Some interval-valued hesitant uncertain linguistic Bonferroni mean operators and their application in multiple attribute group decision making

Abstract

The aim of this paper is to propose some novel multiple attribute group decision making (MAGDM) methods to deal with MAGDM problems in which the attributes are interactive in the form of interval-valued hesitant uncertain linguistic numbers (IVHULNs). Firstly, some new aggregation operators for IVHULNs based on Bonferroni mean (BM) are proposed, which are the interval-valued hesitant uncertain linguistic BM (IVHULBM) operator, the normalized weighted IVHULBM (NWIVHULBM) operator, the interval-valued hesitant uncertain linguistic geometric BM (IVHULGBM) operator and the normalized weighted IVHULGBM (NWIVHULGBM) operator. The advantages of the proposed operators are that it cannot only effectively aggregate IVHULNs, but it can also consider the interactive characteristics among attributes. At the same time, some special cases of these operators are discussed. Then, this paper demonstrates that these presented operators are able to meet four desirable properties, which are reducibility, idempotency, monotonicity, and boundedness. Moreover, to solve MAGDM problems, two approaches on the basis of the NWIVHULBM and NWIVHULGBM operators are put forward. Finally, the proposed methods are applied to a decision making problem regarding online service quality evaluation. It provides us with a useful way for MAGDM with IVHULNs.

Keywords

Multiple attribute group decision making interval-valued hesitant uncertain linguistic set Bonferroni mean geometric Bonferroni mean

1 Introduction

Multiple attribute group decision making (MAGDM) can be used to choose the most eligible alternative(s) from a set of available alternatives in different criteria assessed by decision makers, and it has been successfully applied in many areas, such as the personnel selection [29], cloud computing service evaluation [38], warehouse location selection [32], investment selection [33] and so on. In MAGDM problems, experts often evaluate attributes in the form of numerical values. However, under some circumstances, it is inadequate or insufficient for experts to provide evaluation values by means of numerical values, especially for qualitative aspects, while it is easy to express the evaluation values by means of linguistic variables (LVs). For example, when the moral character of students, the computer performance and so on are evaluated, they are easy to be expressed by the LVs, such as medium, high, very high. So far, the research on the MAGDM with linguistic information has made many achievements [3 , 45].

The linguistic variable (LV) firstly proposed by Zedeh [44] is a very useful tool to express qualitative information. Some extensions of linguistic set (LS), such as uncertain linguistic set (ULS) [40], triangular fuzzy linguistic set [39], 2-dimension uncertain linguistic set [23], Neutrosophic uncertain linguistic set [19], hesitant fuzzy linguistic term set [30], were presented. LS and its extensions utilized distance measures [13], similarity measures [14], cross entropy, entropy measures [5], correlation measures [15], and aggregation operators [26 , 40] to handle MAGDM problems. Linguistic VIKOR method [17], 2-dimension uncertain linguistic TODIM Method [22], hesitant linguistic ELECTRE I method [4] were proposed for solving the MAGDM problems.

Above LS and its extensions are only able to provide the qualitative evaluation value and are not able to take advantage of the information about the possible confidence degrees for the provided evaluation value. Hence, the concept of hesitant fuzzy uncertain linguistic set (HFULS) was introduced by Lin et al. [16]. HFULS is able to depict two fuzzy attributes of an object: an ULV and a HFS. The former describes an evaluation value, such as [medium, very high] or [low, high]. The latter gives the hesitancy for the provided evaluation value and denotes the membership degrees associated with the specific ULV. Ju and Liu [12] generalized the notion of HFULS to that of interval-valued hesitant uncertain linguistic set (IVHULS) in which the membership degrees of an element to an ULV are represented by differing interval values within [0, 1] instead of specific values. For instance, three experts discuss the membership degrees of x to an ULV [low, medium]. They want to assign [0.4, 0.5], [0.4, 0.6] and [0.5, 0.7], respectively, and they are not able to persuade each other to change their opinions, thus the membership degrees of x to the ULV [low, medium] can be denoted by an interval-valued HFS (IVHFS) {[0.4, 0.5], [0.4, 0.6], [0.5, 0.7]}, which is more precise than the result expressed by the HFS. In this situation, IVHULN is a good choice, and the evaluation value is denoted by < [low, medium], {[0.4, 0.5], [0.4, 0.6], [0.5, 0.7]}>. More and more MAGDM methods with HFULNs or IVHULNs have been developed. Lin et al. [16] proposed some aggregation operators for aggregating HFULNs. Ju and Liu [12] extended average operator and geometric operator to IVHULS, and applied proposed new operators to MAGDM problems. Liu et al. [25] proposed some generalized aggregation operators to aggregate the IVHULNs and developed a MAGDM method with IVHULNs.

It should be noted that the above aggregation operators about HFULNs or IVHULNs are under the supposition that the aggregated arguments are independent which may be not realistic. In real MAGDM problem, the complex relationships among the decision making criteria exist commonly. Hence, Wei [34] extended Choquet integral and power average operator, which can reflect the interaction phenomena among the criteria, to IVHULS, and proposed a wide range of operators for aggregating IVHULNs. In addition, Wei [34] extended the prioritized average operator, which can consider the prioritization phenomenon of the aggregated arguments, to IVHULS. The Bonferroni mean (BM) was proposed by Bonferroni [2], which can capture the interrelationship of the aggregated arguments. The BM is different from Choquet integral and power average operator. The BM concentrates on the input arguments while the Choquet integral and power average operator concentrate on the weight vector. The BM is also different from prioritized operator which mainly considers a special case of relationship of criteria called the prioritized relationship. Due to the advantage of the BM, it has been applied in decision making by many researchers [1 , 42]. The BM was extended to hesitant fuzzy set (HFS) [46], intuitionistic fuzzy set (IFS) [47], 2-tuple linguistic set (2TLS) [10], interval-valued 2TLS (IV2TLS) [24], ULS [36], linguistic intuitionistic fuzzy set [20], intuitionistic linguistic set (ILS) [21], and so on. However, BM fails to aggregate IVHULNs. In order to effectively aggregate IVHULNs, capture their interrelationship, and exactly make decision, this paper extends the BM to IVHULS and proposed a series of operators, including the IVHULBM, NWIVHULBM, IVHULGBM, and NWIVHULGBM operators.

The aim of this paper is to propose two novel MAGDM methods to deal with interval-valued hesitant uncertain linguistic MAGDM problems with interactive attributes based on the proposed interval-valued hesitant uncertain linguistic BM operators. To do so, the rest of this paper is shown as follows. Section 2 briefly reviews some basic concepts of the ULS, IVHULS, BM and GBM. Section 3 develops the IVHULBM, NWIVHULBM, IVHULGBM and NWIVHULGBM operators, and discusses some properties and special cases of the NWIVHULBM and NWIVHULGBM operators. Section 4 puts forward two methods for MAGDM on the basis of the NWIVHULBM and NWIVHULGBM operators, and gives the detail procedures. Section 5 provides an illustrative example to verify the developed methods. Section 6 ends up with some concluding remarks.

2 Preliminaries

2.1 The uncertain linguistic variable

Suppose that S = (s₀, s₁, …, s_l-1) consists of the odd number of discrete linguistic terms, which means l is an odd value. Generally, l is equal to 3, 5, 7, 9, 11, 13 etc. For instance, when l = 7, S = (s₀, s₁, s₂, s₃, s₄, s₅, s₆) = {extremely low, verylow, low, medium, high, veryhigh, extremelyhigh}. Here, s_a (a = 0, 1, 2 … , l - 1) can be called a LV.

Suppose s_α and s_β are any two elements in LS S; then they must meet the following conditions [7, 8]:

If α < β, then s_α < s_β;

Negative operator: neg (s_β) = s_α, where β = l - 1 - α;

Maximum operator: If s_α ≥ s_β, then max(s_α, s_β) = s_α;

Minimum operator: If s_α ≥ s_β, then min(s_α, s_β) = s_β.

Definition 1. [40] Suppose $\tilde{s} = [s_{ϑ}, s_{π}]$ , s_ϑ, s_π ∈ S and π ≥ ϑ ≥ 0, s_ϑ and s_π are the lower limit and upper limit of $\tilde{s}$ , respectively; then $\tilde{s}$ is called an ULV.

