\tilde{r}}_{3}) = s_{1.029}, \\ E ({\tilde{r}}_{4}) = s_{1.2726} . \end{matrix}$

Thus, we can get X₁ ≻ X₂ ≻ X₄ ≻ X₃.

(3) Ranking the alternatives by the proposed method in this paper based on the IULHWG operator.

Now, we use the proposed method in this paper to calculate this case by IULHWG operator. By the first step, we can get the collective decision matrix $\tilde{R}$ shown in Table 11.

Next, use the IULHWG operator to obtain the comprehensive group attribute value ${\tilde{r}}_{i} (i = 1, 2, 3, 4)$ of the alternative X_i, according to the weight vector of attributes ω = (0.32, 0.26, 0.18, 0.24) ^T and W = (0.5, 0.21, 0.16, 0.13) ^T. The results are shown as follows: $\begin{matrix} {\tilde{r}}_{1} = 〈 [s_{4.0103}, s_{4.7985}], (0.3173, 0.5987) 〉, \\ {\tilde{r}}_{2} = 〈 [s_{4.2903}, s_{5.0914}], (0.3490, 0.4266) 〉, \\ {\tilde{r}}_{3} = 〈 [s_{4.0037}, s_{4.4912}], (0.2314, 0.3383) 〉, \\ {\tilde{r}}_{4} = 〈 [s_{4.5013}, s_{4.9107}], (0.3442, 0.4653) 〉 . \end{matrix}$

Then, $E ({\tilde{r}}_{1}) = 0.2638$ , $E ({\tilde{r}}_{2}) = 0.3606$ , $E ({\tilde{r}}_{3}) = 0.3161$ , $E ({\tilde{r}}_{4}) = 0.3446$ .

Thus, we can get X₂ ≻ X₄ ≻ X₃ ≻ X₁.

Table 12

Comparison with different methods for new example

Method	Whether the	Ranking order
operations
are closed
The proposed method	Yes	X₂ ≻ X₄ ≻ X₃ ≻ X₁
in this paper
The method in [20]	No	X₂ ≻ X₁ ≻ X₄ ≻ X₃
The method In [12]	No	X₁ ≻ X₂ ≻ X₄ ≻ X₃

The ranking results from three methods are shown in Table 12.

As we can see from Table 12, the results obtained by the three methods are different. The results produced by the methods in [12, 20] have a discrepancy with reality. Through the first step aggregated operation, from the Tables 9 and 10, we can see that three experts have the same evaluation values for alternative X₁. Therefore, according to the actual situation, the comprehensive attribute value of the alternative X₁ by aggregating the information from the three DMs should still be the IULV itself. However, comprehensive group attribute value ${\tilde{r}}_{1}$ of the alternative X₁ obtained by IULHA and IULHG operators is not equal to the evaluation value for alternative X₁. This is due to the fact that IULHA and IULHG operators is not satisfied with idempotency. Therefore, these two methods in [12, 20] do not fit the actual situation. However, IULHWG operator proposed in this paper satisfies idempotency, and it can be seen from the Table 11 that the result after aggregation by IULHWG operator is in line with the actual situation. So, our method is effective and feasible.

In summary, we can give the advantages of the proposed method in this paper by comparing with the methods in [12, 20].

(1) Compared with Liu and Zhang’s method [20] based on IULWAA and IULHA operators.

We can see that there are some shortcomings in the method proposed by Liu and Zhang [20]. Firstly, one of the weaknesses is that operational rules of IULVs in [20] are directly based on the subscript of LTs and gave a simple transformation from LTs to real numbers, which cannot properly maintain the original vagueness of the evaluation information. By the example 1, we can see that its calculation results exceed the upper limit of S. However, the proposed method in this paper developed a new operational rules of IULVs based on LSFs, which effectively avoid cross-border situation. Secondly, by IULHA operator in [20], we need to calculate ${\tilde{b}}_{k} = n ω_{k} {\tilde{a}}_{k} (k = 1, 2, \dots, n)$ , then to rank ${\tilde{b}}_{k}$ . Obviously, this process is relatively complex. In contrast, based on the proposed IULHWG operator in this paper, because of ω_j and w_ɛ(j) are crisp number, we only need to calculate this formula of $\frac{w_{ɛ (j)} ω_{j}}{\sum_{j = 1}^{n} w_{ɛ (j)} ω_{j}}$ . Obviously, the proposed approach is relatively simple. In addition, the IULHA operator in [20] does not meet idempotency, boundedness and monotonicity. However, these are the most basic properties. In this paper, the new IULHWG operator proposed in this paper can not only weigh the importance of each argument and its ordered position but also maintain these desirable properties. Therefore, our approach deal with the MAGDM problems more reasonably.

