Hesitant interval-valued intuitionistic fuzzy linguistic sets and their applications

Abstract

Hesitant interval-valued intuitionistic fuzzy linguistic set has been developed by extending hesitant fuzzy set, interval-valued intuitionistic fuzzy values and linguistic terms. Some operational laws and distance measures for hesitant interval-valued intuitionistic fuzzy linguistic elements are defined. Several new generalized aggregation operators are proposed for hesitant interval-valued intuitionistic fuzzy linguistic information and some special cases of new operators are studied. Two new multiple attribute decision-making methods have been proposed using new aggregation operators and TOPSIS method. Painting selection problem has been presented to illustrate new methods.

Keywords

Multiple attribute decision making hesitant interval-valued intuitionistic fuzzy linguistic set hesitant linguistic decision making

1 Introduction

Hesitant fuzzy set (HFS) was first developed by Torra [14], which is the generalization of fuzzy set. In HFS, the memberships are permitted to have several possible values. Comparing with other tools to model uncertain and fuzzy information, hesitant fuzzy set is more flexible and powerful. HFS has received broad study and application [2–15 , 17]. In some complex multiple attribute decision making problems, linguistic variables are more reliable and flexible since they can reflect human thinking’s fuzzy nature. Rodríguez et al. [11] extended the HFS to accommodate linguistic terms to propose the hesitant fuzzy linguistic term set (HFLT). Liao et al. [3, 6] presented the mathematical form of HFLT and some correlation coefficients, similarity measures. Chen [12] proposed a new multicriteria hesitant fuzzy linguistic decision making method considering decision makers’ risk attitudes. Zhang and Wu [17] introduced hesitant fuzzy linguistic set (HFLS) by using the HFS and linguistic approach. Wang et al. [8] developed interval-valued HFLS (IVHFLS) by extending interval-valued HFS to accommodate linguistic terms. Beg and Rashid [7] extended TOPSIS to accommodate hesitant fuzzy linguistic values. Liao et al. proposed some distance measures and similarity measures for HFLT in [5]. The hesitant fuzzy linguistic VIKOR method has been studied in [4]. Yang et al. [15] developed linguistic hesitant intuitionistic fuzzy set and some generalized aggregation operators. However, most of existing hesitant fuzzy linguistic methods cannot reflect membership degrees of a linguistic term belonging to a given concept.

Decision makers (DMs) like to evaluate by linguistic terms in decision making. Though several useful hesitant fuzzy linguistic sets have been developed, there still exist cases that can not be solved by existing methods. Due to the complicated decision problems, decision makers think it is proper to use several linguistic terms to evaluate, but the memberships of linguistic terms are different. Interval-valued intuitionistic fuzzy set [9] is the powerful tool to model uncertainty and hesitation. In this paper, memberships of linguistic terms are described by interval-valued intuitionistic fuzzy values (IVIFVs). For complicated decision making problems, experts from different departments and fields. They can evaluate with proper linguistic evaluation values and interval-valued fuzzy memberships. For example, one university wants to evaluate a candidate of a college dean with respect to attributes: management ability, academic ability, decision making ability. One DM thinks his/her management ability belonging to the degree of ‘very good (s₈)’ as ([0.5,0.6],[0.2,0.3]), academic ability belonging to the degree of ‘fair (s₅)’ as ([0.7,0.8],[0.1,0.2]) and decision making ability belonging to the degree of ‘slightly good (s₆)’ as ([0.6,0.7],[0.1,0.3]). Another DM thinks that the above attribute evaluation values are ‘good (s₇) ([0.6,0.7],[0.1,0.2])’, ‘slightly good ([0.7,0.8],[0.1,0.1])’, ‘fair ([0.5,0.6], [0.3,0.4])’, respectively. In order to deal with this situation, we develop hesitant interval-valued intuitionistic fuzzy linguistic set (HIVIFLS), which combines the advantages of both hesitant fuzzy sets, linguistic term sets and interval-valued intuitionistic fuzzy sets. The main advantage of HIVIFLSs is that they can model fuzziness and uncertainty by using the linguistic terms and interval-valued intuitionistic fuzzy memberships. Comparing with existing tools, the new HIVIFLS is more powerful and flexible. So in this paper, we first give the definition of the hesitant interval-valued intuitionistic fuzzy linguistic set. Then we propose two new generalized aggregation operators for hesitant interval-valued intuitionistic fuzzy linguistic information. We propose two new multiple attribute decision making methods based on new aggregation operators and TOPSIS method.

The paper is organized as follows. In Section 2, definition of HIVIFLS is given and some operational laws are defined. In Section 3, some generalized hesitant interval-valued intuitionistic fuzzy linguistic aggregation operators are developed. In Section 4, two new decision making methods considering different cases of attribute weight situations are proposed. In Section 5, an illustrative example to illustrate the new methods is presented. In the last section, the paper is concluded.

2 Hesitant interval-valued intuitionistic fuzzy linguistic set

Definition 1. [14] Let X be a reference set. An HFS A on X is defined as: $A = {< x, h_{A} (x) > ∣ x \in X},$ (1) where h_A (x) is a set having different values belonging to [0, 1]. h_A (x) is the possible membership degrees for element x ∈ X. h_A (x) represents a hesitant fuzzy element (HFE).

Definition 2. [9] Let X be a reference set and D [0, 1] be the set of all closed subintervals in [0, 1]. An interval-valued intuitionistic fuzzy set (IVIFS) $\tilde{A}$ in X is defined as: $\tilde{A} = {< x, ({\tilde{μ}}_{\tilde{A}} (x), {\tilde{ν}}_{\tilde{A}} (x)) > | x \in X},$ (2) where ${\tilde{μ}}_{\tilde{A}} : X \to D [0, 1]$ and ${\tilde{ν}}_{\tilde{A}} : X \to D [0, 1]$ are the membership and nonmembership, respectively.

Definition 3. [3] Let X be a reference set and S = {s₁, s₂, . . . , s_g} be a linguistic term set. A hesitant fuzzy linguistic term sets (HFLSs) $\bar{A}$ on X is defined as $\bar{A} = {< x_{i}, {\bar{h}}_{\bar{A}} (x_{i}) > ∣ x_{i} \in X},$ (3) where ${\bar{h}}_{\bar{A}} (x_{i}) : X \to S$ is the possible linguistic evaluation values of element x_i ∈ X.

Decision makers may want to use several linguistic terms to evaluate in decision making, but different linguistic terms have different membership degrees. The IVIFVs are used to represent the membership of linguistic terms.

Definition 4. Let X be a reference set and $\bar{S} = {s_{α} ∣ s_{0} \leq s_{α} \leq s_{g}}$ be a continuous linguistic term set. The hesitant interval-valued intuitionistic fuzzy linguistic sets (HIVIFLSs) $\overset{˘}{A}$ on X is defined as $\overset{˘}{A} = {< x_{i}, {\overset{˘}{h}}_{\overset{˘}{A}} (x_{i}) > ∣ x_{i} \in X},$ (4) where ${\overset{˘}{h}}_{\overset{˘}{A}} (x_{i}) : X \to Ω$ represents the possible interval-valued intuitionistic fuzzy linguistic evaluation values of x_i ∈ X.

