Models for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators

Abstract

In this paper, we expand the Hamy mean (HM) operator and dual Hamy mean (DHM) operator with IVIFNs to propose the interval -valued intuitionistic fuzzy Hamy mean (IVIFHM) operator, interval-valued intuitionistic fuzzy weighted Hamy mean (IVIFWHM) operator, interval-valued intuitionistic fuzzy dual Hamy mean (IVIFDHM) operator and interval-valued intuitionistic fuzzy weighted dual Hamy mean (IVIFWDHM) operator. Then the MADM models are designed with IVIFWHM and IVIFWDHM operators. Finally, we gave an example for competiveness evaluation of tourist destination to show the proposed models.

Keywords

Multiple attribute decision making (MADM)interval-valued intuitionistic fuzzy numbers (IVIFNs)IVIFWHM operator IVIFWDHM operator competiveness evaluation tourist destination

1. Introduction

The concept of intuitionistic fuzzy sets (IFSs) [1, 2] has been utilized to deal with uncertainty and imprecision. Atanassov and Gargov [3] defined the interval-valued intuitionistic fuzzy sets (IVIFSs). Xu [4, 5] developed some aggregation operators with IVIFNs. Li [6] developed the nonlinear-programming models to solve MADM with IVIFNs. Li [7] designed linear programming models to solve MADM with IVIFNs. Ye [8] gave the fuzzy cross entropy for IVIFNs. Xu [9] investigated the incomplete MADM with IVIFNs. Cheng [10] designed the autocratic MAGDM model for hotel location selection with IVIFNs. Wang & Chen [11] proposed a new MADM model with IVIFSs based on LP methodology and TOPSIS method. Chen & Chiou [12] designed a new MADM model with IVIFSs based on PSO techniques and the evidential reasoning. Chen [13] proposed the likelihood-based model for MADM with IVIFNs. Garg [14] presented the improved score function for IVIFNs. Ren et al. [15] introduced a simplified IVIFN and some operational laws for simplified IVIFN. Beliakov et al. [16] analyzed the median aggregation operator for IVIFNs. Chen et al. [17] proposed new ranking IVIFNs. Lakshmana Gomathi Nayagam et al. [18] introduced the non-hesitance score function of IVIFSs by using illustrative examples and established a new MADM algorithm. Chen [19] developed the QUALIFLEX method for MAGDM with IVIFNs. Zhang et al. [20] developed new entropy measures for IVIFNs. Until now, more and more decision making theories of IFSs, IVIFSs and their extensions are studied in the MADM [21 –33].

Although, IFSs and IVIFSs have been effectively used in some areas, however, all the existed approaches are unsuitable to depict the interrelationships among any number of IVIFNs assigned by a variable vector. Then, we propose some HM operator to overcome this limits. Thus, how to aggregate these IVIFNs based the traditional HM operators [34 –40] is an interesting issue. Thus, we propose some HM and DHM operators with IVIFNs.

In order to do so, the rest of this paper is organized as follows. In Section 2, we introduce the IVIFNs. In Section 3, we develop some HM operators with IVIFNs. In Section 4, we present a example for competiveness evaluation of tourist destination with IVIFNs. Section 5 ends this paper with some comments.

2. Preliminaries

2.1. IFSs and IVIFSs

The concept of IFSs and IVIFSs are introduced.

Definition 1. [1, 2] An IFS Q in X is designed by $Q = {〈 x, θ_{Q} (x), ϑ_{Q} (x) 〉 | x \in X}$ (1)

where θ_Q : X → [0, 1] and ϑ_Q : X → [0, 1], and 0 ≤ θ_Q (x) + ϑ_Q (x) ≤ 1, ∀ x ∈ X. The number θ_Q (x) and ϑ_Q (x) represents the membership degree and non- membership degree of the element x to the set Q, respectively.

Definition 2. [3] Let X be an universe of discourse, An IVIFS $\tilde{Q}$ over X is an object having the form: $\tilde{Q} = {〈 x, {\tilde{θ}}_{\tilde{Q}} (x), {\tilde{ϑ}}_{\tilde{Q}} (x) 〉 | x \in X}$ (2)

Where ${\tilde{θ}}_{\tilde{Q}} (x) \subset [0, 1]$ and ${\tilde{ϑ}}_{\tilde{Q}} (x) \subset [0, 1]$ are interval numbers, and $0 \leq sup ({\tilde{θ}}_{\tilde{Q}} (x)) + sup ({\tilde{ϑ}}_{\tilde{Q}} (x)) \leq 1$ , ∀ x ∈ X. For convenience, let ${\tilde{θ}}_{\tilde{Q}} (x) = [e, f]$ , ${\tilde{ϑ}}_{\tilde{Q}} (x) = [g, h]$ , so $\tilde{φ} = ([e, f], [g, h])$ is an IVIFNs.

Definition 3. [4, 5] Let $\tilde{φ} = ([e, f], [g, h])$ be an IVIFN, a score function S can be defined: $S (\tilde{φ}) = \frac{e - g + f - h}{2}, S (\tilde{φ}) \in [- 1, 1] .$ (3)

Definition 4. [4, 5] Let $\tilde{φ} = ([e, f], [g, h])$ be an IVIFN, an accuracy function H can be defined: $H (\tilde{φ}) = \frac{e + f + g + h}{2}, H (\tilde{φ}) \in [0, 1] .$ (4) to evaluate the degree of accuracy of the IVIFN $\tilde{φ} = ([e, f], [g, h])$ .

