Information Aggregation and Psychological Risk Dual-Driven Sustainable Supplier Selection Method Based on Extended Fuzzy Set and Choquet Integral

Abstract

A novel sustainable supplier selection (SSS) method is proposed to address the interrelation among attributes and the psychological state and risk attitude of decision-makers (DMs). The method integrates proportional interval type-2 hesitant fuzzy sets (PIT2HFSs), a generalized Shapley-based aggregation operator, and a modified regret theory combined with a normalized bidirectional projection (NBP) measure. The aggregation operators handle the correlations among attributes, while the NBP and regret theory reflect DMs’ risk preferences by considering both the best and worst alternatives. An application case study in a manufacturing enterprise, along with sensitivity and comparative analyses, demonstrates the effectiveness and robustness of the proposed approach. The results indicate that the method outperforms existing approaches in handling attribute interdependencies, decision uncertainty, and human risk behavior, providing a comprehensive and practical framework for sustainable supplier selection in the manufacturing industry.

1. Introduction

With the global environmental challenges and increasing awareness of social responsibility, sustainable supply chain management (SSCM) has gradually become an important strategy for enterprises to cope with market changes, resource depletion and environmental pressures [1]. This study focuses on the decision-making problem of sustainable supplier selection (SSS) in sustainable supply chain management, with the core being the integration of multiple sustainable development goals, such as environment, society, and economy, in the evaluation and selection of suppliers [2,3]. However, the existing SSS methods face two major limitations: one is the difficulty in accurately handling the multidimensional assessment information from different experts under uncertainty; the other is the lack of a scientific decision-making framework that can simultaneously balance environmental and social sustainability, economic benefits, and risk tolerance [3,4,5]. To address this, this study aims to construct a more stable and flexible SSS decision model to effectively cope with uncertainty and achieve scientific trade-offs among multiple objectives, thereby providing decision support for enterprises to balance economic benefits and social responsibility and enhance supply chain resilience and sustainable performance in practice [2,3,4,5].

In the process of SSS, it is often necessary for a group of experts from various fields to evaluate suppliers based on given attributes, such as cost, product quality, and service performance [6]. Therefore, the SSS problem can be viewed as a multi-attribute group decision-making (MAGDM) problem [7]. Various MAGDM methods have been utilized to solve the SSS problem, such as the MULTIMOORA-based method [7], the TOPSIS-based method [8], the Choquet integral-based method [9,10], and the TODIM-based method [11], among others [12,13,14,15].

Table 1. The comparison between the proposed method and other SSS methods.

Reference	Preference Information	Methods	Interrelation Between Attributes	Psychological State of DMs	Risk Attitudes of DMs
Liu et al. [7]	ILRN	MULTIMOORA	No	No	No
Yu et al. [8]	IVPFS	TOPSIS	No	No	No
Büyüközkan et al. [9]	PFS	CI	Yes	No	No
Xing et al. [10]	IT2FS	CI	Yes	No	No
Qin et al. [11]	IT2FS	TODIM	No	Yes	Yes
Rashidi and Cullinane [12]	TFS	DEA + TOPSIS	No	No	No
Liu et al. [13]	TFS	AHP + TOPSIS	No	No	No
Stević et al. [14]	AN	MARCOS + nonlinear model	Yes	No	No
Liu et al. [15]	IVIULS	BWM + AQM	No	No	No
Our method	PIT2HFS	CI + Regret theory	Yes	Yes	Yes

presents a comparison between the method proposed in this paper and other SSS methods in group decision-making (GDM) in recent years. A detailed analysis of these methods is provided in the following two paragraphs of this section.

With the increasing complexity and uncertainty in real SSS process, most of the assessment detailed information is unknown, many factors are influenced by uncertainty, and human thinking is always ambiguous [16]. To overcome the uncertainty and ambiguity in the SSS process, fuzzy set theory [17] is widely used in SSS problems, such as Intuitionistic fuzzy sets (IFSs) [15], Pythagorean fuzzy sets (PFSs) [9], Hesitant fuzzy sets (HFSs) [18] and triangular fuzzy number (TFN) [12,13]. However, with the increasing complexity of the SSS process, type-1 fuzzy sets (T1FSs) are not sufficient to depict the information with high uncertainty [11]. To address this issue, Zadeh [19] defined the concept of type-2 fuzzy set (T2FSs), whose membership is represented by T1FS, as an extension of T1FSs. T2FSs own a wide range of advantages. However, the huge amount of calculation impedes their progress in practical application. To overcome this drawback, Mendel et al. [20] defined interval type-2 fuzzy sets (IT2FSs) which only take primary memberships into account and set all secondary memberships equal to 1. Compared with T2FSs, the computational burden of IT2FSs is greatly reduced [21]. In recent years, in order to enhance the ability to express uncertain information, research has gradually expanded towards multi-level and multi-structured forms of fuzzy sets, such as intuitive trapezoidal dense fuzzy sets [22] and spherical hesitant fuzzy sets [23]. These methods integrate the advantages of different fuzzy structures to handle fuzziness and hesitation in the decision-making process in a more flexible manner. In practical applications, various decision-making models based on fuzzy sets have emerged, including cyclic picture fuzzy values [24], intuitionistic fuzzy numbers [25], normalization methods [26], various integration operators, etc. These models have achieved specific applications in areas such as brand evaluation, transportation tool selection, environmental quality assessment, and sustainable supplier selection. Although existing research has made progress in dealing with uncertain information, there are still significant limitations: on the one hand, most methods fail to systematically consider the intrinsic relationships and dependencies among evaluation attributes; on the other hand, there is also a lack of sufficient integration of the actual psychological behaviors of decision-makers (such as risk aversion, reference dependence, etc.), thereby restricting the explanatory power and adaptability of the models in complex SSS environments.

Although the above methods could effectively solve uncertainty and ambiguity in the SSS process, there are still shortcomings. The above methods assume that each attribute is independent of the others, namely, there is no correlation between attributes. However, in practical application, there is often a correlation between attributes [27], for example, there is a correlation between cost and profit. The Choquet integral (CI) [28] can capture the heterogeneous relationship of the attributes because it is based on the fuzzy measure [29], which is an effective tool for modeling the positive interaction, negative interaction or independence of the attributes. Hence, the CI has been widely applied to solve MAGDM problems under a type-1 fuzzy environment [30,31,32,33]. To describe higher uncertainty where the exact membership degrees of type-1 fuzzy sets are difficult to determine, some scholars [10,34] extend the CI to an IT2F environment. Xing et al. [10] proposed a CI-based IT2F-MAGDM method. To comprehensively capture the relation between attributes, Tang et al. [34] proposed Banzhaf interval type-2 Fuzzy Archimedean Choquet (BIT2FAC) operator. Existing research has developed several representative approaches to overcome the limitations of traditional methods: first, introducing more flexible fuzzy sets at the information representation level and combining them with corresponding operators to capture the hesitation and emotional factors in decision-making [35]; second, extending aggregation operators in specific mathematical environments (such as q-ROF environments) and applying them to practical decision-making problems [36]; and third, enhancing the robustness and interpretability of multi-criteria trade-off models in complex situations by integrating multiple grey decision-making methods [37]. These advancements have been made along the paths of information foundation, operator construction, and model integration, but they have not simultaneously addressed the integration of attribute partition correlation and decision-makers’ psychological factors within a unified framework. Therefore, integrating the advantages of existing approaches and constructing a decision-making model that can take into account both attribute structural characteristics and behavioral factors has become the key focus and innovative direction of this study.

As we have stated before, the risk attitude and psychological state of DMs are very important factors in the SSS process. In order to effectively describe the risk attitude of DMs, the prospect theory-based method and TODIM method are put forward to solve MAGDM problems under the IT2F environment. However, Wang et al. [38] noted that when decision-makers (DMs) realize they have missed the optimal plan, they experience not only risk but also regret. As a representative behavioral theory, regret theory can effectively address this issue as it proposes a projection-based regret theory approach in IT2F environments. From the perspective of decision-making psychology, the theory of regret has a solid foundation. Bell and Loomis pointed out that decision-makers experience joy or regret by comparing outcomes [39]. The theory of risk tolerance further explains that individuals pay particular attention to the better outcomes that might be lost by not choosing other options, which is highly consistent with the actual psychology of investors [40]. Although scholars have increasingly focused on the limited rationality in decision-making, the systematic role of regret psychology in multi-attribute group decision-making still requires further exploration [38,41]. Pan et al. [42] proposed a regret-based risk decision-making method for investing in renewable energy. Then, Wang et al. [43] proposed a three-way decision-making framework based on regret theory in a IT2F environment. However, Chen et al. [44] show that these methods only consider the regret of a single choice in isolation. Since they measure regret outcomes from a utility perspective, they do not take into account worst-case scenarios, which can lead to incorrect decisions. Hence, they improved the regret–rejoice function to make up for the above shortcoming.

In summary, the existing studies, when integrating interval type-2 fuzzy sets (IT2FSs), choquet integral (CI), and regret theory to solve the problem of sustainable supplier selection (SSS), still have three main limitations. First, at the information expression level, the traditional IT2FS is unable to fully capture the multiple hesitations and their probability distributions commonly found in expert evaluations, resulting in the loss of some uncertain information. Second, at the attribute interaction modeling level, the existing aggregation methods based on the CI usually assume that all attributes are correlated, but they fail to depict the heterogeneity structure of attributes that are partitioned in a “within-group correlated, between-group independent” manner in reality. Third, at the behavioral psychological modeling level, classical regret theory only compares with the optimal solution when calculating the regret–happiness value, ignoring the two-way reference to the worst solution and thus failing to fully reflect the dual motivations of decision-makers to “avoid regret” and “seek happiness”.

Based on the above analysis, the main research motivation of this paper can be summarized as the following three points:

(1) The current SSS process faces the challenge of information ambiguity and uncertainty, while existing fuzzy set structures such as T1FS, IT2FS and IT2HFS still suffer from information loss and insufficient expressive power when dealing with complex information. In contrast, proportional interval type-2 hesitation fuzzy sets (PIT2HFSs) have stronger expressive power and can portray the multidimensional interactions of expert hesitation, trustworthiness and uncertainty in more complex cognitive environments, so it is of great theoretical value and practical significance to introduce PIT2HFSs into SSS problems.

(2) Existing CI-based methods suffer from the problems of failing to consider inter-attribute correlation and ignoring the nonlinear effects of different attribute groups on overall utility. Therefore, it is necessary to improve the existing CI methods and construct aggregation operators that can reflect partition correlation and heterogeneous interaction characteristics.

(3) The existing method only combines regret theory with the best alternative for comparison but ignores the worst alternative when calculating the regret–joy value, so it is necessary to adopt the improved regret theory to solve the SSS problem.

The structure of this paper is organized as follows. In Section 2, we briefly review several preliminary definitions and concepts of PIT2HFSs, CI, generalized Shapley value and regret theory. In Section 3, we propose a proportional interval type-2 hesitant fuzzy partitioned bidirectional Choquet integral (PIT2HFPBCI) operator and a generalized Shapley proportional interval type-2 hesitant fuzzy partitioned bidirectional Choquet integral (GS-PIT2HFPBCI) operator and study the properties and special cases of these novel operators. In Section 4, we define the projection measure and normalized bidirectional projection measure of PIT2HFSs. In Section 5, we proposed a novel PIT2HF-MAGDM method with modified regret theory. In Section 6, an example of SSS is presented to demonstrate the application of the proposed method. Also, some sensitivity analyses and comparison analyses are conducted. In Section 7, we conclude the study and elaborate on future studies.

2. Preliminaries

In this section, some basic concepts of PIT2HFSs, CI, the generalized Shapley value, and the regret theory method that will be used in subsequent sections are introduced.

2.1. PIT2HFSs

Definition 1

[43]. Let $\mathrm{X}$ be a universal discourse; the trapezoidal IT2FN $\mathrm{A}$ defined on $\mathrm{X}$ can be represented as follows:

$$\mathrm{A}=\left({\mathrm{A}}^{\mathrm{u}},{\mathrm{A}}^{\mathrm{l}}\right)=\left(\left({\mathrm{a}}_1^{\mathrm{u}},{\mathrm{a}}_2^{\mathrm{u}},{\mathrm{a}}_3^{\mathrm{u}},{\mathrm{a}}_4^{\mathrm{u}};\mathrm{h}\left({\mathrm{A}}^{\mathrm{u}}\right)\right),\left({\mathrm{a}}_1^{\mathrm{l}},{\mathrm{a}}_2^{\mathrm{l}},{\mathrm{a}}_3^{\mathrm{l}},{\mathrm{a}}_4^{\mathrm{l}};\mathrm{h}\left({\mathrm{A}}^{\mathrm{l}}\right)\right)\right)$$

where ${\mathrm{A}}^{\mathrm{u}}$ and ${\mathrm{A}}^{\mathrm{l}}$ are both trapezoidal fuzzy numbers, ${\mathrm{a}}_{\mathrm{j}}^{\nabla }\left(\nabla =\mathrm{u},\mathrm{l};\mathrm{j}=1,2,3,4\right)$ are reference points of trapezoidal IT2FN $\mathrm{A}$, subject to $0\le {\mathrm{a}}_1^{\mathrm{u}}\le {\mathrm{a}}_2^{\mathrm{u}}\le {\mathrm{a}}_3^{\mathrm{u}}\le {\mathrm{a}}_4^{\mathrm{u}}\le 1$ and $0\le {\mathrm{a}}_1^{\mathrm{l}}\le {\mathrm{a}}_2^{\mathrm{l}}\le {\mathrm{a}}_3^{\mathrm{l}}\le {\mathrm{a}}_4^{\mathrm{l}}\le 1$. $\mathrm{h}\left({\mathrm{A}}^{\mathrm{u}}\right)$ and $\mathrm{h}\left({\mathrm{A}}^{\mathrm{l}}\right)$ denote the membership functions of the elements ${\mathrm{a}}_{\mathrm{i}+1}^{\mathrm{u}}$ and ${\mathrm{a}}_{\mathrm{i}+1}^{\mathrm{l}}$, respectively, and $\mathrm{h}\left({\mathrm{A}}^{\mathrm{u}}\right),\mathrm{h}\left({\mathrm{A}}^{\mathrm{l}}\right)\in \left[0,1\right]$.