${\overset{˘}{h}}_{\overset{˘}{A}} (x_{i})$ represents a hesitant interval-valued intuitionistic fuzzy linguistic element (HIVIFLE)

$\begin{matrix} {\overset{˘}{h}}_{\overset{˘}{A}} (x_{i}) & = & {(s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}], [ν_{i}^{L}, ν_{i}^{U}])) ∣ (s_{θ_{i}}, \\ ([μ_{i}^{L}, μ_{i}^{U}], [ν_{i}^{L}, ν_{i}^{U}])) \in {\overset{˘}{h}}_{\overset{˘}{A}} (x_{i})}, \end{matrix}$ (5) here $s_{θ_{i}} \in \bar{S}$ is a linguistic term and $([μ_{i}^{L}, μ_{i}^{U}], [ν_{i}^{L}, ν_{i}^{U}])$ is the IVIFV. $[μ_{i}^{L}, μ_{i}^{U}]$ and $[ν_{i}^{L}, ν_{i}^{U}]$ are the interval-valued membership and non-membership of linguistic argument s_{θ
_i} satisfying the x_i, respectively. $(s_{θ_{i}}, ([μ_{i}^{L},$ $μ_{i}^{U}],$ $[ν_{i}^{L}, ν_{i}^{U}]))$ is interval-valued intuitionistic fuzzy linguistic element (IVIFLE). If $μ_{i}^{L} = μ_{i}^{U} = μ_{i}$ and $ν_{i}^{L} = ν_{i}^{U} = ν_{i}$ , the IVIFLE reduces to the intuitionistic fuzzy linguistic element (s_{θ
_i}, (μ_i, ν_i)). Ω is the set of HIVIFLEs.

Definition 5. Let ${\overset{˘}{h}}_{1} = {{\overset{˘}{a}}_{i} ∣ {\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L},$ $μ_{i}^{U}], [ν_{i}^{L}, ν_{i}^{U}])) \in {\overset{˘}{h}}_{1}}$ , ${\overset{˘}{h}}_{2} = {{\overset{˘}{a}}_{j} ∣ {\overset{˘}{a}}_{j} = (s_{θ_{j}}, ([μ_{j}^{L}, μ_{j}^{U}],$ $[ν_{j}^{L}, ν_{j}^{U}])) \in {\overset{˘}{h}}_{2}}$ be HIVIFLEs, λ ∈ [0, 1], we have

$(1) {\overset{˘}{h}}_{1} \oplus {\overset{˘}{h}}_{2} = ⋃_{{\overset{˘}{a}}_{i} \in {\overset{˘}{h}}_{1}, {\overset{˘}{a}}_{j} \in {\overset{˘}{h}}_{2}} {(s_{(θ_{i} + θ_{j})}, ([μ_{i}^{L} + μ_{j}^{L} - μ_{i}^{L} μ_{j}^{L}, μ_{i}^{U} + μ_{j}^{U} - μ_{i}^{U} μ_{j}^{U}], [ν_{i}^{L} ν_{j}^{L}, ν_{i}^{U} ν_{j}^{U}]))}$ ,

$(2) {\overset{˘}{h}}_{1} \otimes {\overset{˘}{h}}_{2} = ⋃_{{\overset{˘}{a}}_{i} \in {\overset{˘}{h}}_{1}, {\overset{˘}{a}}_{j} \in {\overset{˘}{h}}_{2}} {(s_{θ_{i} θ_{j}}, ([μ_{i}^{L} μ_{j}^{L}, μ_{i}^{U} μ_{j}^{U}], [ν_{i}^{L} + ν_{j}^{L} - ν_{i}^{L} ν_{j}^{L}, ν_{i}^{U} + ν_{j}^{U} - ν_{i}^{U} ν_{j}^{U}]))}$ ,

$(3) λ {\overset{˘}{h}}_{1} = ⋃_{{\overset{˘}{a}}_{i} \in {\overset{˘}{h}}_{1}, {\overset{˘}{a}}_{j} \in {\overset{˘}{h}}_{2}} {(s_{λ θ_{i}}, ([1 - (1 - μ_{i}^{L})^{λ}, 1 - (1 - μ_{i}^{U})^{λ}], [(ν_{i}^{L})^{λ}, (ν_{i}^{U})^{λ}]))}$ ,

$(4) {\overset{˘}{h}}_{1}^{λ} = ⋃_{{\overset{˘}{a}}_{i} \in {\overset{˘}{h}}_{1}} {(s_{(θ_{i})^{λ}}, ([(μ_{i}^{L})^{λ}, (μ_{i}^{U})^{λ}], [1 - (1 - ν_{i}^{L})^{λ}, 1 - (1 - ν_{i}^{U})^{λ}]))}$ .

Theorem 1. Let ${\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, \overset{˘}{h}$ be HIVIFLEs, then

${\overset{˘}{h}}_{1} \oplus {\overset{˘}{h}}_{2} = {\overset{˘}{h}}_{2} \oplus {\overset{˘}{h}}_{1}$ ,

${\overset{˘}{h}}_{1} \otimes {\overset{˘}{h}}_{2} = {\overset{˘}{h}}_{2} \otimes {\overset{˘}{h}}_{1}$ ,

$λ ({\overset{˘}{h}}_{1} \oplus {\overset{˘}{h}}_{2}) = λ {\overset{˘}{h}}_{1} \oplus λ {\overset{˘}{h}}_{2}, λ \in [0, 1]$ ,

$(λ_{1} + λ_{2}) \overset{˘}{h} = λ_{1} \overset{˘}{h} \oplus λ_{2} \overset{˘}{h}, λ_{1}, λ_{2} \in [0, 1]$ ,

$({\overset{˘}{h}}_{1} \otimes {\overset{˘}{h}}_{2})^{λ} = {\overset{˘}{h}}_{1}^{λ} \otimes {\overset{˘}{h}}_{2}^{λ}, λ \in [0, 1]$ ,

${\overset{˘}{h}}^{λ_{1} + λ_{2}} = {\overset{˘}{h}}^{λ_{1}} \otimes {\overset{˘}{h}}^{λ_{2}}, λ_{1}, λ_{2} \in [0, 1]$ .

Definition 6. Let ${\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}], [ν_{i}^{L}, ν_{i}^{U}]))$ be an IVIFLE. g is the linguistic term number in S. We define the score function $s ({\overset{˘}{a}}_{i})$ of element ${\overset{˘}{a}}_{i}$ as $s ({\overset{˘}{a}}_{i}) = \frac{θ_{i}}{2 g} (μ_{i}^{L} + μ_{i}^{U} - ν_{i}^{L} - ν_{i}^{U}),$ the accuracy function $h ({\overset{˘}{a}}_{i})$ of element ${\overset{˘}{a}}_{i}$ as $h ({\overset{˘}{a}}_{i}) = \frac{θ_{i}}{2 g} (μ_{i}^{L} + μ_{i}^{U} + ν_{i}^{L} + ν_{i}^{U}),$ the membership uncertainty function $t ({\overset{˘}{a}}_{i})$ of element ${\overset{˘}{a}}_{i}$ as $t ({\overset{˘}{a}}_{i}) = \frac{θ_{i}}{2 g} (μ_{i}^{U} + ν_{i}^{L} - μ_{i}^{L} - ν_{i}^{U}),$ and the hesitation uncertainty function $g ({\overset{˘}{a}}_{i})$ of element ${\overset{˘}{a}}_{i}$ as $g ({\overset{˘}{a}}_{i}) = \frac{θ_{i}}{2 g} (μ_{i}^{U} + ν_{i}^{U} - μ_{i}^{L} - ν_{i}^{L}) .$

Then we present the following method to compare two IVIFLEs.

