The Interval-Valued Intuitionistic Fuzzy Optimized Weighted Bonferroni Means and Their Application

Abstract

We investigate and propose two new Bonferroni means, that is, the optimized weighted BM (OWBM) and the generalized optimized weighted BM (GOWBM), whose characteristics are to reflect the preference and interrelationship of the aggregated arguments and can satisfy the basic properties of the aggregation techniques simultaneously. Further, we propose the interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (IIFOWBM) and the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (GIIFOWBM) and detailed study of their desirable properties such as idempotency, monotonicity, transformation, and boundary. Finally, based on IIFOWBM and GIIFOWBM, we give an approach to group decision making under the interval-valued intuitionistic fuzzy environment and utilize a practical case involving the assessment of a set of agroecological regions in Hubei Province, China, to illustrate the developed methods.

1. Introduction

As a useful aggregation technique, the Bonferroni mean (BM) can capture the interrelationship between input arguments and has been a hot research topic recently. Bonferroni [1] originally introduced a mean-type aggregation operator, called the Bonferroni mean, whose prominent characteristic is that it cannot only consider the importance of each criterion but also reflect the interrelationship of the individual criterion. Recently, Yager [2] extended the BM by two mean-type operators, such as the Choquet integral operator [3] and the ordered weighted averaging operator [4], as well as associates differing importance with the arguments. Mordelová and Rückschlossová [5] also investigated the generalizations of BM referred to as ABC-aggregation functions. Beliakov et al. [6] further extended the BM by considering the correlations of any three aggregated arguments instead of any two and proposed the generalized Bonferroni mean (GBM).

Nevertheless, the arguments suitable to be aggregated by the BM and GBM can only take the forms of crisp numbers. In the real world, due to the increasing complexity of the socioeconomic environment and the lack of knowledge and data, crisp data are sometimes unavailable. Thus, the input arguments may be more suitable with representation of fuzzy formats, such as fuzzy number [7], interval-valued fuzzy number [8], intuitionistic fuzzy value [9], interval-valued intuitionistic fuzzy value [10], and hesitant fuzzy element [11]. Thus, Xu and Yager [12] introduced the intuitionistic fuzzy Bonferroni mean (IFBM) and the intuitionistic fuzzy weighted Bonferroni mean (IFWBM). Xu and Chen [13] further proposed the interval-valued intuitionistic fuzzy Bonferroni mean (IIFBM) and the interval-valued intuitionistic fuzzy weighted Bonferroni mean (IIFWBM). Xia et al. [14] proposed the generalized intuitionistic fuzzy Bonferroni means. Zhou and He [15] developed some geometric Bonferroni Means. Furthermore, Beliakov and James [16] defined Bonferroni means over lattices which is a new viewpoint. Recently, Zhou and He [17] constructed an intuitionistic fuzzy weighted Bonferroni mean, Xia et al. [18] further investigated the generalized geometric Bonferroni means, Beliakov and James [19] extend the generalized Bonferroni means to Atanassov orthopairs, and so on.

The desirable characteristic of the BM is its capability to capture the interrelationship between input arguments. However, the classical BM and GBM, even the extended BMs, cannot reflect the interrelationship between the individual criterion and other criteria. To deal with these issues, in this paper, we propose the optimized weighted Bonferroni mean (OWBM) and the generalized optimized weighted Bonferroni mean (GOWBM), whose characteristics are to reflect the preference and interrelationship of the aggregated arguments and can satisfy the basic properties of the aggregation techniques simultaneously. Further, we have proposed the interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (IIFOWBM) and the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (GIIFOWBM) and detailed study of their desirable properties such as idempotency, monotonicity, transformation, and boundary.

The remainder of this paper is organized as follows. We briefly review some basic definitions of the interval-valued intuitionistic fuzzy values and Bonferroni mean. Then, in Section 3, we propose two BMs including OWBM and GOWBM and study their properties. Furthermore, in Section 4, we develop the IIFOWBM and GIIFOWBM operators, and their idempotency, monotonicity, transformation, and boundary are also investigated. A practical example is provided in Section 5 to demonstrate the application of the OWBM, GOWBM, and two interval-valued intuitionistic fuzzy BMs. The paper ends in Section 6 with concluding remarks.