Definition 2

[43]. Let $\mathrm{A}$ be a trapezoidal IT2FN. The ranking value of $\mathrm{A}$ can be defined as follows:

$$\mathrm{R}\left(\mathrm{A}\right)=\left({\displaystyle \frac{{\mathrm{a}}_1^{\mathrm{u}}+{\mathrm{a}}_4^{\mathrm{u}}}{2}}+{\displaystyle \frac{\mathrm{h}\left({\mathrm{A}}^{\mathrm{u}}\right)+\mathrm{h}\left({\mathrm{A}}^{\mathrm{l}}\right)}{2}}\right)\times {\displaystyle \frac{{\displaystyle \sum _{\nabla =\mathrm{l}}^{\mathrm{u}}{\displaystyle \sum _{\mathrm{j}=1}^4{\mathrm{a}}_{\mathrm{j}}^{\nabla }}}}{8}}$$

(1)

Definition 3

[45]. Let $\mathrm{X}$ be a universal discourse; the PIT2HFS defined on $\mathrm{X}$ can be defined as follows:

$$\mathrm{H}=\left\{〈\mathrm{x},⌈{\mathrm{h}}_{\mathrm{E}}\left(\mathrm{x}\right),{\mathrm{p}}_{\mathrm{H}}\left(\mathrm{x}\right)⌉〉|\mathrm{x}\in \mathrm{X}\right\}$$

where ${\mathrm{h}}_{\mathrm{E}}\left(\mathrm{x}\right)$ is a set of some IT2FNs, denoting the possible membership degrees of the element $\mathrm{x}\in \mathrm{X}$ to the set, and ${\mathrm{p}}_{\mathrm{H}}\left(\mathrm{x}\right)$ is a set of proportions associated with ${\mathrm{h}}_{\mathrm{E}}\left(\mathrm{x}\right)$. For convenience, we call

$$⌈{\mathrm{h}}_{\mathrm{E}}\left(\mathrm{x}\right),{\mathrm{p}}_{\mathrm{H}}\left(\mathrm{x}\right)⌉=\mathrm{h}\left(\mathrm{p}\right)=\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}⌉|{\mathrm{A}}_{\mathrm{i}}\in {\mathrm{h}}_{\mathrm{E}}\left(\mathrm{x}\right),{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}\in \left[0,1\right],{\displaystyle \sum _{{\mathrm{A}}_{\mathrm{i}}\in {\mathrm{h}}_{\mathrm{E}}\left(\mathrm{x}\right)}{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}}=1\right\}$$

a PIT2HFN, where ${\mathrm{A}}_{\mathrm{i}}=\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}},{\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}}\right)=\left(\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{u}},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{u}},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{u}},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{u}};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}}\right)\right),\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{l}},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{l}},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{l}},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{l}};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}}\right)\right)\right)$ is a trapezoidal IT2FN.

Definition 4

[45]. Let $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{k}\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}⌉\right\}\left(\mathrm{k}=1,2\right)}$ be two PIT2HFNs. Given a T-norm $\mathrm{T}\left(\mathrm{x},\mathrm{y}\right)$ with additive generator function:

(1)

$\mathrm{h}{\left(\mathrm{p}\right)}_1\otimes \mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{1i}}\otimes {\mathrm{A}}_{2\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{ki}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{ki}}}}}\right)⌉\right\}}$, where

$${\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}}=\left(\begin{array}{l}\left({\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}\right)}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\right),\\ \left({\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^2\mathrm{g}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}\right)}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\right)\end{array}\right)$$

(2)

(2)

${\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)}^{\eta }={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta },{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$, where

$${\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta }=\left(\begin{array}{l}\left({\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)\right),{\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)\right),{\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)\right),{\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)\right);\mathrm{h}\left({\mathrm{A}}_1^{\mathrm{u}}\right)\right),\\ \left({\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)\right),{\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)\right),{\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)\right),{\mathrm{g}}^{-1}\left(\eta \mathrm{g}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)\right);\mathrm{h}\left({\mathrm{A}}_1^{\mathrm{l}}\right)\right)\end{array}\right)$$

(3)

Note that Equations (2) and (3) are the general formulations with additive generator $\mathrm{g}\left(\mathrm{t}\right)$. In what follows, we provide some special cases when using different additive generators in

Table 2. T-norms and corresponding additive generators.

Name	T-Norms	Additive Generators
Algebraic	${\mathrm{T}}_{\mathrm{A}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}\mathrm{y}$	$\mathrm{g}\left(\mathrm{t}\right)=-\log \left(\mathrm{t}\right)$
Einstein	${\mathrm{T}}_{\mathrm{E}}\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \frac{\mathrm{x}\mathrm{y}}{1+\left(1-\mathrm{x}\right)\left(1-\mathrm{y}\right)}}$	$\mathrm{g}\left(\mathrm{t}\right)=-\log \left({\displaystyle \frac{2-\mathrm{t}}{\mathrm{t}}}\right)$
Hamacher	${\mathrm{T}}_{\mathrm{H},\gamma }\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \frac{\mathrm{x}\mathrm{y}}{\gamma +\left(1-\gamma \right)\left(\mathrm{x}+\mathrm{y}-\mathrm{x}\mathrm{y}\right)}}$	$\mathrm{g}\left(\mathrm{t}\right)=-\log \left({\displaystyle \frac{\gamma +\left(1-\gamma \right)\mathrm{t}}{\mathrm{t}}}\right),\gamma >0$
Frank	${\mathrm{T}}_{\mathrm{F},\tau }\left(\mathrm{x},\mathrm{y}\right)={\log }_{\tau }\left(1+{\displaystyle \frac{\left({\tau }^{\mathrm{x}}-1\right)\left({\tau }^{\mathrm{y}}-1\right)}{\tau -1}}\right)$	$\mathrm{g}\left(\mathrm{t}\right)=\left\{\begin{array}{l}-\log \left(\mathrm{t}\right),\tau =1\\ -\log \left({\displaystyle \frac{\tau -1}{{\tau }^{\mathrm{t}}-1}}\right),\tau \ne 1\end{array}\right.$

.

(1)

If $\mathrm{g}\left(\mathrm{t}\right)=-\log \left(\mathrm{t}\right)$, the operational laws for the PIT2HFNs based on Algebraic T-norm can be defined as follows:

$\mathrm{h}{\left(\mathrm{p}\right)}_1\otimes \mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{1i}}\otimes {\mathrm{A}}_{2\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{ki}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{ki}}}}}\right)⌉\right\}}$, where

$${\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}}=\left(\begin{array}{l}\left({\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\right),\\ \left({\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\right)\end{array}\right)$$

(4)

${\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)}^{\eta }={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta },{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$, where

$${\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta }=\left(\begin{array}{l}\left({\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)}^{\eta },{\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)}^{\eta },{\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)}^{\eta },{\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)}^{\eta };\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{u}}\right)\right),\\ \left({\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)}^{\eta },{\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)}^{\eta },{\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)}^{\eta },{\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)}^{\eta };\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}}\right)\right)\end{array}\right)$$

(5)

(2)

If $\mathrm{g}\left(\mathrm{t}\right)=-\log \left({\displaystyle \frac{2-\mathrm{t}}{\mathrm{t}}}\right)$, the operational laws for the PIT2HFNs based on Einstein T-norm can be defined as follows:

$\mathrm{h}{\left(\mathrm{p}\right)}_1\otimes \mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}}\right)⌉\right\}}$, where

$${\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}}=\left(\begin{array}{l}\left(\begin{array}{l}{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{u}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{u}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{u}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{u}}\right)}},\\ {\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{u}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{u}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{u}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{u}}\right)}};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\end{array}\right),\\ \left(\begin{array}{l}{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{l}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{l}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{l}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{l}}\right)}},\\ {\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{l}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{l}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{l}}}{1+\left(1-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)\left(1-{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{l}}\right)}};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\end{array}\right)\end{array}\right)$$

(6)

${\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)}^{\eta }={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta },{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$, where

$${\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta }=\left(\begin{array}{l}\left(\begin{array}{l}{\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)}^{\eta }}},{\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)}^{\eta }}},\\ {\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)}^{\eta }}},{\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)}^{\eta }}};\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{u}}\right)\end{array}\right),\\ \left(\begin{array}{l}{\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)}^{\eta }}},{\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)}^{\eta }}},\\ {\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)}^{\eta }}},{\displaystyle \frac{2{\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)}^{\eta }}{{\left(2-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)}^{\eta }+{\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)}^{\eta }}};\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}}\right)\end{array}\right)\end{array}\right)$$

(7)

(3)

If $\mathrm{g}\left(\mathrm{t}\right)=-\log \left({\displaystyle \frac{\gamma +\left(1-\gamma \right)\mathrm{t}}{\mathrm{t}}}\right)$, the operational laws for the PIT2HFNs based on Hamacher T-norm can be defined as follows:

$\mathrm{h}{\left(\mathrm{p}\right)}_1\otimes \mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}}\right)⌉\right\}}$, where

$${\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}}=\left(\begin{array}{l}\left(\begin{array}{l}{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{u}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}+{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{u}}-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{u}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{u}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}+{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{u}}-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{u}}\right)}},\\ {\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{u}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}+{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{u}}-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{u}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{u}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}+{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{u}}-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{u}}\right)}}\\ ;\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\end{array}\right),\\ \left(\begin{array}{l}{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{l}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}+{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{l}}-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{l}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{l}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}+{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{l}}-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{l}}\right)}},\\ {\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{l}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}+{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{l}}-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{l}}\right)}},{\displaystyle \frac{{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{l}}}{\gamma +\left(1-\gamma \right)\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}+{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{l}}-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{l}}\right)}}\\ ;\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\end{array}\right)\end{array}\right)$$

(8)

${\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)}^{\eta }={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta },{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$, where

$${\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta }=\left(\begin{array}{l}\left(\begin{array}{l}{\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}\right)}^{\eta }}},{\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}\right)}^{\eta }}},\\ {\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}\right)}^{\eta }}},{\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}\right)}^{\eta }}}\\ ;\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{u}}\right)\end{array}\right),\\ \left(\begin{array}{l}{\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}\right)}^{\eta }}},{\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}\right)}^{\eta }}},\\ {\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}\right)}^{\eta }}},{\displaystyle \frac{\gamma {\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)}^{\eta }}{{\left(1+\left(\gamma -1\right)\left(1-{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)\right)}^{\eta }+\left(\gamma -1\right){\left({\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}\right)}^{\eta }}}\\ ;\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}}\right)\end{array}\right)\end{array}\right)$$

(9)

(4)

If $\mathrm{g}\left(\mathrm{x}\right)=-\log \left({\displaystyle \frac{\tau -1}{{\tau }^{\mathrm{x}}-1}}\right)$, the operational laws for the PIT2HFNs based on Frank T-norm can be defined as follows:

$\mathrm{h}{\left(\mathrm{p}\right)}_1\otimes \mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^2{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}}\right)⌉\right\}}$, where

$${\mathrm{A}}_{1\mathrm{i}}\otimes {\mathrm{A}}_{2\mathrm{i}}=\left(\begin{array}{l}\left(\begin{array}{l}{\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}}-1\right)}}{\tau -1}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}}-1\right)}}{\tau -1}}\right),\\ {\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}}-1\right)}}{\tau -1}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}}-1\right)}}{\tau -1}}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\end{array}\right),\\ \left(\begin{array}{l}{\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}}-1\right)}}{\tau -1}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}}-1\right)}}{\tau -1}}\right),\\ {\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}}-1\right)}}{\tau -1}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\displaystyle \prod _{\mathrm{k}=1}^2\left({\tau }^{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}}-1\right)}}{\tau -1}}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\end{array}\right)\end{array}\right)$$

(10)

${\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)}^{\eta }={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta },{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$, where

$${\left({\mathrm{A}}_{1\mathrm{i}}\right)}^{\eta }=\left(\begin{array}{l}\left(\begin{array}{l}{\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}}-1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}}-1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right),\\ {\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}}-1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}}-1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right);\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}}\right)\end{array}\right)\\ \left(\begin{array}{l}{\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}}-1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}}-1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right),\\ {\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}}-1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right),{\log }_{\tau }\left(1+{\displaystyle \frac{{\left({\tau }^{{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}},-,1\right)}^{\eta }}{{\left(\tau -1\right)}^{\eta -1}}}\right);\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}}\right)\end{array}\right)\end{array}\right)$$

(11)

There is a premise to multiplication operation of two PIT2HFNs, that is, the length of them should be equal. However, in practical application, different PIT2HFNs may have different lengths. Hence, we need to adjust the lengths of the two PIT2HFNs before utilizing multiplication operation. According to the operation laws in Definition 4 and the length adjustment algorithm based on the proportion distribution shown in [43], we give the following examples.

Example 1.