Definition 5. [4] Let ${\tilde{φ}}_{1} = ([e_{1}, f_{1}], [g_{1}, h_{1}])$ and ${\tilde{φ}}_{2} = ([e_{2}, f_{2}], [g_{2}, h_{2}])$ be two IVIFNs, $s ({\tilde{φ}}_{1}) = \frac{e_{1} - f_{1} + g_{1} - h_{1}}{2}$ and $s ({\tilde{φ}}_{2}) = \frac{e_{2} - f_{2} + g_{2} - h_{2}}{2}$ be the scores of ${\tilde{φ}}_{1}$ and ${\tilde{φ}}_{2}$ , respectively, and let $H ({\tilde{φ}}_{1}) = \frac{e_{1} + f_{1} + g_{1} + h_{1}}{2}$ and $H ({\tilde{φ}}_{2}) = \frac{e_{2} + f_{2} + g_{2} + h_{2}}{2}$ be the accuracy degrees of ${\tilde{φ}}_{1}$ and ${\tilde{φ}}_{2}$ , respectively, then if, then ${\tilde{φ}}_{1} < {\tilde{φ}}_{2}$ ; if $S ({\tilde{φ}}_{1}) = S ({\tilde{φ}}_{2})$ , then (1) if $H ({\tilde{φ}}_{1}) = H ({\tilde{φ}}_{2})$ , then ${\tilde{φ}}_{1} = {\tilde{φ}}_{2}$ ; (2) if $H ({\tilde{φ}}_{1}) < H ({\tilde{φ}}_{2})$ , then ${\tilde{φ}}_{1} < {\tilde{φ}}_{2}$ .

Definition 6. [4, 5] For two IVIFNs ${\tilde{φ}}_{1} = ([e_{1}, f_{1}], [g_{1}, h_{1}])$ and ${\tilde{φ}}_{2} = ([e_{2}, f_{2}], [g_{2}, h_{2}])$ , the operational laws are defined: $\begin{matrix} (1) {\tilde{φ}}_{1} \oplus {\tilde{φ}}_{2} = ([e_{1} + e_{2} - e_{1} e_{2}, f_{1} + f_{2} - f_{1} f_{2}], [g_{1} g_{2}, h_{1} h_{2}]); \\ (2) {\tilde{φ}}_{1} \otimes {\tilde{φ}}_{2} = ([e_{1} e_{2}, f_{1} f_{2}], [g_{1} + g_{2} - g_{1} g_{2}, h_{1} + h_{2} - h_{1} h_{2}]); \\ (3) λ {\tilde{φ}}_{1} = ([1 - {(1 - e_{1})}^{λ}, 1 - {(1 - f_{1})}^{λ}], [g_{1}^{λ}, h_{1}^{λ}]), λ > 0; \\ (4) {({\tilde{φ}}_{1})}^{λ} = ([e_{1}^{λ}, f_{1}^{λ}], [1 - {(1 - g_{1})}^{λ}, 1 - {(1 - h_{1})}^{λ}]), λ > 0 . \end{matrix}$

2.2. HM operator

Hara et al. [34] proposed the HM operator.

Definition 7. [34]. The HM operator is defined: ${HM}^{(x)} (φ_{1}, φ_{2}, \dots, φ_{n}) = \frac{\sum_{1 \leq i_{1} < \dots < i_{x} \leq n} {(\prod_{j = 1}^{x} φ_{i_{j}})}^{\frac{1}{x}}}{C_{n}^{x}}$ (5) where x is a parameter, x = 1, 2, …, n, i₁, i₂, …, i_x are x integer values taken from the set {1, 2, … , n } of k integer values, $C_{n}^{x}$ is the binomial coefficient, $C_{n}^{x} = \frac{n!}{x! (n - x)!}$ .

3. Some Hamy mean operators with IVIFNs

3.1. The IVIFHM operator

Based on the HM operator, the IVIFHM operator is defined:

Definition 8. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. The IVIFHM operator is: ${IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = \frac{\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {\tilde{φ}}_{i_{j}})}^{\frac{1}{x}}}{C_{n}^{x}}$ (6)

Theorem 1. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. The fused value by IVIFHM operators is also an IVIFN where

$\begin{matrix} {IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = \frac{\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {\tilde{φ}}_{i_{j}})}^{\frac{1}{x}}}{C_{n}^{x}} \\ = {\begin{matrix} [1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} f_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}], \\ [{(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - g_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - h_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}] \end{matrix}} \end{matrix}$ (7)

Proof. $\otimes_{j = 1}^{x} {\tilde{φ}}_{i_{j}} = {[\prod_{j = 1}^{x} e_{i_{j}}, \prod_{j = 1}^{x} f_{i_{j}}], [1 - \prod_{j = 1}^{x} (1 - g_{i_{j}}), 1 - \prod_{j = 1}^{x} (1 - h_{i_{j}})]}$ (8)

Thus, ${(\otimes_{j = 1}^{x} {\tilde{φ}}_{i_{j}})}^{\frac{1}{x}} = {[{(\prod_{j = 1}^{x} (e_{i_{j}}))}^{\frac{1}{x}}, {(\prod_{j = 1}^{x} (f_{i_{j}}))}^{\frac{1}{x}}], [1 - {(\prod_{j = 1}^{x} (1 - g_{i_{j}}))}^{\frac{1}{x}}, 1 - {(\prod_{j = 1}^{x} (1 - h_{i_{j}}))}^{\frac{1}{x}}]}$ (9)

Thereafter, $\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {\tilde{φ}}_{i_{j}})}^{\frac{1}{x}} = {\begin{matrix} [1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}), 1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} f_{i_{j}})}^{\frac{1}{x}})], \\ [\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} g_{i_{j}})}^{\frac{1}{x}}), \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} h_{i_{j}})}^{\frac{1}{x}})] \end{matrix}}$ (10)

Therefore,

Thus (7) is right.