Definition 7. Let ${\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}], [ν_{i}^{L}, ν_{i}^{U}]))$ and ${\overset{˘}{a}}_{j} = (s_{θ_{j}}, ([μ_{j}^{L}, μ_{j}^{U}], [ν_{j}^{L}, ν_{j}^{U}]))$ be two IVIFLEs, then

if $s ({\overset{˘}{a}}_{i}) < s ({\overset{˘}{a}}_{j})$ , then ${\overset{˘}{a}}_{i} < {\overset{˘}{a}}_{j}$ ,

if $s ({\overset{˘}{a}}_{i}) = s ({\overset{˘}{a}}_{j})$ , and

if $h ({\overset{˘}{a}}_{i}) < h ({\overset{˘}{a}}_{j})$ , then ${\overset{˘}{a}}_{i} < {\overset{˘}{a}}_{j}$ ,

if $h ({\overset{˘}{a}}_{i}) = h ({\overset{˘}{a}}_{j})$ , then

if $t ({\overset{˘}{a}}_{i}) < t ({\overset{˘}{a}}_{j})$ , then ${\overset{˘}{a}}_{i} > {\overset{˘}{a}}_{j}$ ,

if $t ({\overset{˘}{a}}_{i}) = t ({\overset{˘}{a}}_{j})$ , then

if $g ({\overset{˘}{a}}_{i}) < g ({\overset{˘}{a}}_{j})$ , then ${\overset{˘}{a}}_{i} > {\overset{˘}{a}}_{j}$ ,

if $g ({\overset{˘}{a}}_{i}) = g ({\overset{˘}{a}}_{j})$ and s_θ(i) < s_θ(j), then ${\overset{˘}{a}}_{i} <$ ${\overset{˘}{a}}_{j}$ ,

if $g ({\overset{˘}{a}}_{i}) = g ({\overset{˘}{a}}_{j})$ and s_θ(i) = s_θ(j), then ${\overset{˘}{a}}_{i} =$ ${\overset{˘}{a}}_{j}$ .

Definition 8. Let $\overset{˘}{h} = {{\overset{˘}{a}}_{i} ∣ {\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}], [ν_{i}^{L},$ $ν_{i}^{U}])) \in \tilde{h}}$ be HIVIFLE, g be the cardinality of linguistic set S and l be the count of IVIFLEs in $\overset{˘}{h}$ . The expectation function $E (\overset{˘}{h})$ of $\overset{˘}{h}$ is defined as $E (\overset{˘}{h}) = \frac{1}{l} \sum_{i = 1}^{l} \frac{θ_{i}}{2 g} (μ_{i}^{L} - ν_{i}^{L} + μ_{i}^{U} - ν_{i}^{U}) .$ The variance function $V (\overset{˘}{h})$ of $\overset{˘}{h}$ is defined as $V (\overset{˘}{h}) = \frac{1}{l} \sum_{i = 1}^{l} (\frac{θ_{i}}{2 g} (μ_{i}^{L} + μ_{i}^{U} + ν_{i}^{L} + ν_{i}^{U}) - E (\overset{˘}{h}))^{2} .$

Based on the expectation function $E (\overset{˘}{h})$ and the variance function $V (\overset{˘}{h})$ of HIVIFLEs, we present the method to compare them as follows. Let ${\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}$ be two HIVIFLEs, then

if $E ({\overset{˘}{h}}_{1}) > E ({\overset{˘}{h}}_{2})$ , then ${\overset{˘}{h}}_{1} > {\overset{˘}{h}}_{2}$ ,

if $E ({\overset{˘}{h}}_{1}) = E ({\overset{˘}{h}}_{2})$ , then

if $V ({\overset{˘}{h}}_{1}) > V ({\overset{˘}{h}}_{2})$ , then ${\overset{˘}{h}}_{1} < {\overset{˘}{h}}_{2}$ ,

if $V ({\overset{˘}{h}}_{1}) = V ({\overset{˘}{h}}_{2})$ , then ${\overset{˘}{h}}_{1} \sim {\overset{˘}{h}}_{2}$ .

Different HIVIFLEs may have different number of IVIFLEs. To define distance measures more accurately, we extend the HIVIFLEs until they have the same number of IVIFLEs. The HIVIFLEs can be extended according to DMs’ risk attitude. If DMs are risk- seek, the largest IVIFLE can be added. If DMs are risk-averse, the smallest IVIFLE can be added. If DMs are risk-neutral, the average value of IVIFLEs can be added.

The distance measures are very important since they are the basis of many well-known methods including TOPSIS, VIKOR, ELECTRE. We develop Euclidean distance for HIVIFLEs.

Definition 9. Let ${\overset{˘}{h}}_{1} = {{\overset{˘}{a}}_{i}^{(1)}} = {(s_{θ_{i}^{(1)}}, ([μ_{i}^{L (1)},$ $μ_{i}^{U (1)}], [ν_{i}^{L (1)}, ν_{i}^{U (1)}]))}$ , ${\overset{˘}{h}}_{2} = {{\overset{˘}{a}}_{i}^{(2)}} = {(s_{θ_{i}^{(2)}}, ([μ_{i}^{L (2)}, μ_{i}^{U (2)}], [ν_{i}^{L (2)}, ν_{i}^{U (2)}]))}$ be two HIVIFLEs, then the Euclidean distance $d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2})$ between ${\overset{˘}{h}}_{1}$ and ${\overset{˘}{h}}_{2}$ is defined as

$\begin{matrix} d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}) \\ = (\frac{1}{5 l} \sum_{i = 1}^{l} (∣ θ_{σ (i)}^{(1)} - θ_{σ (i)}^{(2)} ∣^{2} / g^{2} + ∣ μ_{σ (i)}^{L (1)} \\ - μ_{σ (i)}^{L (2)} ∣^{2} + ∣ μ_{σ (i)}^{U (1)} - μ_{σ (i)}^{U (2)} ∣^{2} + ∣ ν_{σ (i)}^{L (1)} \\ - ν_{σ (i)}^{L (2)} ∣^{2} + ∣ ν_{σ (i)}^{U (1)} - ν_{σ (i)}^{U (2)} ∣^{2}))^{1 / 2}, \end{matrix}$ (6) where l is the count of IVIFLEs in HIVIFLE and g is the linguistic term number in linguistic set S. ${\overset{˘}{a}}_{σ (i)}^{(j)} = (s_{θ_{σ (i)}^{(j)}}, ([μ_{σ (i)}^{L (j)}, μ_{σ (i)}^{U (j)}], [ν_{σ (i)}^{L (j)}, ν_{σ (i)}^{U (j)}]))$ is the ith largest ${\overset{˘}{a}}_{k}^{(j)} (k = 1, 2, . . ., l)$ . The distance for extended HIVIFLEs ${\overset{˘}{h}}_{j}$ has the following properties:

$0 \leq d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}) \leq 1,$

$d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}) = d ({\overset{˘}{h}}_{2}, {\overset{˘}{h}}_{1})$ ,

$d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}) = 0$ if and only if ${\overset{˘}{h}}_{1} = {\overset{˘}{h}}_{2}$ ,

$d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}) \leq d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{3}) + d ({\overset{˘}{h}}_{3}, {\overset{˘}{h}}_{2})$ .