Let $\mathrm{h}{\left(\mathrm{p}\right)}_1=\left\{⌈{\mathrm{A}}_{11},0.3⌉,⌈{\mathrm{A}}_{12},0.3⌉,⌈{\mathrm{A}}_{13},0.4⌉\right\}$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2=\left\{⌈{\mathrm{A}}_{21},0.3⌉,⌈{\mathrm{A}}_{22},0.7⌉\right\}$ be two PIT2HFNs. Then, $⌈{\mathrm{A}}_{22},0.7⌉$ should be divided into two IT2FNs distributed with the same proportion, namely, $⌈{\mathrm{A}}_{22},0.35⌉$ and $⌈{\mathrm{A}}_{22},0.35⌉$ . Hence, the adjusted $\mathrm{h}{\left(\mathrm{p}\right)}_2$ is $\left\{⌈{\mathrm{A}}_{21},0.3⌉,⌈{\mathrm{A}}_{22},0.35⌉,⌈{\mathrm{A}}_{22},0.35⌉\right\}$.

Example 2.

Let $\mathrm{h}{\left(\mathrm{p}\right)}_1=\left\{⌈{\mathrm{A}}_{11},0.3⌉,⌈{\mathrm{A}}_{12},0.3⌉,⌈{\mathrm{A}}_{13},0.4⌉\right\}$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2=\left\{⌈{\mathrm{A}}_{21},0.2⌉,⌈{\mathrm{A}}_{22},0.2⌉,⌈{\mathrm{A}}_{23},0.6⌉\right\}$ be two PIT2HFNs. Then, according to Definition 4, we have

$$\begin{array}{ll}\mathrm{h}{\left(\mathrm{p}\right)}_1\otimes \mathrm{h}{\left(\mathrm{p}\right)}_2& =\left\{⌈{\mathrm{A}}_{11}\otimes {\mathrm{A}}_{21},{\displaystyle \frac{6}{36}}⌉,⌈{\mathrm{A}}_{12}\otimes {\mathrm{A}}_{22},{\displaystyle \frac{6}{36}}⌉,⌈{\mathrm{A}}_{13}\otimes {\mathrm{A}}_{23},{\displaystyle \frac{24}{36}}⌉\right\}\\ & =\left\{⌈{\mathrm{A}}_{11}\otimes {\mathrm{A}}_{21},{\displaystyle \frac{1}{6}}⌉,⌈{\mathrm{A}}_{12}\otimes {\mathrm{A}}_{22},{\displaystyle \frac{1}{6}}⌉,⌈{\mathrm{A}}_{13}\otimes {\mathrm{A}}_{23},{\displaystyle \frac{2}{3}}⌉\right\}.\end{array}$$

In the rest of this paper, we assume that the lengths of all PIT2HFNs are adjusted based on the length adjustment algorithm in the literature [46].

Definition 5

[45]. Let $\mathrm{h}\left(\mathrm{p}\right)={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}⌉\right\}}$ be a PIT2HFN. The score function of $\mathrm{h}\left(\mathrm{p}\right)$ can be defined as follows:

$$\mathrm{S}\left(\mathrm{h}\left(\mathrm{p}\right)\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\mathrm{R}\left({\mathrm{A}}_{\mathrm{i}}\right)\times {\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}}$$

(12)

2.2. Choquet Integral and Generalized Shapley Value

The fuzzy measure plays an important role in modeling the interrelationships among input arguments in practical decision situations.

Definition 6

[46]. Given a universe of discourse $\mathrm{Y}=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,\dots ,{\mathrm{y}}_{\mathrm{n}}\right\}$ let $\mathrm{P}\left(\mathrm{Y}\right)$ be the power set of $\mathrm{Y}$. A fuzzy measure g on Y is a set function $\rho :\mathrm{P}\left(\mathrm{Y}\right)\to \left[0,1\right]$ that satisfies the following conditions:

• $\rho \left(\varnothing \right)=0,\rho \left(\mathrm{Y}\right)=1$;
• If $\mathrm{B},\mathrm{C}\in \mathrm{Y}$ and $\mathrm{B}\subseteq \mathrm{C}$, then $\rho \left(\mathrm{B}\right)\le \rho \left(\mathrm{C}\right)$;
• If $\mathrm{B},\mathrm{C}\in \mathrm{Y}$, then $\rho \left(\mathrm{B}\cup \mathrm{C}\right)=\rho \left(\mathrm{B}\right)+\rho \left(\mathrm{C}\right)+\lambda \rho \left(\mathrm{B}\right)\rho \left(\mathrm{C}\right)$, $\lambda \in \left[-1,\infty \right)$ and $\mathrm{B}\cap \mathrm{C}=\varnothing$.

Moreover, the $\lambda$-fuzzy measure $\rho$ is described as:

$$\rho \left(\mathrm{Y}\right)={\displaystyle \frac{1}{\lambda }}\left({\displaystyle \prod _{\mathrm{j}=1}^{\mathrm{n}}\left(1+\lambda \rho \left({\mathrm{y}}_{\mathrm{j}}\right)\right)}-1\right)$$

(13)

Let $\rho \left(\mathrm{Y}\right)=1$. Then, Equation (13) can be rewritten as:

$$\lambda \rho \left(\mathrm{Y}\right)={\displaystyle \prod _{\mathrm{j}=1}^{\mathrm{n}}\left(1+\lambda \rho \left({\mathrm{y}}_{\mathrm{j}}\right)\right)}-1$$

(14)

Example 3.

Given the fuzzy measure ${\rho }_1$ based on a universe of discourse ${\mathrm{Y}}_1=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3\right\}$, suppose ${\rho }_1\left({\mathrm{y}}_1\right)=0.28$, ${\rho }_1\left({\mathrm{y}}_2\right)=0.31$, and ${\rho }_1\left({\mathrm{y}}_3\right)=0.29$.

According to Equation (14), we have ${\lambda }_1=\left(1+{\lambda }_1\rho \left({\mathrm{y}}_1\right)\right)\left(1+{\lambda }_1\rho \left({\mathrm{y}}_2\right)\right)\left(1+{\lambda }_1\rho \left({\mathrm{y}}_3\right)\right)-1$. Solving the equation yields ${\lambda }_1=0.45$.

Further, we have $\lambda \rho \left(\left\{{\mathrm{y}}_1,{\mathrm{y}}_2\right\}\right)=\left(1+\lambda \rho \left({\mathrm{y}}_1\right)\right)\left(1+\lambda \rho \left({\mathrm{y}}_2\right)\right)-1$. Solving the equation yields $\rho \left(\left\{{\mathrm{y}}_1,{\mathrm{y}}_2\right\}\right)=0.6291$.

Similarly, we can get $\rho \left(\left\{{\mathrm{y}}_1,{\mathrm{y}}_3\right\}\right)=0.6065$, $\rho \left(\left\{{\mathrm{y}}_2,{\mathrm{y}}_3\right\}\right)=0.6405$ , and $\rho \left(\left\{{\mathrm{y}}_{1,}{\mathrm{y}}_2,{\mathrm{y}}_3\right\}\right)=1$.

Example 4.

Given the fuzzy measure ${\rho }_2$ based on a universe of discourse ${\mathrm{Y}}_2=\left\{{\mathrm{y}}_4,{\mathrm{y}}_5,{\mathrm{y}}_6\right\}$ , suppose ${\rho }_2\left({\mathrm{y}}_4\right)=0.34$ , ${\rho }_2\left({\mathrm{y}}_5\right)=0.25$ , and ${\rho }_2\left({\mathrm{y}}_6\right)=0.33$.

According to Equation (14), we have ${\lambda }_2=\left(1+{\lambda }_2\rho \left({\mathrm{y}}_4\right)\right)\left(1+{\lambda }_2\rho \left({\mathrm{y}}_5\right)\right)\left(1+{\lambda }_2\rho \left({\mathrm{y}}_6\right)\right)-1$. Solving the equation yields ${\lambda }_2=0.29$.

Further, we have ${\lambda }_2\rho \left(\left\{{\mathrm{y}}_4,{\mathrm{y}}_5\right\}\right)=\left(1+{\lambda }_2\rho \left({\mathrm{y}}_4\right)\right)\left(1+{\lambda }_2\rho \left({\mathrm{y}}_5\right)\right)-1$. Solving the equation yields $\rho \left(\left\{{\mathrm{y}}_4,{\mathrm{y}}_5\right\}\right)=0.6291$.

Similarly, we can get $\rho \left(\left\{{\mathrm{y}}_4,{\mathrm{y}}_6\right\}\right)=0.7025$, $\rho \left(\left\{{\mathrm{y}}_5,{\mathrm{y}}_6\right\}\right)=0.6039$, and $\rho \left(\left\{{\mathrm{y}}_{4,}{\mathrm{y}}_5,{\mathrm{y}}_6\right\}\right)=1$.

In the process of measuring the importance of elements, $\rho \left({\mathrm{y}}_{\mathrm{j}}\right)$ cannot completely reflect the function ${\mathrm{y}}_{\mathrm{j}}\in \mathrm{P}\left(\mathrm{Y}\right)$. The importance of each measure is not only obtained by itself but also affected by others. Shapley values can reflect the overall contribution of each element based on others, which is one of the important tools in processing the interactions among elements.

Definition 7

[47]. Given the fuzzy measure $\rho$ based on a universe of discourse $\mathrm{Y}=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,\dots ,{\mathrm{y}}_{\mathrm{n}}\right\}$, the generalized Shapely value for $\forall \mathrm{S}\in \mathrm{Y}$ is

$${\phi }_{\mathrm{i}}\left(\rho ,\mathrm{S}\right)={\displaystyle \sum _{\mathrm{T}\in \mathrm{Y}\backslash \mathrm{S}}\frac{\left(\mathrm{n}-\left|\mathrm{T}\right|-\left|\mathrm{S}\right|\right)!\left|\mathrm{T}\right|!}{\left(\mathrm{n}-\left|\mathrm{S}\right|+1\right)!}\left(\mathrm{g}\left(\mathrm{S}\cup \mathrm{T}\right)-\mathrm{g}\left(\mathrm{T}\right)\right)}$$

(15)

where |T| and |S| are the number of elements in the subsets T and S.

Example 5.

Following Examples 3 and 4, we can compute ${\phi }_{\left\{{\mathrm{y}}_1\right\}}\left(\rho ,\mathrm{S}\right)$ according to Equation (15); the calculation process is as follows:

$$\begin{array}{l}{\phi }_{\left\{{\mathrm{y}}_1\right\}}\left(\rho ,\mathrm{S}\right)={\displaystyle \frac{\left(3-1-2\right)!2!}{\left(3-1+1\right)!}}\left(\rho \left(\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3\right\}\right)-\rho \left(\left\{{\mathrm{y}}_2,{\mathrm{y}}_3\right\}\right)\right)+{\displaystyle \frac{\left(3-1-1\right)!1!}{\left(3-1+1\right)!}}\left(\rho \left(\left\{{\mathrm{y}}_1,{\mathrm{y}}_2\right\}\right)-\rho \left(\left\{{\mathrm{y}}_2\right\}\right)\right)\\ +{\displaystyle \frac{\left(3-1-1\right)!1!}{\left(3-1+1\right)!}}\left(\rho \left(\left\{{\mathrm{y}}_1,{\mathrm{y}}_3\right\}\right)-\rho \left(\left\{{\mathrm{y}}_3\right\}\right)\right)+{\displaystyle \frac{\left(3-1-0\right)!0!}{\left(3-1+1\right)!}}\left(\rho \left(\left\{{\mathrm{y}}_1\right\}\right)-\rho \left(\left\{\varnothing \right\}\right)\right)=0.3191\end{array}$$

Similarly, we can get ${\phi }_{\left\{{\mathrm{y}}_2\right\}}\left(\rho ,\mathrm{S}\right)=0.3511$, ${\phi }_{\left\{{\mathrm{y}}_3\right\}}\left(\rho ,\mathrm{S}\right)=0.3298$, ${\phi }_{\left\{{\mathrm{y}}_1,{\mathrm{y}}_2\right\}}\left(\rho ,\mathrm{S}\right)=0.6695$, ${\phi }_{\left\{{\mathrm{y}}_1,{\mathrm{y}}_3\right\}}\left(\rho ,\mathrm{S}\right)=0.6483$, ${\phi }_{\left\{{\mathrm{y}}_2,{\mathrm{y}}_3\right\}}\left(\rho ,\mathrm{S}\right)=0.6802$, ${\phi }_{\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3\right\}}\left(\rho ,\mathrm{S}\right)=1$, ${\phi }_{\left\{{\mathrm{y}}_4\right\}}\left(\rho ,\mathrm{S}\right)=0.3682$, ${\phi }_{\left\{{\mathrm{y}}_5\right\}}\left(\rho ,\mathrm{S}\right)=0.2739$, ${\phi }_{\left\{{\mathrm{y}}_6\right\}}\left(\rho ,\mathrm{S}\right)=0.3579$, ${\phi }_{\left\{{\mathrm{y}}_4,{\mathrm{y}}_5\right\}}\left(\rho ,\mathrm{S}\right)=0.6423$, ${\phi }_{\left\{{\mathrm{y}}_4,{\mathrm{y}}_6\right\}}\left(\rho ,\mathrm{S}\right)=0.7263$, ${\phi }_{\left\{{\mathrm{y}}_5,{\mathrm{y}}_6\right\}}\left(\rho ,\mathrm{S}\right)=0.6320$, and ${\phi }_{\left\{{\mathrm{y}}_4,{\mathrm{y}}_5,{\mathrm{y}}_6\right\}}\left(\rho ,\mathrm{S}\right)=1$.