Suppose ${\tilde{s}}_{1} = [s_{ϑ (h_{1})}, s_{π (h_{1})}]$ and ${\tilde{s}}_{2} = [s_{ϑ (h_{2})}, s_{π (h_{2})}]$ are any two ULVs; then the operation rules are defined as follows [40, 41]: $(1) {\tilde{s}}_{1} \oplus {\tilde{s}}_{2} = [s_{ϑ (h_{1}) + ϑ (h_{2})}, s_{π (h_{1}) + π (h_{2})}],$ (1) $(2) {\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = [s_{ϑ (h_{1}) \times ϑ (h_{2})}, s_{π (h_{1}) \times π (h_{2})}],$ (2) $(3) λ {\tilde{s}}_{1} = [s_{λ \times ϑ (h_{1})}, s_{λ \times π (h_{1})}] (λ \geq 0),$ (3) $(4) {\tilde{s}}_{1}^{λ} = [s_{(ϑ (h_{1}))^{λ}}, s_{(π (h_{1}))^{λ}}] (λ \geq 0) .$ (4)

2.2 The interval-valued hesitant uncertain linguistic set

Definition 2. [12] Suppose X is a reference set and $[s_{ϑ (χ)}, s_{π (χ)}] \in \tilde{S} .$ An IVHULS in X is an object: $H = (< χ, [s_{ϑ (χ)}, s_{π (χ)}], ψ (χ)) > | χ \in X),$ (5) where ψ (χ) is a set of some interval values in [0,1], representing the possible membership degrees of the element χ ∈ X to the ULV [s_ϑ(χ), s_π(χ)] . From now on, it will be convenient to call h =< [s_ϑ(h), s_π(h)] , ψ (h) > an IVHULN.

Suppose h₁ =< [s_ϑ(h₁), s_π(h₁)] , ψ (h₁) > and h₂ =< [s_ϑ(h₂), s_π(h₂)] , ψ (h₂) > are any two IVHULNs; then the operation rules are defined as follows [12]: $\begin{matrix} h_{1} \oplus h_{2} & = & < [s_{ϑ (h_{1}) + ϑ (h_{2})}, s_{π (h_{1}) + π (h_{2})}], \\ \cup \tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \in ψ (h_{2}) \\ {r_{h_{1}}^{L} + r_{h_{2}}^{L} - r_{h_{1}}^{L} r_{h_{2}}^{L}, r_{h_{1}}^{U} \\ + r_{h_{2}}^{U} - r_{h_{1}}^{U} r_{h_{2}}^{U}} >, \end{matrix}$ (6) $\begin{matrix} h_{1} \otimes h_{2} & = & < [s_{ϑ (h_{1}) \times ϑ (h_{2})}, s_{π (h_{1}) \times π (h_{2})}], \\ \cup \tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \\ \in ψ (h_{2}) {r_{h_{1}}^{L} r_{h_{2}}^{L}, r_{h_{1}}^{U} r_{h_{2}}^{U}} >, \end{matrix}$ (7) $\begin{matrix} λ h_{1} & = & < [s_{λ \times ϑ (h_{1})}, s_{λ \times π (h_{1})}], \cup \tilde{r} (h_{1}) \in ψ (h_{1}) \\ {1 - (1 - r_{h_{1}}^{L})^{λ}, 1 - (1 - r_{h_{1}}^{U})^{λ}} > \\ (λ \geq 0), \end{matrix}$ (8) $\begin{matrix} h_{1}^{λ} & = & < [s_{ϑ (h_{1})^{λ}}, s_{π (h_{1})^{λ}}], \cup \tilde{r} (h_{1}) \\ \in ψ (h_{1}) {(r_{h_{1}}^{L})^{λ}, (r_{h_{1}}^{U})^{λ}} > (λ \geq 0) . \end{matrix}$ (9)

Suppose h, h₁ and h₂ are any three IVHULNs; then the characteristics of IVHULNs can be proved as follows [12]: $(1) h \oplus h_{1} = h_{1} \oplus h,$ (10) $(2) h \otimes h_{1} = h_{1} \otimes h,$ (11) $(3) h \oplus h_{1} \oplus h_{2} = h \oplus (h_{1} \oplus h_{2}),$ (12) $(4) h \otimes h_{1} \otimes h_{2} = h \otimes (h_{1} \otimes h_{2}),$ (13) $(5) λ (h \oplus h_{1}) = λ h \oplus λ h_{1} (λ \geq 0),$ (14) $(6) (λ_{1} + λ_{2}) h = λ_{1} h \oplus λ_{2} h (λ_{1}, λ_{2} \geq 0),$ (15) $(7) (h \otimes h_{1})^{λ} = h^{λ} \otimes h_{1}^{λ} \geq 0 .$ (16)

Definition 3. [12] Let h =< [s_ϑ(h), s_π(h)] , ψ (h) > be an IVHULN; then the score function S (h) of h can be defined as follows: $S (h) = s_{\frac{(ϑ (h) + π (h)) \times (Σ_{\tilde{r} (h) \in ψ (h)} (r_{h}^{L} + r_{h}^{U}))}{4 \times # ψ}},$ (17) where # ψ is the number of the interval numbers in ψ (h).

Definition 4. [12] Suppose h and h₁ are any two IVHULNs; then the comparison method of IVHULNs is expressed as follows:

If S (h) > S (h₁) , then h > h₁.

If S (h) = S (h₁) , then h = h₁.

If S (h) < S (h₁) , then h < h₁.

2.3 The Bonferroni mean (BM) operator

The BM operator can consider the interrelationship of input arguments, which was introduced by Bonferroni [2].

Definition 5. [2] Suppose p, q ≥ 0 and {x₁, x₂, …, x_n} is a set of nonnegative numbers. The BM operator is defined as

$\begin{matrix} {BM}^{p, q} (x_{1}, x_{2}, \dots, x_{n}) \\ = {(\frac{1}{n (n - 1)} \sum_{i = 1}^{n} \sum_{κ = 1, κ \neq i}^{n} x_{i}^{p} x_{κ}^{q})}^{\frac{1}{p + q}} . \end{matrix}$ (18)

The BM operator has a shortcoming that it ignores the importance of each input argument itself. However, in most practical situations, the weight of input data is also an important parameter. So, a normalized weighted Bonferroni mean (NWBM) operator was introduced by Zhou [47], which can overcome the shortcoming.

Definition 6. [47] Suppose p, q ≥ 0 and {x₁, x₂, …, x_n} is a set of nonnegative numbers, and w_i is the weight of x_i (i = 1, 2, …, n) , satisfying w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1 .$ A NWBM operator is defined as

$\begin{matrix} {NWBM}^{p, q} (x_{1}, x_{2}, \dots, x_{n}) \\ = {(\sum_{i = 1}^{n} \sum_{κ = 1, κ \neq i}^{n} \frac{w_{i} w_{κ}}{1 - w_{i}} x_{i}^{p} x_{κ}^{q})}^{1 / - p + q} . \end{matrix}$ (19)

Zhou [47] has proved that the NWBM operator possesses four desirable properties which are reducibility, idempotency, monotonicity and boundedness.

2.4 The geometric Bonferroni mean (GBM) operator

Similar to the BM operator, the GBM operator can also consider the interrelationship of input arguments.

Definition 7. [37] Suppose p, q ≥ 0 and {x₁, x₂, …, x_n} is a set of nonnegative numbers. The GBM operator is defined as

$\begin{matrix} {GBM}^{p, q} (x_{1}, x_{2}, \dots, x_{n}) \\ = \frac{1}{p + q} \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} ({px}_{i} + {qx}_{κ})^{\frac{1}{n (n - 1)}} \end{matrix}$ (20)

Similar to the BM operator, the GBM operator also has the shortcoming that it cannot take the weights of the aggregated arguments into account. Sun and Liu [31] further defined the NWGBM operator, which can overcome the shortcoming.

Definition 8. [31] Suppose p, q ≥ 0 and {x₁, x₂, …, x_n} is a set of nonnegative numbers; w_i is the weight of x_i (i = 1, 2, …, n) , satisfying w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1 .$ A NWGBM operator is defined as

$\begin{matrix} {NWGBM}^{p, q} (x_{1}, x_{2}, \dots, x_{n}) \\ = \frac{1}{p + q} \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} ({px}_{i} + {qx}_{κ})^{\frac{w_{i} w_{κ}}{1 - w_{i}}} . \end{matrix}$ (21)

Sun and Liu [31] have proved that the NWGBM operator possesses four desirable properties which are reducibility, idempotency, monotonicity and boundedness.