Proof. We only prove the triangle inequality. $\begin{matrix} d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}) \\ = (\frac{1}{5 l} \sum_{i = 1}^{l} (\frac{1}{g^{2}} ∣ θ_{σ (i)}^{(1)} - θ_{σ (i)}^{(2)} ∣^{2} + ∣ μ_{σ (i)}^{L (1)} \\ - μ_{σ (i)}^{L (2)} ∣^{2} + ∣ μ_{σ (i)}^{U (1)} - μ_{σ (i)}^{U (2)} ∣^{2} + ∣ ν_{σ (i)}^{L (1)} \\ - ν_{σ (i)}^{L (2)} ∣^{2} + ∣ ν_{σ (i)}^{U (1)} - ν_{σ (i)}^{U (2)} ∣^{2}))^{\frac{1}{2}} \\ = (\frac{1}{5 l} \sum_{i = 1}^{l} (\frac{1}{g^{2}} ∣ (θ_{σ (i)}^{(1)} - θ_{σ (i)}^{(3)}) + (θ_{σ (i)}^{(3)} - θ_{σ (i)}^{(2)}) ∣^{2} \\ + ∣ (μ_{σ (i)}^{L (1)} - μ_{σ (i)}^{L (3)}) + (μ_{σ (i)}^{L (3)} - μ_{σ (i)}^{L (2)}) ∣^{2} \\ + ∣ (μ_{σ (i)}^{U (1)} - μ_{σ (i)}^{U (3)}) + (μ_{σ (i)}^{U (3)} - μ_{σ (i)}^{U (2)}) ∣^{2} \\ + ∣ (ν_{σ (i)}^{L (1)} - ν_{σ (i)}^{L (3)}) + (ν_{σ (i)}^{L (3)} - ν_{σ (i)}^{L (2)}) ∣^{2} \\ + ∣ (ν_{σ (i)}^{U (1)} - ν_{σ (i)}^{U (3)}) + (ν_{σ (i)}^{U (3)} - ν_{σ (i)}^{U (2)}) ∣^{2}))^{\frac{1}{2}} \\ \leq \frac{1}{\sqrt{5 l}} (\sum_{i = 1}^{l} (\frac{1}{g} ∣ θ_{σ (i)}^{(1)} - θ_{σ (i)}^{(3)} ∣ + \frac{1}{g} ∣ θ_{σ (i)}^{(3)} - θ_{σ (i)}^{(2)} ∣)^{2} \\ + \sum_{i = 1}^{l} (∣ μ_{σ (i)}^{L (1)} - μ_{σ (i)}^{L (3)} ∣ + ∣ μ_{σ (i)}^{L (3)} - μ_{σ (i)}^{L (2)} ∣)^{2} \\ + \sum_{i = 1}^{l} (∣ μ_{σ (i)}^{U (1)} - μ_{σ (i)}^{U (3)} ∣ + ∣ μ_{σ (i)}^{U (3)} - μ_{σ (i)}^{U (2)} ∣)^{2} \\ + \sum_{i = 1}^{l} (∣ ν_{σ (i)}^{L (1)} - ν_{σ (i)}^{L (3)} ∣ + ∣ ν_{σ (i)}^{L (3)} - ν_{σ (i)}^{L (2)} ∣)^{2} \\ + \sum_{i = 1}^{l} (∣ ν_{σ (i)}^{U (1)} - ν_{σ (i)}^{U (3)} ∣ + ∣ ν_{σ (i)}^{U (3)} - ν_{σ (i)}^{U (2)} ∣)^{2})^{\frac{1}{2}} . \end{matrix}$

Let $b_{5 i - 4} = \frac{1}{g} ∣ θ_{σ (i)}^{(1)} - θ_{σ (i)}^{(3)} ∣, c_{5 i - 4} = \frac{1}{g} ∣ θ_{σ (i)}^{(3)} - θ_{σ (i)}^{(2)} ∣,$ $b_{5 i - 3} = ∣ μ_{σ (i)}^{L (1)} - μ_{σ (i)}^{L (3)} ∣,$ $c_{5 i - 3} = ∣ μ_{σ (i)}^{L (3)} - μ_{σ (i)}^{L (2)} ∣,$ $b_{5 i - 2} = ∣ μ_{σ (i)}^{U (1)} - μ_{σ (i)}^{U (3)} ∣,$ $c_{5 i - 2} = ∣ μ_{σ (i)}^{U (3)} - μ_{σ (i)}^{U (2)} ∣,$ $b_{5 i - 1} = ∣ ν_{σ (i)}^{L (1)} - ν_{σ (i)}^{L (3)} ∣,$ $c_{5 i - 1} = ∣ ν_{σ (i)}^{L (3)} - ν_{σ (i)}^{L (2)} ∣,$ $b_{5 i} = ∣ ν_{σ (i)}^{U (1)} - ν_{σ (i)}^{U (3)} ∣,$ $c_{5 i} = ∣ ν_{σ (i)}^{U (3)} - ν_{σ (i)}^{U (2)} ∣,$ i = 1, 2, ·· · , l. By using Minkowski inequality, we can get $\begin{matrix} d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}) \\ \leq \frac{1}{\sqrt{5 l}} {(\sum_{m = 1}^{5 l} (b_{m} + c_{m})^{2})}^{\frac{1}{2}} \\ \leq \frac{1}{\sqrt{5 l}} {(\sum_{m = 1}^{5 l} b_{m}^{2})}^{\frac{1}{2}} + \frac{1}{\sqrt{5 l}} {(\sum_{m = 1}^{5 l} c_{m}^{2})}^{\frac{1}{2}} \\ = \frac{1}{\sqrt{5 l}} (\sum_{i = 1}^{l} (\frac{1}{g^{2}} ∣ θ_{σ (i)}^{(1)} - θ_{σ (i)}^{(3)} ∣^{2} \\ + ∣ μ_{σ (i)}^{L (1)} - μ_{σ (i)}^{L (3)} ∣^{2} + ∣ μ_{σ (i)}^{U (1)} - μ_{σ (i)}^{U (3)} ∣^{2} \\ {+ ∣ ν_{σ (i)}^{L (1)} - ν_{σ (i)}^{L (3)} ∣^{2} + ∣ ν_{σ (i)}^{U (1)} - ν_{σ (i)}^{U (3)} ∣^{2}))}^{\frac{1}{2}} \\ + \frac{1}{\sqrt{5 l}} (\sum_{i = 1}^{l} (\frac{1}{g^{2}} ∣ θ_{σ (i)}^{(3)} - θ_{σ (i)}^{(2)} ∣^{2} \\ + ∣ μ_{σ (i)}^{L (3)} - μ_{σ (i)}^{L (2)} ∣^{2} + ∣ μ_{σ (i)}^{U (3)} - μ_{σ (i)}^{U (2)} ∣^{2} \\ {+ ∣ ν_{σ (i)}^{L (3)} - ν_{σ (i)}^{L (2)} ∣^{2} + ∣ ν_{σ (i)}^{U (3)} - ν_{σ (i)}^{U (2)} ∣^{2}))}^{\frac{1}{2}} \\ = d ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{3}) + d ({\overset{˘}{h}}_{3}, {\overset{˘}{h}}_{2}) . \end{matrix}$

