Some generalized Shapely interval-valued dual hesitant fuzzy uncertain linguistic Choquet geometric operators and their application to multiple attribute decision making

Abstract

To handle more complex multiple attribute decision making (MADM) problems, we define a novel type of fuzzy sets called interval-valued dual hesitant fuzzy uncertain linguistic (IVDHFUL) sets (IVDHFULSs) by combining interval-valued dual hesitant fuzzy sets with uncertain linguistic term sets. Then, the operations, the comparison method and the Hamming distance measure of IVDHFUL elements (IVDHFULEs) are defined. Besides, we propose the IVDHFUL Choquet geometric (IVDHFULCG) operator. This operator considers the interaction between adjacent coalitions of attributes. To overall capture the correlations among attributes, the Shapely IVDHFULCG (SIVDHFULCG) operator is proposed. In order to simplify the complexity of handling fuzzy measures, we further propose the 2-order additive SIVDHFULCG (2OA-SIVDHFULCG) operator. Moreover, when the weighting information is partly known, models for the optimal fuzzy measures are constructed. Finally, we propose two novel MADM methods and provide two cases to illustrate the specific application of the developed methods.

Keywords

Multiple attribute decision making (MADM)interval-valued dual hesitant fuzzy uncertain linguistic sets Hamming distance measure Choquet integral generalized Shapely function 2-order additive fuzzy measure

1 Introduction

Multiple attribute decision making (MADM) means that the most eligible alternative is found from the finite alternatives set according to the different attributes. In the past few decades, many MADM methods have been presented, such as the COPRAS method [17], the MOORA method [22], the ELECTRE TRI-NC method [1] and the MADM method based on aggregation operators [32]. In real MADM, people sometimes may feel hard to judge the membership degree (MD) of an element to a set because of the ambiguity between several crisp values. To address such cases, Torra [24] presented the notion of hesitant fuzzy sets (HFSs) that use a set of crisp values to denote the MD of an element. Later, some MADM methods with HFSs are presented in [4 , 40]. Additionally, Chen et al. [2] extends the MD of HFS to interval values and defined the interval-valued HFSs (IVHFSs). While HFSs give the MDs of an element in a given set, dual HFSs (DHFSs) [41] give both the MDs and no-membership degrees (NDs), which are expressed by two sets of crisp values. Obviously, DHFSs can reflect the hesitancy more objectively than HFSs. The applications of decision making with DHFSs can be seen in [7 , 34]. Motivated by the ideas of HFSs to IVHFSs, Ju et al. [9] introduced the notion of interval-valued DHFSs (IVDHFS), which permits MDs and NDs having two sets of possible interval values. So, it is more adequate and sufficient to describe the preference information by utilizing IVDHFSs. Ju et al. [9] also proposed the operations of IVDHFEs and developed some operators of IVDHFEs to solve MADM problems.

Linguistic variable (LV) [35] is a powerful tool to depict qualitative information. For example, when the mobile phone performance is evaluated, it is easy to be described by the LV, such as extremely poor, poor and very good. Since the appearance of LV, it and its variations such as uncertain linguistic variable (ULV) [31], intuitionistic ULV (IULV) [13], and interval-valued hesitant fuzzy LV (IVHFLV) [25], have attracted more and more attentions. More concretely, Xu [31] introduced the concept of ULV and proposed two uncertain linguistic aggregation operators. To overcome the drawback of ULV that it implies that the MD is one and ND degree is zero, Liu et al. [13] presented the notion of IULV and proposed several geometric operators of IULVs. Wang et al. [25] presented the notion of IVHFLV and proposed several prioritized operators of IVHFLVs for MADM. Furthermore, dual hesitant fuzzy linguistic set (DHFLS) introduced by Yang and Ju [33] combines LV with DHFS, which is better to express decision makers’ hesitance. Since ULVs and IVDHFSs depict uncertain information more easily than LV and DHFS, respectively, the concept of IVDHFULS combining ULV with IVDHFS is defined in this paper. The IVDHFULSs outperform the ULVs and a IVDHFSs. Compared to ULV, the IVDHFULE considers much more information. For example, E-service quality is perhaps evaluated to be lower than ‘good’ but higher ‘fair’. If this result is represented with ULV, it will be [fair, good]. If the result is represented with IVDHFULE, it may be < [fair, good], {([0.5, 0.7], [0.1, 0.2]), ([0.4, 0.6], [0.1, 0.2])}>, which is more accurate than the result represented by ULV. When compared to IVDHFS, IVDHFULS makes the MDs and NDs no longer with respect to a fuzzy concept, but to an ULV, which makes the decision makers to depict uncertain information more easily and precisely.

Since the fuzzy linguistic information can be aggregated by fuzzy linguistic aggregation operators into a collective one, many researchers devoted themselves into studying fuzzy linguistic MADM problems based on fuzzy linguistic aggregation operators. For instance, Xu [30, 31] investigated uncertain linguistic aggregation operators. Liu and Jin [13] studied geometric operator of IULVs. Yang and Ju [33] discussed several aggregation operators of DHFLSs. However, these fuzzy linguistic operators are based on the assumption that aggregated arguments are independent. They are unable consider the relationship of the aggregated arguments. To deal with such case, Zhang et al. [37], Liu et al. [14] and Liu et al. [11] presented some Bonferroni mean operators, partitioned Heronian mean operators and Maclaurin symmetric mean operators of fuzzy linguistic information, respectively. In addition, Liu et al. [12], Liu and Tang [15] and Zhang et al. [36] presented some fuzzy linguistic Choquet operators by utilizing the Choquet integral [3], fuzzy measures (FMs) [23] and the generalized Shapely function [21]. Since FMs [23] can be utilized to measure interactions, the Choquet integral [3] is a useful tool to aggregate fuzzy information, and the generalized Shapely function [21] denotes the expected value of the overall interaction between coalition and every coalition removing front coalition from the union set, the operators presented in [12 , 36] reflect redundant, complementary or independent characteristics among the criteria.

IVDHFULSs can more precisely describe the fuzzy information, and can be utilized to handle more complex MADM problems. However, all above-mentioned aggregation operators fail to aggregate IVDHFULSs. From above analysis, we find that the aggregation operators by utilizing the Choquet integral, the FM and the generalized Shapely function, could overall reflect the correlations among elements. So, we extend these operators to the IVDHFULSs, and propose the IVDHFULCG operator, which considers the existing interactions between two adjacent coalitions of criteria, the SIVDHFULCG operator, which overall captures the correlations among criteria, and the 2OA-SIVDHFULCG operator, which simplifies the complexity of handling FMs. Furthermore, when the weighting information is partly known, the models based on the maximizing deviation method are constructed. As a series of development, two approaches to IVDHFUL MADM with interactive attributes and partly known weight information are proposed. To do so, the organization is offered as:

Section 2 recalls two basic concepts, including IVDHFSs and DHFLSs. Section 3 defines the concept, the operations, the comparison method and the Hamming distance of IVDHFULEs. Section 4 proposes three geometric aggregation operators of IVDHFULSs that are the IVDHFULCG operator, the SIVDHFULCG operator and the 2OA-SIVDHFULCG operator. Section 5 develops two approaches to MADM with IVDHFULSs. Section 6 gives two cases to demonstrate the developed MADM methods. Section 7 concludes the study.

2 Preliminaries

2.1 The IVDHFS

Definition 1. [9] An IVDHFS in the fixed set Y is expressed by $DIS = {< y, H (y), G (y) > | y \in Y},$ (1) where H (y) = ∪ _[r^f,r^b] {[r^f, r^b]} and G (x) = ∪ _[t^f,t^b] {[t^l, t^u]} are the possible interval-valued MDs and interval-valued NDs of the element y ∈ Y to DIS, respectively, with the conditions that [r^f, r^b] ⊂ [0, 1] , [t^f, t^b] ⊂ [0, 1] , 0 ≤ (r^b) ⁺ + (t^b) ⁺ ≤ 1, where [r^f, r^b] ∈ H (y) , [t^f, t^b] ∈ G (y) , (r^b) ⁺ ∈ H⁺ $(y) = \cup_{[r^{f}, r^{b}] \in H_{(y)}} \max {r^{b}}, {(t^{b})}^{+} \in G^{+} (y) = \cup_{[t^{f}, t^{b}] \in G (y)} \max {t^{b}}$ for all y ∈ Y .

About the operational rules and characteristics of IVDHFEs, please refer to reference [9].

2.2 The DHFLS

Definition 2. [33] A DHFLS in the fixed set Y is expressed by ${DIS}^{'} = {< y, s_{θ (y)}, H (y), G (y) > | y \in Y},$ (2) where s_θ(y) is a LV defined by Zadeh [35], H (y) and G (y) are the possible MDs and NDs of the element y ∈ Y to the LV s_θ(y), respectively, with the conditions that 0 < t, r < 1, 0 ≤ t⁺ + r⁺ ≤ 1, where t ∈ G (y) , r ∈ H (y) , t⁺ ∈ G⁺ (y) = ∪ _t∈G(y) max {t} , r⁺ ∈ H⁺ (y) = ∪ _r∈H(y) max {r} for all y ∈ Y.

About the operational rules and characteristics of DHFLSs, please refer to reference [33].

3 IVDHFLSs and their operations

We shall introduce the concept of IVDHFLSs based on ULVs [30, 31] and IVDHFSs [9], and propose the operations, the comparison method and the Hamming distance of IVDHFULEs.

3.1 The IVDHFULS

Definition 3. An IVDHFLS in the fixed set Y is expressed by ${DIS}^{″} = {< y, [s_{θ (y)}, s_{τ (y)}], H (y), G (y) > | y \in Y},$ (3) where [s_θ(y), s_τ(y)] is an ULV defined by Xu [30, 31] H (y) = ∪ _[r^f,r^b] {[r^f, r^b]} and H (y) = ∪ _[t^f,t^b] {[t^f, t^b]} are the possible interval-valued MDs and interval-valued NDs of the element y ∈ Y to the ULV [s_θ(y), s_τ(y)], respectively, with the conditions that [r^f, r^b] ⊂ [0, 1] , [t^f, t^b] ⊂ [0, 1] , 0 ≤ (r^b) ⁺ + (t^b) ⁺ ≤ 1, where [ r^f, r^b ] ∈ H (y) , [ t^f , t^b ] ∈ G (y) , (r^b) ⁺ ∈ H⁺ (y) = ∪ [ r^f, r^b ] ∈ H(y) max { r^b } , (t^b) ⁺ ∈ G⁺ (y)=∪ [ t^f, t^b] ∈ G(y) max { t^b } for all y ∈ Y .

For simplicity, IL =< [s_θ, s_τ] , H, G > is called an IVDHFULE.