3 Some Bonferroni mean operators based on the IVHULNs

This section extends the BM and GBM operators to the IVHULNs, and proposes a series of operators, including the IVHULBM, NWIVHULBM, IVHULGBM, and NWIVHULGBM operators.

3.1 The NWIVHULBM operator

Definition 9. Suppose p, q ≥ 0, and h_i =< [s_{ϑ(h_i)}, s_{π(h_i)}] , ψ (h_i) > (i = 1, 2, …, n) is a set of IVHULNs; then an IVHULBM operator is defined as

$\begin{matrix} {IVHULBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = {(\frac{1}{n (n - 1)} \oplus_{i = 1}^{n} \oplus_{κ = 1, κ \neq i}^{n} h_{i}^{p} \otimes h_{κ}^{q})}^{\frac{1}{p + q}} . \end{matrix}$ (22)

In the IVHULBM operator, all of aggregating IVHULNs are equally important. If these IVHULNs are assigned to different weights, then the NWIVHULBM operator is defined as follows:

Definition 10. Suppose p, q ≥ 0, h_i = < [s_{ϑ(h_i)}, s_{π(h_i)}] , ψ (h_i) > (i = 1, 2, …, n) is a set of IVHULNs, and w_i (i = 1, 2, …, n) is the weight of h_i (i = 1, 2, …, n) , satisfying w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1;$ then a NWIVHULBM operator is defined as

$\begin{matrix} {NWIVHULBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = {(\oplus_{i = 1}^{n} \oplus_{κ = 1, κ \neq i}^{n} \frac{w_{i} w_{κ}}{1 - w_{i}} h_{i}^{p} \otimes h_{κ}^{q})}^{\frac{1}{p + q}} . \end{matrix}$ (23)

By the operation rules of the IVHULNs, Theorem 9 can be derived.

Theorem 9.Supposep, q ≥ 0 andh_i = < [s_{ϑ(h_i)}, s_{π(h_i)}] , ψ (h_i) > (i = 1, 2, …, n) is a set of IVHULNs; then the aggregating value by the NWIVHULBM operator is still an IVHULN, and

$\begin{array}{l} N W I V H U L B M (h_{1}, h_{2}, ..., h_{n}) \\ = 〈 [s_{_{{(\sum_{i = 1}^{n} \sum_{κ = 1, κ \neq i}^{n} \frac{w_{i} w_{κ}}{1 - w_{i}} . ϑ {(h_{i})}^{p} . ϑ {(h_{κ})}^{q})}^{\frac{1}{p + q}}}},] \\ s_{{(\sum_{i = 1}^{n} \sum_{κ = 1, κ \neq i}^{n} \frac{w_{i} w_{κ}}{1 - w_{i}} . π {(h_{i})}^{p} . π {(h_{κ})}^{q})}^{\frac{1}{p + q}}}], \\ \cup_{\tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \in ψ (h_{2}), ..., \tilde{r} (h_{n}) \in ψ (h_{n})} \\ {{(1 - \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} {(1 - {(r_{h_{i}}^{L})}^{p} . {(r_{h_{κ}}^{L})}^{q})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} {(1 - {(r_{h_{i}}^{U})}^{p} . {(r_{h_{κ}}^{U})}^{q})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}})}^{\frac{1}{p + q}}} 〉 . \end{array}$ (24)

Proof. By the operation rules of the IVHULNs, we have $\begin{matrix} h_{i}^{p} \\ = < [s_{ϑ (h_{i})}^{p}, s_{π (h_{i})}^{p}], \cup_{\tilde{r} (h_{i}) \in ψ (h_{i})} {(r_{h_{i}}^{L})^{p}, (r_{h_{i}}^{U})^{p}} >, \\ h_{κ}^{q} \\ = < [s_{ϑ (h_{κ})}^{q}, s_{π (h_{κ})}^{q}], \cup_{\tilde{r} (h_{κ}) \in ψ (h_{κ})} {(r_{h_{κ}}^{L})^{q}, (r_{h_{κ}}^{U})^{q}} > \end{matrix}$ and $\begin{matrix} h_{i}^{p} \otimes h_{κ}^{q} = < [s_{ϑ (h_{i})}^{p} ._{ϑ (h_{κ})}^{q}, S_{π (h_{i})}^{p} ._{π (h_{κ})}^{q}], \cup_{\tilde{r} (h_{i}) \in ψ (h_{i}), \tilde{r} (h_{κ}) \in ψ (h_{κ})} {(r_{h_{i}}^{L})^{p} . (r_{h_{κ}}^{L})^{q}, (r_{h_{i}}^{U})^{p} . (r_{h_{κ}}^{U})^{q}} > . \end{matrix}$

Then, we can get $\begin{matrix} \frac{w_{i} w_{κ}}{1 - w_{i}} h_{i}^{p} \otimes h_{κ}^{q} & = & 〈 [s_{\frac{w_{i} w_{κ}}{1 - w_{i}} . ϑ (h_{i})^{p} . ϑ (h_{κ})^{q}}, s_{\frac{w_{i} w_{κ}}{1 - w_{i}} . π (h_{i})^{p} . π (h_{κ})^{q}}], \\ \cup_{\tilde{r} (h_{i}) \in ψ (h_{i}), \tilde{r} (h_{κ}) \in ψ (h_{κ})} {1 - (1 - (r_{h_{i}}^{L})^{p} . (r_{h_{κ}}^{L})^{q})^{\frac{w_{i} w_{κ}}{1 - w_{i}}}, 1 - (1 - (r_{h_{i}}^{U})^{p} . (r_{h_{κ}}^{U})^{q})^{\frac{w_{i} w_{κ}}{1 - w_{i}}}} 〉 . \end{matrix}$

Thus, we can obtain $\begin{array}{l} {(\oplus_{i = 1}^{n} \oplus_{κ = 1, κ \neq i}^{n} \frac{w_{i} w_{κ}}{1 - w_{i}} h_{i}^{p} \otimes h_{κ}^{q})}^{\frac{1}{p + q}} = \\ 〈 [s_{_{{(\sum_{i = 1}^{n} \sum_{κ = 1, κ \neq i}^{n} \frac{w_{i} w_{κ}}{1 - w_{i}} . ϑ {(h_{i})}^{p} . ϑ {(h_{κ})}^{q})}^{\frac{1}{p + q}}}}, s_{{(\sum_{i = 1}^{n} \sum_{κ = 1, κ \neq i}^{n} \frac{w_{i} w_{κ}}{1 - w_{i}} . π {(h_{i})}^{p} . π {(h_{κ})}^{q})}^{\frac{1}{p + q}}}], \\ \cup_{\tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \in ψ (h_{2}), ..., \tilde{r} (h_{n}) \in ψ (h_{n})} \\ {{(1 - \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} {(1 - {(r_{h_{i}}^{L})}^{p} . {(r_{h_{κ}}^{L})}^{q})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} {(1 - {(r_{h_{i}}^{U})}^{p} . {(r_{h_{κ}}^{U})}^{q})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}})}^{\frac{1}{p + q}}} 〉 . \end{array}$

Theorem 10. (Reducibility)Suppose all the weights ofh_i (i = 1, 2, …, n) are equal, i.e., $w_{i} = \frac{1}{n};$ then $\begin{matrix} {NWIVHULBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = {IVHULBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) . \end{matrix}$

Theorem 11. (Idempotency)Leth_i = h =< [s_ϑ(h), s_π(h)] , ψ (h) > (i = 1, 2, …, n) ; then ${NWIVHULBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) = h .$

Theorem 12. (Monotonicity)Suppose (h₁, h₂, …, h_n) and (g₁, g₂, …, g_n) are two sets of IVHULNs. Ifs_{ϑ(h_(i))} ≤ s_{ϑ(g_(i))}, s_{π(h_(i))} ≤ s_{π(g_(i))}, $r_{h_{(i)}}^{L} \leq r_{g_{(i)}}^{L}$ and $r_{g_{(i)}}^{U} \leq r_{g_{(i)}}^{U}$ for all i = 1, 2, …, n, then $\begin{matrix} {NWIVHULBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ \leq {NWIVHULBM}^{p, q} (g_{1}, g_{2}, \dots, g_{n}) . \end{matrix}$