Then we need to show that (7) is an IVIFN.

ding1720 ≤ e ≤ 1, 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1 ding173 0 ≤ f + h ≤ 1

Let $\begin{matrix} e = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ f = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} f_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ g = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - g_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ h = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - h_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$

Proof. ding172 Since 0 ≤ e_{i
_j} ≤ 1, we get $0 \leq \prod_{j = 1}^{x} e_{i_{j}} \leq 1 and 0 \leq 1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}} \leq 1$ (12)

Then,

$0 \leq \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}) \leq 1$ (13) $0 \leq 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \leq 1$ (14)

That means 0 ≤ e ≤ 1, so ding172 is maintained, similarly, we can get 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1.

ding173 Since 0 ≤ e ≤ 1, 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1, 0 ≤ f + h ≤ 1 .

Then we list some properties of IVIFMM operator.

Property 1. (Idempotency) If ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}]) (j = 1, 2, \dots, n)$ are equal, then ${IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = \tilde{φ}$ (15)

Proof. Since ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}]) (j = 1, 2, \dots, n)$ , then $\begin{matrix} {IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ = {\begin{matrix} [1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} f_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}], \\ [{(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - g_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - h_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}] \end{matrix}} \\ = {\begin{matrix} [1 - {({(1 - {(e^{x})}^{\frac{1}{x}})}^{\frac{1}{C_{n}^{x}}})}^{\frac{1}{C_{n}^{x}}}, 1 - {({(1 - {(f^{x})}^{\frac{1}{x}})}^{\frac{1}{C_{n}^{x}}})}^{\frac{1}{C_{n}^{x}}}], \\ [{({(1 - {({(1 - g)}^{x})}^{\frac{1}{x}})}^{\frac{1}{C_{n}^{x}}})}^{\frac{1}{C_{n}^{x}}}, {({(1 - {({(1 - h)}^{x})}^{\frac{1}{x}})}^{\frac{1}{C_{n}^{x}}})}^{\frac{1}{C_{n}^{x}}}] \end{matrix}} = ([e, f], [g, h]) = \tilde{φ} \end{matrix}$

Property 2. (Monotonicity) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) and ${\tilde{θ}}_{j} = ([r_{j}, s_{j}], [m_{j}, n_{j}])$ (j = 1, 2, …, n) be two sets of IVIFNs. If e_j ≤ r_j, f_j ≤ s_j and g_j ⩾ m_j, h_j ⩾ n_j hold for all j, then $\begin{matrix} {IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ \leq {IVIFHM}^{(x)} ({\tilde{θ}}_{1}, {\tilde{θ}}_{2}, \dots, {\tilde{θ}}_{n}) \end{matrix}$ (16)

Proof. Let ${IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = ([e, f], [g, h])$ and ${IVIFHM}^{(x)} ({\tilde{θ}}_{1}, {\tilde{θ}}_{2}, \dots, {\tilde{θ}}_{n}) = ([r, s], [m, n])$ , given that e_j ≤ r_j, and we can obtain $\prod_{j = 1}^{x} e_{i_{j}} \leq \prod_{j = 1}^{x} r_{i_{j}}$ (17) $1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}} ⩾ 1 - {(\prod_{j = 1}^{x} r_{i_{j}})}^{\frac{1}{x}}$ (18)

Thereafter, $\begin{matrix} {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \\ ⩾ {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} r_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$ (19)

Furthermore,

$\begin{matrix} 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \\ \leq 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} r_{i_{j}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$ (20)

That’s e ≤ r. Similarly, we can obtain f ≤ s, g ⩾ m and h ⩾ n.

If e < r and f < s, g > m and h > n . $\begin{matrix} {IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ < {IVIFHM}^{(x)} ({\tilde{θ}}_{1}, {\tilde{θ}}_{2}, \dots, {\tilde{θ}}_{n}) \end{matrix}$

If e = r and f = s, g = m and h = n . $\begin{matrix} {IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ = {IVIFHM}^{(x)} ({\tilde{θ}}_{1}, {\tilde{θ}}_{2}, \dots, {\tilde{θ}}_{n}) \end{matrix}$

So property 2 is right.

Property 3. (Boundedness) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}]) (j = 1, 2, \dots, n)$ be a set of IVIFNs. If ${\tilde{φ}}_{i}^{+} = ((\begin{matrix} [max_{i} (a_{j}), max_{i} (b_{j})], \\ [min_{i} (c_{j}), min_{i} (d_{j})] \end{matrix}))$

and ${\tilde{φ}}_{i}^{-} = (\begin{matrix} [min_{i} (a_{j}), min_{i} (b_{j})], \\ [max_{i} (c_{j}), max_{i} (d_{j})] \end{matrix})$

then ${\tilde{φ}}^{-} \leq {IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \leq {\tilde{φ}}^{+}$ (21)

From property 1, $\begin{matrix} {IVIFHM}^{(x)} ({\tilde{φ}}_{1}^{-}, {\tilde{φ}}_{2}^{-}, \dots, {\tilde{φ}}_{n}^{-}) = {\tilde{φ}}^{-} \\ {IVIFHM}^{(x)} ({\tilde{φ}}_{1}^{+}, {\tilde{φ}}_{2}^{+}, \dots, {\tilde{φ}}_{n}^{+}) = {\tilde{φ}}^{+} \end{matrix}$

From property 2, ${\tilde{φ}}^{-} \leq {IVIFHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \leq {\tilde{φ}}^{+}$

3.2. The IVIFWHM operator

In real MADM, it’s important to pay attention to attribute weights. Thus we propose IVIF weighted Hamy mean (IVIFWHM) operator.