Definition 8

[34]. Let the given $\mathrm{Y}=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,\dots ,{\mathrm{y}}_{\mathrm{n}}\right\}$ be a universe of discourse, let $\mathrm{h}$ be a positive real-valued function on $\mathrm{Y}$ , and let $\rho$ be a fuzzy measure on $\mathrm{Y}$. The CI is represented as follows:

$$\mathrm{C}{\mathrm{I}}_{\rho }\left(\mathrm{h}\left({\mathrm{y}}_1\right),\mathrm{h}\left({\mathrm{y}}_2\right),\dots ,\mathrm{h}\left({\mathrm{y}}_{\mathrm{n}}\right)\right)={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\left(\rho \left({\mathrm{C}}_{\tau \left(\mathrm{k}\right)}\right)-\rho \left({\mathrm{C}}_{\tau \left(\mathrm{k}+1\right)}\right)\right)\mathrm{h}\left({\mathrm{y}}_{\left(\mathrm{k}\right)}\right)}$$

(16)

where $\left\{\left(1\right),\left(2\right),\dots ,\left(\mathrm{n}\right)\right\}$ is a permutation of $\left\{1,2,\dots ,\mathrm{n}\right\}$, which satisfies $\mathrm{h}\left({\mathrm{y}}_{\left(1\right)}\right)\le \mathrm{h}\left({\mathrm{y}}_{\left(2\right)}\right)\le \dots \le \mathrm{h}\left({\mathrm{y}}_{\left(\mathrm{n}\right)}\right)$ and ${\mathrm{C}}_{\tau \left(\mathrm{k}\right)}=\left\{{\mathrm{y}}_{\left(\mathrm{k}\right)},{\mathrm{y}}_{\left(\mathrm{k}+1\right)},\dots ,{\mathrm{y}}_{\left(\mathrm{n}\right)}\right\}$ with ${\mathrm{C}}_{\tau \left(\mathrm{k}+1\right)}=\varnothing$.

Example 6.

Following Example 3, let $\mathrm{Y}=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3\right\}$ be a universe of discourse and $\mathrm{h}$ be a positive real-valued function on $\mathrm{Y}$ where $\mathrm{h}\left({\mathrm{y}}_1\right)=0.5$, $\mathrm{h}\left({\mathrm{y}}_2\right)=0.6$, $\mathrm{h}\left({\mathrm{y}}_3\right)=0.7$, $\mathrm{h}\left({\mathrm{y}}_4\right)=0.8$, $\mathrm{h}\left({\mathrm{y}}_5\right)=0.9$, and $\mathrm{h}\left({\mathrm{y}}_6\right)=1.0$. According to Equation (16), we can obtain

$$\begin{array}{l}\mathrm{C}{\mathrm{I}}_{\rho }\left(\mathrm{h}\left({\mathrm{y}}_1\right),\mathrm{h}\left({\mathrm{y}}_2\right),\mathrm{h}\left({\mathrm{y}}_3\right)\right)=\left(\rho \left(\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}\right)-\rho \left(\left\{{\mathrm{C}}_1,{\mathrm{C}}_2\right\}\right)\right)\mathrm{h}\left({\mathrm{y}}_3\right)\\ +\left(\rho \left(\left\{{\mathrm{C}}_2,{\mathrm{C}}_3\right\}\right)-\rho \left(\left\{{\mathrm{C}}_3\right\}\right)\right)\mathrm{h}\left({\mathrm{y}}_3\right)+\left(\rho \left(\left\{{\mathrm{C}}_3\right\}\right)-\rho \left(\left\{\varnothing \right\}\right)\right)\mathrm{h}\left({\mathrm{y}}_1\right)=0.6081\end{array}$$

$$\begin{array}{l}\mathrm{C}{\mathrm{I}}_{\rho }\left(\mathrm{h}\left({\mathrm{y}}_4\right),\mathrm{h}\left({\mathrm{y}}_5\right),\mathrm{h}\left({\mathrm{y}}_6\right)\right)=\left(\rho \left(\left\{{\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right\}\right)-\rho \left(\left\{{\mathrm{C}}_4,{\mathrm{C}}_5\right\}\right)\right)\mathrm{h}\left({\mathrm{y}}_6\right)\\ +\left(\rho \left(\left\{{\mathrm{C}}_5,{\mathrm{C}}_6\right\}\right)-\rho \left(\left\{{\mathrm{C}}_6\right\}\right)\right)\mathrm{h}\left({\mathrm{y}}_5\right)+\left(\rho \left(\left\{{\mathrm{C}}_6\right\}\right)-\rho \left(\left\{\varnothing \right\}\right)\right)\mathrm{h}\left({\mathrm{y}}_4\right)=0.9041\end{array}$$

Meng et al. [48] further introduced the bidirectional Choquet integral (BDCI), which is a convex combination of the CI and the reverse Choquet integral (RCI).

Definition 9

[48]. Let the given $\mathrm{Y}=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,\dots ,{\mathrm{y}}_{\mathrm{n}}\right\}$ be a universe of discourse, let $\mathrm{h}$ be a positive real-valued function on $\mathrm{Y}$, and let $\rho$ be a fuzzy measure on $\mathrm{Y}$. The BDCI is presented as follows:

$$\begin{array}{l}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\rho }\left(\mathrm{h}\left({\mathrm{y}}_1\right),\mathrm{h}\left({\mathrm{y}}_2\right),\dots ,\mathrm{h}\left({\mathrm{y}}_{\mathrm{n}}\right)\right)\\ ={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\left(\sigma \left(\rho \left({\mathrm{C}}_{\tau \left(\mathrm{k}\right)}\right)-\rho \left({\mathrm{C}}_{\tau \left(\mathrm{k}+1\right)}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\mathrm{R}{\mathrm{C}}_{\tau \left(\mathrm{k}\right)}\right)-\rho \left(\mathrm{R}{\mathrm{C}}_{\tau \left(\mathrm{k}-1\right)}\right)\right)\right)\mathrm{h}\left({\mathrm{y}}_{\left(\mathrm{k}\right)}\right)}\end{array}$$

(17)

where $\left\{\left(1\right),\left(2\right),\dots ,\left(\mathrm{n}\right)\right\}$ is a permutation of $\left\{1,2,\dots ,\mathrm{n}\right\}$, which satisfies $\mathrm{h}\left({\mathrm{y}}_{\left(1\right)}\right)\le \mathrm{h}\left({\mathrm{y}}_{\left(2\right)}\right)\le \dots \le \mathrm{h}\left({\mathrm{y}}_{\left(\mathrm{n}\right)}\right)$, ${\mathrm{C}}_{\tau \left(\mathrm{k}\right)}=\left\{{\mathrm{y}}_{\left(\mathrm{k}\right)},{\mathrm{y}}_{\left(\mathrm{k}+1\right)},\dots ,{\mathrm{y}}_{\left(\mathrm{n}\right)}\right\}$ with ${\mathrm{C}}_{\tau \left(\mathrm{n}+1\right)}=\varnothing$, and $\mathrm{R}{\mathrm{C}}_{\tau \left(\mathrm{k}\right)}=\left\{{\mathrm{y}}_{\left(1\right)},{\mathrm{y}}_{\left(2\right)},\dots ,{\mathrm{y}}_{\left(\mathrm{k}\right)}\right\}$ with $\mathrm{R}{\mathrm{C}}_{\tau \left(0\right)}=\varnothing$.

Example 7.

Following Example 6, according to Equation (17), we have

$$\begin{array}{l}{\mathrm{BCI}}_{\rho }\left(\mathrm{h}\left({\mathrm{y}}_1\right),\mathrm{h}\left({\mathrm{y}}_2\right),\dots ,\mathrm{h}\left({\mathrm{y}}_{\mathrm{n}}\right)\right)\\ =\left(\sigma \left(\rho \left(\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}\right)-\rho \left(\left\{{\mathrm{C}}_2,{\mathrm{C}}_3\right\}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\left\{{\mathrm{C}}_1\right\}\right)-\rho \left(\left\{\varnothing \right\}\right)\right)\right)\mathrm{h}\left({\mathrm{y}}_3\right)\\ +\left(\sigma \left(\rho \left(\left\{{\mathrm{C}}_2,{\mathrm{C}}_3\right\}\right)-\rho \left(\left\{{\mathrm{C}}_3\right\}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\left\{{\mathrm{C}}_1,{\mathrm{C}}_2\right\}\right)-\rho \left(\left\{{\mathrm{C}}_1\right\}\right)\right)\right)\mathrm{h}\left({\mathrm{y}}_2\right)\\ +\left(\sigma \left(\rho \left(\left\{{\mathrm{C}}_3\right\}\right)-\rho \left(\left\{\varnothing \right\}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}\right)-\rho \left(\left\{{\mathrm{C}}_1,{\mathrm{C}}_2\right\}\right)\right)\right)\mathrm{h}\left({\mathrm{y}}_1\right)\end{array}$$

$$\begin{array}{l}{\mathrm{BCI}}_{\rho }\left(\mathrm{h}\left({\mathrm{y}}_4\right),\mathrm{h}\left({\mathrm{y}}_5\right),\mathrm{h}\left({\mathrm{y}}_6\right)\right)\\ =\left(\sigma \left(\rho \left(\left\{{\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right\}\right)-\rho \left(\left\{{\mathrm{C}}_5,{\mathrm{C}}_6\right\}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\left\{{\mathrm{C}}_4\right\}\right)-\rho \left(\left\{\varnothing \right\}\right)\right)\right)\mathrm{h}\left({\mathrm{y}}_6\right)\\ +\left(\sigma \left(\rho \left(\left\{{\mathrm{C}}_4,{\mathrm{C}}_5\right\}\right)-\rho \left(\left\{{\mathrm{C}}_6\right\}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\left\{{\mathrm{C}}_4,{\mathrm{C}}_5\right\}\right)-\rho \left(\left\{{\mathrm{C}}_4\right\}\right)\right)\right)\mathrm{h}\left({\mathrm{y}}_5\right)\\ +\left(\sigma \left(\rho \left(\left\{{\mathrm{C}}_6\right\}\right)-\rho \left(\left\{\varnothing \right\}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\left\{{\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right\}\right)-\rho \left(\left\{{\mathrm{C}}_4,{\mathrm{C}}_5\right\}\right)\right)\right)\mathrm{h}\left({\mathrm{y}}_4\right)\end{array}$$

2.3. Regret Theory

Regret theory is a type of decision-making theory first proposed by Bell [49] in 1982. According to this theory, decision-makers evaluate outcomes not only for their absolute utility, but by comparing them to the outcomes of other choices. Regret arises when people realize that another choice could have led to a more favorable outcome. On the other hand, people may also feel satisfaction or pleasure.

Definition 10

[38]. Let $\mathrm{s}$ be the attribute value; then, the utility function $\mathrm{u}\left(\mathrm{s}\right)$ can be defined as follows:

$$\mathrm{u}\left(\mathrm{s}\right)={\mathrm{s}}^{\alpha },0<\alpha <1$$

(18)

where $\alpha$ denotes the risk aversion coefficient of decision-maker; the smaller the value of $\alpha$, the lager the risk aversion value is. On the contrary, a lager value of $\alpha$ indicates that the decision-maker prefers risk. The specific function graph with different $\alpha$ values is shown in Figure 1.

Definition 11

[38]. Let ${\mathrm{s}}_1$ and ${\mathrm{s}}_2$ denote the evaluation value of objects ${\mathrm{o}}_1$ and ${\mathrm{o}}_2$, respectively. The regret–rejoice function $\mathrm{R}\left(\Delta \mathrm{u}\right)$ can be defined as follows:

$$\mathrm{R}\left(\Delta \mathrm{u}\right)=1-\exp \left\{-\beta \left(\Delta \mathrm{u}\right)\right\},\Delta \mathrm{u}=\mathrm{u}\left({\mathrm{s}}_1\right)-\mathrm{u}\left({\mathrm{s}}_2\right)$$

(19)

where $\beta$ denotes the regret aversion coefficient of decision-maker and a larger $\beta$ represents a lager risk aversion value. The difference between utility values of ${\mathrm{o}}_1$ and ${\mathrm{o}}_2$ is denoted by $\Delta \mathrm{u}$. The specific function graph with different $\beta$ is shown in Figure 2.

Regret theory was initially used to tackle binomial alternative problems, but MAGDM problems in reality usually consist of multiple alternatives. Hence, some scholars [38,42,44,50] modified and extended the regret theory to apply it to MAGDM problems. Liang [50] further defined the regret value and rejoice value of alternative $o_{\mathrm{i}}$ as follows:

• Regret value:

  $$\mathrm{R}\mathrm{E}{\mathrm{G}}_{\mathrm{i}}=1-\exp \left\{-\alpha \left(\mathrm{u}\left({\mathrm{o}}_{\mathrm{i}}\right)-\underset{\mathrm{i}}{\max }\left(\mathrm{u}\left({\mathrm{o}}_{\mathrm{i}}\right)\right)\right)\right\}$$

  (20)
• Rejoice value:

  $$\mathrm{R}\mathrm{E}{\mathrm{J}}_{\mathrm{i}}=1-\exp \left\{-\alpha \left(\mathrm{u}\left({\mathrm{o}}_{\mathrm{i}}\right)-\underset{\mathrm{i}}{\min }\left(\mathrm{u}\left({\mathrm{o}}_{\mathrm{i}}\right)\right)\right)\right\}$$

  (21)

Then, the regret–rejoice value of alternative $o_{\mathrm{i}}$ is presented as follows:

$$ℝ\left({\mathrm{o}}_{\mathrm{i}}\right)=\mathrm{R}\mathrm{E}{\mathrm{G}}_{\mathrm{i}}+\mathrm{R}\mathrm{E}{\mathrm{J}}_{\mathrm{i}}$$

(22)

3. PIT2HFBCI Operator Based on Archimedean T-Norms

In this section, we propose the PIT2HFPBCI operator and the GS-PIT2HFPBCI operator. Furthermore, we study the properties and special cases of these novel operators.

3.1. The PIT2HFPBCI Operator

Definition 12.