Suppose IL₁ =< [s_θ1, s_τ1] , H₁, G₁ > and IL₂ =< [s_θ2, s_τ2] , H₂, G₂ > are two IVDHFULEs. Their operations are defined as: $\begin{matrix} (1) {IL}_{1} \oplus I L_{2} = < [s_{θ 1 + θ 2}, s_{τ 1 + τ 2}], \\ \cup_{[{r_{1}}^{f}, {r_{1}}^{b}] \in H_{1}, [{t_{1}}^{f}, {t_{1}}^{b}] \in G_{1}, [{r_{2}}^{f}, {r_{2}}^{b}] \in H_{2}, [{t_{2}}^{f}, {t_{2}}^{b}] \in G_{2}} \\ {{[{r_{1}}^{f} + {r_{2}}^{f} - {r_{1}}^{f} {r_{2}}^{f}, {r_{1}}^{b} + {r_{2}}^{b} - {r_{1}}^{b} {r_{2}}^{b}]}, \\ {[{t_{1}}^{f} {t_{2}}^{f}, {t_{1}}^{b} {t_{2}}^{b}]}} >, \end{matrix}$ (4) $\begin{matrix} (2) {IL}_{1} \otimes I L_{2} = < [s_{θ 1 θ 2}, s_{τ 1 τ 2}], \\ \cup_{[{r_{1}}^{f}, {r_{1}}^{b}] \in H_{1}, [{t_{1}}^{f}, {t_{1}}^{b}] \in G_{1}, [{r_{2}}^{f}, {r_{2}}^{b}] \in H_{2}, [{t_{2}}^{f}, {t_{2}}^{b}] \in G_{2}} \\ {{[{r_{1}}^{f} {r_{2}}^{f}, {r_{1}}^{b} {r_{2}}^{b}]}, \\ {[{t_{1}}^{f} + {t_{2}}^{f} - {t_{1}}^{f} {t_{2}}^{f}, {t_{1}}^{b} + {t_{2}}^{b} - {t_{1}}^{b} {t_{2}}^{b}]}} >, \end{matrix}$ (5)

$\begin{matrix} (3) (I L_{1})^{λ} = < [s_{{(θ 1)}^{λ}}, s_{{(τ 1)}^{λ}}], \cup_{[{r_{1}}^{l}, {r_{1}}^{u}] \in H_{1}, [{t_{1}}^{l}, {t_{1}}^{u}] \in G_{1}} \\ {{[({r_{1}}^{f})^{λ}, ({r_{1}}^{b})^{λ}]}, \\ {[1 - (1 - {t_{1}}^{f})^{λ}, 1 - (1 - {t_{1}}^{b})^{λ}]}} >, \end{matrix}$ (6) $\begin{matrix} (4) λ \cdot I L_{1} = < [s_{λ θ 1}, s_{λ τ 1}], \cup_{[{r_{1}}^{f}, {r_{1}}^{b}] \in H_{1}, [{t_{1}}^{f}, {t_{1}}^{b}] \in G_{1}} \\ {{[1 - (1 - {r_{1}}^{f})^{λ}, 1 - (1 - {r_{1}}^{b})^{λ}]}, \\ {[({t_{1}}^{f})^{λ}, ({t_{1}}^{b})^{λ}]}} > . \end{matrix}$ (7)

Theorem 1. Suppose IL₁ =< [s_θ1, s_τ1] , H₁, G₁ > and IL₂ =< [s_θ2, s_τ2] , H₂, G₂ > are two IVDHFULEs, and λ₁, λ₂ ≥ 0; then we have: $(1) I L_{1} \oplus I L_{2} = I L_{2} \oplus I L_{1},$ (8) $(2) I L_{1} \otimes I L_{2} = I L_{2} \otimes I L_{1},$ (9) $(3) λ_{1} (I L_{1} \oplus I L_{2}) = λ_{1} I L_{1} \oplus λ_{1} I L_{2},$ (10) $(4) λ_{1} I L_{1} \oplus λ_{2} I L_{1} = (λ_{1} + λ_{2}) I L_{1},$ (11) $(5) I {L_{1}}^{λ_{1}} \otimes I {L_{2}}^{λ_{1}} = (I L_{2} \otimes I L_{1})^{λ_{1}},$ (12)

$(6) I {L_{1}}^{λ_{1}} \otimes I {L_{1}}^{λ_{2}} = (I L_{1})^{λ_{1} + λ_{2}} .$ (13)

Proof.

According to Equations (4 and 5), Equations (8 and 9) are right.

For Equation (10):

$\begin{matrix} λ_{1} (I L_{1} \oplus I L_{2}) = λ_{1} < [s_{θ 1 + θ 2}, s_{τ 1 + τ 2}], \\ \cup_{[{r_{1}}^{f}, {r_{1}}^{b}] \in H_{1}, [{t_{1}}^{f}, {t_{1}}^{b}] \in G_{1}, [{r_{2}}^{f}, {r_{2}}^{b}] \in H_{2}, [{t_{2}}^{f}, {t_{2}}^{b}] \in G_{2}} \\ {{[{r_{1}}^{f} + {r_{2}}^{f} - {r_{1}}^{f} {r_{2}}^{f}, {r_{1}}^{b} + {r_{2}}^{b} - {r_{1}}^{b} {r_{2}}^{b}]}, \\ {[{t_{1}}^{f} {t_{2}}^{f}, {t_{1}}^{b} {t_{2}}^{b}]}} > \\ = < [s_{λ_{1} θ 1 + λ_{1} θ 2}, s_{λ_{1} τ 1 + λ_{1} τ 2}], \\ \cup_{[{r_{1}}^{f}, {r_{1}}^{b}] \in H_{1}, [{t_{1}}^{f}, {t_{1}}^{b}] \in G_{1}, [{r_{2}}^{f}, {r_{2}}^{b}] \in H_{2}, [{t_{2}}^{f}, {t_{2}}^{b}] \in G_{2}} \\ {{[1 - {(1 - {r_{1}}^{f})}^{λ_{1}} {(1 - {r_{2}}^{f})}^{λ_{1}}, 1 - {(1 - {r_{1}}^{b})}^{λ_{1}} {(1 - {r_{2}}^{b})}^{λ_{1}}]}, \\ {[({t_{1}}^{f})^{λ_{1}} {({t_{2}}^{f})}^{λ_{1}}, ({t_{1}}^{b})^{λ_{1}} {({t_{2}}^{b})}^{λ_{1}}]}} > \\ = < [s_{λ_{1} θ 1}, s_{λ_{1} τ 1}], \cup_{[{r_{1}}^{f}, {r_{1}}^{b}] \in H_{1}, [{t_{1}}^{f}, {t_{1}}^{b}] \in G_{1}} \\ {{[1 - {(1 - {r_{1}}^{f})}^{λ_{1}}, 1 - {(1 - {r_{1}}^{b})}^{λ_{1}}]}, {[({t_{1}}^{f})^{λ_{1}}, ({t_{1}}^{b})^{λ_{1}}]}} > \\ \oplus < [s_{λ_{1} θ 2}, s_{λ_{1} τ 2}], \cup_{[{r_{2}}^{f}, {r_{2}}^{b}] \in H_{2}, [{t_{2}}^{f}, {t_{2}}^{b}] \in G_{2}} \\ {{[1 - {(1 - {r_{2}}^{f})}^{λ_{1}}, 1 - {(1 - {r_{2}}^{b})}^{λ_{1}}]}, {[({t_{2}}^{f})^{λ_{1}}, ({t_{2}}^{b})^{λ_{1}}]}} > \\ = λ_{1} < [s_{θ 1}, s_{τ 1}], \cup_{[{r_{1}}^{f}, {r_{1}}^{b}] \in H_{1}, [{t_{1}}^{f}, {t_{1}}^{b}] \in G_{1}} \\ {{[{r_{1}}^{f}, {r_{1}}^{b}]}, {[{t_{1}}^{f}, {t_{1}}^{b}]}} > \\ \oplus λ_{1} < [s_{θ 2}, s_{τ 2}], \cup_{[{r_{2}}^{f}, {r_{2}}^{b}] \in H_{2}, [{t_{2}}^{f}, {t_{2}}^{b}] \in G_{2}} \\ {{[{r_{2}}^{f}, {r_{2}}^{b}]}, {[{t_{2}}^{f}, {t_{2}}^{b}]}} > = λ_{1} I L_{1} \oplus λ_{1} I L_{2}, \end{matrix}$ the proof of Equation (10) is completed.

Similar to proof of Equation (10), it can be easily proved that the Equations (11–13) are right.

3.2 Comparison of two IVDHFULEs

Definition 4. Suppose IL =< [s_θ, s_τ] , H, G > is an IVDHFULE, and suppose # H and # G represent the counts of interval values in # H and # G, respectively. The score function S (IL) and the accuracy function P (IL) of IL are defined as: $\begin{matrix} S (IL) = \frac{θ + τ}{4} (\frac{1}{# H} \sum_{[r^{l}, r^{u}] \in H} r^{f} + \frac{1}{# H} \sum_{[r^{l}, r^{u}] \in H} r^{b} \\ - \frac{1}{# G} \sum_{[t^{l}, t^{u}] \in G} t^{f} - \frac{1}{# G} \sum_{[t^{l}, t^{u}] \in G} t^{b}), \end{matrix}$ (14) $\begin{matrix} P (IL) = \frac{θ + τ}{4} (\frac{1}{# H} \sum_{[r^{l}, r^{u}] \in H} r^{f} + \frac{1}{# H} \sum_{[r^{l}, r^{u}] \in H} r^{b} \\ + \frac{1}{# G} \sum_{[t^{l}, t^{u}] \in G} t^{f} + \frac{1}{# G} \sum_{[t^{l}, t^{u}] \in G} t^{b}) . \end{matrix}$ (15)

Theorem 2. Suppose IL₁ =< [s_θ1, s_τ1] , H₁, G₁ > and IL₂ =< [s_θ2, s_τ2] , H₂, G₂ > are any two IVDHFULEs; the comparison method is shown as follows:

If S (IL₁) > S (IL₂), then IL₁ > IL₂;

If S (IL₁) = S (IL₂) and P (IL₁) > P (IL₂), then IL₁ > IL₂;

If S (IL₁) = S (IL₂) and P (IL₁) = P (IL₂), then IL₁ = IL₂.

3.3 The Hamming distance between two IVDHFULEs

Definition 5. Suppose IL₁, IL₂ and IL₃ are three IVDHFULEs. If dis (IL₁, IL₂) meets the conditions as follows:

0 ≤ dis (IL₁, IL₂) ≤1;

dis (IL₁, IL₂) =0 If and only if IL₁ = IL₂;

dis (IL₁, IL₂) = dis (IL₂, IL₁);

dis (IL₁, IL₂) + dis (IL₂, IL₃) > dis (IL₁, IL₃);

then dis (IL₁, IL₂) is called the distance between IL₁ and IL₂.

Definition 6. Suppose IL₁ =< [s_θ1, s_τ1] , H₁, G₁ > and IL₂ =< [s_θ2, s_τ2] , H₂, G₂ > are any two IVDHFULEs; then the Hamming distance dis (IL₁, IL₂) between IL₁ and IL₂ is defined as follows:

$\begin{array}{l} d i s (I L_{1}, I L_{2}) = \frac{1}{8 (l - 1)} (\frac{1}{l_{1}} \sum_{i = 1}^{l_{1}} | θ_{1} r_{1 σ (i)}^{f} - θ_{2} r_{2} {_{σ (i)}}^{f} | \\ + \frac{1}{l_{1}} \sum_{i = 1}^{l_{1}} | θ_{1} r_{1} {_{σ (i)}}^{b} - θ_{2} r_{2} {_{σ (i)}}^{b} | + \frac{1}{l_{2}} \sum_{j = 1}^{l_{2}} | θ_{1} t_{1} {_{σ (j)}}^{f} - θ_{2} t_{2} {_{σ (j)}}^{f} | \\ + \frac{1}{l_{2}} \sum_{j = 1}^{l_{2}} | θ_{1} t_{1} {_{σ (j)}}^{b} - θ_{2} t_{2} {_{σ (j)}}^{b} | + \frac{1}{l_{1}} \sum_{i = 1}^{l_{1}} | τ_{1} r_{1} {_{σ (i)}}^{f} - τ_{2} r_{2} {_{σ (i)}}^{f} | \\ + \frac{1}{l_{1}} \sum_{i = 1}^{l_{1}} | τ_{1} r_{1} {_{σ (i)}}^{b} - τ_{2} {\tilde{γ}}_{2} {_{σ (i)}}^{b} + \frac{1}{l_{2}} \sum_{j = 1}^{l_{2}} | τ_{1} t_{1} {_{σ (j)}}^{f} - τ_{2} t_{2} {_{σ (j)}}^{f} | | \\ + \frac{1}{l_{2}} \sum_{j = 1}^{l_{2}} | τ_{1} t_{1} {_{σ (j)}}^{b} - τ_{2} t_{2} {_{σ (j)}}^{b} |), \end{array}$

where $[r_{1} σ (i) f, r_{1} σ (i) b]$ and $[r_{2} σ (i) f, r_{2} σ (i) b]$ are the ith largest value in H₁ and H₂, respectively; $[t_{1} σ (j) f, t_{1} σ (j) b]$ and $[t_{2} σ (j) f, t_{2} σ (j) b]$ are the jth largest value in G₁ and G₂, respectively; # H₁, # G₁, # H₂ and # G₂ are the counts of interval values in H₁, G₁, H₂ and G₂, respectively; l₁ = max(# H₁, # H₂), l₂ = max(# G₁, # G₂); l is the cardinality of linguistic set defined by Zadeh [35]. In most cases, # H₁ ≠ # H₂, # G₁ ≠ # G₂. To get the distance measure between IL₁ and IL₂, the shorter one need to be extended until the interval-valued MDs and interval-valued NDs of both IVDHFULEs have the same length.