Definition 9. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs with their weight vector be w_i = (w₁, w₂, …, w_n) ^T, thereby satisfying w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the IVIFWHM operator is:

$\begin{matrix} {IVIFWHM}_{w}^{(x)} (φ_{1}, φ_{2}, \dots, φ_{n}) \\ = \frac{\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {({\tilde{φ}}_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}}{C_{n}^{x}} \end{matrix}$ (22)

Theorem 2. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. The fused value by IVIFHM operators is also a IVIFN where $\begin{matrix} {IVIFWHM}_{w}^{(x)} (φ_{1}, φ_{2}, \dots, φ_{n}) = \frac{\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {({\tilde{φ}}_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}}{C_{n}^{x}} \\ = {\begin{matrix} [1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} f_{i_{j}}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}], \\ [{(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}] \end{matrix}} \end{matrix}$ (23)

Proof. ${({\tilde{φ}}_{i_{j}})}^{w_{i_{j}}} = ([{(e_{i_{j}})}^{w_{i_{j}}}, {(f_{i_{j}})}^{w_{i_{j}}}], [1 - {(1 - g_{i_{j}})}^{w_{i_{j}}}, 1 - {(1 - h_{i_{j}})}^{w_{i_{j}}}])$ (24)

Thus, $\otimes_{j = 1}^{x} {({\tilde{φ}}_{i_{j}})}^{w_{i_{j}}} = ([\prod_{j = 1}^{x} {(e_{i_{j}})}^{w_{i_{j}}}, \prod_{j = 1}^{x} {(f_{i_{j}})}^{w_{i_{j}}}], [1 - \prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}}, 1 - \prod_{j = 1}^{x} {(1 - h_{i_{j}})}^{w_{i_{j}}}])$ (25)

Therefore, ${(\otimes_{j = 1}^{x} {({\tilde{φ}}_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}} = (\begin{matrix} [{(\prod_{j = 1}^{x} {(e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}, {(\prod_{j = 1}^{x} {(f_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}], \\ [1 - {(\prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}, 1 - {(\prod_{j = 1}^{x} {(1 - h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}] \end{matrix})$ (26)

Thereafter, $\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {({\tilde{φ}}_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}} = (\begin{matrix} [1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}), 1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(f_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}})], \\ [\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}), \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}})] \end{matrix})$ (27)

Therefore, $\begin{matrix} {IVIFWHM}_{w}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = \frac{\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {({\tilde{φ}}_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}}{C_{n}^{x}} \\ = {\begin{matrix} [1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} f_{i_{j}}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}], \\ [{(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}] \end{matrix}} \end{matrix}$ (28)

Hence, (23) is kept.

Then we need to prove that (23) is an IVIFN.

ding172! [e, f] ⊂ [0, 1] , [g, h] ⊂ [0, 1] ding173 0 ≤ f + h ≤ 1

Proof. Let $\begin{matrix} e = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} e_{i_{j}}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ f = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} f_{i_{j}}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ g = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ h = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$

ding172 Since 0 ≤ e_{i
_j} ≤ 1, we get $0 \leq \prod_{j = 1}^{x} {(e_{i_{j}})}^{w_{i_{j}}} \leq 1 and 0 \leq 1 - {(\prod_{j = 1}^{x} {(e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}} \leq 1$ (29)

Then $0 \leq \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}) \leq 1$ (30) $0 \leq 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \leq 1$ (31)

That’s means 0 ≤ e ≤ 1, so ding172 is maintained. Similarly, we can get 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1.

ding173 Since 0 ≤ e ≤ 1, 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1, 0 ≤ f + h ≤ 1 .

Then we list some properties of IVIFWMM operator.

Property 4. (Monotonicity) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}]) (j = 1, 2, \dots, n)$ and ${\tilde{θ}}_{j} = ([r_{j}, s_{j}], [m_{j}, n_{j}])$ (j = 1, 2, …, n) be two sets of IVIFNs. If e_j ≤ r_j, f_j ≤ s_j and g_j ⩾ m_j, h_j ⩾ n_j hold for all j, then $\begin{matrix} {IVIFWHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ \leq {IVIFWHM}^{(x)} ({\tilde{θ}}_{1}, {\tilde{θ}}_{2}, \dots, {\tilde{θ}}_{n}) (32) \end{matrix}$ (32)

The proof is similar to IVIFMM, it’s omitted here.

Property 5. (Boundedness) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}]) (j = 1, 2, \dots, n)$ be a set of IVIFNs. If ${\tilde{φ}}_{i}^{+} = ((\begin{matrix} [max_{i} (e_{j}), max_{i} (f_{j})], \\ [min_{i} (g_{j}), min_{i} (h_{j})] \end{matrix}))$

and ${\tilde{φ}}_{i}^{-} = (\begin{matrix} [min_{i} (e_{j}), min_{i} (f_{j})], \\ [max_{i} (g_{j}), max_{i} (h_{j})] \end{matrix})$

then ${\tilde{φ}}^{-} \leq {IVIFWHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \leq {\tilde{φ}}^{+}$ (33)