Let $\mathrm{C}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$ be a set of $\mathrm{n}$ attributes that can be divided into different partitions ${\mathrm{P}}_1,{\mathrm{P}}_2,\dots ,{\mathrm{P}}_{\mathrm{d}}$, and $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be $\mathrm{n}$ PIT2HFNs. The PIT2HFPBCI operator is defined as follows: 

$$\begin{array}{l}\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\\ ={\left(\underset{\mathrm{r}=1}{\overset{\mathrm{d}}{\otimes }}\left(\left(\underset{{\mathrm{k}}_{\mathrm{r}}=1}{\overset{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}\right)}^{\left(\sigma \left(\rho \left({\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}\right)-\rho \left({\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}+1\right)}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}\right)-\rho \left(\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}-1\right)}\right)\right)\right)}\right)\right)\right)}^{{\displaystyle \frac{1}{\mathrm{d}}}}\end{array}$$

(23)

where ${\mathrm{C}}_{\tau (\mathrm{k})}=\left\{{\mathrm{C}}_{\mathrm{k}},{\mathrm{C}}_{\mathrm{k}+1},\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$ with ${\mathrm{C}}_{\tau \left(\mathrm{n}+1\right)}=\varnothing$, $\mathrm{R}{\mathrm{C}}_{\tau \left(\mathrm{k}\right)}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{k}}\right\}$ with $\mathrm{R}{\mathrm{C}}_{\tau \left(0\right)}=\varnothing$, $\mathrm{h}{\left(\mathrm{p}\right)}_{\left(1\right)}\le \mathrm{h}{\left(\mathrm{p}\right)}_{\left(2\right)}\le \dots \le \mathrm{h}{\left(\mathrm{p}\right)}_{\left(\left|{\mathrm{P}}_{\mathrm{r}}\right|\right)}$, ${\rho }_{\lambda }$ is an fuzzy measure on $\mathrm{C}$ and $\sigma \in \left[0,1\right]$. $\left|{\mathrm{P}}_{\mathrm{r}}\right|$ denotes the number of attributes in partition ${\mathrm{P}}_{\mathrm{r}}$.

Based on the generalized operations of PIT2HFNs, we can obtain the following results.

Theorem 1.

The aggregated result of $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ based on the PIT2HFPBCI operator is the following PIT2HFN:

$$\begin{array}{l}\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\\ ={\left(\underset{\mathrm{r}=1}{\overset{\mathrm{d}}{\otimes }}\left(\left(\underset{{\mathrm{k}}_{\mathrm{r}}=1}{\overset{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}\right)}^{{\omega }_{{\mathrm{k}}_{\mathrm{r}}}}\right)\right)\right)}^{{\displaystyle \frac{1}{\mathrm{d}}}}\\ ={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\mathrm{P}}_{{\mathrm{A}}_{\mathrm{i}}}⌉\right\}}\end{array}$$

(24)

where ${\omega }_{{\mathrm{k}}_{\mathrm{r}}}=\sigma \left(\rho \left({\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}\right)-\rho \left({\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}+1\right)}\right)\right)+\left(1-\sigma \right)\left(\rho \left(\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}\right)-\rho \left(\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}-1\right)}\right)\right)$,

$${\mathrm{P}}_{{\mathrm{A}}_{\mathrm{i}}}={\displaystyle \frac{{\displaystyle \prod _{\mathrm{r}=1}^{\mathrm{d}}\left(\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)}\right)\right)}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \prod _{\mathrm{r}=1}^{\mathrm{d}}\left(\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)}\right)\right)}\right)}}},$$

$${\mathrm{A}}_{\mathrm{i}}=\left(\begin{array}{l}\left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}1}^{\mathrm{u}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}2}^{\mathrm{u}}\right)}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{h}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}3}^{\mathrm{u}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}4}^{\mathrm{u}}\right)}\right)}\right);\underset{\mathrm{r}}{\min }\left\{\underset{{\mathrm{k}}_{\mathrm{r}}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}^{\mathrm{u}}\right)\right\}\right\}\end{array}\right),\\ \left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{h}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}1}^{\mathrm{l}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}2}^{\mathrm{l}}\right)}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{h}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}3}^{\mathrm{l}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}4}^{\mathrm{l}}\right)}\right)}\right);\underset{\mathrm{r}}{\min }\left\{\underset{{\mathrm{k}}_{\mathrm{r}}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}^{\mathrm{l}}\right)\right\}\right\}\end{array}\right)\end{array}\right)$$

Proof. 

Please see Appendix A. □

Corollary 1.

If all attributes ${\mathrm{C}}_{\mathrm{k}}\left(\mathrm{k}=1,2,\dots ,\mathrm{n}\right)$ in $\mathrm{C}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$ are independent, namely, there is only one partition, and $\rho \left(\mathrm{B}\right)={\displaystyle \sum _{\mathrm{k}=1}^{\left|\mathrm{B}\right|}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)}$ , which indicates that the fuzzy measure is an additive measure. Then, the PIT2HFBCI operator is simplified into a proportional interval type-2 hesitant fuzzy Archimedean weighted average (PIT2HFAWA) operator:

$$\begin{array}{l}\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{A}\mathrm{W}{\mathrm{A}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\\ ={\left(\underset{\mathrm{r}=1}{\overset{\mathrm{d}}{\otimes }}\left(\left(\underset{{\mathrm{k}}_{\mathrm{r}}=1}{\overset{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}\right)}^{{\omega }_{{\mathrm{k}}_{\mathrm{r}}}}\right)\right)\right)}^{{\displaystyle \frac{1}{\mathrm{d}}}}\\ ={\left(\underset{\mathrm{k}=1}{\overset{\mathrm{n}}{\otimes }}\mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{k}\right)}\right)}^{\rho \left({\mathrm{C}}_{\mathrm{k}}\right)}\\ ={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}}\right)⌉\right\}}\end{array}$$

(25)

where ${\mathrm{A}}_{\mathrm{i}}=\left(\begin{array}{l}\left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\rho \left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{u}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{u}}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\rho \left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{u}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{u}}\right)}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}}^{\mathrm{u}}\right)\right\}\end{array}\right),\\ \left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{m}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\rho \left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{l}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{l}}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{l}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{l}}\right)}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}}^{\mathrm{l}}\right)\right\}\end{array}\right)\end{array}\right)$.

Corollary 2.

If $\mathrm{g}\left(\mathrm{B}\right)={\displaystyle \sum _{\mathrm{k}=1}^{\left|\mathrm{B}\right|}{\mathrm{w}}_{\mathrm{k}}}$ for any $\mathrm{B}\subset \mathrm{Z}$ , where $\left|\mathrm{B}\right|$ denotes the cardinality of $\mathrm{B}$, then the PIT2HFBCI operator is simplified into a proportional interval type-2 hesitant fuzzy Archimedean ordered weighted average (PIT2HFAOWA) operator:

$$\begin{array}{l}\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\\ ={\left(\underset{\mathrm{r}=1}{\overset{\mathrm{d}}{\otimes }}\left(\left(\underset{{\mathrm{k}}_{\mathrm{r}}=1}{\overset{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}\right)}^{{\omega }_{{\mathrm{k}}_{\mathrm{r}}}}\right)\right)\right)}^{{\displaystyle \frac{1}{\mathrm{d}}}}\\ =\underset{\mathrm{k}=1}{\overset{\mathrm{n}}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{k}\right)}\right)}^{{\mathrm{w}}_{\mathrm{k}}}\\ ={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}}\right)⌉\right\}}\end{array}$$

(26)

where${\mathrm{A}}_{\mathrm{i}}=\left(\begin{array}{l}\left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{u}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{u}}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{u}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{u}}\right)}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)}^{\mathrm{u}}\right)\right\}\end{array}\right),\\ \left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{m}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{l}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{l}}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{l}}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{k}}\mathrm{g}\left({\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{l}}\right)}\right);\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)}^{\mathrm{l}}\right)\right\}\end{array}\right)\end{array}\right)$.

Theorem 2.

(Commutativity) Suppose ${\mathrm{H}}^{\diamond }=\left\{\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right\}$ and $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ are two collections of PIT2HFNs. If ${\mathrm{H}}^{\diamond }$ is an any permutation of $\mathrm{H}$, then, we have 

$${\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right)=\left({\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}\mathrm{I}}_{\mathrm{g}},\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right)$$

Proof. 

Because ${\mathrm{H}}^{\diamond }$ is an any permutation of $\mathrm{H}$, we can easily find this conclusion based on Theorem 1. □

Theorem 3.

(Monotonicity) Suppose ${\mathrm{H}}^{\diamond }=\left\{\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right\}$ and $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ are two collections of PIT2HFNs. If ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{u}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{u}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{u}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{u}\diamond }$, $\mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{u}}\right)\le \mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{u}\diamond }\right)$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{l}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{l}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{l}}\le {\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{l}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{l}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{l}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{l}\diamond }$, $\mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{l}}\right)\le \mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{l}\diamond }\right)$, and $\mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{k}\right)}$ ${\mathrm{p}}_{{\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}}={\mathrm{p}}_{{\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\diamond }}$ for any and $\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\left(\mathrm{k}\right)}$, where $\mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{k}\right)}$ and $\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\left(\mathrm{k}\right)}$ are kth smallest elements in $\mathrm{H}$ and ${\mathrm{H}}^{\diamond }$, respectively, then we have

$$\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\le \mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right)$$

Proof. 

Please see Appendix B. □

Theorem 4.

(Boundness) Let ${\mathrm{H}}^{+}=\left\{\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_{\mathrm{n}}\right\}$, ${\mathrm{H}}^{+}=\left\{\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_{\mathrm{n}}\right\}$, and $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be three collections of PIT2HFNs, where ${\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{+}}={\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{-}}={\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}$ and

$$\begin{array}{ll}{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{-}& =\left(\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{u}-},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{u}-},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{u}-},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{u}-};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}-}\right)\right),\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{l}-},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{l}-},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{l}-},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{l}-};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}-}\right)\right)\right)\\ & =\left(\begin{array}{l}\left(\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}\right\};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\right),\\ \left(\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}\right\};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\right)\end{array}\right)\end{array},$$

$$\begin{array}{l}{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{+}=\left(\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{u}+},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{u}+},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{u}+},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{u}+};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}+}\right)\right),\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{l}+},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{l}+},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{l}+},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{l}+};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}+}\right)\right)\right)\\ =\left(\begin{array}{l}\left(\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}\right\};\underset{\mathrm{k}}{\max }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\right),\\ \left(\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}\right\};\underset{\mathrm{k}}{\max }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\right)\end{array}\right)\end{array}.$$

for all $\mathrm{h}{\left({\mathrm{p}}^{\nabla }\right)}_{\mathrm{k}}\left(\nabla =+,-;\mathrm{k}=1,2,\dots ,\mathrm{n}\right)$. Then, we have

$$\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_{\mathrm{n}}\right)\le \mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right),$$

$$\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\right)\le \mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_{\mathrm{n}}\right)$$

Proof. 

Please see Appendix C. □

3.2. GSPIT2HFBCI Operator Based on Archimedean T-Norms

The PIT2HFBCI operator can only describe the relationship between two adjacent ordered combinations. For example, suppose $\mathrm{C}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right\}$, $\mathrm{h}{\left(\mathrm{p}\right)}_1\le \mathrm{h}{\left(\mathrm{p}\right)}_2\le \mathrm{h}{\left(\mathrm{p}\right)}_3\le \mathrm{h}{\left(\mathrm{p}\right)}_4$ and there is an interrelation between these four attributes. Then, according to Definition 12, we can obtain

$$\begin{array}{l}{\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_4\right)\\ =\underset{\mathrm{k}=1}{\overset{4}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{k}\right)}\right)}^{{\omega }_{\left(\mathrm{k}\right)}}\\ ={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\displaystyle \prod _{\mathrm{k}=1}^4{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\displaystyle \prod _{\mathrm{k}=1}^4{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}}}\right)⌉\right\}}\end{array}$$

(27)

where

$${\omega }_1={\sigma }_1\left(\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right)-\rho \left({\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_1\right)-\rho \left(\varnothing \right)\right);$$

$${\omega }_2={\sigma }_1\left(\rho \left({\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right)-\rho \left({\mathrm{C}}_3,{\mathrm{C}}_4\right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2\right)-\rho \left({\mathrm{C}}_1\right)\right);$$

$${\omega }_3={\sigma }_1\left(\rho \left({\mathrm{C}}_3,{\mathrm{C}}_4\right)-\rho \left({\mathrm{C}}_4\right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right)-\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2\right)\right);$$

$${\omega }_4={\sigma }_1\left(\rho \left({\mathrm{C}}_4\right)-\rho \left(\varnothing \right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right)-\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right)\right).$$

Obviously, the proposed PIT2HFBCI operator can only indicate the relationships of combinations of attributes $\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right\}$, $\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}$$\left\{{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right\}$, $\left\{{\mathrm{C}}_3,{\mathrm{C}}_4\right\}$, $\left\{{\mathrm{C}}_1,{\mathrm{C}}_2\right\}$, $\left\{{\mathrm{C}}_1\right\}$, and $\left\{{\mathrm{C}}_4\right\}$, but it ignores $\left\{{\mathrm{C}}_3\right\}$, $\left\{{\mathrm{C}}_4\right\}$, $\left\{{\mathrm{C}}_1,{\mathrm{C}}_3\right\}$, $\left\{{\mathrm{C}}_1,{\mathrm{C}}_4\right\}$, $\left\{{\mathrm{C}}_2,{\mathrm{C}}_4\right\}$ and $\left\{{\mathrm{C}}_2,{\mathrm{C}}_3,{\mathrm{C}}_4\right\}$. To consider the interactions among multiple attributes, we introduce the GSPIT2HFBCI operator based on the generalized Shapley value.

Definition 13.