3 Generalized hesitant interval-valued intuitionistic fuzzy linguistic aggregation operators

Definition 10. Let ${\overset{˘}{h}}_{i} = {{\overset{˘}{a}}_{i} ∣ {\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}],$ $[ν_{i}^{L}, ν_{i}^{U}])) \in {\overset{˘}{h}}_{i}} (i = 1, 2, . . ., n)$ be HIVIFLEs. A generalized hesitant interval-valued intuitionistic fuzzy linguistic ordered weighted averaging (GHIVIFLOWA) operator is defined as: Ωⁿ → Ω

$\begin{matrix} GHIVIFLOWA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = (w_{1} {\overset{˘}{h}}_{σ (1)}^{λ} \oplus w_{2} {\overset{˘}{h}}_{σ (2)}^{λ} \oplus . . . \oplus w_{n} {\overset{˘}{h}}_{σ (n)}^{λ})^{1 / λ}, \end{matrix}$ (7) where λ > 0, ${\overset{˘}{h}}_{σ (j)}$ is the jth largest ${\overset{˘}{h}}_{j}$ , w_j (j = 1, 2, . . . , n) are weights satisfying w_j ≥ 0 and $\sum_{j = 1}^{n} w_{j}$ = 1.

Theorem 2. Let ${\overset{˘}{h}}_{i} = {{\overset{˘}{a}}_{i} ∣ {\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}],$ $[ν_{i}^{L}, ν_{i}^{U}])) \in {\overset{˘}{h}}_{i}}$ be HIVIFLEs. The value of GHIVIFLOWA operator is an HIVIFLE and $\begin{matrix} GHIVIFLOWA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = ⋃_{{\overset{˘}{a}}_{σ (i)} \in {\overset{˘}{h}}_{σ (i)}} {(s_{(\sum_{j = 1}^{n} w_{j} (θ_{σ (j)})^{λ})^{1 / λ}}, \\ ([(1 - \prod_{j = 1}^{n} (1 - (μ_{σ (j)}^{L})^{λ})^{w_{j}})^{1 / λ}, \\ (1 - \prod_{j = 1}^{n} (1 - (μ_{σ (j)}^{U})^{λ})^{w_{j}})^{1 / λ}], \\ [1 - (1 - \prod_{j = 1}^{n} (1 - (1 - ν_{σ (j)}^{L})^{λ})^{w_{j}})^{1 / λ}, \\ 1 - (1 - \prod_{j = 1}^{n} (1 - (1 - ν_{σ (j)}^{U})^{λ})^{w_{j}})^{1 / λ}]))} . \end{matrix}$

Theorem 3. (Commutativity) Let ${\overset{˘}{h}}_{i}$ and ${\overset{˘}{h}}_{i}^{'} (i = 1, 2, . . ., n)$ be HIVIFLEs. If $({\overset{˘}{h}}_{1}^{'}, {\overset{˘}{h}}_{2}^{'}, . . ., {\overset{˘}{h}}_{n}^{'})$ is any permutation of $({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2},$ $. . ., {\overset{˘}{h}}_{n})$ , then $GHIVIFLOWA ({\overset{˘}{h}}_{1},$ ${\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) = GHIVIFLOWA ({\overset{˘}{h}}_{1}^{'}, {\overset{˘}{h}}_{2}^{'}, . . ., {\overset{˘}{h}}_{n}^{'}) .$

Theorem 4. (Idempotency) Let ${\overset{˘}{h}}_{j} (j = 1, 2, . . ., n)$ be HIVIFLEs. If all ${\overset{˘}{h}}_{j}$ are equal, ${\overset{˘}{h}}_{i} = \overset{˘}{h}$ , ${\overset{˘}{a}}_{i} = \overset{˘}{a} = (s_{θ}, ([μ^{L}, μ^{U}], [ν^{L}, ν^{U}])) \in \overset{˘}{h}$ , then $GHIVIFLOWA ({\overset{˘}{h}}_{1},$ ${\overset{˘}{h}}_{2},$ $. . ., {\overset{˘}{h}}_{n}) = \overset{˘}{h} .$

Theorem 5. (Boundary) Let ${\overset{˘}{h}}_{j} (j = 1, 2, . . ., n)$ be HIVIFLEs, ${\overset{˘}{h}}^{+} = {(s_{α}^{+}, ([μ^{L +}, μ^{U +}],$ $[ν^{L +}, ν^{U +}]))} = {(max_{i} s_{α_{i}}, ([max_{i} μ_{i}^{L}, max_{i} μ_{i}^{U}],$ $[min_{i} ν_{i}^{L}, min_{i} ν_{i}^{U}]))}$ and ${\overset{˘}{h}}^{-} = {(min_{i} s_{α_{i}},$ $([min_{i} μ_{i}^{L},$ $min_{i} μ_{i}^{U}], [max_{i} ν_{i}^{L},$ $max_{i} ν_{i}^{U}]))} .$ Then ${\overset{˘}{h}}^{-} \leq GHIVIFLOWA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n})$ $\leq {\overset{˘}{h}}^{+} .$

We investigate several special cases of the GHIVIFLOWA operator.

If λ = 1, the GHIVIFLOWA operator degenerates to the hesitant interval-valued intuitionistic fuzzy linguistic ordered weighted averaging (HIVIFLOWA) operator $\begin{matrix} HIVIFLOWA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = w_{1} {\overset{˘}{h}}_{σ (1)} \oplus w_{2} {\overset{˘}{h}}_{σ (2)} \oplus . . . \oplus w_{n} {\overset{˘}{h}}_{σ (n)} . \end{matrix}$

If $μ_{j}^{L} = μ_{j}^{U} = μ_{j}$ and $ν_{j}^{L} = ν_{j}^{U} = ν_{j}$ in this case, HIVIFLEs reduce to the hesitant intuitionistic fuzzy linguistic elements ${\overset{˘}{h}}_{i} = {(s_{θ_{i}}, (μ_{i}, ν_{i}))} (i = 1, 2, . . ., n)$ . The HIVIFLOWA operator reduces to the hesitant intuitionistic fuzzy linguistic ordered weighted averaging (HIFLOWA) operator as follows $\begin{matrix} HIFLOWA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = ⋃_{(s_{θ_{σ (i)}}, (μ_{σ (i)}, ν_{σ (i)})) \in {\overset{˘}{h}}_{σ (i)}} {(s_{(\sum_{j = 1}^{n} w_{j} θ_{σ (j)})}, \\ (1 - \prod_{j = 1}^{n} (1 - μ_{σ (j)})^{w_{j}}, \prod_{j = 1}^{n} (ν_{σ (j)})^{w_{j}}))} . \end{matrix}$

Definition 11. Let ${\overset{˘}{h}}_{i} = {{\tilde{a}}_{i} ∣ {\tilde{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}],$ $[ν_{i}^{L}, ν_{i}^{U}])) \in {\overset{˘}{h}}_{i}} (i = 1, 2, . . ., n)$ be HIVIFLEs. A generalized hesitant interval-valued intuitionistic fuzzy linguistic ordered weighted geometric averaging (GHIVIFLOWGA) operator is a mapping:Ωⁿ → Ω

$\begin{matrix} GHIVIFLOWGA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = \frac{1}{λ} (\otimes (λ {\overset{˘}{h}}_{σ (j)})^{w_{j}}), \end{matrix}$ (8) where λ > 0, ${\overset{˘}{h}}_{σ (j)}$ is the jth largest ${\overset{˘}{h}}_{j}$ , (w₁, w₂, . . . , w_n) is the weight vector with w_j ≥ 0 and $\sum_{j = 1}^{n} w_{j} = 1$ .