From theorem 2, we get $\begin{matrix} {IVIFWHM}_{w}^{(x)} ({\tilde{φ}}_{1}^{-}, {\tilde{φ}}_{2}^{-}, \dots, {\tilde{φ}}_{n}^{-}) = \frac{\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {(min {\tilde{φ}}_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}}{C_{n}^{x}} \\ = {\begin{matrix} [\begin{matrix} 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(min (e_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(min (f_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}], \\ [\begin{matrix} {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - max (g_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - max (h_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}] \end{matrix}} \end{matrix}$ (34) $\begin{matrix} {IVIFWHM}_{w}^{(x)} ({\tilde{φ}}_{1}^{+}, {\tilde{φ}}_{2}^{+}, \dots, {\tilde{φ}}_{n}^{+}) = \frac{\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\oplus} {(\otimes_{j = 1}^{x} {(max {\tilde{φ}}_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}}{C_{n}^{x}} \\ = (\begin{matrix} [\begin{matrix} 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(max (e_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(max (f_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}], \\ [\begin{matrix} {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - min (g_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - min (h_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}] \end{matrix}) \end{matrix}$ (35)

From property 4, we get ${\tilde{φ}}^{-} \leq {IVIFWHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \leq {\tilde{φ}}^{+}$ (36)

3.3. The IVIFDHM operator

Wu et al. [35] proposed the dual HM (DHM) operator.

Definition 10. The DHM operator is:

${DHM}^{(x)} (φ_{1}, φ_{2}, \dots, φ_{n}) = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (\frac{\sum_{j = 1}^{x} φ_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}}$ (37) where x is a parameter and x = 1, 2, …, n, i₁, i₂, …, i_x are x integer values taken from the set {1, 2, … , n } of k integer values, $C_{n}^{x}$ denotes the binomial coefficient and $C_{n}^{x} = \frac{n!}{x! (n - x)!}$

In this section, we will propose the interval-valued intuitionistic fuzzy DHM (IVIFDHM)operator.

Definition 11. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. The IVIFDHM operator is:

$\begin{matrix} {IVIFDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ = {(\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} {\tilde{φ}}_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$ (38)

Theorem 3. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. The fused value by IVIFDHM operators is also an IVIFN where $\begin{matrix} {IVIFDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = {(\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} {\tilde{φ}}_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}} \\ = (\begin{matrix} [{(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - f_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}], \\ [1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (g_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (h_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}] \end{matrix}) \end{matrix}$ (39)

Proof. $\oplus_{j = 1}^{x} {\tilde{φ}}_{i_{j}} = ([1 - \prod_{j = 1}^{x} (1 - e_{i_{j}}), 1 - \prod_{j = 1}^{x} (1 - f_{i_{j}})], [\prod_{j = 1}^{x} g_{i_{j}}, \prod_{j = 1}^{x} h_{i_{j}}])$ (40)

Thus, $\frac{\oplus_{j = 1}^{x} {\tilde{φ}}_{i_{j}}}{x} = ([1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}}, 1 - {(\prod_{j = 1}^{x} (1 - f_{i_{j}}))}^{\frac{1}{x}}], [{(\prod_{j = 1}^{x} (g_{i_{j}}))}^{\frac{1}{x}}, {(\prod_{j = 1}^{x} (h_{i_{j}}))}^{\frac{1}{x}}])$ (41)

Therefore, $\underset{σ \in S_{n}}{\otimes} \underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} {\tilde{φ}}_{i_{j}}}{x}) = (\begin{matrix} [\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}}), \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - f_{i_{j}}))}^{\frac{1}{x}})], \\ [1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (g_{i_{j}}))}^{\frac{1}{x}}), 1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (h_{i_{j}}))}^{\frac{1}{x}})] \end{matrix})$ (42)

Therefore, $\begin{matrix} {IVIFDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = {(\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} {\tilde{φ}}_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}} \\ = (\begin{matrix} [{(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - f_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}], \\ [1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (g_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (h_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}] \end{matrix}) \end{matrix}$ (43)

Thus, (39) is right.

Then we shall prove that (39) is an IVIFN.

ding172 0 ≤ e ≤ 1, 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1;

ding173 0 ≤ f + h ≤ 1.

Let $\begin{matrix} e = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ f = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - f_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ g = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (g_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ h = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (h_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$

ding172 Since 0 ≤ e_{i
_j} ≤ 1, we get $0 \leq \prod_{j = 1}^{x} (1 - e_{i_{j}}) \leq 1 and$ ∥ $0 \leq 1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}} \leq 1$ (44) ∥ $0 \leq \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}}) \leq 1$ (45)

Then, $0 \leq {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} (1 - e_{i_{j}}))}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \leq 1$ (46)

That means 0 ≤ e ≤ 1, so ding172 is maintained.

Similarly, we can get 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1.

ding173 Since 0 ≤ e ≤ 1, 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1 0 ≤ f + h ≤ 1 .

The IVIFDHM operator has following properties.

Property 6. (Idempotency) If ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) are equal, then ${IVIFDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = \tilde{φ}$ (47)

Property 7. (Monotonicity) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}]) (j = 1, 2, \dots, n)$ and ${\tilde{θ}}_{j} = ([r_{j}, s_{j}], [m_{j}, n_{j}]) (j = 1, 2, \dots, n)$ be two sets of IVIFNs. If e_j ≤ r_j, f_j ≤ s_j and g_j ⩾ m_j, h_j ⩾ n_j hold for all j, then $\begin{matrix} {IVIFDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ \leq {IVIFDHM}^{(x)} ({\tilde{θ}}_{1}, {\tilde{θ}}_{2}, \dots, {\tilde{θ}}_{n}) \end{matrix}$ (48)

Property 8. (Boundedness) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. If ${\tilde{φ}}_{i}^{+} = ((\begin{matrix} [max_{i} (e_{j}), max_{i} (f_{j})], \\ [min_{i} (g_{j}), min_{i} (h_{j})] \end{matrix}))$

and ${\tilde{φ}}_{i}^{-} = (\begin{matrix} [min_{i} (e_{j}), min_{i} (f_{j})], \\ [max_{i} (g_{j}), max_{i} (h_{j})] \end{matrix})$

then ${\tilde{φ}}^{-} \leq {IVIFDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \leq {\tilde{φ}}^{+}$ (49)

3.4. The IVIFWDMM operator

In practical MADM, it’s important to pay attention to attribute weights; we propose the interval-valued intuitionistic weighted DHM (IVIFWDHM) operator.