Let $\mathrm{C}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$ be a set of $\mathrm{n}$ attributes and $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be $\mathrm{n}$ PIT2HFNs. The GS-PIT2HFBCI operator is defined as follows:

$$\begin{array}{l}\mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\\ ={\left(\underset{\mathrm{r}=1}{\overset{\mathrm{d}}{\otimes }}\left(\underset{{\mathrm{k}}_{\mathrm{r}}=1}{\overset{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}\right)}^{\left(\sigma \left({\phi }_{{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}}\left(\rho ,\mathrm{S}\right)-{\phi }_{{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}+1\right)}}\left(\rho ,\mathrm{S}\right)\right)+\left(1-\sigma \right)\left({\phi }_{\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}}\left(\rho ,\mathrm{S}\right)-{\phi }_{\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}-1\right)}}\left(\rho ,\mathrm{S}\right)\right)\right)}\right)\right)}^{{\displaystyle \frac{1}{\mathrm{d}}}}\end{array}$$

(28)

where ${\mathrm{C}}_{\tau (\mathrm{k})}=\left\{{\mathrm{C}}_{\mathrm{k}},{\mathrm{C}}_{\mathrm{k}+1},\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$ with ${\mathrm{C}}_{\tau \left(\mathrm{n}+1\right)}=\varnothing$, $\mathrm{R}{\mathrm{C}}_{\tau \left(\mathrm{k}\right)}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{k}}\right\}$ with $\mathrm{R}{\mathrm{C}}_{\left(0\right)}=\varnothing$, $\mathrm{h}{\left(\mathrm{p}\right)}_{\left(1\right)}\le \mathrm{h}{\left(\mathrm{p}\right)}_{\left(2\right)}\le \dots \le \mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{n}\right)}$, $\rho$ is an fuzzy measure on $\mathrm{C}$, ${\phi }_{{\mathrm{C}}_{\tau \left(\mathrm{k}\right)}}\left(\rho ,\mathrm{S}\right)$ and ${\phi }_{\mathrm{R}{\mathrm{C}}_{\tau \left(\mathrm{k}\right)}}\left(\rho ,\mathrm{S}\right)$ are the generalized Shapley values of ${\mathrm{C}}_{\tau (\mathrm{k})}$ and $\mathrm{R}{\mathrm{C}}_{\tau (\mathrm{k})}$, and $\sigma \in \left[0,1\right]$. $\left|{\mathrm{P}}_{\mathrm{r}}\right|$ denotes the number of attributes in partition ${\mathrm{P}}_{\mathrm{r}}$.

Theorem 5.

The aggregated result of $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ based on the GS-PIT2HFBCI operator is the following PIT2HFN:

$$\begin{array}{ll}\mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)& ={\left(\underset{\mathrm{r}=1}{\overset{\mathrm{d}}{\otimes }}\left(\underset{{\mathrm{k}}_{\mathrm{r}}=1}{\overset{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\otimes }}{\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}\right)}^{{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}}\right)\right)}^{{\displaystyle \frac{1}{\mathrm{d}}}}\\ & ={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\mathrm{P}}_{{\mathrm{A}}_{\mathrm{i}}}⌉\right\}}\end{array}$$

(29)

where

$${\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}=\sigma \left({\phi }_{{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}+1\right)}}\left({\rho }_{\lambda },\mathrm{S}\right)\right)+\left(1-\sigma \right)\left({\phi }_{\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}\right)}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\mathrm{R}{\mathrm{C}}_{\tau \left({\mathrm{k}}_{\mathrm{r}}-1\right)}}\left({\rho }_{\lambda },\mathrm{S}\right)\right),$$

$${\mathrm{P}}_{{\mathrm{A}}_{\mathrm{i}}}={\displaystyle \frac{{\displaystyle \prod _{\mathrm{r}=1}^{\mathrm{d}}\left(\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)}\right)\right)}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \prod _{\mathrm{r}=1}^{\mathrm{d}}\left(\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)/\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \prod _{{\mathrm{k}}_{\mathrm{r}=1}}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\mathrm{P}}_{{\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}}}}\right)}\right)\right)}\right)}}},$$

$${\mathrm{A}}_{\mathrm{i}}=\left(\begin{array}{l}\left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}1}^{\mathrm{u}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}2}^{\mathrm{u}}\right)}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{h}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}3}^{\mathrm{u}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}4}^{\mathrm{u}}\right)}\right)}\right);\underset{\mathrm{r}}{\min }\left\{\underset{{\mathrm{k}}_{\mathrm{r}}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}^{\mathrm{u}}\right)\right\}\right\}\end{array}\right),\\ \left(\begin{array}{l}{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{h}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}1}^{\mathrm{l}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}2}^{\mathrm{l}}\right)}\right)}\right),\\ {\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{h}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}3}^{\mathrm{l}}\right)}\right)}\right),{\mathrm{g}}^{-1}\left({\displaystyle \frac{1}{\mathrm{d}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{d}}\left({\displaystyle \sum _{{\mathrm{k}}_{\mathrm{r}}=1}^{\left|{\mathrm{P}}_{\mathrm{r}}\right|}{\omega }_{{\mathrm{k}}_{\mathrm{r}}}^{\mathrm{G}}\mathrm{g}\left({\mathrm{a}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)\mathrm{i}4}^{\mathrm{l}}\right)}\right)}\right);\underset{\mathrm{r}}{\min }\left\{\underset{{\mathrm{k}}_{\mathrm{r}}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\left({\mathrm{k}}_{\mathrm{r}}\right)}^{\mathrm{l}}\right)\right\}\right\}\end{array}\right)\end{array}\right).$$

Proof. 

Similar to Theorem 1, so omitted here. □

Corollary 3.

If all attributes ${\mathrm{C}}_{\mathrm{k}}\left(\mathrm{k}=1,2,\dots ,\mathrm{n}\right)$ in $\mathrm{C}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$ are independent and $\rho \left(\mathrm{B}\right)={\displaystyle \sum _{\mathrm{k}=1}^{\left|\mathrm{B}\right|}\rho \left({\mathrm{C}}_{\mathrm{k}}\right)}$, namely, the fuzzy measure is an additive measure, then, the GS-PIT2HFBCI operator is simplified into a proportional interval type-2 hesitant fuzzy Archimedean weighted average (PIT2HFAWA) operator.

Proof. 

Please see Appendix D. □

Corollary 4.

If $\rho \left(\mathrm{B}\right)={\displaystyle \sum _{\mathrm{k}=1}^{\left|\mathrm{B}\right|}{\mathrm{w}}_{\mathrm{k}}}$ for any $\mathrm{B}\subset \mathrm{Z}$, where $\left|\mathrm{B}\right|$ denotes the cardinality of $\mathrm{B}$, then, the GS-PIT2HFBCI operator is simplified into a proportional interval type-2 hesitant fuzzy Archimedean ordered weighted average (PIT2HFAOWA) operator.

Proof. 

Similar to Corollary 3, so omitted here. □

Theorem 6.

(Commutativity) If ${\mathrm{H}}^{\diamond }=\left\{\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right\}$ is an any permutation of $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$, then we have

$$\begin{array}{l}\mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\\ =\mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right).\end{array}$$

Proof. 

Similar to Theorem 2, so omitted here. □

Theorem 7.

(Monotonicity) Suppose ${\mathrm{H}}^{\diamond }=\left\{\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right\}$ and $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ are two collections of PIT2HFNs. If ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{u}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{u}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{u}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{u}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{u}\diamond }$, $\mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{u}}\right)\le \mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{u}\diamond }\right)$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{l}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}1}^{\mathrm{l}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}2}^{\mathrm{l}}\le {\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{l}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}3}^{\mathrm{l}\diamond }$, ${\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{l}}\le {\mathrm{a}}_{\left(\mathrm{k}\right)\mathrm{i}4}^{\mathrm{l}\diamond }$, $\mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{u}}\right)\le \mathrm{h}\left({\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\mathrm{u}\diamond }\right)$, and ${\mathrm{p}}_{{\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}}={\mathrm{p}}_{{\mathrm{A}}_{\left(\mathrm{k}\right)\mathrm{i}}^{\diamond }}$ for any $\mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{k}\right)}$ and $\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\left(\mathrm{k}\right)}$, where $\mathrm{h}{\left(\mathrm{p}\right)}_{\left(\mathrm{k}\right)}$ and $\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\left(\mathrm{k}\right)}$ are kth smallest elements in $\mathrm{H}$ and ${\mathrm{H}}^{\diamond }$, respectively, then we have

$$\mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\le \mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{\diamond }\right)}_{\mathrm{n}}\right)$$

Proof. 

Similar to Theorem 3, so omitted here. □

Theorem 8.

(Boundness) Let ${\mathrm{H}}^{+}=\left\{\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_{\mathrm{n}}\right\}$, ${\mathrm{H}}^{+}=\left\{\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_{\mathrm{n}}\right\}$, and $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be three collections of PIT2HFNs, where ${\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{+}}={\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{-}}={\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}$ and

$$\begin{array}{ll}{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{-}& =\left(\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{u}-},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{u}-},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{u}-},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{u}-};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}-}\right)\right),\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{l}-},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{l}-},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{l}-},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{l}-};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}-}\right)\right)\right)\\ & =\left(\begin{array}{l}\left(\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}\right\};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\right),\\ \left(\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\min }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}\right\};\underset{\mathrm{k}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\right)\end{array}\right)\end{array},$$

$$\begin{array}{l}{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{+}=\left(\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{u}+},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{u}+},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{u}+},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{u}+};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}+}\right)\right),\left({\mathrm{a}}_{\mathrm{i}1}^{\mathrm{l}+},{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{l}+},{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{l}+},{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{l}+};\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}+}\right)\right)\right)\\ =\left(\begin{array}{l}\left(\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}\right\};\underset{\mathrm{k}}{\max }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right\}\right),\\ \left(\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}\right\},\underset{\mathrm{k}}{\max }\left\{{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}\right\};\underset{\mathrm{k}}{\max }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right\}\right)\end{array}\right)\end{array}.$$

for all $\mathrm{h}{\left({\mathrm{p}}^{\nabla }\right)}_{\mathrm{k}}\left(\nabla =+,-;\mathrm{k}=1,2,\dots ,\mathrm{n}\right)$. Then, we have

$$\mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{-}\right)}_{\mathrm{n}}\right)\le \mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right),$$

$$\mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right)\right)\le \mathrm{G}\mathrm{S}-\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_1,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_2,\dots ,\mathrm{h}{\left({\mathrm{p}}^{+}\right)}_{\mathrm{n}}\right).$$

Proof. 

Similar to Theorem 3, so omitted here. □

We also derive the special cases of the proposed GS-PIT2HFPBCI operator when using the different additive generator functions in Table 2. As the probability ${\mathrm{P}}_{{\mathrm{A}}_{\mathrm{i}}}$ is not affected by the additive generator, we only present the aggregated results of ${\mathrm{A}}_{\mathrm{i}}$. The specific forms of this operator obtained when using different generators are detailed in Appendix H.

Example 8.

Following Example 3, let $\mathrm{C}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_6\right\}$ be a set of six attributes that can be divided into two partitions ${\mathrm{P}}_1=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}$ and ${\mathrm{P}}_2=\left\{{\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right\}$. $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_6\right\}$ is a set of six PIT2HFNs, where $\mathrm{h}{\left(\mathrm{p}\right)}_1=\left\{\right(\mathrm{B},1/2), (\mathrm{M},1/2\left)\right\}$, $\mathrm{h}{\left(\mathrm{p}\right)}_2=\left\{\right(\mathrm{B},1/2), (\mathrm{B},1/2\left)\right\}$, $\mathrm{h}{\left(\mathrm{p}\right)}_3=\left\{\right(\mathrm{M},1/2), (\mathrm{M},1/2\left)\right\}$, $\mathrm{h}{\left(\mathrm{p}\right)}_4=\left\{\right(\mathrm{M},1/2), (\mathrm{LG},1/2\left)\right\}$, $\mathrm{h}{\left(\mathrm{p}\right)}_5=\left\{\right(\mathrm{LB},1/2), (\mathrm{M},1/2\left)\right\}$, and $\mathrm{h}{\left(\mathrm{p}\right)}_6=\left\{\right(\mathrm{M},1/2), (\mathrm{LG},1/2\left)\right\}$. The linguistic terms are obtained from

Table 3. Linguistic terms and their corresponding IT2FSs.

Linguistic Terms	IT2FS
very bad (VB)	[(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.8)]
bad (B)	[(0, 0.05, 0.2, 0.25; 1), (0.05, 0.1, 0.15, 0.2; 0.8)]
little bad (LB)	[(0.15, 0.25, 0.4, 0.5; 1), (0.2, 0.3, 0.35, 0.45; 0.8)]
medium (M)	[(0.35, 0.45, 0.55, 0.65; 1), (0.4, 0.5, 0.5, 0.6; 0.8)]
little good (LG)	[(0.5, 0.6, 0.75, 0.85; 1), (0.55, 0.65, 0.7, 0.8; 0.8)]
good (G)	[(0.75, 0.8, 0.95, 1; 1), (0.8, 0.85, 0.9, 0.95; 0.8)]
very good (VG)	[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.8)]

. Then, according to the PIT2HFPBCI operator, we have

$$\begin{array}{l}\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{A}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_1\right)\\ ={\left(\begin{array}{l}\left({\left(\mathrm{h}{\left(\mathrm{p}\right)}_3\right)}^{{\omega }_1}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)}^{{\omega }_2}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_2\right)}^{{\omega }_3}\right)\\ \otimes \left({\left(\mathrm{h}{\left(\mathrm{p}\right)}_6\right)}^{{\omega }_4}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_4\right)}^{{\omega }_5}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_5\right)}^{{\omega }_6}\right)\end{array}\right)}^{{\displaystyle \frac{1}{2}}}\\ =\left\{\begin{array}{l}⌈\left(\left(0,0.199,0.375,0.455;1\right),\left(0.181,0.272,0.318,0.399;0.8\right)\right),0.5⌉,\\ ⌈\left(\left(0,0.339,0.515,0.606;1\right),\left(0.311,0.414,0.457,0.549;0.8\right)\right),0.5⌉\end{array}\right\}\end{array}$$

where

$${\sigma }_1={\sigma }_2=0.5,$$

$${\omega }_1={\sigma }_1\left(\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right)-\rho \left({\mathrm{C}}_2,{\mathrm{C}}_3\right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_1\right)-\rho \left(\varnothing \right)\right),$$

$${\omega }_2={\sigma }_1\left(\rho \left({\mathrm{C}}_2,{\mathrm{C}}_3\right)-\rho \left({\mathrm{C}}_3\right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2\right)-\rho \left({\mathrm{C}}_1\right)\right),$$

$${\omega }_3={\sigma }_1\left(\rho \left({\mathrm{C}}_3\right)-\rho \left(\varnothing \right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right)-\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2\right)\right),$$

$${\omega }_4={\sigma }_1\left(\rho \left({\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right)-\rho \left({\mathrm{C}}_4,{\mathrm{C}}_5\right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_4\right)-\rho \left(\varnothing \right)\right),$$

$${\omega }_5={\sigma }_1\left(\rho \left({\mathrm{C}}_5,{\mathrm{C}}_6\right)-\rho \left({\mathrm{C}}_5\right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_4,{\mathrm{C}}_5\right)-\rho \left({\mathrm{C}}_4\right)\right),$$

$${\omega }_6={\sigma }_1\left(\rho \left({\mathrm{C}}_6\right)-\rho \left(\varnothing \right)\right)+{\sigma }_2\left(\rho \left({\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right)-\rho \left({\mathrm{C}}_4,{\mathrm{C}}_5\right)\right).$$

Example 9.