2. Some Basic Concepts

2.1. Interval-Valued Intuitionistic Fuzzy Values

In the following, we introduce the basic concepts and operations related to interval-valued intuitionistic fuzzy value [20], which is an extended definition of the intuitionistic fuzzy value and interval-valued intuitionistic fuzzy set [21, 22].

Definition 1 (see [10]). Let 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛) be fixed. An interval-valued intuitionistic fuzzy set (IIFS) 𝐴 in 𝑋 can be defined as 𝜇𝐴(𝑥),𝜈𝐴𝐴=𝑥,(𝑥)∣𝑥∈𝑋, (1) where 𝜇𝐴(𝑥)⊂[0,1] and 𝜈𝐴(𝑥)⊂[0,1] satisfy 𝜇𝐴𝜈𝐴sup(𝑥)+sup(𝑥)≤1 for all 𝑥_𝑖∈𝑋 and 𝜇𝐴(𝑥) and 𝜈𝐴(𝑥) are, respectively, called the degree of membership and the degree of nonmembership of the element 𝑥_𝑖∈𝑋 to 𝐴.

Definition 2 (see [20]). Let 𝜇𝐴𝜈𝐴𝐴={(𝑥,(𝑥),(𝑥))∣𝑥∈𝑋} be an IIFS; the pair (𝜇𝐴𝜈𝐴(𝑥),(𝑥)) is called an interval-valued intuitionistic fuzzy value (IIFV).
For computational convenience, an IIFV can be denoted by ([𝑎,𝑏],[𝑐,𝑑]), with the condition that [𝑎,𝑏]⊂[0,1], [𝑐,𝑑]⊂[0,1], and 𝑏+𝑑≤1. Furthermore, based on the operations of interval-valued intuitionistic fuzzy values [21, 22], Xu [23] defined some operations of IIFNs as follows.

Definition 3 (see [23]). Letting 𝛼=([𝑎,𝑏],[𝑐,𝑑]), 𝛼₁=([𝑎₁,𝑏₁],[𝑐₁,𝑑₁]), and 𝛼₂=([𝑎₂,𝑏₂],[𝑐₂,𝑑₂]) be three IIFVs, then the following operational laws are valid:(1)𝛼₁⊕𝛼₂=([𝑎₁+𝑎₂−𝑎₁𝑎₂,𝑏₁+𝑏₂−𝑏₁𝑏₂],[𝑐₁𝑐₂,𝑑₁𝑑₂]),(2)𝛼₁⊗𝛼₂=([𝑎₁𝑎₂,𝑏₁𝑏₂],[𝑐₁+𝑐₂−𝑐₁𝑐₂,𝑑₁+𝑑₂−𝑑₁𝑑₂]),(3)𝜆𝛼=([1−(1−𝑎)^𝜆,1−(1−𝑏)^𝜆],[𝑐^𝜆,𝑑^𝜆]), 𝜆>0,(4)𝛼^𝜆=([𝑎^𝜆,𝑏^𝜆],[1−(1−𝑐)^𝜆,1−(1−𝑑)^𝜆]), 𝜆>0.
Then, Xu [23] introduced the score function 𝑠(𝛼)=(𝑎−𝑐+𝑏−𝑑)/2 and the accuracy function ℎ(𝛼)=(𝑎+𝑏+𝑐+𝑑)/2 to calculate the score value and accuracy degree of IIFV 𝛼=([𝑎,𝑏],[𝑐,𝑑]) and gave an order relation between two IIFNs 𝛼₁ and 𝛼₂ as follows: (1)if 𝑠(𝛼₁)<𝑠(𝛼₂), then 𝛼₁<𝛼₂; (2)if 𝑠(𝛼₁)=𝑠(𝛼₂), then (i)if ℎ(𝛼₁)<ℎ(𝛼₂), then 𝛼₁<𝛼₂; (ii)if ℎ(𝛼₁)=ℎ(𝛼₂), then 𝛼₁∼𝛼₂.

2.2. Bonferroni Means

Definition 4 (see [1]). Let 𝑝,𝑞≥0 and 𝑎_𝑖(𝑖=1,2,…,𝑛) a collection of nonnegative numbers. If BM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1𝑖≠𝑗}𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}, (2) then BM^𝑝,𝑞 is called the Bonferroni mean (BM).