Definition 12. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs with their weight vector be w_i = (w₁, w₂, …, w_n) ^T, thereby satisfying w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ .

$\begin{matrix} {IVIFWDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ = {(\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} w_{i_{j}} {\tilde{φ}}_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$ (50)

Theorem 4. Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. The fused value by IVIFWDHM operators is also an IVIFN where $\begin{matrix} {IVIFWDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = {(\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} w_{i_{j}} {\tilde{φ}}_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}} \\ = ([\begin{matrix} {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}], [\begin{matrix} 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(f_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}]) \end{matrix}$ (51)

Proof. $w_{i_{j}} φ_{i_{j}} = ([1 - {(1 - e_{i_{j}})}^{w_{i_{j}}}, 1 - {(1 - f_{i_{j}})}^{w_{i_{j}}}], [{(g_{i_{j}})}^{w_{i_{j}}}, {(h_{i_{j}})}^{w_{i_{j}}}])$ (52)

Then, $\oplus_{j = 1}^{x} w_{i_{j}} {\tilde{φ}}_{i_{j}} = ([1 - \prod_{j = 1}^{x} ({(1 - e_{i_{j}})}^{w_{i_{j}}}), 1 - \prod_{j = 1}^{x} ({(1 - f_{i_{j}})}^{w_{i_{j}}})], [(\prod_{j = 1}^{x} {(g_{i_{j}})}^{w_{i_{j}}}), (\prod_{j = 1}^{x} {(h_{i_{j}})}^{w_{i_{j}}})])$ (53)

Thus, $\frac{\oplus_{j = 1}^{x} w_{i_{j}} {\tilde{φ}}_{i_{j}}}{x} = ([1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}, 1 - {(\prod_{j = 1}^{x} {(1 - f_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}], [{(\prod_{j = 1}^{x} {(g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}, {(\prod_{j = 1}^{x} {(h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}])$ (54)

Therefore, $\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} w_{i_{j}} {\tilde{φ}}_{i_{j}}}{x}) = (\begin{matrix} [\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}), \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - f_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}})], \\ [1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}), 1 - \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}})] \end{matrix})$ (55)

Therefore, $\begin{matrix} {IVIFWDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) = {(\underset{1 \leq i_{1} < \dots < i_{x} \leq n}{\otimes} (\frac{\oplus_{j = 1}^{x} w_{i_{j}} {\tilde{φ}}_{i_{j}}}{x}))}^{\frac{1}{C_{n}^{x}}} \\ = ([\begin{matrix} {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}], [\begin{matrix} 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(f_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}]) \end{matrix}$ (56)

Thus, (51) is right.

Then we shall prove that (51) is an IVIFN.

ding172 0≤ e ≤ 1, 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1 ;

ding173 0 ≤ f + h ≤ 1 .

Proof. Let $\begin{matrix} e = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ f = {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - f_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \end{matrix}$ $\begin{matrix} g = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(g_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ h = 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(h_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}$

ding172 Since 0 ≤ e_{i
_j} ≤ 1, we can get $\begin{matrix} 0 & \leq & \prod_{j = 1}^{x} (1 - e_{i_{j}}) \leq 1 and \\ 0 & \leq & 1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}} \leq 1 \end{matrix}$ (57)

Then $0 \leq \prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}) \leq 1$ (58)

$0 \leq {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - e_{i_{j}})}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \leq 1$ (59)

That’s means 0 ≤ e ≤ 1, so ding172 is maintained.

Similarly, we can get 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1.

ding173 Since 0 ≤ e ≤ 1, 0 ≤ f ≤ 1, 0 ≤ g ≤ 1, 0 ≤ h ≤ 1, 0 ≤ f + h ≤ 1 .

Then we give some properties of IVIFWMM operator.

Property 9. (Monotonicity) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) and ${\tilde{θ}}_{j} = ([r_{j}, s_{j}], [m_{j}, n_{j}])$ (j = 1, 2, …, n) be two sets of IVIFNs. If e_j ≤ r_j, f_j ≤ s_j and g_j ⩾ m_j, h_j ⩾ n_j hold for all j, then $\begin{matrix} {IVIFWDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \\ \leq {IVIFWDHM}^{(x)} ({\tilde{θ}}_{1}, {\tilde{θ}}_{2}, \dots, {\tilde{θ}}_{n}) \end{matrix}$ (60)

The proof is similar to IVIFWHM, it’s omitted here.