It is easy to see that the Equation (16) satisfies the properties in Definition 5.

4 Some SIVDHFULCG aggregation operators

This section proposes the IVDHFULCG operator, the SIVDHFULCG operator and the 2OA-SIVDHFULCG operator. At the same time, various special cases and desirable properties of these given operators are researched.

4.1 The fuzzy measures and the Choquet integral operator

Definition 7. [23] A FM on Y is a set function v : P (Y) → [0, 1]. It satisfies the following restrictions:

Boundary: v (φ) =0, v (Y) =1,

Monotonicity: if C, D ∈ P (Y) and C ⊂ D, then v (C) ≤ v (D), where P (Y) is the power set of Y.

In MADM, v (M) is often regarded as the weight of the criteria set M.

The discrete Choquet integral with regard to FM can be used to integrate fuzzy interactive information. The formulation of the well-known Choquet integral was proposed by Grabisch [3].

Definition 8. [3] Suppose m is a positive real-valued function on Y, and v is a FM on Y. The formulation of Chouqet integral on discrete sets is defined as follows: $\begin{matrix} C_{v} (m (y_{(1)}), m (y_{(2)}), \dots, m (y_{(n)})) \\ = \sum_{j = 1}^{n} m (y_{(j)}) (v (L_{(j)}) - v (L_{(j + 1)})), \end{matrix}$ (16) where (.) signifies a permutation on Y so that m (y₍₁₎) ≤ m (y₍₂₎) ≤ , …, ≤ m (y_(n)) and L_(j) = {y_(j), …, y_(n)} with L_(n+1) = φ.

4.2 The IVDHFULCG operator

Based on IVDHFULEs and the Choquet integral, we shall develop the IVDHFULCG operator, which reflects the interactions among criteria.

Definition 9. Suppose that Y = {y₁, y₂, …, y_n} is a set of attributes and suppose IL_i =< [s_θj, s_τj] , H_j, G_j > (j = 1, 2, …, n) is a set of IVDHFULEs on Y. An IVDHFULCG operator is defined by

$\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) = \\ \otimes_{j = 1}^{n} I {L_{(i)}}^{v (L_{(j)}) - v (L_{(j + 1)})}, \end{matrix}$ (17) where IL_(j) represents the jth smallest element in the set {IL₍₁₎, IL₍₂₎, …, IL_(n) }, y_(j) is the attribute corresponding to IL_(j), L_(j) = {y_(j), y_(j+1), …, y_(n)} with L_(n+1) = φ, and v (L_(j)) (j = 1, 2, …, n) represents the FM on L_(j) described in Definition 7.

Theorem 3. The aggregated value of the IL_j (j = 1, 2, …, n) obtained by the IVDHFULCG operator is also an IVDHFULE, where $\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \\ = < [s_{\prod_{j = 1}^{n} θ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}, s_{\prod_{j = 1}^{n} τ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}], \\ \cup_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(j)}, [{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} \\ {{[\prod_{j = 1}^{n} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \prod_{j = 1}^{n} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}, \\ {[1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ 1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}} > . \end{matrix}$ (18)

Proof.

We first prove Equation (19) is right by using mathematical induction.

For n = 1, it is obvious Equation (19) is right.

For n = 2, we have $\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}) \\ = I {L_{1}}^{v (L_{(1)}) - μ (L_{(2)})} \otimes I {L_{2}}^{v (L_{(2)}) - v (L_{(3)})} \\ = < [s_{\prod_{j = 1}^{2} θ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}, s_{\prod_{j = 1}^{2} τ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}], \\ \cup_{[{r_{j}}^{f}, {r_{j}}^{t}] \in H_{j}, [{t_{j}}^{f}, {t_{j}}^{t}] \in G_{j}} \end{matrix}$

$\begin{matrix} {{[\prod_{j = 1}^{2} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \prod_{j = 1}^{2} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}, \\ {[1 - \prod_{j = 1}^{2} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ 1 - \prod_{j = 1}^{2} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}} >, \end{matrix}$

Hence, the Equation (19) holds for n = 2.

Suppose Equation (19) still holds for n = k, i.e., $\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{k}) \\ = < [s_{\prod_{j = 1}^{k} θ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}, s_{\prod_{j = 1}^{k} τ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}], \\ \cup_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(j)}, [{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} \\ {{[\prod_{j = 1}^{k} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \prod_{j = 1}^{k} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}, \\ {[1 - \prod_{j = 1}^{k} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ 1 - \prod_{j = 1}^{k} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}} > . \end{matrix}$

Then, for n = k + 1, we have $\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, I L_{k + 1}) \\ = < [s_{\prod_{j = 1}^{k} θ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}, s_{\prod_{j = 1}^{k} τ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}], \\ \cup_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(j)}, [{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} \\ {{[\prod_{j = 1}^{k} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \prod_{j = 1}^{k} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}, \\ {[1 - \prod_{j = 1}^{k} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ 1 - \prod_{j = 1}^{k} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}} > \\ \otimes < [s_{θ {(k + 1)}^{v (L_{(k + 1)}) - v (L_{(k + 2)})}}, s_{τ {(k + 1)}^{v (L_{(k + 1)}) - v (L_{(k + 2)})}}], \\ \cup_{[{r_{(k + 1)}}^{f}, {r_{(k + 1)}}^{b}] \in H_{(k + 1)}, [{t_{(k + 1)}}^{f}, {t_{(k + 1)}}^{b}] \in G_{(k + 1)}} \\ {{[{r^{f}}_{(k + 1)}, {r^{b}}_{(k + 1)}]}, {[{t^{f}}_{(k + 1)}, {t^{b}}_{(k + 1)}]}} > \\ = < [s_{\prod_{j = 1}^{k + 1} θ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}, s_{\prod_{j = 1}^{k + 1} τ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}}], \\ \cup_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(i)}, [{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} \\ {{[\prod_{j = 1}^{k + 1} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \end{matrix}$

$\begin{matrix} \prod_{j = 1}^{k + 1} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}, \\ {[1 - \prod_{j = 1}^{k + 1} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ 1 - \prod_{j = 1}^{k + 1} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}} > \end{matrix}$

Next, we prove Equation (19) is an IVDHFULE. The following inequalities can be easily proved: $\begin{matrix} min_{j} (θ (j)) \leq \prod_{j = 1}^{n} θ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq min_{j} (θ (j)), \\ min_{j} (τ (j)) \leq \prod_{j = 1}^{n} τ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq min_{j} (τ (j)), \\ \prod_{j = 1}^{n} θ {(i)}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq \prod_{j = 1}^{n} τ {(j)}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ [\prod_{j = 1}^{n} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ \prod_{j = 1}^{n} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}] \subseteq [0, 1], \\ [1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ 1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}] \subseteq [0, 1], \\ 0 \leq \prod_{j = 1}^{n} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})} \\ + 1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq 1 . \end{matrix}$

Thus, it is an IVDHFULE and the result is obtained.

To illustrate how to utilize the proposed operator, we provide the following example.

Example 2.1. An expert evaluates an alternatives a₁ with respect to the criteria y₁ and y₂. Assume that the FM of the criteria set {y₁, y₂} is shown as follows: v (φ) =0, v ({y₁}) =0.4, v ({y₂}) =0.5, v ({y₁, y₂}) =1, and assume that the evaluating values IL_i = < [s_θj, s_τj] , H_j, H_j > (j = 1, 2) of the alternative a₁ about the criteria y₁ and y₂ are shown as follows: $\begin{matrix} I L_{1} = < [s_{2}, s_{4}], {{[0.3, 0.5]}, {[0.3, 0.5], [0.2, 0.4]}} >, \\ I L_{2} = < [s_{3}, s_{5}], {{[0.2, 0.4]}, {[0.1, 0.3], [0.3, 0.4]}} > . \end{matrix}$

Then, we can obtain IVDHFULCG_v (IL₁, IL₂) by Equation (19), shown as follows:

First, by utilizing Equation (14) to get the score value S (IL_j) of each evaluating value IL_j (j = 1, 2), we get $\begin{matrix} S (I L_{1}) = \frac{2 + 4}{4} \times (0.3 - \frac{0.3 + 0.2}{2} + 0.5 - \frac{0.5 + 0.4}{2}) = 0.15, \\ S (I L_{2}) = \frac{3 + 5}{4} \times (0.2 - \frac{0.1 + 0.3}{2} + 0.4 - \frac{0.3 + 0.4}{2}) = 0.1 . \end{matrix}$

Thus, by Theorem 2, we can obtain IL₁ > IL₂.

Second, by utilizing the IVDHFULCG operator shown in Equation (19) to obtain the comprehensive IVDHFUL value IVDHFULCG_v (IL₁, IL₂) of alternative a₁, we have $\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}) = I {L_{2}}^{v ({y_{1}, y_{2}}) - v ({y_{1}})} \otimes I {L_{1}}^{v ({y_{1}})} \\ = < [s_{3}, s_{5}], {{[0.2, 0.4]}, {[0.1, 0.3], [0.3, 0.4]}} >^{1 - 0.4} \\ \otimes < [s_{2}, s_{4}], {{[0.3, 0.5]}, {[0.3, 0.5], [0.2, 0.4]}} >^{0.4 - 0} \\ = < [s_{3^{0.6} \times 2^{0.4}}, s_{5^{0.6} \times 4^{0.4}}], {{[{0.2}^{0.6} \times {0.3}^{0.4}, {0.4}^{0.6} \times {0.5}^{0.4}]}, \\ {[1 - {(1 - 0.1)}^{0.6} \times {(1 - 0.3)}^{0.4}, 1 - {(1 - 0.3)}^{0.6} \times {(1 - 0.5)}^{0.4}], \\ [1 - {(1 - 0.3)}^{0.6} \times {(1 - 0.4)}^{0.4}, 1 - {(1 - 0.2)}^{0.6} \times {(1 - 0.4)}^{0.4}], \\ [1 - {(1 - 0.1)}^{0.6} \times {(1 - 0.3)}^{0.4}, 1 - {(1 - 0.2)}^{0.6} \times {(1 - 0.4)}^{0.4}], \\ [1 - {(1 - 0.3)}^{0.6} \times {(1 - 0.4)}^{0.4}, 1 - {(1 - 0.3)}^{0.6} \times {(1 - 0.5)}^{0.4}]}} > \\ = < [s_{2.5508}, s_{4.5731}], {{[0.235 2, 0.437 3]}, {[0.186 1, 0.388 1], \\ [0.2616, 0.4], [0.1414, 0.3419], [0.3, 0.4422]}} > . \end{matrix}$

In the following, we shall give some propositions which are proved to be special cases of the proposed IVDHFULCG operator.