Theorem 13. (Boundedness)Leth_i (i = 1, 2, …, n) be a set of IVHULNs. If $\begin{matrix} s_{ϑ -} & = & min_{1 \leq i \leq n} {s_{ϑ (h_{i})}}, \\ s_{π -} & = & min_{1 \leq i \leq n} {s_{π (h_{i})}}, \\ s_{ϑ +} & = & max_{1 \leq i \leq n} {s_{ϑ (h_{i})}}, \\ s_{π +} & = & max_{1 \leq i \leq n} {s_{π (h_{i})}}, \\ r^{L -} & = & min_{1 \leq i \leq n} {r_{h_{i}}^{L} | [r_{h_{i}}^{L}, r_{h_{i}}^{U}] \in ψ (h_{i})}, \\ r^{U -} & = & min_{1 \leq i \leq n} {r_{h_{i}}^{U} | [r_{h_{i}}^{L}, r_{h_{i}}^{U}] \in ψ (h_{i})}, \\ h^{-} & = & [s_{ϑ -}, s_{π -}], {[r^{L -}, r^{U -}]}, \\ r^{L +} & = & max_{1 \leq i \leq n} {r_{h_{i}}^{L} | [r_{h_{i}}^{L}, r_{h_{i}}^{U}] \in ψ (h_{i})}, \\ r^{U +} & = & max_{1 \leq i \leq n} {r_{h_{i}}^{U} | [r_{h_{i}}^{L}, r_{h_{i}}^{U}] \in ψ (h_{i})}, \\ h^{+} & = & [s_{ϑ +}, s_{π +}], {[r^{L +}, r^{U +}]}; \end{matrix}$ then $h^{-} \leq {NWIVHULBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \leq h^{+} .$

In what follows, some special cases of the NWIVHULBM operator are discussed.

(1) If q = 0, then $\begin{array}{l} N W I V H U L B M^{p, 0} (h_{1}, h_{2}, ..., h_{n}) \\ = 〈 [s_{(\sum_{i = 1}^{n} w_{i} ϑ {(h_{i})}^{p})^{\frac{1}{p}}}, s_{(\sum_{i = 1}^{n} w_{i} π {(h_{i})}^{p})^{\frac{1}{p}}}], \\ \cup_{\tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \in ψ (h_{2}), ..., \tilde{r} (h_{n}) \in ψ (h_{n})} \\ {{(1 - \prod_{i = 1}^{n} (1 - {(r_{h_{^{i}}}^{L})}^{p})^{w_{i}})}^{\frac{1}{p}}, \\ {(1 - \prod_{i = 1}^{n} (1 - {(r_{h_{^{i}}}^{U})}^{p})^{w_{i}})}^{\frac{1}{p}}} 〉 . \end{array}$

(2) If p = 1 and q = 0, then $\begin{array}{l} N W I V H U L B M^{1, 0} (h_{1}, h_{2}, ..., h_{n}) = \oplus_{i = 1}^{n} w_{i} h_{i} \\ = 〈 [s_{\sum_{i = 1}^{n} w_{i} ϑ (h_{i})}, s_{\sum_{i = 1}^{n} w_{i} π (h_{i})}], \\ \cup_{\tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \in ψ (h_{2}), ..., \tilde{r} (h_{n}) \in ψ (h_{n})} \\ {1 - {\prod_{i = 1}^{n} (1 - r_{h_{^{i}}}^{L})}^{w_{i}}, 1 - {\prod_{i = 1}^{n} (1 - r_{h_{^{i}}}^{U})}^{w_{i}}} 〉 . \end{array}$

Now, an example shall be provided to shown the aggregation process.

Example 1. Let h₁ =< [s₄, s₅] , {[0.5, 0.7] , [0.7, 0.8]} > and h₂ =< [s₃, s₅] , {[0.6, 0.8]} > be two IVHULNs, suppose the weight vector of the IVHULNs is w = {w₁, w₂} ^T = {0.4, 0 . 6} ^T, and suppose there exists relationships among these aggregated arguments.

(1) The proposed NWIVHULBM operator is utilized to aggregate these two IVHULNs (let p = q = 2) to get a comprehensive value NWIVHULBM^2,2 (h₁, h₂) , shown as follows:

By the operational rules of the IVHULNs, we have $\begin{matrix} h_{1}^{2} & = & < [s_{16}, s_{25}], {[0.25, 0.49], [0.49, 0.64]} >, \\ h_{2}^{2} & = & < [s_{9}, s_{25}], {[0.36, 0.64]} > \end{matrix}$ and $\begin{matrix} h_{1}^{2} \otimes h_{2}^{2} \\ = < [s_{144}, s_{625}], {[0.09, 0.31], [0.18, 0.41]} > . \end{matrix}$

Then, we have $\begin{matrix} \frac{w_{1} w_{2}}{1 - w_{1}} h_{1}^{2} \otimes h_{2}^{2} \\ = 0.4 \cdot < [s_{144}, s_{625}], {[0.09, 0.31], \\ [0.18, 0.41]} > \\ = < [s_{57.6}, s_{250}], {[0.04, 0.14], [0.07, 0.19]} > \end{matrix}$ and $\begin{matrix} \frac{w_{2} w_{1}}{1 - w_{2}} h_{2}^{2} \otimes h_{1}^{2} = 0.6 \cdot < [s_{144}, s_{625}], \\ {[0.09, 0.31], [0.18, 0.41]} > \\ = < [s_{86.4}, s_{375}], {[0.06, 0.20], \\ [0.11, 0.27]} > . \end{matrix}$

Thus, we have $\begin{matrix} {NWIVHULBM}^{2, 2} (h_{1}, h_{2}) \\ = {(\oplus_{i = 1}^{2} \oplus_{κ = 1, κ \neq i}^{2} \frac{w_{i} w_{κ}}{1 - w_{i}} h_{i}^{2} \otimes h_{κ}^{2})}^{\frac{1}{4}} \\ = {((\frac{w_{1} w_{2}}{1 - w_{1}} h_{1}^{2} \otimes h_{1}^{2}) \oplus (\frac{w_{2} w_{1}}{1 - w_{2}} h_{2}^{2} \otimes h_{1}^{2}))}^{\frac{1}{4}} \\ = {(< [s_{144}, s_{625}], {[0.53, 0.81], [0.68, 0.88]} >)}^{\frac{1}{4}} \\ = < [s_{3.46}, s_{5.00}], {[0.55, 0.75], [0.65, 0.80]} > . \end{matrix}$

(2) If the uncertain linguistic weighted BM (ULWBM) operator proposed by Wei et al. [36] is utilized to deal with Example 1. We need to transform IVHULNs into ULVs by omitting the interval-valued hesitant fuzzy part. Then, suppose p = q = 2, and we can get ${ULWBM}^{2, 2} (h_{1}, h_{2}) = [s_{3.46}, s_{5.00}] .$

(3) If the generalized interval-valued hesitant uncertain linguistic weighted average (GIVHULWA) operator proposed by Liu et al. [25] is utilized to deal with Example 1. Suppose λ = 1, and we can get $\begin{matrix} {GIVHULWA}_{1} (h_{1}, h_{2}) \\ = < [s_{3.46}, s_{5.00}], {[0.56, 0.76], [0.64, 0.80]} > . \end{matrix}$

Note that both the proposed NWIVHULBM operator and the ULWBM operator [36] can consider the relationships among the aggregated arguments. The GIVHULWA operator [25] is under the supposition that the aggregated arguments are independent. There exist relationships among these aggregated arguments in Example 1. Thus, the GIVHULWA operator [25] is less reasonable to deal with Example 1. Both the proposed NWIVHULBM operator and the GIVHULWA operator [25] can effectively aggregate the IVHULNs. The ULWBM operator [36] can aggregate ULVs rather than IVHULNs. Thus, the ULWBM operator [36] is also less reasonable to deal with Example 1.