Theorem 6. Let ${\overset{˘}{h}}_{i} = {{\overset{˘}{a}}_{i} ∣ {\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}],$ $[ν_{i}^{L}, ν_{i}^{U}])) \in {\overset{˘}{h}}_{i}} (i = 1, 2, . . ., n)$ be HIVIFLEs. The value of GHIVIFLOWGA operator is an HIVIFLE $\begin{matrix} GHIVIFLOWGA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = ⋃_{{\overset{˘}{a}}_{σ (i)} \in {\overset{˘}{h}}_{σ (i)}} {(s_{\prod_{j = 1}^{n} (θ_{σ (j)})^{w_{j}}}, ([1 - (1 - \prod_{j = 1}^{n} \\ (1 - (1 - μ_{σ (j)}^{L})^{λ})^{w_{j}})^{1 / λ}, 1 - (1 - \prod_{j = 1}^{n} \\ (1 - (1 - μ_{σ (j)}^{U})^{λ})^{w_{j}})^{1 / λ}], \\ [(1 - \prod_{j = 1}^{n} (1 - (ν_{σ (j)}^{L})^{λ})^{w_{j}})^{1 / λ}, \\ (1 - \prod_{j = 1}^{n} (1 - (ν_{σ (j)}^{U})^{λ})^{w_{j}})^{1 / λ}]))} . \end{matrix}$

The GHIVIFLOWGA operator is also commutative, bounded, idempotent. These properties are similar to that of the GHIVIFLOWA operator and are omitted.

If λ = 1, the GHIVIFLOWGA operator becomes the hesitant interval-valued intuitionistic fuzzy linguistic ordered weighted geometric averaging (HIVIFLOWGA) operator $\begin{matrix} HIVIFLOWGA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = ({\overset{˘}{h}}_{σ (1)})^{w_{1}} \otimes ({\overset{˘}{h}}_{σ (2)})^{w_{2}} \otimes . . . \otimes ({\overset{˘}{h}}_{σ (n)})^{w_{n}} . \end{matrix}$

If λ = 1, $μ_{j}^{L} = μ_{j}^{U} = μ_{j}$ and $ν_{j}^{L} = ν_{j}^{U} = ν_{j}$ , j = 1, 2, . . . , n, then ${\overset{˘}{h}}_{i} = {(s_{θ_{i}}, (μ_{i}, ν_{i}))}$ and the HIVIFLOWGA operator becomes the following hesitant intuitionistic fuzzy linguistic ordered weighted geometric averaging (HIFLOWGA) operator $\begin{matrix} HIFLOWGA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = ⋃_{(s_{θ_{σ (i)}}, (μ_{σ (i)}, ν_{σ (i)})) \in {\overset{˘}{h}}_{σ (i)}} {(s_{\prod_{j = 1}^{n} (θ_{σ (j)})^{w_{j}}}, \\ (\prod_{j = 1}^{n} (μ_{σ (j)})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - ν_{σ (j)})^{w_{j}}))} . \end{matrix}$

Definition 12. Let ${\overset{˘}{h}}_{i} = {{\overset{˘}{a}}_{i} ∣ {\overset{˘}{a}}_{i} = (s_{θ_{i}}, ([μ_{i}^{L}, μ_{i}^{U}],$ $[ν_{i}^{L}, ν_{i}^{U}])) \in {\overset{˘}{h}}_{i}} (i = 1, 2, . . ., n)$ be HIVIFLEs. A quasi-arithmetic hesitant interval-valued intuitionistic fuzzy linguistic ordered weighted averaging (QHIVIFLOWA) operator is a mapping: Ωⁿ → Ω

$\begin{matrix} QHIVIFLOWA ({\overset{˘}{h}}_{1}, {\overset{˘}{h}}_{2}, . . ., {\overset{˘}{h}}_{n}) \\ = g^{- 1} (\oplus_{j = 1}^{n} w_{j} g ({\overset{˘}{h}}_{σ (j)})), \end{matrix}$ (9) where λ > 0, ${\overset{˘}{h}}_{σ (j)}$ is the jth largest ${\overset{˘}{h}}_{j}$ , w_j (j = 1, 2, . . . , n) are the weights satisfying w_j ≥ 0 and $\sum_{j = 1}^{n} w_{j} = 1$ , g (x) is a strictly monotonic continuous function.

4 Decision making with the new aggregation operators

Let {A₁, A₂, . . . , A_m} be alternative set, {C₁, C₂, . . . , C_n} be attribute set and {E₁, E₂, . . . , E_s} be expert set. The experts give hesitant interval-valued intuitionistic fuzzy linguistic information by evaluating the alternatives with IVIFLEs. The decision matrix is formed as $D = ({\overset{˘}{h}}_{ij})_{m \times n}$ , where ${\overset{˘}{h}}_{ij} = {(s_{θ_{ij}^{t}}, ([μ_{ij}^{Lt}, μ_{ij}^{Ut}], [ν_{ij}^{Lt}, ν_{ij}^{Ut}]))}$ is IVIFLE. The weights of attributes are w_j satisfying w_j ≥ 0 and $\sum_{j = 1}^{n} w_{j} = 1$ .

In the process of decision making, there are situations that attribute weights are known partly. According to information theory, an attribute should have a large weight if it has large deviation value and it should have a small weight if it has small deviation value. We can determine attribute weights by using maximum deviation method. Extend the HIVIFLEs until they have the same number of IVIFLEs according to decision makers’ risk attitude. Then the deviation value d_j (j = 1, 2, . . . , n) of evaluation values for attribute C_j is defined as $d_{j} = \sum_{k = 1}^{m} \sum_{i = 1}^{m} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj}) .$ (10)

Then the weighted deviation value of all the attributes is calculated as $d = \sum_{j = 1}^{n} w_{j} d_{j} = \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sum_{i = 1}^{m} w_{j} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj}) .$ (11)

Reasonable weight vector (w₁, w₂, . . . , w_n) should make the deviation value as large as possible. Then the following model (M-1) is set up $\begin{matrix} (M - 1) & max \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sum_{i = 1}^{m} w_{j} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj}) \\ s . t . w \in H, w_{j} \geq 0, j = 1, 2, . . ., n, \\ w_{1} + w_{2} + . . . + w_{n} = 1 . \end{matrix}$

The model (M-1) is a linear programming model, which can be solved easily by using many existing methods.