Definition 5 (see [6]). Let 𝑝,𝑞,𝑟≥0 and 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers. If GBM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑛(𝑛−1)(𝑛−2)^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}, (3) then GBM^{𝑝,𝑞,𝑟} is called the generalized Bonferroni mean (GBM).
To deal with intuitionistic fuzzy value and hesitant fuzzy value, Xu and Yager [12] extended these BMs to fuzzy environment and gave the following concepts.

Definition 6 (see [12]). Let 𝛼_𝑖(𝑖=1,2,…,𝑛) be a set of intuitionistic fuzzy values. The intuitionistic fuzzy Bonferroni mean (IFBM), the intuitionistic fuzzy weighted Bonferroni mean (IFWBM), and the generalized intuitionistic fuzzy weighted Bonferroni mean (GIFWBM) are, respectively, defined as IFBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)},IFWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝛼^𝑝_𝑖⊗𝑤_𝑗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)},GIFBM^{𝑝,𝑞,𝑟}𝛼₁,𝛼₂,…,𝛼_𝑛=1𝑛(𝑛−1)(𝑛−2_{𝑛𝑖,𝑗,𝑘=1,𝑖≠𝑗≠𝑘})𝑤_𝑖𝛼^𝑝_𝑖⊗𝑤_𝑗𝛼^𝑞_𝑗⊗𝑤_𝑘𝛼^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}. (4)
However, it is noted that the above BMs could not effectively aggregate the general fuzzy value, that is, interval-valued intuitionistic fuzzy value. On the other hand, the above BMs just consider the whole correlationship between the criterion 𝑎_𝑖 and all criteria ∑^𝑛_𝑗=1𝑎^𝑞_𝑗⋅∑^𝑛_𝑘=1𝑎^𝑟_𝑘 and cannot reflect the interrelationship between the individual criterion 𝑎_𝑖 and other criteria 𝑣_𝑖^𝑞,𝑟 which is the main advantage of the BM. To overcome this drawback, we propose the following OWBM, GOWBM, IIFOWBM, and GIIFOWBM operators.

3. The Optimized Weighted BM and Its Generalized Form

The BM and GBM, including IFWBM and GIFWBM, just consider the whole correlationship between the criterion 𝑎_𝑖 and all criteria and cannot reflect the interrelationship between the individual criterion 𝑎_𝑖 and other criteria 𝑣_𝑖^𝑞,𝑟 which is the main advantage of the BM. To deal with these issues, in the following subsections, we propose the optimized weighted versions of BM and its generalized form, that is, the optimized weighted BM (OWBM) and the generalized optimized weighted BM (GOWBM). Based on the Bonferroni mean, we can define the following optimized weighted BM (OWBM) and generalized optimized weighted BM (GOWBM).

Definition 7. Let 𝑝,𝑞≥0 and 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) such that 𝑤_𝑖≥0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. Then the optimized weighted Bonferroni mean (OWBM) can be defined as follows: OWBM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_{𝑖,𝑗=1𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}. (5)