3.2 The NWIVHULGBM operator

The IVHULGBM operator is proposed based on the GBM operator and the IVHULS, which is shown as follows:

Definition 11. Suppose p, q ≥ 0, and h_i = < [s_{ϑ(h_i)}, s_{π(h_i)}] , ψ (h_i) > (i = 1, 2, …, n) is a set of IVHULNs; then an IVHULGBM operator is defined as

$\begin{matrix} {IVHULGBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = \frac{1}{p + q} \otimes_{i = 1}^{n} \otimes_{κ = 1, κ \neq i}^{n} {({ph}_{i} \oplus {qh}_{κ})}^{\frac{1}{n (n - 1)}} . \end{matrix}$ (25)

The IVHULGBM operator ignores the weights of IVHULNs. To solve this drawback, the NWIVHULGBM operator is proposed as follows:

Definition 12. Suppose p, q ≥ 0, h_i = < [s_{ϑ(h_i)}, s_{π(h_i)}] , ψ (h_i) > (i = 1, 2, …, n) is a set of IVHULNs, and w_i (i = 1, 2, …, n) is the weight of h_i (i = 1, 2, …, n) , satisfying w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1;$ then a NWIVHULGBM operator is defined as

$\begin{matrix} {NWIVHULGBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = \frac{1}{p + q} \otimes_{i = 1}^{n} \otimes_{κ = 1, κ \neq i}^{n} {({ph}_{i} \oplus {qh}_{j})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}} \end{matrix}$ (26)

By the operation rules of the IVHULNs, we can derive the result presented thus.

Theorem 14.Supposep, q ≥ 0 andh_i = < [s_{ϑ(h_i)}, s_{π(h_i)}] , ψ (h_i) > (i = 1, 2, …, n) is a set of IVHULNs; then the aggregating value by the NWIVHULGBM operator is still an IVHULN, and $\begin{matrix} NWIVHULGBM (h_{1}, h_{2}, \dots, h_{n}) \\ = \frac{1}{p + q} \otimes_{i = 1}^{n} \otimes_{κ = 1, κ = i}^{n} {({ph}_{i} \oplus {qh}_{κ})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}} \\ = 〈 [s_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} (p . ϑ (h_{i}) + q . ϑ (h_{κ}))^{\frac{w_{i} w_{κ}}{1 - w_{i}}}}, s_{\frac{1}{p + q} \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} (p . π (h_{i}) + q . π (h_{κ}))^{\frac{w_{i} w_{κ}}{1 - w_{i}}}}] \\ \cup_{\tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \in ψ (h_{2}), \dots, \tilde{r} (h_{n}) \in ψ (h_{n})} \end{matrix}$

$\begin{matrix} {1 - {(1 - \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} {(1 - (1 - r_{h_{i}}^{L})^{p} . (1 - r_{h_{κ}}^{L})^{q})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - \prod_{i = 1}^{n} \prod_{κ = 1, κ \neq i}^{n} {(1 - (1 - r_{h_{i}}^{U})^{p} . (1 - r_{h_{κ}}^{U})^{q})}^{\frac{w_{i} w_{κ}}{1 - w_{i}}})}^{\frac{1}{p + q}}} 〉 . \end{matrix}$ (27)

Proof. Similar to Theorem 9, the proof of Theorem 14 is omitted.

Obviously, the NWIVHULGBM operator also has reducibility, idempotency, monotonicity and boundedness, and the proofs are omitted here.

In what follows, some special cases of the NWIVHULGBM operator are discussed.

(1) If q = 0, then $\begin{matrix} NWIVHULGBM (h_{1}, h_{2}, \dots, h_{n})^{p, 0} = \frac{1}{p} \otimes_{i = 1}^{n} {({ph}_{i})}^{w_{i}} \\ = 〈 [s_{\frac{1}{p} \prod_{i = 1}^{n} (p . ϑ (h_{i}))^{w_{i}}}, s_{\frac{1}{p} \prod_{i = 1}^{n} (p . π (h_{i}))^{w_{i}}}], \\ \cup_{\tilde{r} (h_{i}) \in ψ (h_{i}), \tilde{r} (h_{j}) \in ψ (h_{j})} \\ {1 - {(1 - \prod_{i = 1}^{n} {(1 - (1 - r_{h_{i}}^{L})^{p})}^{w_{i}})}^{\frac{1}{p}}, \\ 1 - {(1 - \prod_{i = 1}^{n} {(1 - (1 - r_{h_{i}}^{U})^{p})}^{w_{i}})}^{\frac{1}{p}}} 〉 . \end{matrix}$

(2) If p = 1 and q = 0, then $\begin{matrix} NWIVHULGBM (h_{1}, h_{2}, \dots, h_{n})^{1, 0} = \otimes_{i = 1}^{n} h_{i}^{w_{i}} \\ = 〈 [s_{\prod_{i = 1}^{n} ϑ (h_{i})^{w_{i}}}, s_{\prod_{i = 1}^{n} π (h_{i})^{w_{i}}}], . \\ \cup_{\tilde{r} (h_{1}) \in ψ (h_{1}), \tilde{r} (h_{2}) \in ψ (h_{2}), \dots, \tilde{r} (h_{n}) \in ψ (h_{n})} \\ {\prod_{i = 1}^{n} {(r_{h_{i}}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(r_{h_{i}}^{U})}^{w_{i}}} 〉 \end{matrix}$

4 Two decision making methods with IVHULNs

4.1 The description of the decision making problems

For a MAGDM problem with IVHULNs: Suppose A = {A₁, A₂, …, A_m} is the set of alternatives, C = {C₁, C₂, …, C_n} is a discrete set of n attributes, and w_i is the weight of the attribute C_j (j = 1, 2, …, n) , satisfying w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1 .$ Suppose D = {D₁, D₂, …, D_s} is the set of s experts, ω_e is the weight of experts D_e (e = 1, 2, …, s) and ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1 .$ Assume that $T^{(e)} = (t_{ij}^{(e)})_{m \times n}$ is a decision matrix provided by D_e (e = 1, 2, …, s) , where $t_{ij}^{(e)} = 〈 [s_{ϑ (t_{ij}^{(e)})}, s_{π (t_{ij}^{(e)})}], {ψ (t_{ij}^{(e)})} 〉$ is an IVHULN, given by expert D_k (k = 1, 2, …, s) for the rating of the alternative A_i (i = 1, 2 … , m) in regard to the attribute C_j (j = 1, 2, …, n) , with $θ (t_{ij}^{(e)}) \leq τ (t_{ij}^{(e)})$ , $s_{θ (t_{ij}^{(e)})}, s_{τ (t_{ij}^{(e)})} \in S = {s_{0}, s_{1}, s_{2}, \dots, s_{l - 1}},$ and $ψ (t_{ij}^{(e)}) = \cup \tilde{r} (t_{ij}^{(e)}) \in ψ (t_{ij}^{(e)}) {[{\tilde{r}}_{(t_{ij}^{(e)})}^{L}, {\tilde{r}}_{(t_{ij}^{(e)})}^{U}]} .$

4.2 The detailed steps of decision making

Step 1. Aggregate the assessment information of all individual experts to collective information on the basis of the NWIVHULBM or NWIVHULGBM operators as follows: $t_{ij} = {NWIVHULBM}^{p, q} (t_{ij}^{1}, t_{ij}^{2}, \dots, t_{ij}^{s}) .$ (28) $t_{ij} = {NWIVHULGBM}^{p, q} (t_{ij}^{1}, t_{ij}^{2}, \dots, t_{ij}^{s}) .$ (29)

Step 2. Aggregate the assessment information of all attributes to comprehensive information on the basis of the NWIVHULBM or NWIVHULGBM operators. $t_{i} = {NWIVHULBM}^{p, q} (t_{ij}^{1}, t_{ij}^{2}, \dots, t_{ij}^{s}) .$ (30) $t_{i} = {NWIVHULGBM}^{p, q} (t_{ij}^{1}, t_{ij}^{2}, \dots, t_{ij}^{s}) .$ (31)

Step 3. Calculate the score values S (t_i) of the comprehensive information t_i (i = 1, 2, …, m).

Step 4. Rank all the alternatives { A₁, A₂, …, A_n } by Definition 4, and select the most eligible alternative(s).

Step 5. End.