If the weights are completely unknown, we construct a non-linear programming model as $\begin{matrix} (M - 2) & max \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sum_{i = 1}^{m} w_{j} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj}) \\ s . t . \sum_{j = 1}^{n} w^{2} = 1, \\ w_{j} \geq 0, j = 1, 2, . . ., n . \end{matrix}$

The Lagrange function L (w, ξ) is defined to solve the model,

$\begin{matrix} L (w, ξ) & = & \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sum_{i = 1}^{m} w_{j} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj}) \\ + \frac{ξ}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1), \end{matrix}$ (12) where ξ is the Lagrange multiplier variable. Calculate the partial derivatives of L (w, ξ) and let partial derivatives be zero to get ${\begin{matrix} \frac{\partial L}{\partial w_{j}} = \sum_{k = 1}^{m} \sum_{i = 1}^{m} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj}) + ξ w_{j} = 0, \\ \frac{\partial L}{\partial ξ} = \frac{1}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1) = 0 . \end{matrix}$ (13)

By solving the Equation (13) to get $w_{j} = \frac{\sum_{k = 1}^{m} \sum_{i = 1}^{m} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj})}{\sqrt{\sum_{j = 1}^{n} (\sum_{k = 1}^{m} \sum_{i = 1}^{m} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{kj}))^{2}}} .$ (14)

Normalize w_j (j = 1, 2, . . . , n) using the equation as follows to get $w_{j}^{'} (j = 1, 2, . . ., n)$ $w_{j}^{'} = \frac{w_{j}}{\sum_{j = 1}^{n} w_{j}} .$ (15)

We give new algorithms for hesitant interval-valued intuitionistic fuzzy linguistic information considering different attribute weight situations.

Algorithm I

Step 1. The alternative A_i is evaluated with respect to the attribute C_j by using IVIFLE to get hesitant interval-valued intuitionistic fuzzy linguistic evaluation value ${\overset{˘}{h}}_{ij}$ and the decision matrix is got as $D = ({\overset{˘}{h}}_{ij})_{m \times n}$ .

Step 2. (w₁, w₂, . . . , w_n) is attribute weight vector satisfying w_j ≥ 0, $\sum_{j = 1}^{n} w_{j} = 1$ . If attribute weights are known exactly, go to next step directly. If attribute weights are known partly, solve model (M-1) to determine them. If attribute weights are unknown completely, Equations (14)–(15) are used to calculate.

Step 3. Aggregate alternatives’ evaluation values into collective ones by using the GHIVIFLOWA operator or the GHIVIFLOWGA operator.

Step 4. Calculate the expectation function $E ({\overset{˘}{h}}_{j})$ and the variance function $V ({\overset{˘}{h}}_{j})$ by using Definition 8. Rank alternatives’ collective evaluation values using the method of Definition 8.

Step 5. Rank alternatives according to alternative collective evaluation values’ ranking.

TOPSIS method was first developed by Hwang and Yoon [1], which has been studied and applied extensively. We generalize the TOPSIS method to accommodate hesitant interval-valued intuitionistic fuzzy linguistic information. The main steps are as follows.

Algorithm II

Step 1. As for algorithm I.

Step 2. As for algorithm I.

Step 3. Determine hesitant interval-valued intuitionistic fuzzy linguistic positive ideal solution (HIVIFLPIS) as

$\begin{matrix} {\overset{˘}{h}}^{+} & = & ({\overset{˘}{h}}_{1}^{+}, {\overset{˘}{h}}_{2}^{+}, . . ., {\overset{˘}{h}}_{n}^{+}) \\ = & (max {\overset{˘}{h}}_{i 1}, max {\overset{˘}{h}}_{i 2}, . . ., max {\overset{˘}{h}}_{in}), \end{matrix}$ (16) and hesitant interval-valued intuitionistic fuzzy linguistic negative ideal solution (HIVIFLNIS) as

$\begin{matrix} {\overset{˘}{h}}^{-} & = & ({\overset{˘}{h}}_{1}^{-}, {\overset{˘}{h}}_{2}^{-}, . . ., {\overset{˘}{h}}_{n}^{-}) \\ = & (min {\overset{˘}{h}}_{i 1}, min {\overset{˘}{h}}_{i 2}, . . ., min {\overset{˘}{h}}_{in}) . \end{matrix}$ (17)

Step 4. Calculate the weighted distances of each alternative hesitant interval-valued intuitionistic fuzzy linguistic evaluation values to HIVIFLPIS and HIVIFLNIS, respectively. $d_{i}^{+} = \sum_{j = 1}^{n} w_{j} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{j}^{+}),$ (18) $d_{i}^{-} = \sum_{j = 1}^{n} w_{j} d ({\overset{˘}{h}}_{ij}, {\overset{˘}{h}}_{j}^{-}) .$ (19)

Step 5. Calculate the relative closeness coefficient CC_i (i = 1, 2, . . . , m) of alternatives as ${CC}_{i} = \frac{d_{i}^{-}}{d_{i}^{+} + d_{i}^{-}} .$ (20)

Step 6. Rank alternatives according to the ranking of CC_i. Select the optimal alternative with the largest CC_i.

5 Numerical example

A numerical example adapted from Yang and Chen [16] is used to illustrate the feasibility and practical advantages of new methods. There are five paintings in Xi’an subway are selected to evaluate: A₁-Feng Ming Chao Yang, A₂-Qin Ling Si Bao, A₃-Da Qin Qiang, A₄-Ying Bin Tu, A₅-Ji Xiang Zhuan Diao. The following attributes are considered: C₁-affinity, C₂-artistry, C₃-diversity, C₄-acceptability. Linguistic terms are as s₁-extremely poor, s₂-very poor, s₃-poor, s₄-slightly poor, s₅-fair, s₆-slightly good, s₇-good, s₈-very good, s₉-extremely good.

Step 1. Multiple decision makers evaluate alternatives with linguistic terms and interval-valued intuitionistic fuzzy values. Hesitant interval-valued intuitionistic fuzzy linguistic decision matrix $D = ({\overset{˘}{h}}_{ij})_{4 \times 5}$ is formed as in Table 1.

Step 2. Assume that the attribute weights are known exactly as W₁ = (0.20, 0.30, 0.35, 0.15). Go to next step directly.

Step 3. We can aggregate the evaluation values by using the GHIVIFLOWA operator or the GHIVIFLOWGA operator for different λ. For example, if λ = 1, we use the HIVIFLOWA operator. The collective evaluation value ${\overset{˘}{h}}_{1}$ for alternative A₁ can be got as ${\overset{˘}{h}}_{1} = {(s_{5.05}, ([0.6001, 0.7268], [0.1366,$ 0.2400]) , (s_5.25, ([0.6260, 0.7282] , [0.1569, 0.2603]))} . Other collective evaluation values can be got similarly.

Step 4. Calculate expectations of ${\overset{˘}{h}}_{i}$ got by using the HIVIFLOWA operator as $E ({\overset{˘}{h}}_{1}) = 0.2390, E ({\overset{˘}{h}}_{2}) = 0.2095, E ({\overset{˘}{h}}_{3}) = 0.2636, E ({\overset{˘}{h}}_{4}) = 0.2158, E ({\overset{˘}{h}}_{5}) = 0.2644 .$ If other aggregation operators are used, we calculate as above and results are also shown in Table 2.

Step 5. According to the ranking of expectations, we can rank alternatives. The results are also shown in Table 2.