Definition 8. Let 𝑝,𝑞,𝑟≥0 and 𝑎_𝑖(𝑖=1,2,…,𝑛) a collection of nonnegative numbers with the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) such that 𝑤_𝑖≥0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. Then the generalized optimized weighted Bonferroni mean (GOWBM) can be defined as follows: GOWBM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}𝑤_𝑖𝑤_𝑗𝑤_𝑘1−𝑤_𝑖1−𝑤_𝑖−𝑤_𝑗𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}. (6)
Furthermore, we can transform the OWBM and GOWBM into the interrelationship between OWBM and GOWBM forms as follows: OWBM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_𝑖=1𝑤_𝑖𝑎^𝑝_𝑖^𝑛_{𝑗=1𝑗≠𝑖}𝑤_𝑗1−𝑤_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)},GOWBM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_𝑖=1𝑤_𝑖𝑎^𝑝_𝑖^𝑛_{𝑗=1𝑗≠𝑖}𝑤_𝑗1−𝑤_𝑖𝑎^𝑞_𝑗^𝑛_{𝑘=1𝑘≠𝑖≠𝑗}𝑤_𝑘1−𝑤_𝑖−𝑤_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}. (7)(8)
According to (6)-(7), we see that the terms ∑^𝑛_{𝑗=1,𝑗≠𝑖}(𝑤_𝑗/(1−𝑤_𝑖))𝑎^𝑞_𝑗 and ∑^𝑛_{𝑖,𝑗,𝑘=1,𝑖≠𝑗≠𝑘}(𝑤_𝑖𝑤_𝑗𝑤_𝑘/(1−𝑤_𝑖)(1−𝑤_𝑖−𝑤_𝑗))𝑎^𝑞_𝑗𝑎^𝑟_𝑘 are the weighted power average satisfaction of all criteria except 𝐴_𝑖, and ∑^𝑛_{𝑗=1,𝑗≠𝑖}(𝑤_𝑗/(1−𝑤_𝑖))=1 and ∑^𝑛_{𝑖,𝑗,𝑘=1,𝑖≠𝑗≠𝑘}(𝑤_𝑖𝑤_𝑗𝑤_𝑘/(1−𝑤_𝑖)(1−𝑤_𝑖−𝑤_𝑗))=1. If we, respectively, denote the above terms as 𝑢^𝑞_𝑖 and 𝑢^𝑞_𝑖𝑣^𝑟_𝑖, thus OWBM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_𝑖=1𝑤_𝑖𝑎^𝑝_𝑖𝑢^𝑞_𝑖^{1/(𝑝+𝑞)},GOWBM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_𝑖=1𝑤_𝑖𝑎^𝑝_𝑖𝑢^𝑞_𝑖𝑣^𝑟_𝑖^{1/(𝑝+𝑞+𝑟)}. (9)
Here then 𝑢^𝑞_𝑖 and 𝑢^𝑞_𝑖𝑣^𝑟_𝑖 are the weighted power average satisfaction to all criteria except 𝐴_𝑖, which represents the interrelationship between the individual criterion 𝑎_𝑖 and other criteria 𝑎_𝑗(𝑗≠𝑖). This is similar to the BM.

4. Two Interval-Valued Intuitionistic Fuzzy BM Operators Based on OWBM and GOWBM

To aggregate the fuzzy information, Xu and Yager [12] proposed the IFBM and IFWBM. However, it is noted that the above BMs could not effectively aggregate the general fuzzy value, that is, interval-valued intuitionistic fuzzy value. Thus, we further propose two interval-valued intuitionistic fuzzy BM operators to aggregate the interval-valued intuitionistic fuzzy correlated information based on the optimized weighted and generalized optimized weighted BMs, respectively, that is, IIFOWBM and GIIFOWBM.

Definition 9. Let 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖]) be a collection of IIFVs with the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) such that 𝑝,𝑞≥0, 𝑤_𝑖≥0, and ∑^𝑛_𝑖=1𝑤_𝑖=1. If IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)}, (10) then IIFOWBM^𝑝,𝑞 is called the interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (IIFOWBM).

On the basis of the operational laws of IIFVs, we have the following.

Theorem 10. Letting 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of IIFVs with the weight vector (𝑤₁,𝑤₂,…,𝑤_𝑛), such that 𝑝,𝑞≥0, 𝑤_𝑖≥0, and ∑^𝑛_𝑖=1𝑤_𝑖=1, then the aggregated value by using the IIFOWBM is also an IIFV, and 𝐼𝐼𝐹𝑂𝑊𝐵𝑀^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)}=1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)},1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)},1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑐_𝑖^𝑝×1−𝑐_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)},1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝×1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}. (11)

Proof. By the operational laws for IIFVs, we get 𝛼^𝑝_𝑖=𝑎^𝑝_𝑖,𝑏^𝑝_𝑖,1−1−𝑐_𝑖^𝑝,1−1−𝑑_𝑖^𝑝,𝛼^𝑞_𝑗=𝑎^𝑞_𝑗,𝑏^𝑞_𝑗,1−1−𝑐_𝑗^𝑞,1−1−𝑑_𝑗^𝑞,𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗=𝑎^𝑝_𝑖𝑎^𝑞_𝑗,𝑏^𝑝_𝑖𝑏^𝑞_𝑗,1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^𝑞,1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^𝑞,𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗=1−1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)},1−1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)},1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)},1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)},_{𝑛𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗=1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)},1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)},_{𝑛𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)},_{𝑛𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)}. (12) Therefore, IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)}=1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)},1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)},1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)},1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝×1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)},0≤1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}≤1,0≤1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}≤1. (13) We have 0≤1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}≤1,0≤1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}≤1,1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}−+11−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}+≤11−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^{𝑞𝑤𝑖𝑤𝑗/(1−𝑤𝑖)1/(𝑝+𝑞)}=1 (14) which completes the proof.