5 An application example

Suppose that four social commerce websites {A₁, A₂, A₃, A₄} are chosen to evaluate online service quality. Five attributes are taken into account, including the efficiency (C₁), the fulfillment (C₂), the responsiveness (C₃), the contact (C₄), and the security (C₅). w = {0.0625, 0 . 25, 0 . 375, 0 . 25, 0.0625} ^T is the weight vector of the attributes C_j (j = 1, 2, …, n) . A committee of three experts { D₁, D₂, D₃ } , whose weight vector is ω = { 0.25, 0.5, 0.25 } ^T, evaluates the four possible alternatives A_i (i = 1, 2, 3, 4) under above five attributes C_j (j = 1, 2, 3, 4, 5) by utilizing the LS: S = (s₀, s₁, s₂, s₃, s₄, s₅, s₆) = {extremelylow, verylow, low, medium, high, veryhigh, extremelyhigh}. These experts provide the decision matrices $T^{(k)} = (t_{ij}^{(k)})_{4 \times 5}$ (k = 1, 2, 3) listed in Tables 1–3. Then, to evaluate the online service quality of these four social commerce websites A₁, A₂, A₃ and A₄, the ranking isneeded.

Table 1
Decision matrix given by D₁

C ₁ C ₂ C ₃ C ₄ C ₅

A ₁ < [s₄,s₅], {, [0.7,0.8]}> < [s₃,s₅], {[0.6,0.8]}> < [s₄,s₅], {[0.4,0.7]> < [s₅,s₆], {[0.3,0.4]}> < [s₂,s₄], {[0.5,0.8]>

A ₂ < [s₃,s₅], {[0.3,0.6]}> < [s₂,s₄], {[0.4,0.6] [0.5,0.7]}> < [s₂,s₄], {[0.4,0.6], [0.5,0.7]}> < [s₁,s₂], {[0.4,0.5], [0.6,0.7]}> < [s₃,s₄], {[0.2,0.3], [0.4,0.6]}>

A ₃ < [s₁,s₂], {[0.1,0.3]}> < [s₃,s₅], {[0.1,0.4]}> < [s₂,s₅], {[0.7,0.8]}> < [s₃,s₅], {[0.7,0.8]}> < [s₁,s₂], {[0.3,0.4]}>

A ₄ < [s₃,s₃], {[0.4,0.5]}> < [s₁,s₃], {[0.3,0.5]}> < [s₃,s₄], {[0.3,0.5]}> < [s₂,s₄], {[0.1,0.3]}> < [s₂,s₃], {[0.4,0.7], [0.6,0.8],[0.7,0.9]}>

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	< [s₄,s₅], {, [0.7,0.8]}>	< [s₃,s₅], {[0.6,0.8]}>	< [s₄,s₅], {[0.4,0.7]>	< [s₅,s₆], {[0.3,0.4]}>	< [s₂,s₄], {[0.5,0.8]>
A ₂	< [s₃,s₅], {[0.3,0.6]}>	< [s₂,s₄], {[0.4,0.6] [0.5,0.7]}>	< [s₂,s₄], {[0.4,0.6], [0.5,0.7]}>	< [s₁,s₂], {[0.4,0.5], [0.6,0.7]}>	< [s₃,s₄], {[0.2,0.3], [0.4,0.6]}>
A ₃	< [s₁,s₂], {[0.1,0.3]}>	< [s₃,s₅], {[0.1,0.4]}>	< [s₂,s₅], {[0.7,0.8]}>	< [s₃,s₅], {[0.7,0.8]}>	< [s₁,s₂], {[0.3,0.4]}>
A ₄	< [s₃,s₃], {[0.4,0.5]}>	< [s₁,s₃], {[0.3,0.5]}>	< [s₃,s₄], {[0.3,0.5]}>	< [s₂,s₄], {[0.1,0.3]}>	< [s₂,s₃], {[0.4,0.7], [0.6,0.8],[0.7,0.9]}>

Table 2

Decision matrix given by D₂

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	< [s₃,s₄], {}>	< [s₄,s₆],{[0.3,0.5], [0.4,0.6]}>	< [s₃,s₄], {[0.8,1.0]>	< [s₄,s₅], {[0.4,0.6], [0.7,0.8]}>	< [s₂,s₃], {[0.3,0.5],[0.7,0.9]>
A ₂	< [s₄,s₆], {[0.4,0.5], [0.6,0.7]}>	< [s₃,s₄], {[0.2,0.4]}>	< [s₃,s₅], {[0.2,0.4]}>	< [s₀,s₁], {[0.3,0.5]}>	< [s₁,s₂], {[0.5,0.7]}>
A ₃	< [s₂,s₃], {[0.2,0.4]}>	< [s₁,s₃], {[0.5,0.8]}>	< [s₄,s₅], {[0.5,0.6]}>	< [s₃,s₄], {[0.2,0.4]}>	< [s₃,s₄],{[0.0,0.3], [0.2,0.4],[0.3,0.5]}>
A ₄	< [s₁,s₃], {[0.5,0.8]}>	< [s₂,s₅], {[0.4,0.6], [0.7,0.8]}>	< [s₁,s₄], {[0.1,0.3], [0.5,0.6]}>	< [s₃,s₅], {[0.6,0.7]}>	< [s₁,s₂], {[0.2,0.4]}>

Table 3

Decision matrix given by D₃

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	< [s₄,s₅], {}>	< [s₃,s₄], {[0.5,0.7]}>	< [s₄,s₅], {[0.3,0.5]>	< [s₃,s₄], {[0.2,0.4], [0.4,0.7]}>	< [s₂,s₄], {[0.5,0.7], [0.8,1.0]>
A ₂	< [s₂,s₃], {[0.1,0.3], [0.5,0.6]}>	< [s₂,s₃], {[0.4,0.5]}>	< [s₅,s₅], {[0.4,0.6], [0.7,0.9]}>	< [s₀₁,s₁₂], {[0.3,0.4]}>	< [s₃,s₄], {[0.4,0.6]}>
A ₃	< [s₃,s₄], {[0.5,0.6], [0.7,0.8]}>	< [s₄,s₅], {[0.5,0.8]}>	< [s₄,s₅], {[0.5,0.6]}>	< [s₄,s₅], {[0.5,0.6], [0.7,0.9]}>	< [s₁,s₂], {[0.3,0.5]}>
A ₄	< [s₁,s₃], {[0.5,0.8]}>	< [s₂,s₅], {[0.4,0.6], [0.7,0.8]}>	< [s₁,s₄], {[0.1,0.3], [0.5,0.6]}>	< [s₂,s₅], {[0.4,0.5]}>	< [s₁,s₃], {[0.6,0.7]}>

5.1 The evaluation steps by NWIVHULBM operator

To get the best alternative(s), the following steps are involved.

Step 1. Aggregate the assessment information of all individual experts to collective information on the basis of the NWIVHULBM operator, which is listed in Table 4. (suppose p = q = 1).

Table 4
Collective decision matrix

C ₁ C ₂ C ₃ C ₄ C ₅

A ₁ < [s_3.56,s_4.56], {,[0.54,0.69]}> < [s_3.39,s_5.08], {[0.44,0.64],[0.48,0.69]}> < [s_3.56,s_4.56], {[0.51,0.76]}> < [s_3.98,s_4.98], {[0.31,0.48], [0.40,0.54],[0.37,0.57], [0.48,0.65]}> < [s_2.00,s_3.56], {[0.41,0.64], [0.67,0.91],[0.48,0.73], [0.58,0.81]}>

A ₂ < [s_3.06,s_4.74], {[0.27,0.47],[0.48,0.64], [0.40,0.56],[0.33,0.55]}> < [s_2.38,s_3.70], {[0.31,0.48],[0.33,0.51]}> < [s_3.23,s_4.70], {[0.33,0.54], [0.40,0.62]}> < [s_0.41,s_1.53], {[0.33,0.56], [0.46,0.65]}> < [s_2.00,s_3.06], {[0.37,0.55], [0.44,0.64]}>

A ₃ < [s_1.96,s_2.97], {[0.24,0.43],[0.28,0.48]}> < [s_2.22,s_4.08], {[0.15,0.47],[0.37,0.48], [0.30,0.54],[0.26,0.51]}> < [s_3.06,s_5.00], {[0.56,0.69], [0.59,0.69]}> < [s_3.28,s_4.56], {[0.40,0.56], [0.46,0.65]}> < [s_1.63,s_2.71], {[0.12,0.38], [0.26,0.43],[0.30,0.47]}>