If attribute weights are known partly as shown as $H = {0.2 \leq w_{1} \leq 0.30, 0.15 \leq w_{2} \leq 0.20, 0.10 \leq w_{3} \leq 0.20, 0.10 \leq w_{4} \leq 0.35, 2 w_{3} \leq w_{4}, w_{1} + w_{2} + w_{3} + w_{4} = 1} .$ We first extend the decision matrix to calculate distances more accurately. Assume decision makers are risk averse, then the smallest IVIFLE can be added until all HIVIFLEs having the same number of IVIFLE. We then set up the following model to determine attribute weight vector. $\begin{matrix} (M - 3) \max & 3.3624 w_{1} + 1.4597 w_{2} + 1.0963 w_{3} \\ + 3.7464 w_{4} \\ s . t . & 0.2 \leq w_{1} \leq 0.25, 0.15 \leq w_{2} \leq 0.20, \\ 0.10 \leq w_{3} \leq 0.20, 0.10 \leq w_{4} \leq 0.35, \\ 2 w_{3} \leq w_{4}, w_{1} + w_{2} + w_{3} + w_{4} = 1 . \end{matrix}$

The attribute vector W₂ = (0.30, 0.20, 0.15, 0.35) can get by solving the model (M-3). For completely unknown attribute weights, we can determine them by using Equations (14)–(15) to get W₃ = (0.3466, 0.1505, 0.1130, 0.3899). For other steps, we can calculate as the same as that attribute weights are completely known and the results areomitted.

If algorithm II is used to rank alternatives, we should extend decision matrix first according to the risk attitude of decision makers. Assume decision makers are risk averse and extend decision matrix by adding the minimum IVIFLE. Determine the HIVIFLPIS as ${\overset{˘}{h}}^{+} = ({\overset{˘}{h}}_{1}^{+}, {\overset{˘}{h}}_{2}^{+}, . . ., {\overset{˘}{h}}_{n}^{+})$ and HIVIFLNIS as ${\overset{˘}{h}}^{-} = ({\overset{˘}{h}}_{1}^{-}, {\overset{˘}{h}}_{2}^{-}, . . .,$ ${\overset{˘}{h}}_{n}^{-})$ . The IVIFLEs are (s₉, ([1, 1] , [0, 0])) in ${\overset{˘}{h}}_{j}^{+}$ and (s₁, ([0, 0] , [1, 1])) in ${\overset{˘}{h}}_{j}^{-}$ . Calculate distances of each evaluation value ${\overset{˘}{h}}_{ij}$ to ${\overset{˘}{h}}_{j}^{+}$ and ${\overset{˘}{h}}_{j}^{-}$ by using Equation (6). The attribute weight vector is taken as W₁ = (0.20, 0.30, 0.35, 0.15) to facilitate comparison. Calculate weighted distances of alternatives as $d_{1}^{+} = 0.4352, d_{2}^{+} = 0.4484,$ $d_{3}^{+} = 0.4435, d_{4}^{+} = 0.4250, d_{5}^{+} = 0.4157, d_{1}^{-} = 0.6134, d_{2}^{-} = 0.5949,$ $d_{3}^{-} = 0.6124, d_{4}^{-} = 0.6169,$ $d_{5}^{-} = 0.6322 .$ Calculate closeness coefficients of alternatives by using Equation (20) to get CC₁ = 0.5850, CC₂ = 0.5702, CC₃ = 0.5799, CC₄ = 0.5921, CC₅ = 0.6033 . Rank closeness coefficients to get CC₅ > CC₄ > CC₁ > CC₃ > CC₂ and rank alternatives accordingly as A₅ > A₄ > A₁ > A₃ > A₂. A₅ is the optimal alternative and A₂ is ranked last. The results are similar to that of methods based on aggregation operators.

We further compare the new methods with the method of Yang et al. [13], in which each linguistic term has one intuitionistic fuzzy membership. If we only consider $μ_{ij}^{L}$ and $ν_{ij}^{L}$ in Table 1, the HIVIFLEs reduce to hesitant linguistic intuitionistic fuzzy elements. Assume decision makers are risk averse, we first extend the decision matrix by adding the smallest linguistic intuitionistic fuzzy element (s₂, (0.5, 0.3)). Determine the hesitant linguistic intuitionistic fuzzy positive ideal solution as $h^{+} = (h_{1}^{+}, h_{2}^{+}, . . ., h_{n}^{+})$ and the hesitant linguistic intuitionistic fuzzy negative ideal solution as $h^{-} = (h_{1}^{-}, h_{2}^{-}, . . .,$ $h_{n}^{-})$ . The linguistic intuitionistic fuzzy elements are (s₉, (1, 0)) in $h_{j}^{+}$ and (s₁, (0, 1)) in $h_{j}^{-}$ . The attribute weight vector is also taken as W₁ = (0.20, 0.30, 0.35, 0.15) to facilitate comparison. Calculate weighted distances to get $d_{1}^{+} = 0.4453, d_{2}^{+} = 0.4497, d_{3}^{+} = 0.4717, d_{4}^{+} = 0.4219, d_{5}^{+} = 0.4203, d_{1}^{-} = 0.6007, d_{2}^{-} = 0.5796, d_{3}^{-} = 0.6033, d_{4}^{-} = 0.6154, d_{5}^{-} = 0.6364$ . The close-ness coefficients can be calculated as CC₁ = 0.5743, CC₂ = 0.5631, CC₃ = 0.5613, CC₄ = 0.5933, CC₅ = 0.6022 . Then CC₅ > CC₄ > CC₁ > CC₂ > CC₃ and alternatives can be ranked as A₅ > A₄ > A₁ > A₂ > A₃. We can get similar ranking results with the proposed method based on TOPSIS. But there are differences in the ranking of A₂ and A₃ due to different types evaluation values and more information has been used in proposed method. From the analysis presented above, the new proposed approach has the following advantages. First, HIVIFLEs used in the new method can model uncertain and fuzzy information more flexibly. Each HIVIFLE has several linguistic terms and each linguistic term has its interval-valued intuitionistic fuzzy membership. HIVIFLE can avoid effect of unduly low or unduly high evaluation values since decision makers can refuse to give evaluation values due to the unfamiliar attribute, which makes the decision results are more corresponding with decision-making problems. By using linguistic terms in evaluation process, inherent thoughts of decision makers can be reflected. Hesitation and fuzzy thought can be more accurately modeled by interval-valued intuitionistic fuzzy memberships. Second, the new method can deal with different weight situations including the weights are completely known, partly known and completely unknown. Finally, by using the GHIVIFLOWA operator or the GHIVIFLOWGA operator in aggregation process, the risk attitudes of the decision makers can be reflected.

6 Conclusions

In this paper, considering hesitation in decision making when use linguistic terms to evaluate, we develop hesitant interval-valued intuitionistic fuzzy linguistic set by extending hesitant fuzzy set using linguistic arguments and interval-valued intuitionistic fuzzy memberships. The prominent feature of the new set is that it can provide a flexible and powerful way to model uncertain and fuzzy information in the decision making process. Then the GHIVIFLOWA operator and the GHIVIFLOWGA operator are developed. The new method using proposed aggregation operator and the method based on TOPSIS method have been developed for solving decision-making problems with hesitant interval-valued intuitionistic fuzzy linguistic information considering different situations of attribute weight information.

Footnotes

Acknowledgments

This work is partly supported by National Natural Science Foundation of China (Nos. 11401457, 61403298), Postdoctoral Science Foundation of China (2015M582624).

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