Proposition 1. If μ (L) = ∑_{y_j∈L}v ({y_j}) for all L ⊆ Y, then v (y_(j)) = v (L_(j)) - v (L_(j+1)) , j = 1, 2, …, n . In this case, IVDHFULCG operator reduces to the weighted geometric (WG) operator of IVDHFULSs (IVDHFULWG): $\begin{matrix} IVDHFULWG (I L_{1}, I L_{2}, \dots, {IL}_{n}) \\ = < [s_{\prod_{j = 1}^{n} θ j^{v ({y_{i}})}}, s_{\prod_{j = 1}^{n} τ j^{v ({y_{i}})}}], \cup_{[{r_{j}}^{f}, {r_{j}}^{b}] \in H_{j}, [{t_{i}}^{f}, {t_{i}}^{b}] \in G_{j}} \\ {{[\prod_{j = 1}^{n} {({r_{j}}^{f})}^{v ({y_{i}})}, \prod_{j = 1}^{n} {({r_{j}}^{b})}^{v ({y_{i}})}]}, \\ {[1 - \prod_{j = 1}^{n} {(1 - {t_{j}}^{f})}^{v ({y_{i}})}, 1 - \prod_{j = 1}^{n} {(1 - {t_{j}}^{b})}^{v ({y_{i}})}]}} > . \end{matrix}$ (19)

Proposition 2. If $v (L) = \sum_{i = 1}^{| L |} w_{i}$ for all L ⊆ Y, where |L| indicates the count of the elements in the set L, then w_j = v (L_(j)) - v (L_(j+1)) , j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ . In this case, the IVDHFULCG operator degenerates to the ordered WG operator of IVDHFULSs (IVDHFULOWG): $\begin{matrix} IVDHFULOWG (I L_{1}, I L_{2}, \dots, {IL}_{n}) \\ = < [s_{\prod_{j = 1}^{n} θ {(j)}^{w_{j}}}, s_{\prod_{j = 1}^{n} τ {(j)}^{w_{j}}}], \cup_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(j)}, [{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} \\ {{[\prod_{j = 1}^{n} {({r_{(j)}}^{f})}^{w_{j}}, \prod_{j = 1}^{n} {({r_{(j)}}^{b})}^{w_{j}}]}, \\ {[1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{f})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{b})}^{w_{j}}]}} > . \end{matrix}$ (20)

Further, we can prove the IVDHFULCG operator has the commutativity, the idempotency and the boundedness.

Theorem 4. Suppose that Y = {y₁, y₂, …, y_n} is a set of criteria, IL_i = < [s_θj, s_τj] , H_j, G_j > (j = 1, 2, …, n) is a set of IVDHFULEs on Y, IL_(j) denotes the jth smallest element in the set {IL₍₁₎, IL₍₂₎, …, IL_(n) }, y_(j) denotes the criterion corresponding to IL_(j), L_(j) = {y_(j), y_(j+1), …, y_(n)} with L_(n+1) = φ, and v (L_(j)) (j = 1, 2 … , n) is the FM on L_(j). Then , we get:

Commutativity. Suppose $I L_{j}^{'} (j = 1, 2, \dots, n)$ is any permutation of IL_j; then

$\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \\ = IVDHFULC G_{v} (I {L^{'}}_{1}, I {L^{'}}_{2}, \dots, {IL}_{n}^{'}) \end{matrix}$

Idempotency. Suppose for all j, IL_j = IL =< [s_θ, s_τ] , {{[r^f, r^b]} , {[t^f, t^b]}} >; then

$IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) = IL .$

Boundedness. Suppose that θ - = min {θj} , τ - = min {τj} , r^f- = min {r^f} , r^b- = min {r^b} , t^f- = min {t^f} , t^b- = min {t^b} , θ + = max {θj} , τ + = max {τj} , r^f+ = max {r^f} , r^b+ = max {r^b} , t^f+ = max {t^f} , t^b+ = max {t^b} , and suppose IL^- = < [s_θ-, s_τ-] , {{r^f-, r^b-} , {t^f+, t^b+} } > , IL⁺ = < [s_θ+, s_τ+] , {{ [r^f+, r^b+]} , {[t^f-, t^b-]}}>; then

$I L^{-} \leq IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \leq I L^{+} .$

Proof.

Since IL_j (j = 1, 2, …, n) is a permutation ofIL_j, by Definition 9, we can easily proof it.

Since IL_j = < [s_θ, s_τ] , {{[r^f, r^b]} , {[t^f, t^b]}} > and $\sum_{j = 1}^{n} v (L_{(j)}) - v (L_{(j + 1)}) = 1$ , by Theorem 3, we get

$\begin{matrix} IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \\ = < [s_{\prod_{j = 1}^{n} θ^{v (L_{(j)}) - v (L_{(j + 1)})}}, s_{\prod_{j = 1}^{n} τ^{v (L_{(j)}) - v (L_{(j + 1)})}}], \\ {{[\prod_{j = 1}^{n} {(γ^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \prod_{j = 1}^{n} {(γ^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}, \\ {[1 - \prod_{j = 1}^{n} {(1 - η^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ 1 - \prod_{j = 1}^{n} {(1 - η^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})}]}} > \\ = < [s_{θ^{\sum_{j = 1}^{n} (v (L_{(j)}) - v (L_{(j + 1)}))}}, s_{τ^{\sum_{j = 1}^{n} (v (L_{(j)}) - v (L_{(j + 1)}))}}], \\ {{[{(r^{f})}^{\sum_{j = 1}^{n} (v (L_{(j)}) - v (L_{(j + 1)}))}, {(r^{b})}^{\sum_{j = 1}^{n} (v (L_{(j)}) - v (L_{(j + 1)}))}]}, \\ {[1 - {(1 - t^{f})}^{\sum_{j = 1}^{n} (v (L_{(j)}) - v (L_{(j + 1)}))}, \\ 1 - {(1 - t^{b})}^{\sum_{j = 1}^{n} (v (L_{(j)}) - v (L_{(j + 1)}))}]}} > \\ = < [s_{θ}, s_{τ}], {{[r^{f}, r^{b}]}, {[t^{f}, 1 - 1 - t^{b}]}} > = IL . \end{matrix}$

Since θ - = min {θj} , τ - = min {τj} , r^f- = min {r^f} , r^b- = min {r^b} , t^f+ = max {t^f} , t^b+ = max {t^b} , IL^- = < [s_θ-, s_τ-] , {{r^f-, r^b-} , {t^f+, t^b+}} > , v (L_(j)) - v (L_(j+1)) ≥0 and $\sum_{j = 1}^{n} v (L_{(j)}) - v (L_{(j + 1)}) = 1$ , we have

$θ - = \prod_{j = 1}^{n} {(θ -)}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq \prod_{j = 1}^{n} {(θ j)}^{v (L_{(j)}) - v (L_{(j + 1)})},$ (21) $τ - = \prod_{j = 1}^{n} {(τ -)}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq \prod_{j = 1}^{n} {(τ j)}^{v (L_{(j)}) - v (L_{(j + 1)})},$ (22) $r^{f -} = \prod_{j = 1}^{n} {(r^{f -})}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq \prod_{j = 1}^{n} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})},$ (23)

$r^{b -} = \prod_{j = 1}^{n} {(r^{b -})}^{v (L_{(j)}) - v (L_{(j + 1)})} \leq \prod_{j = 1}^{n} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})},$ (24)

$\begin{matrix} t^{f +} = 1 - \prod_{j = 1}^{n} {(1 - t^{f +})}^{v (L_{(j)}) - v (L_{(j + 1)})} \\ \geq 1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})}, \end{matrix}$ (25) $\begin{matrix} t^{b +} = 1 - \prod_{j = 1}^{n} {(1 - t^{b +})}^{v (L_{(j)}) - v (L_{(j + 1)})} \\ \geq 1 - \prod_{i = 1}^{n} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})} . \end{matrix}$ (26)

Then, suppose $\prod_{j = 1}^{n} {({r_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})} = {r_{(j)}^{'}}^{f},$ $\prod_{j = 1}^{n} {({r_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})} = {r_{(j)}^{'}}^{b},$ $1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{f})}^{v (L_{(j)}) - v (L_{(j + 1)})} = {t_{(j)}^{'}}^{f},$ $1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{b})}^{v (L_{(j)}) - v (L_{(j + 1)})} = {t_{(j)}^{'}}^{b},$ we get $\begin{matrix} \frac{(θ -) + (τ -)}{2} \leq \frac{\prod_{j = 1}^{n} θ j + \prod_{j = 1}^{n} τ j}{2}, \\ r^{f -} - η^{f +} + r^{u -} - η^{u +} \leq \frac{1}{# H} \sum_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(j)}} \prod_{j = 1}^{n} {r^{'}}_{(j)}^{f} \\ - \frac{1}{# G} \sum_{[{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} {η^{'}}_{(j)}^{f} + \frac{1}{# H} \sum_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(j)}} {r^{'}}_{(j)}^{b} \\ - \frac{1}{# G} \sum_{[{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} {t^{'}}_{(j)}^{b}, \end{matrix}$ where # H and # G are the counts of the MDs and NDs in IVDHFULCG_v (IL₁, IL₂, …, IL_n) operator, respectively.

Furthermore, by Definition 4 and Theorem 2, we obtain $\begin{matrix} S (IVDHFULC G_{v} (I L^{-}, I L^{-}, \dots, I L^{-})) \\ \leq S (IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n})) . \end{matrix}$

Then, we need to analyze the following situations:

If S (IVDHFULCG_v (IL^-, IL^-, …, IL^-)) < S (IVDHFULCG_v (IL₁, IL₂, …, IL_n)) , according to Theorem 2, we have

$\begin{matrix} IVDHFULC G_{v} (I L^{-}, I L^{-}, \dots, I L^{-}) \\ \leq IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) . \end{matrix}$

If S (IVDHFULCG_v (IL^-, IL^-, … , IL^-)) = S (IVDHFULCG_v (IL₁, IL₂, …, IL_n)) , by Equations (22) – (27), we have

$\begin{matrix} θ - = \prod_{j = 1}^{n} {(θ j)}^{v (L_{(j)}) - v (L_{(j + 1)})}, τ - = \prod_{j = 1}^{n} {(τ j)}^{v (L_{(j)}) - v (L_{(j + 1)})}, \\ r^{f -} = {r^{'}}_{(j)}^{f}, r^{b -} = {r^{'}}_{(j)}^{b}, t^{f -} = {t^{'}}_{(j)}^{f}, t^{b -} = {t^{'}}_{(j)}^{b} . \end{matrix}$

Then, according to Theorem 2, we have $\begin{array}{l} P (I V D H F U L C G_{υ} (I L^{-}, I L^{-}, ..., I L^{-})) \\ = P (I V D H F U L C G_{υ} (I L_{1}, I L_{2}, ..., I L_{n})) . \end{array}$

Thus, according to the idempotency of the IVDHFULCG operator, we have $I L^{-} \leq IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) .$

Similarly, we have $IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \leq I L^{+} .$

Therefore, we have $I L^{-} \leq IVDHFULC G_{v} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \leq I L^{+} .$

4.3 The SIVDHFULCG operator

From Definition 9, we know the IVDHFULCG operator only takes into account the interaction between two adjacent combinations L_(j) and L_(j+1) (j = 1, 2, …, n). When there exist complex correlative characteristics among criteria, it is more or less unreasonable to only take into account interaction between two adjacent combinations. To overall capture the interactions among criteria, we extend the generalized Shapley function [21] to the proposed IVDHFULCG operator and propose the SIVDHFULCG operator.

The generalized Shapely function was proposed by Shapely [21], which is shown as follows: $φ (v (S)) = \sum_{T \subseteq Y ∖ S} \frac{(n - s - t)! t!}{(n - s + 1)!} (v (S \cup T) - v (T)) \forall S \subseteq Y,$ (27) where v indicates a FM on the set Y = {y₁, y₂, …, y_n}, S is any subset of Y, “Y ∖ S” is the difference set between the set Y and the set S, and T is any subset of Y ∖ S. Besides, n, s and t represent the cardinalities of the coalitions Y, S and T, respectively.