Property 1 (idempotency). If all IIFVs 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) are equal, that is, 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])=𝛼=([𝑎,𝑏],[𝑐,𝑑]), for all 𝑖, we have IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=𝛼=([𝑎,𝑏],[𝑐,𝑑]). (15)

Proof. Since 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])=𝛼=([𝑎,𝑏],[𝑐,𝑑]), then IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝⊗𝛼^{𝑞1/(𝑝+𝑞)}=𝛼^𝑝⊗𝛼^{𝑞1/(𝑝+𝑞)}=𝛼_𝑖 (16) which completes the proof of Property 1.

Corollary 11. If 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) is a collection of the largest IIFVs, that is, 𝛼_𝑖=𝛼=([1,1],[0,0]), for all 𝑖, and (𝑤_𝜎(1),𝑤_𝜎(2),…,𝑤_𝜎(𝑛)) is precise weight vector of 𝛼_𝑖, then 𝐼𝐼𝐹𝑂𝑊𝐵𝑀^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)}=1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}(1−1)^{1/(𝑝+𝑞)},1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}(1−1)^{1/(𝑝+𝑞)},1−(1)^{1/(𝑝+𝑞)},1−(1)^{1/(𝑝+𝑞)}=([1,1],[0,0]) (17) which is also the largest IIFV.

Property 2 (monotonicity). Let 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖]) and 𝛽_𝑗=([𝑎_𝑗,𝑏_𝑗],[𝑐_𝑗,𝑑_𝑗])(𝑖,𝑗=1,2,…,𝑛) be two collections of IIFVs; if 𝑎_𝑖≤𝑎_𝑗, 𝑏_𝑖≤𝑏_𝑗, 𝑐_𝑖≤𝑐_𝑗 and 𝑑_𝑖≤𝑑_𝑗, for all 𝑖, then IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛≤IIFOWBM^𝑝,𝑞𝛽₁,𝛽₂,…,𝛽_𝑛. (18)

Proof. The proof of Property 2 is similar to Property 10 in [15].

Property 3 (transformation). Letting 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a set of IIFVs and (𝛼₁,𝛼₂,…,𝛼_𝑛) the permutation of (𝛼₁,𝛼₂,…,𝛼_𝑛), then IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛. (19)

Proof. Since (𝛼₁,𝛼₂,…,𝛼_𝑛) is a permutation of (𝛼₁,𝛼₂,…,𝛼_𝑛), then ^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)}=^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖𝑤_𝑗1−𝑤_𝑖𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗^{1/(𝑝+𝑞)}. (20) Thus, IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=IIFOWBM^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛. (21)

Definition 12. Let 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of IIFVs with the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) such that 𝑝,𝑞,𝑟≥0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. If GIIFOWBM^{𝑝,𝑞,𝑟}𝛼₁,…,𝛼_𝑛=^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}𝑤_𝑖𝑤_𝑗𝑤_𝑘1−𝑤_𝑖1−𝑤_𝑖−𝑤_𝑗𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗⊗𝛼^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}, (22) then GIIFOWBM^{𝑝,𝑞,𝑟} is called the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (GIIFOWBM).

Theorem 13. Letting 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a set of IIFVs with the weight vector (𝑤₁,𝑤₂,…,𝑤_𝑛), such that 𝑝,𝑞,𝑟≥0 and ∑^𝑛_𝑖=1𝑤_𝑖=1, then the aggregated value by using the GIIFOWBM is also an IIFV and 𝐺𝐼𝐼𝐹𝑂𝑊𝐵𝑀^{𝑝,𝑞,𝑟}𝛼₁,𝛼₂,…,𝛼_𝑛=^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}𝑤_𝑖𝑤_𝑗𝑤_𝑘1−𝑤_𝑖1−𝑤_𝑖−𝑤_𝑗(𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗⊗𝛼^𝑟_𝑘)^{1/(𝑝+𝑞+𝑟)}=1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)},1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗𝑏^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)}×1−1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^𝑞×1−𝑐_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)},1−1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^𝑞×1−𝑑_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)}. (23)