A ₄ < [s_1.47,s_2.69], {[0.37,0.55],[0.44,0.61]}> < [s_1.96,s_4.38], {[0.34,0.51],[0.48,0.65], [0.37,0.57],[0.44,0.58]}> < [s_2.22,s_4.55], {[0.30,0.53], [0.53,0.69]}> < [s_2.38,s_4.70], {[0.37,0.51]}> < [s_1.26,s_2.55], {[0.35,0.57], [0.41,0.59],[0.43,0.62]}>

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	< [s_3.56,s_4.56], {,[0.54,0.69]}>	< [s_3.39,s_5.08], {[0.44,0.64],[0.48,0.69]}>	< [s_3.56,s_4.56], {[0.51,0.76]}>	< [s_3.98,s_4.98], {[0.31,0.48], [0.40,0.54],[0.37,0.57], [0.48,0.65]}>	< [s_2.00,s_3.56], {[0.41,0.64], [0.67,0.91],[0.48,0.73], [0.58,0.81]}>
A ₂	< [s_3.06,s_4.74], {[0.27,0.47],[0.48,0.64], [0.40,0.56],[0.33,0.55]}>	< [s_2.38,s_3.70], {[0.31,0.48],[0.33,0.51]}>	< [s_3.23,s_4.70], {[0.33,0.54], [0.40,0.62]}>	< [s_0.41,s_1.53], {[0.33,0.56], [0.46,0.65]}>	< [s_2.00,s_3.06], {[0.37,0.55], [0.44,0.64]}>
A ₃	< [s_1.96,s_2.97], {[0.24,0.43],[0.28,0.48]}>	< [s_2.22,s_4.08], {[0.15,0.47],[0.37,0.48], [0.30,0.54],[0.26,0.51]}>	< [s_3.06,s_5.00], {[0.56,0.69], [0.59,0.69]}>	< [s_3.28,s_4.56], {[0.40,0.56], [0.46,0.65]}>	< [s_1.63,s_2.71], {[0.12,0.38], [0.26,0.43],[0.30,0.47]}>
A ₄	< [s_1.47,s_2.69], {[0.37,0.55],[0.44,0.61]}>	< [s_1.96,s_4.38], {[0.34,0.51],[0.48,0.65], [0.37,0.57],[0.44,0.58]}>	< [s_2.22,s_4.55], {[0.30,0.53], [0.53,0.69]}>	< [s_2.38,s_4.70], {[0.37,0.51]}>	< [s_1.26,s_2.55], {[0.35,0.57], [0.41,0.59],[0.43,0.62]}>

Step 2. Aggregate the assessment information of all attributes to comprehensive information t_i on the basis of the NWIVHULBM operator. Due to the limited space, the comprehensive values of other alternatives A_i (i = 2, 3, 4) are omitted here (suppose p = q = 1) $\begin{matrix} t_{1} & = & < [s 3 . 5064, s 4 . 7303], {[0 . 4250, 0 . 6392], [0 . 4458, 0 . 6637], [0 . 4307, 0 . 6467], [0 . 4380, 0 . 6539], \\ [0 . 4497, 0 . 6564], [0 . 4701, 0 . 6801], [0 . 4553, 0 . 6637], [0 . 4624, 0 . 6707], [0 . 4425, 0 . 6637], \\ [0 . 4629, 0 . 6871], [0 . 4481, 0 . 6709], [0 . 4553, 0 . 6778], [0 . 4706, 0 . 6852], [0 . 4906, 0 . 7078], \\ [0 . 4760, 0 . 6921], [0 . 4831, 0 . 6987], [0 . 4380, 0 . 6519], [0 . 4587, 0 . 6761], [0 . 4436, 0 . 6593], \\ [0 . 4509, 0 . 6664], [0 . 4626, 0 . 6688], [0 . 4828, 0 . 6922], [0 . 4681, 0 . 6759], [0 . 4752, 0 . 6828], \\ [0 . 4554, 0 . 6760], [0 . 4757, 0 . 6990], [0 . 4609, 0 . 6830], [0 . 4681, 0 . 6898], [0 . 4834, 0 . 6971], \\ [0 . 5033, 0 . 7193], [0 . 4888, 0 . 7038], [0 . 4958, 0 . 7104], [0 . 4301, 0 . 6423], [0 . 4508, 0 . 6666], \\ [0 . 4358, 0 . 6497], [0 . 4431, 0 . 6569], [0 . 4547, 0 . 6594], [0 . 4750, 0 . 6830], [0 . 4602, 0 . 6666], \\ [0 . 4674, 0 . 6736], [0 . 4475, 0 . 6666], [0 . 4679, 0 . 6899], [0 . 4531, 0 . 6738], [0 . 4602, 0 . 6806], \\ [0 . 4755, 0 . 6880], [0 . 4955, 0 . 7105], [0 . 4809, 0 . 6948], [0 . 4880, 0 . 7015], [0 . 4431, 0 . 6549], \\ [0 . 4637, 0 . 6790], [0 . 4487, 0 . 6623], [0 . 4559, 0 . 6693], [0 . 4676, 0 . 6717], [0 . 4877, 0 . 6950], \\ [0 . 4730, 0 . 6788], [0 . 4801, 0 . 6857], [0 . 4604, 0 . 6788], [0 . 4806, 0 . 7018], [0 . 4659, 0 . 6858], \\ [0 . 4730, 0 . 6926], [0 . 4883, 0 . 6998], [0 . 5081, 0 . 7219], [0 . 4937, 0 . 7066], [0 . 5006, 0 . 7131]} > . \end{matrix}$

Step 3. Calculate the score values S (t_i) of the comprehensive information t_i (i = 1, 2, 3, 4) by Definition 3. $\begin{matrix} S (t_{1}) & = & s_{2 . 3585}, S (t_{2}) = s_{1 . 2175}, \\ S (t_{3}) & = & s_{1 . 7405}, S (t_{4}) = s_{1 . 5313} . \end{matrix}$

Step 4. Ranking all the alternatives A_i (i = 1, 2, 3, 4) by Definition 4. $A_{1} ≻ A_{3} ≻ A_{4} ≻ A_{2} .$

Hence, A₁ is the best alternative.

Similar to the calculation steps by the NWIVHULBM operator, we use the method based on the NWIVHULGBM operator to rank the alternatives. The result is also A₁ ≻ A₃ ≻ A₄ ≻ A₂. Thus, A₁ is the best alternative.

5.2 Comparison with the exiting method

To demonstrate the practicality and validity of the proposed methods in this paper, the generalized weighted geometric operator based on the IVHULNs proposed by Liu et al. [25] is used to rank the alternatives. The ranking result is A₁ ≻ A₄ ≻ A₃ ≻ A₂. Thus, A₁ is the best alternative.

Obviously, there is the same best alternative A₁ in all above methods. However, the ranking of the alternatives produced here are slightly different to that obtained by the first two methods. The reason may be that the last method is only suitable for the situations that the input arguments are independent, which seems a bit biased towards this application case. While the first two methods are suitable to solve the situations that the input arguments are interactive, which could consider interaction between the experts and attributes, which is more reasonable to handle this example.

6 Conclusion

The IVHFLS is based on the combination of ULV and IVHFS, which can depict the real preferences of experts and capture their uncertainty, hesitancy and inconsistency. The BM can cope with the interactions between the attributes. The BM has been extended to IVHFLS, and some BM aggregation operators have been proposed based on IVHFLNs in this paper. Firstly, the IVHULBM and IVHULGBM operators have been developed. Considering the weight vector of the arguments, the NWIVHULBM and NWIVHULGBM operators further have been developed. The advantages of the proposed operators are that they cannot only effectively aggregate IVHULNs, but they can also consider the interactive characteristics among attributes. At the same time, some special cases and some desired properties of these operators have been discussed, inclusive of reducibility, idempotency, monotonicity and boundedness. Moreover, two novel approaches to the MCGDM problems with IVHULNs based on the developed operators have been proposed. The proposed approaches can take all the decision arguments and their relationships into account. Finally, an illustrative example regarding emergency alternative evaluation has been presented to show the effectiveness of the developed methods. It provides us with a useful way for MAGDM with IVHULNs.

In future research, we shall concentrate on applying the developed MAGDM methods to cope with some concrete examples in real world, such as supply chain management, data mining, pattern recognition and so on, and extending the traditional MAGDM methods, such as GRA [35], TODIM [6], TOPSIS [43] and so on, to IUHULS.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71771140, 71471172 and 71271124), the Special Funds of Taishan Scholars of Shandong Province (No. ts201511045), Shandong Provincial Social Science Planning Project (Nos. 16CGLJ31 and 16CKJJ27), Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (No. 2015Z057), and Key research and development program of Shandong Province (2016GNC110016).