By Equation (28), we find that the generalized Shapley function represents an expectation value of the overall interaction between the combination S and every combination in Y ∖ S.

By Equation (28), we know when s = y_j, then Equation (28) reduces to the Shapely index [21]. $\begin{matrix} φ (v ({y_{j}})) \\ = \sum_{T \subseteq Y ∖ y_{j}} \frac{(n - 1 - t)! t!}{n!} (v ({y_{j}} \cup T) - v (T)) \forall y_{j} \in Y . \end{matrix}$ (28)

Definition 10. Suppose that Y = {y₁, y₂, …, y_n} is a set of attributes and suppose that IL_i = < [s_θj, s_τj] , H_j, G_j > (j = 1, 2, …, n) is a set of IVDHFULEs on Y. A SIVDHFULCG operator is defined by

$\begin{matrix} SIVDHFULC G_{φ} (I L_{1}, I L_{2}, \dots, {IL}_{n}) = \\ \otimes_{j = 1}^{n} I {L_{(j)}}^{φ (v (L_{(j)})) - φ (v (L_{(j + 1)}))}, \end{matrix}$ (29) where IL_(j) represents the jth smallest element in the set {IL₍₁₎, IL₍₂₎, …, IL_(n) } , y_(j) is the attribute corresponding to IL_(j), L_(j) = {y_(j), y_(j+1), …, y_(n)} with L_(n+1) = φ, and φ (v (L_(i))) (i = 1, 2, …, n) indicates the generalized Shapley function about the FM v (L_(j)) on L_(j) shown in Equation (28).

Theorem 5. The aggregated value of the IL_j (j = 1, 2, …, n) obtained by the SIVDHFULCG operator is also an IVDHFULE, where $\begin{matrix} SIVDHFULC G_{φ} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \\ = < [s_{\prod_{j = 1}^{n} θ {(j)}^{φ (v (L_{(j)})) - φ (v (L_{(j + 1)}))}}, s_{\prod_{j = 1}^{n} τ {(j)}^{φ (v (L_{(j)})) - φ (v (L_{(j + 1)}))}}], \\ \cup_{[{r_{(j)}}^{f}, {r_{(j)}}^{b}] \in H_{(j)}, [{t_{(j)}}^{f}, {t_{(j)}}^{b}] \in G_{(j)}} {{[\prod_{j = 1}^{n} {({r_{(j)}}^{f})}^{φ (v (L_{(j)})) - φ (v (L_{(j + 1)}))}, \\ \prod_{j = 1}^{n} {({r_{(j)}}^{b})}^{φ (v (L_{(j)})) - φ (v (L_{(j + 1)}))}]}, \\ {[1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{f})}^{φ (v (L_{(j)})) - φ (v (L_{(j + 1)}))}, \\ 1 - \prod_{j = 1}^{n} {(1 - {t_{(j)}}^{b})}^{φ (v (L_{(j)})) - φ (v (L_{(j + 1)}))}]}} > . \end{matrix}$ (30)

Proof.

From Equation (28), it is easy to get φ (v (L_(j))) - φ (v (L_(j+1))) ≥ 0 and $\sum_{j = 1}^{n} φ (v (L_{(j)}))$ -φ (v (L_(j+1))) =1 for any permutation (.) of the elements in the set Y. The proof of Theorem 5 is similar to that of Theorem 3 and it is omitted here.

Similar to the properties of the IVDHFULCG operator, we can easily know that the SIVDHFULCG operator also satisfies the idempotency, the commutativity and the boundedness. Their proofs are omitted here.

4.4 The 2OA-SIVDHFULCG operator

From Definition 10, we find that the FM makes the problems exponential because of being defined on the power set. Hence, we cannot easily determine the FM of each coalition in a set when the cardinality of the set Y is large. To increase the practicability of the SIVDHFULCG operator, we shall introduce a special kind of FMs named as 2-order additive FMs (2OAFMs) and propose the 2OA-SIVDHFULCG operator based on this kind of FM.

For 2OAFMs, we know for any S ⊆ Y with |S|≥2: $\begin{matrix} v^{'} (S) = \sum_{j = 1}^{n} e_{j} y_{j} + \sum_{{y_{j}, y_{i}} \subseteq Y} e_{ji} y_{j} y_{i} = \sum_{y_{j} \in S} e_{j} + \sum_{{y_{j}, y_{i}} \subseteq S} e_{ji} \\ = \sum_{{y_{j}, y_{i}} \subseteq S} v^{'} ({y_{j}, y_{i}}) - (| S | - 2) \sum_{y_{j} \in S} v^{'} ({y_{j}}) \end{matrix}$ (31) where e_T = ∑_K⊆T (-1) ^|T∖K|v_K is the Mobius transform coefficient (for simplicity, when T = {y_j}, e_T is denoted as e_j. When T = {y_j, y_i}, e_T is denoted as e_ji), y_j is Boolean variable (y_j = 1 ⇔ y_j ∈ S), v′ ({y_j}) = e_j, v′ ({y_j, y_i}) = e_j + e_i + e_ji.

Obviously, 2OAFM is obtained only by the coefficients v′ ({y_j}) and v′ ({y_j, y_i}). So, n (n + 1)/2 coefficients are needed for 2OAFM.

Grabisch [6] provided the following theorem to obtain a 2OAFM.

Theorem 6. [6] Suppose v′ is a FM on the set Y ; then v′ is a 2OAFM if and only if there exist coefficients v′ ({y_j}) and v′ ({y_j, y_i}) for all y_j, y_i ∈ Y that meet the following restrictions:

v′ ({y_j})≥0 ∀ y_j ∈ Y ;

∑_{{y_j,y_i}⊆Y}v′ ({y_j, y_i}) - (|Y|-2) ∑_{y_i∈Y}v′ ({y_j}) = 1 ;

∑_{{y_j}⊆S∖y_k} (v′ ({y_j, y_k}) - v′ ({y_j})) ≥ (|S|-2) v′ ({y_k}) ∀S ⊆ Y subject to y_k ∈ S and |S|>2.

Meng [16] presented the explicit expression of the Shapely index about 2OAFM as follows.

Theorem 7. [16] Suppose v′ is a 2OAFM on the set Y . The Shapely index given in Equation (29) about the 2OAFM v′ is defined as follows: $\begin{matrix} φ (v^{'} ({y_{j}})) = \frac{3 - n}{2} v^{'} ({y_{j}}) + \\ \sum_{y_{i} \in Y ∖ y_{j}} \frac{1}{2} (v^{'} ({y_{j}, y_{i}}) - v^{'} ({y_{i}})) \forall y_{j} \in Y . \end{matrix}$ (32)

Theorem 8. [16] Suppose v′ is 2OAFM on the set Y and φ is the generalized Shapely index about the 2OAFM v′ . Then, $φ (v^{'} (S \cup {y_{j}})) - φ (v^{'} (S)) = φ (v^{'} ({y_{j}})),$ (33) for any y_j ∈ Y and any S ⊆ Y with y_j ∉ S.

By Theorem 8, the definition of the 2OA-SIVDHFULCG operator is provided as follows.

Definition 11. Suppose that Y = {y₁, y₂, …, y_n} is a set of attributes and suppose that IL_j = < [s_θj, s_τj] , H_j, G_j > (j = 1, 2, …, n) is a set of IVDHFULEs on Y. A 2OA-SIVDHFULCG operator is defined by

$\begin{matrix} 2 OA - SIVDHFULC G_{v^{'}} (I L_{1}, I L_{2}, \dots, {IL}_{n}) = \\ \otimes_{j = 1}^{n} I {L_{j}}^{φ (v^{'} ({y_{j}}))} \end{matrix}$ (34) where y_j is the attribute corresponding to IL_j and φ (v′ ({y_j})) (j = 1, 2, …, n) is the Shapley index about the 2OAFM v′ ({y_j}) on y_j shown in Equation (33).

Theorem 9. The aggregated value of the IL_j (j = 1, 2, …, n) obtained by the 2OA-SIVDHFULCG operator is also an IVDHFULE, where $\begin{matrix} 2 OA - SIVDHFULCG v^{'} (I L_{1}, I L_{2}, \dots, {IL}_{n}) \\ = < [s_{\prod_{j = 1}^{n} θ j^{φ (v^{'} ({y_{j}}))}}, s_{\prod_{j = 1}^{n} τ j^{φ (v^{'} ({y_{j}}))}}], \cup_{[{r_{j}}^{f}, {r_{j}}^{b}] \in H_{j}, [{t_{j}}^{f}, {t_{j}}^{b}] \in G_{j}} \\ {{[\prod_{j = 1}^{n} {({r_{j}}^{f})}^{φ (v^{'} ({y_{j}}))}, \prod_{j = 1}^{n} {({r_{j}}^{b})}^{φ (v^{'} ({y_{j}}))}]}, \\ {[1 - \prod_{j = 1}^{n} {(1 - {t_{j}}^{f})}^{φ (v^{'} ({y_{j}}))}, 1 - \prod_{j = 1}^{n} {(1 - {t_{j}}^{b})}^{φ (v^{'} ({y_{j}}))}]}} > . \end{matrix}$ (35)

Similar to the proofs of Theorems 2 and 4, it is easy to get the conclusion.

Similar to the properties of the IVDHFULCG operator and the SIVDHFULCG operator, we can easily know that the 2OA-SIVDHFULCG operator also satisfies the idempotency, the commutativity and the boundedness. Their proofs are omitted here.

5 An approach to a MADM problem under IVDHFUL environment

For an IVDHFUL MADM problem with interactive criteria: Let Z = {Z₁, Z₂, …, Z_p} be a set of p alternatives and let D = {D₁, D₂, …, D_q} be a set of q criteria. Assume that Z = [z_ij] _p×q represents the decision matrix, where z_ij is an IVDHFUL value for attribute D_j ∈ D with respect to alternative Z_i ∈ Z. In what follows, two approaches shall be developed by using the proposed operators for the MADM problem with partly known weights information under IVDHFUL environment.

5.1 The models for the optimal fuzzy measures

The maximization deviation method as an important approach to determine the attribute weights have been researched by many scholars [27 , 38]. Based on this method [27] and the Shapley index [21], several optimization models are constructed to obtain the FM on the attribute set.

If the attribute’s weights are incompletely unknown, the following model is built to determine the optimal FMs on the attribute set D′ ⊂ D: $\begin{matrix} max fun (φ) = \sum_{j = 1}^{q} \sum_{i = 1}^{p} \sum_{l = 1}^{p} dis (z_{ij}, z_{lj}) φ (v ({D_{j}})) \\ s . t . {\begin{matrix} \begin{matrix} v ({D^{'}}) \in R_{D^{'}}; \\ v (φ) = 0, v (D) = 1; \end{matrix} \\ v (M) \leq v (N), \forall M, N \subseteq D, M \subseteq N, \end{matrix} \end{matrix}$ (36) where dis (z_ij, z_lj) is the Hamming distance between z_ij and z_lj, φ (v ({D_j})) is the Shapely index about the FM v ({D_j}) on attribute D_j (j = 1, 2, …, q), and R_D′ is the range of known weights information of the attribute set D′ ⊂ D.