Proof. By the operational laws for IIFVs, we have 𝛼^𝑝_𝑖=𝑎^𝑝_𝑖,𝑏^𝑝_𝑖,1−1−𝑐_𝑖^𝑝,1−1−𝑑_𝑖^𝑝,𝛼^𝑞_𝑗=𝑎^𝑞_𝑗,𝑏^𝑞_𝑗,1−1−𝑐_𝑗^𝑞,1−1−𝑑_𝑗^𝑞,𝛼^𝑟_𝑘=𝑎^𝑟_𝑘,𝑏^𝑟_𝑘,1−1−𝑐_𝑘^𝑟,1−1−𝑑_𝑘^𝑟,𝑤_𝑖𝑤_𝑗𝑤_𝑘1−𝑤_𝑖1−𝑤_𝑖−𝑤_𝑗𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗⊗𝛼^𝑟_𝑘=1−1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)},1−1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗𝑏^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)},1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^𝑞1−𝑐_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)},1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^𝑞×1−𝑑_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)}. (24) Therefore, GIIFOWBM^{𝑝,𝑞,𝑟}𝛼₁,𝛼₂,…,𝛼_𝑛=^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}𝑤_𝑖𝑤_𝑗𝑤_𝑘1−𝑤_𝑖1−𝑤_𝑖−𝑤_𝑗(𝛼^𝑝_𝑖⊗𝛼^𝑞_𝑗⊗𝛼^𝑟_𝑘)^{1/(𝑝+𝑞+𝑟)}=1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)},1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗𝑏^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)},1−1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^𝑞×1−𝑐_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)},1−1−^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^𝑞×1−𝑑_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)}. (25) In addition, since 0≤1−^𝑛_{𝑖,𝑗,𝑘=1,𝑖≠𝑗≠𝑘}1−𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)}≤1,0≤1−^𝑛_{𝑖,𝑗,𝑘=1,𝑖≠𝑗≠𝑘}1−𝑏^𝑝_𝑖𝑏^𝑞_𝑗𝑏^𝑟_𝑘^{𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)}≤1, (26) then 0≤1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑐_𝑖^𝑝1−𝑐_𝑗^𝑞×1−𝑐_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)}≤1,0≤1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝑑_𝑖^𝑝1−𝑑_𝑗^𝑞×1−𝑑_𝑘^{𝑟𝑤𝑖𝑤𝑗𝑤𝑘/(1−𝑤𝑖)(1−𝑤𝑖−𝑤𝑗)1/(𝑝+𝑞+𝑟)}≤1 (27) which completes the proof of Theorem 13.

Property 4 (idempotency). If all IIFVs 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖], [𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) are equal, that is, 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])=𝛼=([𝑎, 𝑏],[𝑐,𝑑]), for all 𝑖, then GIIFOWBM^{𝑝,𝑞,𝑟}𝛼₁,𝛼₂,…,𝛼_𝑛=𝛼=([𝑎,𝑏],[𝑐,𝑑]). (28)

Proof. The proof of Property 4 is similar to Property 1.

Corollary 14. If 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) is a set of the largest IIFVs, that is, 𝛼_𝑖=𝛼=([1,1],[0,0]), for all 𝑖, and (𝑤_𝜎(1),𝑤_𝜎(2),…,𝑤_𝜎(𝑛)) is precise weight vector of 𝛼_𝑖, then 𝐺𝐼𝐼𝐹𝑂𝑊𝐵𝑀^𝑝,𝑞𝛼₁,𝛼₂,…,𝛼_𝑛=([1,1],[0,0]) (29) which is also the largest IIFV.