Following Example 2 and Example 8, according to the GSPIT2HFPBCI operator, we have

$$\begin{array}{l}\mathrm{G}\mathrm{S}\mathrm{P}\mathrm{I}\mathrm{T}2\mathrm{H}\mathrm{F}\mathrm{A}\mathrm{P}\mathrm{B}\mathrm{C}{\mathrm{I}}_{\mathrm{g}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_1\right)\\ ={\left(\begin{array}{l}\left({\left(\mathrm{h}{\left(\mathrm{p}\right)}_3\right)}^{{\omega }_1^{\mathrm{G}}}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)}^{{\omega }_2^{\mathrm{G}}}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_2\right)}^{{\omega }_3^{\mathrm{G}}}\right)\\ \otimes \left({\left(\mathrm{h}{\left(\mathrm{p}\right)}_6\right)}^{{\omega }_4^{\mathrm{G}}}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_4\right)}^{{\omega }_5^{\mathrm{G}}}\otimes {\left(\mathrm{h}{\left(\mathrm{p}\right)}_5\right)}^{{\omega }_6^{\mathrm{G}}}\right)\end{array}\right)}^{{\displaystyle \frac{1}{2}}}\\ =\left\{\begin{array}{l}⌈\left(\left(0,0.109,0.375,0.456;1\right),\left(0.181,0.272,0.318,0.399;0.8\right)\right),0.5⌉,\\ ⌈\left(\left(0,0.340,0.515,0.606;1\right),\left(0.312,0.415,0.457,0.549;0.8\right)\right),0.5⌉\end{array}\right\}\end{array}$$

where

$${\sigma }_1={\sigma }_2=0.5,$$

$${\omega }_1^{\mathrm{G}}=\sigma \left({\phi }_{\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{{\mathrm{C}}_2,{\mathrm{C}}_3\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right)+\left(1-\sigma \right)\left({\phi }_{\left\{{\mathrm{C}}_1\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{\varnothing \right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right),$$

$${\omega }_2^{\mathrm{G}}=\sigma \left({\phi }_{\left\{{\mathrm{C}}_2,{\mathrm{C}}_3\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{{\mathrm{C}}_3\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right)+\left(1-\sigma \right)\left({\phi }_{\left\{{\mathrm{C}}_1,{\mathrm{C}}_2\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{{\mathrm{C}}_1\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right),$$

$${\omega }_3^{\mathrm{G}}=\sigma \left({\phi }_{\left\{{\mathrm{C}}_3\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{\varnothing \right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right)+\left(1-\sigma \right)\left({\phi }_{\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{{\mathrm{C}}_1,{\mathrm{C}}_2\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right),$$

$${\omega }_4^{\mathrm{G}}=\sigma \left({\phi }_{\left\{{\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{{\mathrm{C}}_5,{\mathrm{C}}_6\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right)+\left(1-\sigma \right)\left({\phi }_{\left\{{\mathrm{C}}_4\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{\varnothing \right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right),$$

$${\omega }_5^{\mathrm{G}}=\sigma \left({\phi }_{\left\{{\mathrm{C}}_5,{\mathrm{C}}_6\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{{\mathrm{C}}_6\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right)+\left(1-\sigma \right)\left({\phi }_{\left\{{\mathrm{C}}_4,{\mathrm{C}}_5\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)-{\phi }_{\left\{{\mathrm{C}}_4\right\}}\left({\rho }_{\lambda },\mathrm{S}\right)\right).$$

4. A Novel Normalized Bidirectional Projection Measure for PIT2HFNs

In this section, first, we define the module and cosine measure of PIT2HFNs. Second, we propose the projection measure of PIT2HFNs based on the module and cosine measure. Finally, as the projection measure of PIT2HFNs has only one reference point, we further define the bidirectional projection measure and normalized bidirectional measure of PIT2HFNs and study some valuable properties of them.

4.1. The Projection Measure of PIT2HFNs

To introduce the bidirectional projection measure of PIT2HFNs, we need to define the cosine measure and projection measure of PIT2HFNs. In the following, we first give the definition of the module of PIT2HFNs.

Definition 14.

Let $\mathrm{h}\left(\mathrm{p}\right)={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}⌉\right\}}$ be a PIT2HFN. The module of $\mathrm{h}\left(\mathrm{p}\right)$ can be defined as follows:

$$\left|\mathrm{h}\left(\mathrm{p}\right)\right|=\sqrt{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \sum _{\mathrm{s}=1}^4{\left({\mathrm{a}}_{\mathrm{i}\mathrm{s}}^{\mathrm{u}}\right)}^2}+{\displaystyle \sum _{\mathrm{k}=1}^4{\left({\mathrm{a}}_{\mathrm{i}\mathrm{s}}^{\mathrm{l}}\right)}^2}\right)}\left(2{\left(\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}}\right)\right)}^2+2{\left(\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}}\right)\right)}^2\right){\left({\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}\right)}^2}$$

(30)

Definition 15.

Let $\mathrm{h}{\left(\mathrm{p}\right)}_1={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{2\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}⌉\right\}}$ be two PIT2HFNs. The inner product of $\mathrm{h}{\left(\mathrm{p}\right)}_1$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2$ can be defined as follows:

$$\mathrm{h}{\left(\mathrm{p}\right)}_1·\mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left(\left({\displaystyle \sum _{\mathrm{s}=1}^4{\mathrm{a}}_{1\mathrm{i}\mathrm{s}}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}\mathrm{s}}^{\mathrm{u}}}+{\displaystyle \sum _{\mathrm{s}=1}^4{\mathrm{a}}_{1\mathrm{i}\mathrm{s}}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}\mathrm{s}}^{\mathrm{l}}}\right)\left(2\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{u}}\right)\mathrm{h}\left({\mathrm{A}}_{2\mathrm{i}}^{\mathrm{u}}\right)+2\mathrm{h}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}}\right)\mathrm{h}\left({\mathrm{A}}_{2\mathrm{i}}^{\mathrm{l}}\right)\right){\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}\right)}$$

(31)

Definition 16.

Let $\mathrm{h}{\left(\mathrm{p}\right)}_1={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{2\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}⌉\right\}}$ be two PIT2HFNs. The cosine of the angle between $\mathrm{h}{\left(\mathrm{p}\right)}_1$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2$ can be defined as follows:

$$\cos \left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2\right)={\displaystyle \frac{\mathrm{h}{\left(\mathrm{p}\right)}_1·\mathrm{h}{\left(\mathrm{p}\right)}_2}{\left|\mathrm{h}{\left(\mathrm{p}\right)}_1\right|\left|\mathrm{h}{\left(\mathrm{p}\right)}_2\right|}}$$

(32)

Definition 17.

Let $\mathrm{h}{\left(\mathrm{p}\right)}_1={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}⌉\right\}}$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{2\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}⌉\right\}}$ be two PIT2HFNs. The projection of $\mathrm{h}{\left(\mathrm{p}\right)}_1$ on $\mathrm{h}{\left(\mathrm{p}\right)}_2$ can be defined as follows:

$$\begin{array}{l}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}_2}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)\\ =\left|\mathrm{h}{\left(\mathrm{p}\right)}_1\right|\cos \left(\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2\right)\\ ={\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left(\left({\displaystyle \sum _{\mathrm{k}=1}^4{\mathrm{a}}_{1\mathrm{i}\mathrm{k}}^{\mathrm{u}}{\mathrm{a}}_{2\mathrm{i}\mathrm{k}}^{\mathrm{u}}}+{\displaystyle \sum _{\mathrm{k}=1}^4{\mathrm{a}}_{1\mathrm{i}\mathrm{k}}^{\mathrm{l}}{\mathrm{a}}_{2\mathrm{i}\mathrm{k}}^{\mathrm{l}}}\right)\left(2{\mathrm{h}}_{\mathrm{k}}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{u}}\right){\mathrm{h}}_{\mathrm{k}}\left({\mathrm{A}}_{2\mathrm{i}}^{\mathrm{u}}\right)+2{\mathrm{h}}_{\mathrm{k}}\left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}}\right){\mathrm{h}}_{\mathrm{k}}\left({\mathrm{A}}_{2\mathrm{i}}^{\mathrm{l}}\right)\right){\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}\right)}}{\sqrt{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\displaystyle \sum _{\mathrm{k}=1}^4{\left({\mathrm{a}}_{2\mathrm{i}\mathrm{k}}^{\mathrm{u}}\right)}^2}+{\displaystyle \sum _{\mathrm{k}=1}^4{\left({\mathrm{a}}_{2\mathrm{i}\mathrm{k}}^{\mathrm{l}}\right)}^2}\right)}\left(2{\left(\mathrm{h}\left({\mathrm{A}}_{2\mathrm{i}}^{\mathrm{u}}\right)\right)}^2+2{\left(\mathrm{h}\left({\mathrm{A}}_{2\mathrm{i}}^{\mathrm{l}}\right)\right)}^2\right){\left({\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}\right)}^2}}}\end{array}$$

(33)

Remark 1.

Compared with the distance measure of PIT2HFNs proposed in [43], the projection measure introduced in this paper considers both the distance and angle between two PIT2HFNs. Generally speaking, the bigger the value of $\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}_2}\left(\mathrm{h}{\left(\mathrm{p}\right)}_1\right)$ , the more similar the two PIT2HFNs are.

4.2. The Normalized Bidirectional Projection Measure of PIT2HFNs

The projection measure is a powerful tool to measure the closeness degree between two PIT2HFNs. However, the single-directional projection measure introduced above also has two limitations. The first one is that the projection measure is a single-directional projection measure, namely, it has only one reference point. The other one is that the value of the projection measure is not a normalized value, namely, the range of it may be larger than one. To overcome these two drawbacks, we propose the normalized bidirectional projection measure of PIT2HFNs.

Definition 18.

Let $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be a collection of PIT2HFNs. The positive ideal point and negative ideal point can be defined as follows:

$$\mathrm{h}{\left(\mathrm{p}\right)}^{+}=\arg \underset{\mathrm{k}}{\max }\left\{\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}|\mathrm{S}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)\right\}$$

(34)

$$\mathrm{h}{\left(\mathrm{p}\right)}^{-}=\arg \underset{\mathrm{k}}{\min }\left\{\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}|\mathrm{S}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)\right\}$$

(35)

Suppose $\mathrm{h}{\left(\mathrm{p}\right)}_1$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2$ are the positive ideal point and negative ideal point of $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$, respectively. Then, inspired by the subtraction operation of IT2FNs, we can compute the difference between $\mathrm{h}{\left(\mathrm{p}\right)}_1$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2$, the difference between $\mathrm{h}{\left(\mathrm{p}\right)}_1$ and $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}$, and the difference between $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}$ and $\mathrm{h}{\left(\mathrm{p}\right)}_2$.

(1)

$\mathrm{h}{\left(\mathrm{p}\right)}_1\mathrm{h}{\left(\mathrm{p}\right)}_2=\mathrm{h}{\left(\mathrm{p}\right)}_1-\mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}}-{\mathrm{A}}_{2\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}/{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}\right)}⌉\right\}}$, where

$${\mathrm{A}}_{1\mathrm{i}}-{\mathrm{A}}_{2\mathrm{i}}=\left(\begin{array}{l}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{u}},{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{u}},{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{u}},{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{u}};\min \left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{u}},{\mathrm{A}}_{2\mathrm{i}}^{\mathrm{u}}\right)\right),\\ \left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{l}},{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{l}},{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{l}},{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{l}};\min \left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}},{\mathrm{A}}_{2\mathrm{i}}^{\mathrm{l}}\right)\right)\end{array}\right).$$

(2)

$\mathrm{h}{\left(\mathrm{p}\right)}_1\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}=\mathrm{h}{\left(\mathrm{p}\right)}_1-\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{1\mathrm{i}}-{\mathrm{A}}_{\mathrm{k}\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}/{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\mathrm{p}}_{{\mathrm{A}}_{1\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}\right)}⌉\right\}}$, where ${\mathrm{A}}_{1\mathrm{i}}-{\mathrm{A}}_{\mathrm{k}\mathrm{i}}=\left(\begin{array}{l}\left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{u}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}},{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{u}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}},{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{u}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}},{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{u}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}};\min \left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{u}},{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}}\right)\right),\\ \left({\mathrm{a}}_{1\mathrm{i}1}^{\mathrm{l}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}},{\mathrm{a}}_{1\mathrm{i}2}^{\mathrm{l}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}},{\mathrm{a}}_{1\mathrm{i}3}^{\mathrm{l}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}},{\mathrm{a}}_{1\mathrm{i}4}^{\mathrm{l}}-{\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}};\min \left({\mathrm{A}}_{1\mathrm{i}}^{\mathrm{l}},{\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}}\right)\right)\end{array}\right)$.