Similarly, the following model is built for the 2OAFMs v′ on the attribute set D′ ⊂ D: $\begin{matrix} max fun (φ^{'}) = \frac{3 - n}{2} \sum_{j = 1}^{q} \sum_{i = 1}^{p} \sum_{l = 1}^{p} d (z_{ij}, z_{lj}) v^{'} ({D_{j}}) \\ + \frac{1}{2} \sum_{j = 1}^{q} \sum_{i = 1}^{p} \sum_{l = 1}^{p} d (z_{ij}, z_{kj}) (\sum_{D_{l} \in D ∖ D_{j}} (v^{'} ({D_{l}, D_{j}}) - v^{'} ({D_{j}}))) \end{matrix}$ (37)

$\begin{matrix} s . t . {\begin{matrix} v^{'} ({D^{'}}) \in H_{D^{'}}, v^{'} ({D_{j}}) \geq 0 j = 1, 2, \dots, q; \\ \sum_{{D_{i}, D_{j}} \subseteq D} v^{'} ({D_{i}, D_{j}}) - (| D | - 2) \sum_{C_{i} \in C} v^{'} ({D_{i}}) = 1; \\ \sum_{{D_{i}} \subseteq S ∖ {D_{j}}} (v^{'} ({D_{i}, D_{j}}) - v^{'} ({D_{j}})) \geq (| S | - 2) v^{'} ({D_{j}}), \\ \forall S \subseteq D, \forall D_{j} \in S, | S | \geq 2, \end{matrix} \end{matrix}$ (38) where dis (z_ij, z_lj) is the Hamming distance between z_ij and z_lj, φ (v ({D_j})) is the Shapely index about the 2OAFM v ({D_j}) on attribute D_j (j = 1, 2, …, q), and R_D′ is the range of known weights information of the attribute set D′ ⊂ D.∥

5.2 The decision procedure

Now, we propose two novel MADM methods to deal with IVDHFUL MADM problems with interactive attributes and incomplete weights information, shown as follows:∥(1) The first proposed MADM method:∥Step 1. Evaluates the alternatives about the criteria to construct the IVDHFUL value matrix Z = [z_ij] _p×q.∥Step 2. Utilize the model shown in Equation (37) to get the optimal FMs v on the criteria set D′ ⊂ D. It should be noted that if the optimal FMs on the criteria set D′ ⊂ D are completely known, then we don’t need to do this step.∥Step 3. Utilize Equation (28) to get the generalized Shapely values of the attribute set D′ ⊆ D.∥Step 4. Utilize the SIVDHFULCG operator shown in Equation (31) to obtain the comprehensive IVDHFUL value z_i of each alternative Z_i (i = 1, 2, …, p).∥Step 5. Utilize Equations (14) and (15) to obtain the score value S (z_i) and accuracy value P (z_i) of each comprehensive value z_i (i = 1, 2, …, p), respectively.∥Step 6. By Theorem 2, rank all the alternatives {Z₁, Z₂, …, Z_p} and obtain the best alternative(s).∥(2) The second proposed MADM method:∥Step 1. Evaluates the alternatives about the criteria to construct the IVDHFUL value matrix Z = [z_ij] _p×q.∥Step 2. Utilize the model shown in Equation (38) to get the 2OAFMs v′ on the attribute set D′ ⊂ D. It should be noted that if the optimal 2OAFMs on the attribute set D′ ⊂ D are completely known, then we don’t need to do this step.∥Step 3. Utilize the Equation (33) to get the Shapely values of the attribute set D′ ⊆ D.∥Step 4. Utilize the 2OA-SIVDHFULCG operator shown in Equation (36) to obtain the comprehensive IVDHFUL value z_i of each alternative Z_i (i = 1, 2, …, p).∥Step 5. Utilize Equations (14) and (15) to obtain the score value S (z_i) and accuracy value P (z_i) of each comprehensive value z_i (i = 1, 2, …, p), respectively.∥Step 6. By Theorem 2, rank all the alternatives {Z₁, Z₂, …, Z_p} and obtain the best alternative(s).∥In spite of the fact that the proposed two MADM methods consider the interactions among criteria, there exist some differences between these two MADM methods as follows:

On the aspect of complexity of computation, the first proposed MADM method based on the SIVDHFULCG operator makes decision problems exponential because 2ⁿ - 2 parameters are needed to determine the optimal FMs. Nevertheless, the second proposed MADM method based on the 2OA-SIVDHFULCG operator simplifies the complexity solving a FM because we only need n (n + 1)/2 parameters to determine the optimal 2OAFMs.

On the aspect of considering the interactions among criteria, the first proposed MADM method based on the SIVDHFULCG operator overall captures the interactions among attributes. Nevertheless, the second proposed MADM method based on the 2OA-SIVDHFULCG operator reflects the interrelationships between any two attributes.

∥In actual MCDM environments, when the experts can endure a long calculation time and want to capture the overall interactions among criteria, the first proposed MADM method is the optimal choice. Otherwise, the second method is the best choice.

6 An illustrate example

In this part, we use two examples to analyze the ranking orders of the alternatives of the proposed approaches and verify the effectiveness and the superiority of the proposed approaches by a comparative analysis of the developed approaches with the existing methods presented in [10, 33].

6.1 The decision procedure of the proposed MADM method

Example 6.1. We use a case adapted from Herrera et al. [8] to explain the procedure of the developed MADM methods. Suppose that four companies Z = {Z₁, Z₂, Z₃, Z₄} are evaluated by an investment company for investment, where three attributes are considered: (1) D₁ denotes the environmental impact analysis; (2) D₂ is the growth situation; (3) D₃ is the risk situation. Suppose the weight vector is W = {0.35, 0.25, 0.40} . Suppose that these four companies Z = {Z₁, Z₂, Z₃, Z₄} are assessed by the linguistic set (LS) S = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} and the assessment results are represented by IVDHFUL information, which is shown in Table 1.

Table 1
IVDHFUL decision matrix

D ₁ D ₂ D ₃

Z ₁ < [s₂, s₄] , {{, [0.4, 0.6] , [0.6, 0.6]} , < [s₃, s₅] , {{[0.2, 0.4] , [0.4, 0.6]} , < [s₃, s₅] , {{[0.2, 0.4] , [0.4, 0.6] , [0.5, 0.7]}

{[0.2, 0.4] , [0.4, 0.4]}}> {[0.1, 0.3] , [0.2, 0.4]}}> {[0.1, 0.1] , [0.1, 0.3]}}>

Z ₂ < [s₁, s₃] , {{[0.3, 0.5] , [0.4, 0.6]} , < [s₄, s₆] , {{[0.3, 0.5] , [0.5, 0.5]} , < [s₂, s₄] , {{[0.1, 0.3] , [0.4, 0.6] , [0.6, 0.6]} ,

{[0.2, 0.4] , [0.4, 0.4]}}> {[0.3, 0.5] , [0.5, 0.5]}}> {[0.1, 0.3] , [0.4, 0.4]}}>

Z ₃ < [s₃, s₅] , {{[0.4, 0.6] , [0.7, 0.7]} , < [s₂, s₄] , {{[0.1, 0.3] , [0.3, 0.5] , [0.4, 0.6]} , < [s₂, s₄] , {{[0.3, 0.5] , [0.4, 0.6] , [0.7, 0.7]} ,

{[0.1, 0.3] , [0.3, 0.3]}}> {[0.2, 0.4] , [0.4, 0.4]}}> {[0.1, 0.3] , [0.3, 0.3]}}>

Z ₄ < [s₃, s₅] , {{[0.3, 0.5] , [0.5, 0.7] , [0.8, 0.8]} , < [s₁, s₃] , {{[0.4, 0.6] , [0.6, 0.6]} , < [s₄, s₆] , {{[0.5, 0.7] , [0.7, 0.7]} ,

{[0.1, 0.1] , [0.2, 0.2]}}> {[0.1, 0.3] , [0.4, 0.4]}}> {[0.1, 0.1] , [0.3, 0.3]}}>

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

(1) The evaluation steps by utilizing the first proposed method

Step 1. Since all the criteria are independent and attribute’s weights D_i (i = 1, 2, 3) are known, the optimal FMs v on the attribute set D′ ⊆ D are obtain as: $\begin{matrix} v ({D_{1}}) = 0.35, v ({D_{2}}) = 0.25, v ({D_{3}}) = 0.40, \\ v ({D_{1}, D_{2}}) = 0.6, v ({D_{1}, D_{3}}) = 0.75, \\ v ({D_{2}, D_{3}}) = 0.65, v ({D_{1}, D_{2}, D_{3}}) = 1 . \end{matrix}$

Step 2. Utilize Equation (28) to get the generalized Shapely values of the attribute set D′ ⊆ D, and have: $\begin{matrix} φ (v ({C_{1}})) = 0.35, φ (v ({C_{2}})) = 0.25, φ (v ({C_{3}})) = 0.40, \\ φ (v ({C_{1}, C_{2}})) = 0.6, φ (v ({C_{1}, C_{3}})) = 0.75, \\ φ (v ({C_{2}, C_{3}})) = 0.65, φ (v ({C_{1}, C_{2}, C_{3}})) = 1 . \end{matrix}$

Step 3. Utilize the SIVDHFULCG operator shown in Equation (31) to obtain the comprehensive IVDHFUL value z_i of each company Z_i (i = 1, 2, 3, 4), and we have: $\begin{matrix} z_{1} = < [s_{2.60}, s_{4.62}], {{[0.23, 0.43], [0.30, 0.51], [0.33, 0.54], [0.27, 0.48], [0.36, 0.56], [0.40, 0.60], \\ [0.25, 0.46], [0.34, 0.54], [0.37, 0.58], [0.30, 0.51], [0.40, 0.60], [0.44, 0.64], [0.29, 0.46], [0.39, 0.54], \\ [0.42, 0.58], [0.35, 0.51], [0.46, 0.60], [0.50, 0.64]}, {[0.14, 0.27], [0.14, 0.34], [0.16, 0.29], [0.16, 0.36], \\ [0.22, 0.27], [0.22, 0.34], [0.24, 0.29], [0.24, 0.36]}} >, \\ z_{2} = < [s_{1.87}, s_{4.00}], {{[0.19, 0.41], [0.34, 0.54], [0.40, 0.54], [0.21, 0.43], [0.37, 0.57], [0.44, 0.57], \\ [0.22, 0.41], [0.38, 0.54], [0.45, 0.54], [0.24, 0.43], [0.42, 0.57], [0.50, 0.57]}, {[0.19, 0.39], [0.31, 0.43], \\ [0.27, 0.39], [0.38, 0.43], [0.25, 0.39], [0.37, 0.43], [0.33, 0.39], [0.43, 0.43]}} >, \\ z_{3} = < [s_{2.30}, s_{4.32}], {{[0.25, 0.47], [0.31, 0.50], [0.28, 0.50], [0.34, 0.53], [0.35, 0.54], [0.43, 0.57], \\ [0.33, 0.53], [0.40, 0.56], [0.37, 0.57], [0.45, 0.61], [0.47, 0.61], [0.57, 0.64], [0.36, 0.56], [0.43, 0.59], \\ [0.40, 0.60], [0.49, 0.63], [0.50, 0.64], [0.61, 0.67]}, {[0.13, 0.33], [0.20, 0.33], [0.21, 0.33], [0.28, 0.33], \\ [0.19, 0.33], [0.26, 0.33], [0.26, 0.33], [0.33, 0.33]}} >, \\ z_{4} = < [s_{2.56}, s_{4.73}], {{[0.40, 0.60], [0.45, 0.60], [0.47, 0.67], [0.54, 0.67], [0.56, 0.71], [0.64, 0.71], \\ [0.44, 0.60], [0.50, 0.60], [0.52, 0.67], [0.60, 0.67], [0.62, 0.71], [0.71, 0.71]}, {[0.10, 0.15], [0.19, 0.24] \\ [0.14, 0.19], [0.22, 0.27], [0.19, 0.19], [0.26, 0.26], [0.22, 0.22], [0.29, 0.29]}} > . \end{matrix}$

Step 4. Utilize Equation (14) to obtain the score value S (z_i) of each comprehensive value z_i (i = 1, 2, 3, 4), and we have $\begin{matrix} S (z_{1}) = 0.7142, S (z_{2}) = 0.1979, \\ S (z_{3}) = 0.7038, S (z_{4}) = 1.4013 . \end{matrix}$

Step 5. Because S (z₄) > S (z₁) > S (z₃) > S (z₂), we get Z₄ > Z₁ > Z₃ > Z₂ and the best company is Z₄.