Property 5 (monotonicity). Let 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖]) and 𝛽_𝑗=([𝑎_𝑗,𝑏_𝑗],[𝑐_𝑗,𝑑_𝑗])(𝑖,𝑗=1,2,…,𝑛) be two sets of IIFVs; if 𝑎_𝑖≤𝑎_𝑗, 𝑏_𝑖≤𝑏_𝑗, 𝑐_𝑖≤𝑐_𝑗 and 𝑑_𝑖≤𝑑_𝑗, for all 𝑖, then GIIFOWBM^{𝑝,𝑞,𝑟}𝛼₁,𝛼₂,…,𝛼_𝑛≤GIIFOWBM^{𝑝,𝑞,𝑟}𝛽₁,𝛽₂,…,𝛽_𝑛. (30)

Proof. The proof of Property 5 is similar to Property 2.

Property 6 (transformation). Letting 𝛼_𝑖=([𝑎_𝑖,𝑏_𝑖],[𝑐_𝑖,𝑑_𝑖])(𝑖=1,2,…,𝑛) be a collection of IIFVs and (𝛼₁,𝛼₂,…,𝛼_𝑛) is any permutation of (𝛼₁,𝛼₂,…,𝛼_𝑛), then GIIFOWBM^{𝑝,𝑞,𝑟}𝛼₁,𝛼₂,…,𝛼_𝑛=GIIFOWBM^{𝑝,𝑞,𝑟}𝛼₁,𝛼₂,…,𝛼_𝑛. (31)

Proof. The proof of Property 6 is similar to Property 3.

5. Case Illustration

In the following, we will apply the IIFOWBM and GIIFOWBM operators to group decision making and utilize a practical case (adapted from Xu, 2007) [23] involving the assessment of a set of agroecological regions in Hubei Province, China, to illustrate the developed methods.

Located in Central China and the middle reaches of the Changjiang (Yangtze) River, Hubei Province is distributed in a transitional belt where physical conditions and landscapes are on the transition from north to south and from east to west. Thus, Hubei Province is well known as “a land of rice and fish” since the region enjoys some of the favorable physical conditions, with diversity of natural resources and the suitability for growing various crops. At the same time, however, there are also some restrictive factors for developing agriculture such as a tight manland relation between a constant degradation of natural resources and a growing population pressure on land resource reserve. Despite cherishing a burning desire to promote their standard of living, people living in the area are frustrated because they have no ability to enhance the power to accelerate economic development because of a dramatic decline in quantity of natural resources and a deteriorating environment. Based on the distinctness and differences in environment and natural resources, Hubei Province can be roughly divided into four agroecological regions: 𝑎₁-Wuhan-Ezhou-Huanggang; 𝑎₂-Northeast of Hubei; 𝑎₃-Southeast of Hubei; and 𝑎₄-West of Hubei. In order to prioritize these agroecological regions 𝑎_𝑖(𝑖=1,2,3,4) with respect to their comprehensive functions, a committee comprised of four experts 𝑒_𝑗(𝑗=1,2,3,4) (whose weight vector is 𝑤=(0.35,0.20,0.15,0.30)) has been set up to provide assessment information on 𝑎_𝑖(𝑖=1,2,3,4). The expert 𝑒_𝑗 compared these four agroecological regions with respect to their comprehensive functions and construct and represented the IIFVs 𝛼_𝑖𝑗=([𝑎_𝑖𝑗,𝑏_𝑖𝑗],[𝑐_𝑖𝑗,𝑑_𝑖𝑗]), where [𝑎_𝑖𝑗,𝑏_𝑖𝑗] indicates the agreement degree and [𝑐_𝑖𝑗,𝑑_𝑖𝑗] indicates the unagreement degree. To get the optimal alternative by the new IIOWBM and GIIFOWBM operators, the following steps are given.

Step 1. We normalize 𝑎_𝑖𝑗 to 𝑟_𝑖𝑗 and construct the normalization interval-valued intuitionistic fuzzy decision matrix 𝑅=(𝑟_𝑖𝑗)_4×5 (see Table 1).

Table 1 

Interval-valued intuitionistic fuzzy decision matrix 𝑅.