References

Blanco-Mesa

, Merigo

J.M.

and Kacprzyk

, Bonferroni means with distance measures and the adequacy coefficient in entrepreneurial group theory, Knowledge-Based Systems111 (2016), 217–227.

Bonferroni

, Sulle medie multiple di potenze, Bolletino Matematica Italiana5 (1950), 267–270.

Dutta

, Guha

and Mesiar

, A model based on linguistic 2-tuples for dealing with heterogeneous relationship among attributes in multi-expert decision making, IEEE Transactions on Fuzzy Systems23(5) (2015), 1817–1831.

Fahmi

, Kahraman

and Bilen

Ü.

, ELECTRE I method using hesitant linguistic term sets: An application to supplier selection, International Journal of Computational Intelligence Systems9(1) (2016), 153–167.

Farhadinia

, Multiple criteria decision-making methods with completely unknown weights in hesitant fuzzy linguistic term setting, Knowledge-Based Systems93 (2015), 135–144.

Gomes

L.F.A.M.

and Lima

M.M.P.P.

, TODIM: Basics and application to multicriteria ranking of projects with environmental impacts, Foundations of Computing and Decision Sciences16 (1992), 113–127.

Herrera

, Herrera-Viedma

and Verdegay

J.L.

, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets and Systems78(1) (1996), 73–87.

Herrera

, Martínez

and Sanchez

P.J.

, Managing non-homogeneous information in group decision-making, European Journal of Operational Research166(1) (2005), 115–132.

, Wang

J.Q.

and Zhang

H.Y.

, Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers, Neural Computing and Applications (2016), 25. doi: 10.1007/s00521–016–2660-6

10.

Jiang

X.P.

and Wei

G.W.

, Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making, Journal of Intelligent & Fuzzy Systems27(5) (2014), 2153–2162.

11.

Y.B.

, A new method for multiple criteria group decision making with incomplete weight information under linguistic environment, Applied Mathematical Modelling38(21-22) (2014), 5256–5268.

12.

Y.B.

, Liu

X.Y.

and Yang

Sh.

, Some generalized interval-valued hesitant uncertain linguistic aggregation operators and their applications to multiple attribute group decision making, Soft Computing20(2) (2016), 495–510.

13.

and Zeng

S.Z.

, Uncertain linguistic aggregation distance measures and their application to group decision making, Journal of Applied Mathematics2013 (2013), 495–496.

14.

Liao

H.C.

and Xu

Z.S.

, Approaches to manage hesitant fuzzy linguistic information based on the cosine distance and similarity measures for HFLTSs and their application in qualitative decision making, Expert Systems with Applications28(12) (2015), 5328–5336.

15.

Liao

H.C.

, Xu

Z.S.

, Zeng

X.J.

and Merigo

J.M.

, Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets, Knowledge-Based Systems76 (2015), 127–138.

16.

Lin

, Zhao

X.F.

and Wei

G.W.

, Models for selecting an ERP system with hesitant fuzzy linguistic information, Journal of Intelligent and Fuzzy Systems26 (2014), 2155–2165.

17.

Liu

H.C.

, You

J.X.

, Fan

X.J.

and Chen

Y.Z.

, Site selection in waste management by the VIKOR method using linguistic assessment, Applied Soft Computing21(5) (2014), 453–461.

18.

Liu

P.D.

, Chen

S.M.

and Liu

J.L.

, Multiple attribute group decision making based on intuitionistic fuzzy interaction partitioned Bonferroni mean operators, Information Sciences411 (2017), 98–121.

19.

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 Applications28(5) (2017), 1079–1093.

20.

Liu

P.D.

and Liu

, Multiple attribute group decision making methods based on linguistic intuitionistic fuzzy power Bonferroni mean operators, Complexity2017 (2017), 15.

21.

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 Making16(3) (2017), 817–850.

22.

Liu

P.D.

and Teng

, An extended TODIM Method for Multiple Attribute Group Decision-making Based on 2-dimension uncertain linguistic variable, Complexity21(5) (2016), 20–30.

23.

Liu

P.D.

, He

and Yu

X.C.

, Generalized hybrid aggregation operators based on the 2-dimension uncertain linguistic information for multiple attribute group decision making, Group Decision and Negotiation25(1) (2016), 103–126.

24.

Liu

, Tao

Z.F.

, Chen

H.Y.

and Zhou

L.G.

, A new interval-valued 2-tuple linguistic bonferroni mean operator and its application to multiattribute group decision making, International Journal of Fuzzy Systems19(1) (2017), 86–108.

25.

Liu

X.Y.

, Ju

Y.B.

and Yang

S.H.

, Some generalized interval-valued hesitant uncertain linguistic aggregation operators and their applications to multiple attribute group decision making, Soft Computing20(2) (2016), 495–510.

26.

Meng

F.Y.

and Chen

X.H.

, An Approach to uncertain linguistic multi-attribute group decision making based on interactive Index, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems23(3) (2015), 319–344.

27.

Merigó

J.M.

, Palacios-Marqués

and Soto-Acosta

, Distance measures, weighted averages, OWA operators and Bonferroni means, Applied Soft Computing50 (2017), 356–366.

28.

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 Systems18(3) (2010), 287–304.

29.

Qin

J.D.

, Liu

X.W.

and Pedrycz

, Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making, Applied Soft Computing41 (2016), 428–452.

30.

Rodríguez

R.M.

, Martínez

and Herrera

, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems20(1) (2012), 109–119.

31.

Sun

and Liu

, Normalized geometric bonferroni operators of hesitant fuzzy sets and their application in multiple attribute decision making, Journal of Information and Computational Science10(9) (2013), 2815–2822.

32.

Temur

G.T.

, A novel multi attribute decision making approach for location decision under high uncertainty, Applied Soft Computing40 (2016), 674–682.

33.

Wan

S.P.

, Wang

, Lin

L.L.

and Dong

J.Y.

, Some new generalized aggregation operators for triangular intuitionistic fuzzy numbers and application to multi-attribute group decision making, Computers & Industrial Engineering93 (2016), 286–301.

34.

Wei

G.W.

, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, International Journal of Machine Learning & Cybernetics7(6) (2016), 1093–1114.

35.

Wei

G.W.

and Wei

, Model of grey relational analysis for interval multiple attribute decision making with preference information on alternatives, Chinese Journal of Management Science16(1) (2008), 158–162.

36.

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 Modeling37(7) (2013), 5277–5285.

37.

Xia

M.M.

, Xu

Z.S.

and Zhu

, Geometric Bonferroni means with their application in multi-criteria decision making, Knowledge-Based Systems40 (2013), 88–100.

38.

, Wan

S.P.

and Dong

J.Y.

, Aggregating decision information into Atanassov’s intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Applied Soft Computing41 (2016), 331–351.

39.

Z.S.

, Group decision making with triangular fuzzy linguistic variables, Intelligent Data Engineering and Automated Learning4881 (2007), 17–26.

40.

Z.S.

, Induced uncertain linguistic OWA operators applied to group decision making, Information Fusion7(2) (2006), 231–238.

41.

Z.S.

, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Information Sciences168(1–4) (2004), 171–184.

42.

Yager

R.R.

, On generalized Bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning50 (2009), 1279–1286.

43.

Yue

Z.L.

, An extended TOPSIS for determining weights of decision makers with interval numbers, Knowledge- Based Systems24(1) (2011), 146–153.

44.

Zadeh

L.A.

, The Concept of a linguistic variable and its applications to approximate reasoning-I, Information Sciences8 (1975), 199–249.

45.

Zhou

, Wang

J.Q.

, Zhang

H.Y.

and Chen

X.H.

, Linguistic hesitant fuzzy multi-criteria decision-making method based on evidential reasoning, International Journal of Systems Science47(2) (2016), 314–327.

46.

Zhou

, On Hesitant Fuzzy Reducible weighted Bonferroni mean and its generalized form for multicriteria aggregation,, Journal of Applied Mathematics2014 (2014), 10.

47.

Zhou

and He

, Intuitionistic fuzzy geometric Bonferroni means and their application in multicriteria decision making, International Journal of Intelligent Systems27 (2012), 995–1019.