(2) The evaluation steps by utilizing the second proposed method

Step 1. Since all the criteria are independent and attributes’ weights D_i (i = 1, 2, 3) are known, the 2OAFMs v′ on the attribute set D′ ⊆ D are obtained as:

$\begin{matrix} v^{'} ({D_{1}}) = 0.35, v^{'} ({D_{2}}) = 0.25, \\ v^{'} ({D_{3}}) = 0.40, v^{'} ({D_{1}, D_{2}}) = 0.6, \\ v^{'} ({D_{1}, D_{3}}) = 0.75, v^{'} ({D_{2}, D_{3}}) = 0.65 . \end{matrix}$

Step 2. Utilize Equation (33) to get the Shapely values of the attribute set D′ ⊆ D, and we have: φ (v′ ({D₁})) =0.35, φ (v′ ({D₂})) =0.45, φ (v′ ({D₃})) =0.4 .

Step 3. Utilize the 2OA-SIVDHFULCG operator shown in Equation (36) to obtain the comprehensive IVDHFUL value z_i of each company Z_i (i = 1, 2, 3, 4), and we have: $\begin{matrix} z_{1} = < [s_{2.60}, s_{4.62}], {{[0.23, 0.43], [0.30, 0.51], [0.33, 0.54], [0.27, 0.48], [0.36, 0.56], [0.40, 0.60], \\ [0.25, 0.46], [0.34, 0.54], [0.37, 0.58], [0.30, 0.51], [0.40, 0.60], [0.44, 0.64], [0.29, 0.46], [0.39, 0.54], \\ [0.42, 0.58], [0.35, 0.51], [0.46, 0.60], [0.50, 0.64]}, {[0.14, 0.27], [0.14, 0.34], [0.16, 0.29], [0.16, 0.36], \\ [0.22, 0.27], [0.22, 0.34], [0.24, 0.29], [0.24, 0.36]}} >, \\ z_{2} = < [s_{1.87}, s_{4.00}], {{[0.19, 0.41], [0.34, 0.54], [0.40, 0.54], [0.21, 0.43], [0.37, 0.57], [0.44, 0.57], \\ [0.22, 0.41], [0.38, 0.54], [0.45, 0.54], [0.24, 0.43], [0.42, 0.57], [0.50, 0.57]}, {[0.19, 0.39], [0.31, 0.43], \\ [0.27, 0.39], [0.38, 0.43], [0.25, 0.39], [0.37, 0.43], [0.33, 0.39], [0.43, 0.43]}} >, \\ z_{3} = < [s_{2.30}, s_{4.32}], {{[0.25, 0.47], [0.31, 0.50], [0.28, 0.50], [0.34, 0.53], [0.35, 0.54], [0.43, 0.57], \\ [0.33, 0.53], [0.40, 0.56], [0.37, 0.57], [0.45, 0.61], [0.47, 0.61], [0.57, 0.64], [0.36, 0.56], [0.43, 0.59], \\ [0.40, 0.60], [0.49, 0.63], [0.50, 0.64], [0.61, 0.67]}, {[0.13, 0.33], [0.20, 0.33], [0.21, 0.33], [0.28, 0.33], \\ [0.19, 0.33], [0.26, 0.33], [0.26, 0.33], [0.33, 0.33]}} >, \\ z_{4} = < [s_{2.56}, s_{4.73}], {{[0.40, 0.60], [0.45, 0.60], [0.47, 0.67], [0.54, 0.67], [0.56, 0.71], [0.64, 0.71], \\ [0.44, 0.60], [0.50, 0.60], [0.52, 0.67], [0.60, 0.67], [0.62, 0.71], [0.71, 0.71]}, {[0.10, 0.15], [0.19, 0.24] \\ [0.14, 0.19], [0.22, 0.27], [0.19, 0.19], [0.26, 0.26], [0.22, 0.22], [0.29, 0.29]}} > . \end{matrix}$

Step 4. Utilize Equation (14) to obtain the score value S (z_i) of each comprehensive value z_i (i = 1, 2, 3, 4), and we get $\begin{matrix} S (z_{1}) = 0.7142, S (z_{2}) = 0.1979, S (z_{3}) = \\ 0.7038, S (z_{4}) = 1.4013 . \end{matrix}$

Step 5. Because S (z₄) > S (z₁) > S (z₃) > S (z₂), we get Z₄ > Z₁ > Z₃ > Z₂, which is the same as that got by the first developed MADM method, where the optimal company is Z₄.

6.2 Comparison with the exiting methods

(1) Verifying the feasibility of the proposed MADM methods.

To verify the feasibility of the developed MADM methods, the proposed MADM methods are compared with the existing methods presented by Yang and Ju [33] and Ju et al. [10]. It should be noted that Yang and Ju’s method [33] and Ju et al.’s method [10] cannot directly aggregate IVDHFULEs. To apply Yang and Ju’s method [33] to deal with Example 6.1, we first need to transform the IVDHFUL information to the DHFL information by means of replacing each interval value with the average value of its upper and lower limits. To apply Ju et al.’s method [10] to deal with Example 6.1, we first need to transform the IVDHFUL information into DHFTL information. The transformation principle is described as follows: the ULV can be transformed into triangular LV by adding the medium value of each ULV. The interval-valued MDs and NDs can be transformed into crisp MDs and NDs by replacing each interval value with the average value of its upper and lower limits. The ranking results of these four companies of Example 6.1 for different approaches are listed in Table 2. From Table 2, we can see that all the methods get the same ranking result, i.e., Z₄ > Z₁ > Z₃ > Z₂ It is obvious that Example 6.1 verify the feasibility of the developed MADM methods.

Table 2
Ranking results by different methods

Methods Score values Ranking results

Yang and Ju’s method [33] S (z₁) =0.1196, S (z₂) =0.0304, S (z₃) =0.0905, S (z₄) =0.1951 Z₄ > Z₁ > Z₃ > Z₂

Ju et al.’s method [10] S (z₁) =0.1196, S (z₂) =0.0303, S (z₃) =0.0904, S (z₄) =0.1940 Z₄ > Z₁ > Z₃ > Z₂

The first proposed method based on the SIVDHFULCG operator S (z₁) =0.7142, S (z₂) =0.1979, S (z₃) =0.7038, S (z₄) =1.4013 Z₄ > Z₁ > Z₃ > Z₂

The second proposed method based on the 2OA-SIVDHFULCG operator S (z₁) =0.7142, S (z₂) =0.1979, S (z₃) =0.7038, S (z₄) =1.4013 Z₄ > Z₁ > Z₃ > Z₂

Methods	Score values	Ranking results
Yang and Ju’s method [33]	S (z₁) =0.1196, S (z₂) =0.0304, S (z₃) =0.0905, S (z₄) =0.1951	Z₄ > Z₁ > Z₃ > Z₂
Ju et al.’s method [10]	S (z₁) =0.1196, S (z₂) =0.0303, S (z₃) =0.0904, S (z₄) =0.1940	Z₄ > Z₁ > Z₃ > Z₂
The first proposed method based on the SIVDHFULCG operator	S (z₁) =0.7142, S (z₂) =0.1979, S (z₃) =0.7038, S (z₄) =1.4013	Z₄ > Z₁ > Z₃ > Z₂
The second proposed method based on the 2OA-SIVDHFULCG operator	S (z₁) =0.7142, S (z₂) =0.1979, S (z₃) =0.7038, S (z₄) =1.4013	Z₄ > Z₁ > Z₃ > Z₂

(2) Demonstrating the superiority of the proposed MADM methods.

To demonstrate the superiority of the developed MADM methods, we utilize the proposed MADM methods, Yang and Ju’s method [33] and Ju et al.’s method [10] to handle the following example and compare the ranking results for different MADM methods.

Example 6.2. Suppose that a university wants to evaluate the ability of four teachers A₁, A₂, A₃ and A₄. Suppose that three attributes are considered, including the teaching skill (D₁), the research capability (D₂) and the education background (D₃). There exists interaction among these three attributes. Suppose that the importance of each attribute is given by 0.1 ≤ v′ ({D₁}) ≤0.2, 0.4 ≤ v′ ({D₂}) ≤0.5, 0.2 ≤ v′ ({D₃}) ≤0.3, v′ ({D₁, D₂}) ≤0.5, v′ ({D₁, D₃}) ≤0.7, v′ ({D₂, D₃}) ≥0.5 . Suppose that these four teachers Z = {Z₁, Z₂, Z₃, Z₄} are assessed by the LS S = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} and the assessment results are represented by IVDHFUL information, which is shown in Table 3. The goal of this practical MADM problem is to select the best teacher for promoting.

Table 3

IVDHFUL decision matrix

	D ₁	D ₂	D ₃
Z ₁	$\begin{matrix} < [s_{4}, s_{6}], {{, [0.4, 0.6]}, \\ {[0.3, 0.5]}} > \end{matrix}$	< [s₂, s₄] , {{[0.3, 0.5]} , {[0.2, 0.3]}}>	< [s₃, s₅] , {{[0.4, 0.6]} , {[0.1, 0.3]}}>
Z ₂	$\begin{matrix} < [s_{4}, s_{6}], {{[0.4, 0.6], [0.5, 0.7]}, \\ {[0.2, 0.2]}} > \end{matrix}$	< [s₁, s₃] , {{[0.6, 0.6]} , {[0.4, 0.4]}}>	< [s₄, s₆] , {{[0.6, 0.6]} , {[0.2, 0.2]}}>
Z ₃	< [s₂, s₄] , {{[0.4, 0.6]} , {[0.3, 0.5]}}>	$\begin{matrix} < [s_{3}, s_{5}], {{[0.4, 0.4]}, {[0.1, 0.1], \\ [0.2, 0.2]}} > \end{matrix}$	< [s₂, s₄] , {{[0.4, 0.6]} , {[0.2, 0.4]}}>
Z ₄	< [s₄, s₆] , {{[0.4, 0.6]} , {[0.3, 0.3]}}>	$\begin{matrix} < [s_{2}, s_{4}], {{[0.3, 0.5]}, {[0.2, 0.2], \\ [0.2, 0.3]}} > \end{matrix}$	< [s₃, s₅] , {{[0.4, 0.6]} , {[0.3, 0.3]}}>

1) The evaluation steps by utilizing the first proposed method

Step 1. Utilize Equation (37) to obtain the optimal FM v on the attribute set D′ ⊆ D, and we have: $\begin{matrix} max fun (φ) = 0.1538 (v ({D_{1}}) - v ({D_{2}, D_{3}})) \\ - 0.2222 (v ({D_{2}}) - v ({D_{1}, D_{3}})) \\ + 0.0684 (v ({D_{3}}) - v ({D_{1}, D_{2}})) + 0.9424 \\ s . t . {\begin{matrix} \begin{matrix} v ({D_{1}}) \in [0.1, 0.2], v ({D_{2}}) \in [0.4, 0.5], \\ v ({D_{3}}) \in [0.2, 0.3], v ({D_{1}, D_{2}}) \leq 0.5, \end{matrix} \\ μ ({D_{1}, D_{3}}) \leq 0.7, μ ({D_{2}, D_{3}}) \geq 0.5, \\ v (E) \leq v (B), \forall E, B \subseteq {D_{1}, D_{2}, D_{3}}, E \subseteq B, \\ v (φ) = 0, v ({D_{1}, D_{2}, D_{3}}) = 1 . \end{matrix} \end{matrix}$