(2) Compared with Liu and Jin’s approach in [12] based on IULWGA and IULHG operators

We can also see that the method proposed by Liu and Jin [12] has also some drawbacks. Similar to Liu and Zhang’s method [20], this method based on IULWGA and IULHG operators also have three drawbacks mentioned above. Firstly, one of the weaknesses is that the operational rules of IULVs in [12] are directly based on the subscript of LTs, and can produce cross-border situation. Secondly, in the IULHG operator, we need to calculate ${\tilde{b}}_{k} = {({\tilde{a}}_{k})}^{n ω_{k}} (k = 1, 2, \dots, n)$ , then to rank ${\tilde{b}}_{k}$ . Obviously, this process is also relatively complex. However, in this paper, by calculating $\frac{w_{ɛ (j)} ω_{j}}{\sum_{j = 1}^{n} w_{ɛ (j)} ω_{j}}$ , we can get the result in a relatively simple way. Lastly, idempotency, boundedness and monotonicity are the most important properties, however, IULHG operator do not meet these basic properties. Thus, the IULHWG operator proposed in this paper is more reasonable and reliable.

6 Conclusion

The traditional aggregation operators are generally suitable for aggregating the information taking the form of crisp values, and yet they fail in dealing with IULVs. In this paper, with respect to MAGDM problems in which the attribute values take the form of IULVs, a new group decision making analysis method is developed. Firstly, we define new operational laws of IULVs based on LSFs. New operational rules have good closure and can solve the cross-border phenomenon of existing operations. At the same time, the expected function and accuracy function of IULVs are developed. Then, an IULWGA operator, an IULOWG operator and an IULHG operator are introduced. The IULHG operator can weigh the importance and the ordered position of the given arguments simultaneously. However, there is a flaw that the IULHG operator do not satisfy some desirable properties, such as idempotency and boundednes. Furthermore, new intuitionistic uncertain linguistic hybrid weighted geometric (IULHWG) operator is proposed. IULHWG operator can not only weigh the importance of each argument and its ordered position but also maintain these desirable properties, i.e., idempotency, boundedness and monotonicity. Based on these operators, we have proposed an approach to MAGDM problems with intuitionistic uncertain linguistic information. Finally, some illustrative examples have been given to show the steps of the developed method. By comparing with the existing two methods, we have confirmed the effectiveness of the new method, as well as greater adaptability and flexibility. In the future research, we can further utilize the proposed operational rules of IULVs and the proposed MAGDM method to deal with other different types of decision-making problems.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71771140,71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).

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Group decision making method based on hybrid aggregation operator for intuitionistic uncertain linguistic variables

Abstract

Keywords

1 Introduction

2.1 Intuitionistic fuzzy set (IFS)

Table 5 Comparison with different methods Method Whether the Ranking order operations are closed The proposed method yes X2 ≻ X4 ≻ X1 ≻ X3 in this paper The method in [20] no X2 ≻ X4 ≻ X1 ≻ X3 The method In [12] no X2 ≻ X4 ≻ X1 ≻ X3

Footnotes

Acknowledgments

References

Table 5
Comparison with different methods

Method Whether the Ranking order

operations

are closed

The proposed method yes X₂ ≻ X₄ ≻ X₁ ≻ X₃

in this paper

The method in [20] no X₂ ≻ X₄ ≻ X₁ ≻ X₃

The method In [12] no X₂ ≻ X₄ ≻ X₁ ≻ X₃