Step 2. Aggregate all the preference values 𝑟_𝑖𝑗(𝑖,𝑗=1,2,3,4) of the 𝑖th line and get the overall performance value 𝑐_𝑖 corresponding to the expert 𝑒_𝑖 by the IIFOWBM operator (here we let 𝑝=𝑞=1): 𝑐₁=],[,𝑐([0.2326,0.64410.1707,0.2532])₂=],[,𝑐([0.2472,0.66290.0000,0.2381])₃=],[,𝑐([0.2702,0.60380.1790,0.2868])₄=([0.2467,0.6432],[0.0000,0.0000]). (32)

Step 3. Calculating the score 𝑠₂(𝑐_𝑖) of 𝑐_𝑖(𝑖=1,2,3,4), respectively, we can get 𝑠₂𝑐₁=0.6132,𝑠₂𝑐₂𝑠=0.6680,₂𝑐₃=0.6021,𝑠₂𝑐₄=0.7225. (33) Therefore, 𝑠₂𝑐₄>𝑠₂𝑐₂>𝑠₂𝑐₁>𝑠₂𝑐₃ (34) and 𝑎₄>𝑎₂>𝑎₁>𝑎₃, and 𝑦₄ is still the optimal alternative.

Based on GOWBM and GIIFOWBM operators, the main steps are as follows.

Step 1^∘. See Step 1.

Step 2^∘. See Step 2.

Step 3^∘. Utilize the GOWBM and GIIFOWBM operators to aggregate all the interval-valued individual intuitionistic fuzzy decision matrices 𝐷_(𝑞)=(𝑑_𝑞^𝑖𝑗)_4×4(𝑞=1,2,3,4) into the collective interval-valued intuitionistic fuzzy decision matrix 𝐷=(𝑑_𝑖𝑗)_4×4 (see Table 2).

Table 2 

Interval-valued intuitionistic fuzzy decision matrix of 𝐷.

Step 4. Aggregate all the preference values 𝑑_𝑖𝑗(𝑖,𝑗=1,2,3,4) of the 𝑖th line and get the overall performance value 𝑐_𝑖 corresponding to the expert 𝑒_𝑖 by the GIIFOWBM operator (here we let 𝑝=𝑞=𝑟=1): 𝑐₁=],[,𝑐([0.1907,0.59990.2086,0.2901])₂=],[,𝑐([0.2093,0.62500.1589,0.2732])₃=],[,𝑐([0.2532,0.55930.2029,0.3137])₄=([0.2179,0.5931],[0.2409,0.3050]). (35)

Step 5. Calculating the score 𝑠₂(𝑐_𝑖) of 𝑐_𝑖(𝑖=1,2,3,4), respectively, we have 𝑠₂𝑐₁=0.5729,𝑠₂𝑐₂𝑠=0.6005,₂𝑐₃=0.5740,𝑠₂𝑐₄=0.5663. (36) Therefore, 𝑠₂𝑐₂>𝑠₂𝑐₃>𝑠₂𝑐₁>𝑠₂𝑐₄ (37) and 𝑎₂>𝑎₃>𝑎₁>𝑎₄, and 𝑦₄ is still the optimal alternative.

Based on the previous analysis, it could be found that the most comprehensive function is the West of Hubei.

It should be noted out that the whole ranking of the alternatives has changed. The IIFOWBM^1,1 produces the ranking of all the alternatives as 𝑎₄>𝑎₂>𝑎₁>𝑎₃, which is slightly different from the ranking of alternatives 𝑎₂>𝑎₃>𝑎₁>𝑎₄, derived by the GIIFOWBM^1,1,1. Therefore, we can see that the value derived by the IIFOWBM or GIIFOWBM operators depends on the choice of the parameters 𝑝, 𝑞, and 𝑟.

6. Concluding Remarks

To further develop the BM, we have proposed the optimized weighted Bonferroni mean (OWBM) and the generalized optimized weighted Bonferroni mean (GOWBM) in this paper. Then, we proposed two new BM operators under the interval-valued intuitionistic fuzzy environment, that is, the interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (IIFOWBM) and the generalized interval-valued intuitionistic fuzzy optimized weighted Bonferroni mean (GIIFOWBM). The new BMs can reflect the preference and interrelationship of the aggregated arguments and can satisfy the basic properties of the aggregation techniques simultaneously. Furthermore, some desirable properties of the IFIOWBM and GIIFOWBM operators are investigated in detail, including idempotency, monotonicity, transformation, and boundary. Finally, based on IIFOWBM and GIIFOWBM, we give a utilized practical case involving the assessment of a set of agroecological regions in China to illustrate these new BMs.

Acknowledgment

This work was supported by National Basic Research Program of China (973 Program, no. 2010 CB328104-02).