Property 10. (Boundedness) Let ${\tilde{φ}}_{j} = ([e_{j}, f_{j}], [g_{j}, h_{j}])$ (j = 1, 2, …, n) be a set of IVIFNs. If ${\tilde{φ}}_{i}^{+} = ((\begin{matrix} [max_{i} (e_{j}), max_{i} (f_{j})], \\ [min_{i} (g_{j}), min_{i} (h_{j})] \end{matrix}))$

and ${\tilde{φ}}_{i}^{-} = (\begin{matrix} [min_{i} (e_{j}), min_{i} (f_{j})], \\ [max_{i} (g_{j}), max_{i} (h_{j})] \end{matrix})$

then ${\tilde{φ}}^{-} \leq {IVIFWDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \leq {\tilde{φ}}^{+}$ (61)

From theorem 4, $\begin{matrix} {IVIFWDHM}^{(x)} (φ_{1}^{+}, φ_{2}^{+}, \dots, φ_{n}^{+}) \\ = (\begin{matrix} [\begin{matrix} {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - max (e_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - max (f_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}], \\ [\begin{matrix} 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(min (g_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(min (h_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}] \end{matrix}) \end{matrix}$ (62) $\begin{matrix} {IVIFWDHM}^{(x)} (φ_{1}^{-}, φ_{2}^{-}, \dots, φ_{n}^{-}) \\ = (\begin{matrix} [\begin{matrix} {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - min (e_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(1 - min (f_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}], \\ [\begin{matrix} 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(max (g_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}}, \\ 1 - {(\prod_{1 \leq i_{1} < \dots < i_{x} \leq n} (1 - {(\prod_{j = 1}^{x} {(max (h_{i_{j}}))}^{w_{i_{j}}})}^{\frac{1}{x}}))}^{\frac{1}{C_{n}^{x}}} \end{matrix}] \end{matrix}) \end{matrix}$ (63)

From property 9, ${\tilde{φ}}^{-} \leq {IVIFWDHM}^{(x)} ({\tilde{φ}}_{1}, {\tilde{φ}}_{2}, \dots, {\tilde{φ}}_{n}) \leq {\tilde{φ}}^{+}$ (64)

4. Example and comparison

4.1. Numerical example

The overall growth of the economy induces an ever more competitive environment of tourism industry, the fundamental competition of the tourism industry lies in the destination. In the tourism development environment, in order to guarantee the steady development of a tourist destination, it is necessary to take measures to develop tourism destination to enhance competitiveness, it’s important to analyze and evaluate the destination competitiveness, which can show the attraction power of a destination, and indicate the direction for the efficient allocation of resources. Therefore, it is an important way for determining the destination development mode and direction to analyze the key factors of tourism destination competitiveness and extract qualitative and quantitative evaluation index, which could uphold long-term competitive advantage and promote the continuous development of destination. Thus, we give an example for competiveness evaluation of tourist destination with IVIFNs. There are five possible tourist destinations A_i (i = 1, 2, 3, 4, 5) to assess. The experts use the above attributes to assess the five tourist destinations: ding172G₁ is the attraction of tourism resources; ding173G₂ is the infrastructure and support for industry development; ding174G₃ is the supporting force of the tourism environment; ding175G₄ is the tourist demand. The five possible tourist destinations are to be assessed with IVIFNs (whose weighting vector ω = (0.3, 0.1, 0.2, 0.4)), as shown in theTable 1.

Table 1
IVIFN decision matrix

G₁ G₂ G₃ G₄

A₁ ([0.6,0.7],[0.2,0.3]) ([0.5,0.7],[0.1,0.3]) ([0.3,0.5],[0.4,0.5]) ([0.6,0.7],[0.2,0.3])

A₂ ([0.4,0.5],[0.1,0.3]) ([0.2,0.6],[0.3,0.4]) ([0.2,0.4],[0.3,0.6]) ([0.4,0.6],[0.2,0.3])

A₃ ([0.7,0.8],[0.1,0.2]) ([0.4,0.7],[0.2,0.3]) ([0.5,0.6],[0.3,0.4]) ([0.5,0.7],[0.2,0.3])

A₄ ([0.2,0.3],[0.3,0.5]) ([0.4,0.5],[0.1,0.2]) ([0.3,0.5],[0.1,0.4]) ([0.5,0.7],[0.1,0.2])

A₅ ([0.3,0.5],[0.4,0.2]) ([0.2,0.3],[0.4,0.6]) ([0.5,0.7],[0.1,0.3]) ([0.2,0.3],[0.4,0.5])

	G₁	G₂	G₃	G₄
A₁	([0.6,0.7],[0.2,0.3])	([0.5,0.7],[0.1,0.3])	([0.3,0.5],[0.4,0.5])	([0.6,0.7],[0.2,0.3])
A₂	([0.4,0.5],[0.1,0.3])	([0.2,0.6],[0.3,0.4])	([0.2,0.4],[0.3,0.6])	([0.4,0.6],[0.2,0.3])
A₃	([0.7,0.8],[0.1,0.2])	([0.4,0.7],[0.2,0.3])	([0.5,0.6],[0.3,0.4])	([0.5,0.7],[0.2,0.3])
A₄	([0.2,0.3],[0.3,0.5])	([0.4,0.5],[0.1,0.2])	([0.3,0.5],[0.1,0.4])	([0.5,0.7],[0.1,0.2])
A₅	([0.3,0.5],[0.4,0.2])	([0.2,0.3],[0.4,0.6])	([0.5,0.7],[0.1,0.3])	([0.2,0.3],[0.4,0.5])

Then, we use the approach developed for selecting the best tourist destinations.

Step 1. According to IVIFNs r_ij (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4), we fuse all IVIFNs r_ij by the IVIFWHM (IVIFWDHM) operator to have the IVIFNs A_i (i = 1, 2, 3, 4, 5) of the tourist destinations A_i. Let x = 2, then the fused results are in Table 2.