By solving the above model using Lingo software [29], we can get: $\begin{matrix} v ({D_{1}}) = 0.2, v ({D_{2}}) = 0.4, v ({D_{3}}) = 0.3, \\ v ({D_{1}, D_{2}}) = 0.4, v ({D_{1}, D_{3}}) = 0.7, \\ v ({D_{2}, D_{3}}) = 0.5, v ({D_{1}, D_{2}, D_{3}}) = 1 . \end{matrix}$

Step 2. Utilize Equation (28) to get the generalized Shapely values of the attribute set D′ ⊆ D, and we have: $\begin{matrix} φ (v ({D_{1}})) = 0.3, φ (v ({D_{2}})) = 0.3, φ (v ({D_{3}})) = 0.4, \\ φ (v ({D_{1}, D_{2}})) = 0.55, φ (v ({D_{1}, D_{3}})) = 0.65, \\ φ (v ({D_{2}, D_{3}})) = 0.65, φ (v ({D_{1}, D_{2}, D 3})) = 1 . \end{matrix}$

Step 3. Utilize the SIVDHFULCG operator shown in Equation (31) to obtain the comprehensive IVDHFUL value z_i of each teacher Z_i (i = 1, 2, 3, 4), and have $\begin{matrix} z_{1} = < [s_{3.00}, s_{5.04}], {{[0.34, 0.54], [0.37, 0.57]}, {[0.20, 0.38]}} >, \\ z_{2} = < [s_{2.46}, s_{4.71}], {{[0.54, 0.60], [0.57, 0.62]}, {[0.28, 0.28]}} >, \\ z_{3} = < [s_{2.26}, s_{4.28}], {{[0.40, 0.53]}, {[0.21, 0.36], [0.24, 0.39]}} >, \\ z_{4} = < [s_{2.84}, s_{4.88}], {{[0.36, 0.56]}, {[0.27, 0.30], [0.27, 0.27]}} > . \end{matrix}$

Step 4. Utilize Equation (14) to obtain the score value S (z_i) of each comprehensive value z_i (i = 1, 2, 3, 4), and we have

$\begin{matrix} S (z_{1}) = 0.6682, S (z_{2}) = 1.1046, S (z_{3}) = 0.5444, \\ S (z_{4}) = 0.7236 . \end{matrix}$

Step 5. Because S (z₂) > S (z₄) > S (z₁) > S (z₃), we get Z₂ > Z₄ > Z₁ > Z₃, where the optimal teacher is Z₂.

2) The evaluation steps by utilizing the first proposed method

Step 1. Utilize Equation (38) to get the optimal 2OAFM v′ on the attribute set D′ ⊆ D, and we have: $\begin{matrix} max fun (φ^{'}) = - 1 . 2500 v^{'} ({D_{1}}) - 0 . 4979 v^{'} ({D_{2}}) \\ - 1 . 0792 v^{'} ({D_{3}}) + 0 . 8740 v^{'} ({D_{1}, D_{2}}) \\ + 1 . 1646 v^{'} ({D_{1}, D_{3}}) + 0 . 7885 v^{'} ({D_{2}, D_{3}}) \\ s . t . {\begin{matrix} v^{'} ({D_{1}}) \in [0.1, 0.2], v^{'} ({D_{2}}) \in [0.4, 0.5], \\ v^{'} ({D_{3}}) \in [0.2, 0.3], v^{'} ({D_{1}, D_{2}}) \leq 0.5, \\ v^{'} ({D_{1}, D_{3}}) \leq 0.7, v^{'} ({D_{2}, D_{3}}) \geq 0.5, \\ v^{'} ({D_{1}, D_{2}}) + v^{'} ({D_{1}, D_{3}}) + v^{'} ({D_{2}, D_{3}}) - \\ v^{'} ({{D_{1}}_{s} quo}) - v^{'} ({D_{2}}_{s} quo) - v^{'} ({{D_{3}}_{s} quo}) = 1, \\ \sum_{{D_{i}} \subseteq E ∖ {D_{j}}} (v^{'} ({D_{i}, D_{j}}) - v^{'} ({D_{j}})) \geq (| E | - 2) v^{'} ({D_{j}}), \\ \forall E \subseteq D, \forall D_{j} \in E, | E | \geq 2 . \end{matrix}} \end{matrix}$

By solving the above model using Lingo software [29], we can obtain: $\begin{matrix} v^{'} ({D_{1}}) = 0.1, v^{'} ({D_{2}}) = 0.5, v^{'} ({D_{3}}) = 0.2, \\ v^{'} ({D_{1}, D_{2}}) = 0.5, v^{'} ({D_{1}, D_{3}}) = 0.7, \\ v^{'} ({D_{2}, D_{3}}) = 0.6 . \end{matrix}$

Step 2. Utilize Equation (33) to get the Shapely values of the attribute set D′ ⊆ D, and we have: $\begin{matrix} φ (v^{'} ({D_{1}})) = 0.5, φ (v^{'} ({D_{2}})) = 0.05, \\ φ (v^{'} ({D_{3}})) = 0.45 . \end{matrix}$

Step 3. Utilize the 2OA-SIVDHFULCG operator shown in Equation (36) to obtain the comprehensive IVDHFUL value z_i of each teacher Z_i (i = 1, 2, 3, 4), and we have: $\begin{matrix} z_{1} = < [s_{3.39}, s_{5.42}], {{[0.34, 0.54], [0.39, 0.59]}, {[0.21, 0.41]}} >, \\ z_{2} = < [s_{3.73}, s_{5.80}], {{[0.49, 0.60], [0.55, 0.65]}, {[0.21, 0.21]}} >, \\ z_{3} = < [s_{2.04}, s_{4.04}], {{[0.40, 0.59]}, {[0.24, 0.45], [0.25, 0.44]}} >, \\ z_{4} = < [s_{3.39}, s_{5.42}], {{[0.39, 0.59]}, {[0.30, 0.30], [0.29, 0.31]}} >, \end{matrix}$

Step 4. Utilize Equation (14) to obtain the score value S (z_i) of each comprehensive value z_i (i = 1, 2, 3, 4), and we have $\begin{matrix} S (z_{1}) = 0.6987, S (z_{2}) = 1.7150, S (z_{3}) = 0.4500, \\ S (z_{4}) = 0.8720 . \end{matrix}$

Step 5. Because S (z₂) > S (z₄) > S (z₁) > S (z₃), we get Z₂ > Z₄ > Z₁ > Z₃, which is the same as that got by the first developed MADM method, where the optimal teacher is Z₂.

It should be noted that Yang and Ju’s method [33] and Ju et al.’s method [10] cannot directly deal with IVDHFUL MCDM problems with incomplete weights. To apply Yang and Ju’s method [33] to handle Example 6.2, we need to transform the IVDHFUL information to the DHFL information and get the attribute weight w_i (i = 1, 2, 3, 4) by the maximum deviation method [28]. To apply Ju et al.’s method [10] to deal with Example 6.2, we need to transform the IVDHFUL information into DHFTL information and then get the attribute weight w_i (i = 1, 2, 3, 4) by the maximum deviation method [28]. The ranking orders of the alternatives of Example 6.2 for different methods are listed in Table 4.

Table 4

Ranking results by different methods

Methods	Score values	Ranking results
Yang and Ju’s method [33]	S (z₁) =0.1105, S (z₂) =0.0834, S (z₃) =0.1346, S (z₄) =0.1189	Z₃ > Z₄ > Z₁ > Z₂,
Ju et al.’s method [10]	S (z₁) =0.1105, S (z₂) =0.0834, S (z₃) =0.1346, S (z₄) =0.1189	Z₃ > Z₄ > Z₁ > Z₂,
The first proposed method based on the SIVDHFULCG operator	S (z₁) =0.6682, S (z₂) =1.1046, S (z₃) =0.5444, S (z₄) =0.7236 .	Z₂ > Z₄ > Z₁ > Z₃,
The second proposed method based on the 2OA-SIVDHFULCG operator	S (z₁) =0.6987, S (z₂) =1.7150, S (z₃) =0.4500, S (z₄) =0.8720 .	Z₂ > Z₄ > Z₁ > Z₃,

From Table 4, we find that the developed two methods derive the same ranking result, i.e., A₂ > A₄ > A₁ > A₃, whereas Yang and Ju’s method [33] and Ju et al.’s method [10] get a different ranking order A₃ > A₄ > A₁ > A₂. The main reason why the ranking results got by Yang and Ju’s method [33] and Ju et al.’s method [10] are different from the ones got by the developed methods is explained as follows. From Example 6.2, we get the following inequalities: v ({D₂}) + v ({D₃}) > v ({D₂, D₃}) and v ({D₁}) + v ({D₃}) < v ({D₁, D₂}). Thus, we can find that the criteria D₂ and D₃ are negative interactive, and the criteria D₁ and D₃ are positive interactive. That is to say, there are complex interactions among the criteria. However, Yang and Ju’s method [33] and Ju et al.’s method [10] only capture the addition of the importance of individual attribute. That is to say, the methods presented in [10, 33] are based on the following inequalities: v ({D₂}) + v ({D₃}) > v ({D₂, D₃}) and v ({D₁}) + v ({D₃}) = v ({D₁, D₃}). Clearly, these equations are invalid in Example 6.2. Thus, both Yang and Ju’s method [33] and Ju et al.’s method [10] obtain unreasonable ranking order of the teachers, as shown in Table 4. However, the proposed two MADM methods get more reasonable ranking order of the teachers and solve the drawback of the methods presented in [10, 33].

From above analysis, the proposed method for the IVDHFUL MADM problem has the following advantages.

Firstly, the defined IVDHFULSs are more flexible and practical than DHFLSs [33]. In spite of the fact that the expression of IVDHFULSs looks complex, they not only describe fuzzy linguistic information closely, but also keep the original data complete and reflect experts’ inherent thoughts, which is the precondition of ensuring precision of decision results.

Secondly, the weights of attributes, which are determined by the established models based on the maximization deviation method [27] and the Shapley index [21], are more objective and reasonable than the completely known weight information about attributes.

Thirdly, the DHFLWG operator in Ju’s method [33] and the DHFTLWG operator in Ju et al.’s method [10] only capture the addition of the importance of individual attribute. Nevertheless, the proposed SIVDHFULCG operator and the proposed 2OA-SIVDHFULCG operator consider the existing complex correlations among the criteria, which could ensure the reasonableness and effectiveness of the decision-making results.

In summary, because the developed IVDHFUL MADM methods are based on the defined IVDHFULSs, the established models for optimal FMs on attributes set and the proposed operators, they not only effectively aggregate IVDHFULSs and objectively determine the attributes weight, but also globally consider correlations among the criteria, which makes the final results more feasible.

7 Conclusions

For solving more complex decision making problems, we define a novel type of fuzzy sets called IVDHFULSs as an extension of DHFLSs. To better integrate fuzzy linguistic information, we apply the Choquet integral operator to the IVDHFUL setting. Then we develop the IVDHFULCG operator, the SIVDHFULCG operator and the 2OA-SIVDHFULCG operator. In particular, the proposed 2OA-SIVDHFULCG operator not only globally takes into account the complex correlations among criteria, but also simplifies the complexity in solving a FM. Meantime, we study several desirable properties of the proposed operators. Furthermore, we develop two MADM methods by utilizing the proposed SIVDHFULCG operator, the proposed 2OA-SIVDHFULCG operator and the established optimization models to handle the IVDHFUL MADM problems. Finally, we use two practical cases to verify the feasibility and the superiority of the developed MADM methods. From the experimental results shown in Examples 6.1 and 6.2, we find that the developed methods overcome the shortcomings of the Yang and Ju’s method [33] and the Ju et al.’s method [10]. In the future, the developed methods shall be applied to other areas.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71771140, 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), the Education Science Planning Project of Beijing (BCFA18051).

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