Table 2

The fused results of the tourist destinations by IVIFWMM (IVIFWDMM) operator

	IVIFWHM	IVIFWDHM
A₁	< ([0.8502,0.9023],[0.0619,0.0977])>	< ([0.1619,0.2270],[0.6941,0.7730])>
A₂	< ([0.7580,0.8548],[0.0536,0.1108])>	< ([0.0901,0.1643],[0.6771,0.7908])>
A₃	< ([0.8499,0.8989],[0.0497,0.0801])>	< ([0.1609,0.2226],[0.6674,0.7424])>
A₄	< ([0.7718,0.8478],[0.0384,0.0929])>	< ([0.0977,0.1583],[0.6291,0.7615])>
A₅	< ([0.7414,0.8209],[0.0919,0.1092])>	< ([0.0821,0.1387],[0.7513,0.7826])>

Step 2. By Table 2 and the score values of the tourist destinations are in Table 3.

Table 3

The score values of tourist destinations

	IVIFWHM	IVIFWDHM
A₁	0.7964	– 0.5390
A₂	0.7242	– 0.6067
A₃	0.8095	– 0.5131
A₄	0.7441	– 0.5673
A₅	0.6806	– 0.6566

Step 3. By Table 3, the order of tourist destinations is listed in Table 4. And, the best tourist destination is A₁.

Table 4

Order of the tourist destinations

	Order
IVIFWHM	A₃ > A₁ > A₄ > A₂ > A₅
IVIFWDHM	A₃ > A₁ > A₄ > A₂ > A₅

4.2. Influence analysis

In order to depict the effects on the ordering by altering parameters of x in IVIFWHM (IVIFWDHM) operators, the analysis results are listed inTables 5–6.

Table 5
Ordering results for IVIFWHM operator with different parameters

S (A₁) S (A₂) S (A₃) S (A₄) S (A₅) Order

x = 1 0.8176 0.7409 0.8295 0.7758 0.7229 A₃ > A₁ > A₄ > A₂ > A₅

x = 2 0.7964 0.7242 0.8095 0.7441 0.6806 A₃ > A₁ > A₄ > A₂ > A₅

x = 3 0.7920 0.7201 0.8003 0.7330 0.6616 A₃ > A₁ > A₄ > A₂ > A₅

x = 4 .7901 0.7182 0.7949 0.7269 0.6515 A₃ > A₁ > A₄ > A₂ > A₅

	S (A₁)	S (A₂)	S (A₃)	S (A₄)	S (A₅)	Order
x = 1	0.8176	0.7409	0.8295	0.7758	0.7229	A₃ > A₁ > A₄ > A₂ > A₅
x = 2	0.7964	0.7242	0.8095	0.7441	0.6806	A₃ > A₁ > A₄ > A₂ > A₅
x = 3	0.7920	0.7201	0.8003	0.7330	0.6616	A₃ > A₁ > A₄ > A₂ > A₅
x = 4	.7901	0.7182	0.7949	0.7269	0.6515	A₃ > A₁ > A₄ > A₂ > A₅

Table 6

Ordering results for IVIFWDHM operator with different parameters

	S (A₁)	S (A₂)	S (A₃)	S (A₄)	S (A₅)	Order
x = 1	–0.5884	– 0.6639	– 0.5665	– 0.6094	– 0.7071	A₃ > A₁ > A₄ > A₂ > A₅
x = 2	– 0.5390	– 0.6067	– 0.5131	– 0.5673	– 0.6566	A₃ > A₁ > A₄ > A₂ > A₅
x = 3	– 0.5193	– 0.5846	– 0.4940	– 0.5450	– 0.6453	A₃ > A₁ > A₄ > A₂ > A₅
x = 4	– 0.5106	– 0.5754	– 0.4853	– 0.5308	– 0.6401	A₃ > A₁ > A₄ > A₂ > A₅

4.3. Comparative analysis

We compare IVIFWHM and IVIFWDHM operator with IVIFWA and IVIFWG proposed by Xu [4, 5]. The results are listed in Table 7.

Table 7
Order of the tourist destinations

Order

IVIFWA A₃ > A₁ > A₄ > A₂ > A₅

IVIFWG A₃ > A₁ > A₄ > A₂ > A₅

	Order
IVIFWA	A₃ > A₁ > A₄ > A₂ > A₅
IVIFWG	A₃ > A₁ > A₄ > A₂ > A₅

From above, we can get the same best tourist destination and two methods’ ranking results are same. However, the existing methods with IVIFNs [4, 5] do not consider the relationship information among the arguments. Our proposed IVIFWHM and IVIFWDHM operators consider the relationship among the aggregated arguments.

5. Conclusion

In this paper, we investigate the MADM problems with IVIFNs. Then, we utilize the HM and DMM operator to design some HM operators with IVIFNs: IVIFHM operator, IVIFWHM operator, IVIFDHM operator and IVIFWDHM operator. The main characteristic of these operators are investigated. Then, we have used IVIFWHM and IVIFWDHM operators to propose two models for MADM problems with IVIFNs. Finally, a real example for competiveness evaluation of tourist destination is used to show the developed approach. In the subsequent studies, the extension and application of IVIFNs needs to be studied in many other fuzzy [41 –48] and uncertain environments [49 –57].

Footnotes

Acknowledgments

The work was supported by the National Social Science Foundation of China under Grant No. 16CGL026.

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Models for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. IFSs and IVIFSs

3.1. The IVIFHM operator

4.1. Numerical example

Table 7 Order of the tourist destinations Order IVIFWA A3 > A1 > A4 > A2 > A5 IVIFWG A3 > A1 > A4 > A2 > A5

Footnotes

Acknowledgments

References

Table 7
Order of the tourist destinations

Order

IVIFWA A₃ > A₁ > A₄ > A₂ > A₅

IVIFWG A₃ > A₁ > A₄ > A₂ > A₅