(3)

$\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}_2=\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}-\mathrm{h}{\left(\mathrm{p}\right)}_2={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{k}\mathrm{i}}-{\mathrm{A}}_{2\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}/{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}\left({\mathrm{p}}_{{\mathrm{A}}_{\mathrm{k}\mathrm{i}}}{\mathrm{p}}_{{\mathrm{A}}_{2\mathrm{i}}}\right)}⌉\right\}}$, where ${\mathrm{A}}_{\mathrm{k}\mathrm{i}}-{\mathrm{A}}_{2\mathrm{i}}=\left(\begin{array}{l}\left({\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{u}},{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{u}},{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{u}},{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{u}}-{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{u}};\min \left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{u}},{\mathrm{A}}_{2\mathrm{i}}^{\mathrm{u}}\right)\right),\\ \left({\mathrm{a}}_{\mathrm{k}\mathrm{i}1}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}4}^{\mathrm{l}},{\mathrm{a}}_{\mathrm{k}\mathrm{i}2}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}3}^{\mathrm{l}},{\mathrm{a}}_{\mathrm{k}\mathrm{i}3}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}2}^{\mathrm{l}},{\mathrm{a}}_{\mathrm{k}\mathrm{i}4}^{\mathrm{l}}-{\mathrm{a}}_{2\mathrm{i}1}^{\mathrm{l}};\min \left({\mathrm{A}}_{\mathrm{k}\mathrm{i}}^{\mathrm{l}},{\mathrm{A}}_{2\mathrm{i}}^{\mathrm{l}}\right)\right)\end{array}\right)$.

In the following, we introduce two types of normalized bidirectional projection measures of PIT2HFNs, which overcome the two limitations explained in Section 4.1.

Definition 19.

Let $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be a collection of PIT2HFNs. Then, the bidirectional projection measure of PIT2HFNs can be defined as follows:

$$\begin{array}{l}\mathrm{B}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)\\ =\left|\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right|\cos \left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-},\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)\\ ={\displaystyle \frac{\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)·\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)}{\left|\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right|}}\end{array}$$

(36)

$$\begin{array}{l}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)\\ =\left|\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right|\cos \left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-},\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)\\ ={\displaystyle \frac{\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)·\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)}{\left|\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right|}}\end{array}$$

(37)

where $\mathrm{h}{\left(\mathrm{p}\right)}^{+}$ and $\mathrm{h}{\left(\mathrm{p}\right)}^{-}$ are the positive ideal point and negative ideal point of $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$, respectively.

Remark 2.

The two-point projection method measures the degree of proximity by taking into account the angular and distance relationships between two reference points. By assessing the relative position with respect to positive and negative reference points, this method also reflects the decision-maker’s psychological preferences, thereby making it more closely aligned with real decision-making situations. Then, using the bi-projection values, it is possible to determine whether ${\mathrm{A}}^{\mathrm{i}}$ is closer to ${\mathrm{A}}^{+}$ or ${\mathrm{A}}^{-}$.

Theorem 9.

The larger the value of $\mathrm{B}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)$, the closer $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}$ is to $\mathrm{h}{\left(\mathrm{p}\right)}^{+}$. Conversely, the smaller the value of $\mathrm{B}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)$, the farther $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}$ is from $\mathrm{h}{\left(\mathrm{p}\right)}^{+}$. Similarly, the larger the value of $\mathrm{B}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)$, the closer $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}$ is to $\mathrm{h}{\left(\mathrm{p}\right)}^{-}$. The smaller the value of $\mathrm{B}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)$, the farther $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}$ is from $\mathrm{h}{\left(\mathrm{p}\right)}^{-}$.

Proof. 

It can be easily deduced based on Definition 19. □

Remark 3.

$\mathrm{B}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)$ and $\mathrm{B}\mathrm{P}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)$ are reverse change.

Inspired by Ji et al. [51,52], we further proposed the normalized bidirectional projection of $\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}$ and $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}$ and the normalized bidirectional projection of $\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}$ and $\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}$.

Definition 20.

Let $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be a collection of PIT2HFNs. Then, the normalized bidirectional projection measure of PIT2HFNs can be defined as follows:

$$\begin{array}{l}\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)\\ ={\displaystyle \frac{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)}{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)+\left|\left|\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right|-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)\right|}}\end{array}$$

(38)

$$\begin{array}{l}\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)\\ ={\displaystyle \frac{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)}{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)+\left|\left|\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right|-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)\right|}}\end{array}$$

(39)

where $\mathrm{h}{\left(\mathrm{p}\right)}^{+}$ and $\mathrm{h}{\left(\mathrm{p}\right)}^{-}$ are the positive ideal point and negative ideal point of $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$, respectively. $\left|\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right|$ denotes the module of $\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}$.

Theorem 10.

Let $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$ be a collection of PIT2HFNs, where $\mathrm{h}{\left(\mathrm{p}\right)}^{+}$ and $\mathrm{h}{\left(\mathrm{p}\right)}^{-}$ are the positive ideal point and negative ideal point of $\mathrm{H}=\left\{\mathrm{h}{\left(\mathrm{p}\right)}_1,\mathrm{h}{\left(\mathrm{p}\right)}_2,\dots ,\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{n}}\right\}$, respectively. Then, we have the following properties:

(1) 

$0\le \mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right),\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)\le 1$;

(2) 

$\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)=1$ when $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}=\mathrm{h}{\left(\mathrm{p}\right)}^{+}$;

(3) 

$\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\mathrm{h}{\left(\mathrm{p}\right)}^{-}\right)=0$ when $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}=\mathrm{h}{\left(\mathrm{p}\right)}^{-}$;

(4) 

$\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)=1$ when $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}=\mathrm{h}{\left(\mathrm{p}\right)}^{-}$;

(5) 

$\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}^{-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}\right)=0$ when $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{k}}=\mathrm{h}{\left(\mathrm{p}\right)}^{+}$.

Proof. 

Please see Appendix E. □

5. The Created Approaches to PIT2HF-MAGDM Problems with Modified Regret Theory

In this section, we combine modified regret theory with PIT2HF information for MAGDM problems. Firstly, the PIT2HF-MAGDM problem is illustrated. In view of this, we utilize the GS-PIT2HFPBCI operator to aggregate the evaluation information under each attribute into a comprehensive opinion. Then, we utilize modified regret theory to solve the PIT2HFMAGDM problem. Finally, we present the decision-making steps of the proposed method.

5.1. Problem Description

Let $\mathrm{O}=\left\{{\mathrm{o}}_1,{\mathrm{o}}_2,\dots ,{\mathrm{o}}_{\mathrm{m}}\right\}$ be a set of alternatives, $\mathrm{C}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$ be a set of attributes, and $\mathrm{D}=\left\{{\mathrm{d}}_1,{\mathrm{d}}_2,\dots ,{\mathrm{d}}_{\mathrm{T}}\right\}$ be a set of DMs. $\mathrm{G}=\left\{\rho \left({\mathrm{C}}_1\right),\rho \left({\mathrm{C}}_2\right),\dots ,\rho \left({\mathrm{C}}_{\mathrm{n}}\right)\right\}$ is the fuzzy measure vector of attributes. $\delta =\left({\delta }_1,{\delta }_2,\dots ,{\delta }_{\mathrm{T}}\right)$ is a weight vector of DMs, which satisfies $0\le {\delta }_{\mathrm{t}}\le 1$ and ${\displaystyle \sum _{\mathrm{t}=1}^{\mathrm{T}}{\delta }_{\mathrm{t}}}=1$. According to the PIT2HF-MAGDM problem, the decision matrix ${\mathrm{R}}_{\mathrm{j}}={\left[\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}}\right]}_{\mathrm{m}\times \mathrm{t}}$ with respect to attribute ${\mathrm{C}}_{\mathrm{j}}$ can be defined as follows:

$${\mathrm{R}}_{\mathrm{j}}=\left[\begin{array}{cccc}\mathrm{h}{\left(\mathrm{p}\right)}_{1\mathrm{j}}^1& \mathrm{h}{\left(\mathrm{p}\right)}_{1\mathrm{j}}^2& \dots & \mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{T}}\\ \mathrm{h}{\left(\mathrm{p}\right)}_{2\mathrm{j}}^1& \mathrm{h}{\left(\mathrm{p}\right)}_{2\mathrm{j}}^2& \dots & \mathrm{h}{\left(\mathrm{p}\right)}_{2\mathrm{j}}^{\mathrm{T}}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{m}\mathrm{j}}^1& \mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{m}\mathrm{j}}^2& \dots & \mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{m}\mathrm{j}}^{\mathrm{T}}\end{array}\right]$$

(40)

where $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}}$ denotes the evaluation value of alternative ${\mathrm{o}}_{\mathrm{i}}$ under attribute ${\mathrm{C}}_{\mathrm{j}}$ given by DM ${\mathrm{d}}_{\mathrm{t}}$$\left(\mathrm{i}=1,2,\dots ,\mathrm{m};\mathrm{j}=1,2,\dots ,\mathrm{n};\mathrm{t}=1,2,\dots ,\mathrm{T}\right)$.

5.2. The PIT2HF-MAGDM Method with Modified Regret Theory

As discussed in Section 1, traditional regret utility functions evaluate outcomes solely by comparing alternatives to the optimal choice, thereby neglecting the impact of worst-case scenarios on regret utility values. In practice, however, decision-makers focus not only on the most favorable outcomes but also on the most unfavorable ones. Therefore, we propose an improved regret utility function that incorporates the effects of both optimal and worst-case scenarios.

Step 1. Normalize and standardize the individual decision matrices.

The PIT2HF decision matrix can exhibit a characteristic, namely, the lack of partial probabilistic information. Moreover, different PIT2HFNs always consist of different IT2FNs. Therefore, to obtain the normalized PIT2HF decision matrix, we should standardize the PIT2HFSs according to Algorithm 1 in [45].

Further, there usually are two types of attributes, namely, benefit type and cost type. Hence, we also need to normalize the decision matrices under each attribute by utilizing Equation (45) before aggregating.

$$\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}}=\left\{\begin{array}{l}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}},\mathrm{for} \mathrm{benefit} \mathrm{type}\\ {\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}}\right)}^{\mathrm{c}},\mathrm{for} \cos \mathrm{t} \mathrm{type}\end{array}\right.$$

(41)

where ${\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}}\right)}^{\mathrm{c}}$ is the complement of $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}}$.

Step 2. Calculate the generalized Shapley value based on Equations (13)–(15).

Step 3. Aggregate the individual decision metrics ${ℝ}_{\mathrm{j}}={\left[\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}\mathrm{j}}^{\mathrm{t}}\right]}_{\mathrm{m}\times \mathrm{T}}$ under different attribute based on the GS-PIT2HFBCI operator. Then, the comprehensive decision matrix $ℝ={\left[\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right]}_{\mathrm{m}\times \mathrm{T}}$ can be built as follows:

$$ℝ=\left[\begin{array}{cccc}\mathrm{h}{\left(\mathrm{p}\right)}_1^1& \mathrm{h}{\left(\mathrm{p}\right)}_1^2& \dots & \mathrm{h}{\left(\mathrm{p}\right)}_1^{\mathrm{t}}\\ \mathrm{h}{\left(\mathrm{p}\right)}_2^1& \mathrm{h}{\left(\mathrm{p}\right)}_2^2& \dots & \mathrm{h}{\left(\mathrm{p}\right)}_2^{\mathrm{t}}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{m}}^1& \mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{m}}^2& \dots & \mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{m}}^{\mathrm{t}}\end{array}\right]$$

(42)

Step 4. Calculate the normalized bidirectional projection measure for all $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}$ in $\mathrm{R}$ based on Equations (47) and (48).

$$\begin{array}{l}\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}\right)\\ ={\displaystyle \frac{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}\right)}{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}\right)+\left|\left|\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}\right|-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}\right)\right|}}\end{array}$$

(43)

$$\begin{array}{l}\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right)\\ ={\displaystyle \frac{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right)}{\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right)+\left|\left|\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}\right|-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right)\right|}}\end{array}$$

(44)

where $\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}=\arg \underset{\mathrm{i}}{\max }\left\{\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}|\mathrm{S}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right)\right\}$, $\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}=\arg \underset{\mathrm{i}}{\min }\left\{\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}|\mathrm{S}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right)\right\}$.

For simplicity, we will use $\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}-}$ and $\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}+}$ to represent $\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}\right)$ and $\mathrm{N}-\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}-}}\left(\mathrm{h}{\left(\mathrm{p}\right)}^{\mathrm{t}+}\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\right)$, respectively.

Step 5. Determine the weights of decision-makers (DMs).

Given variations among decision-makers in expertise, reliability, and consistency, it is necessary to assign appropriate weights to each expert before aggregating evaluation information.

$$\mathrm{R}\left({\mathrm{d}}_{{\mathrm{t}}_1},{\mathrm{d}}_{{\mathrm{t}}_2}\right)={\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}\left(\left|\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_1+}-\frac{1}{\mathrm{m}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_1+}}\right|\ast \left|\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_2+}-\frac{1}{\mathrm{m}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_2+}}\right|\right)}}{\sqrt{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}{\left|\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_1+}-\frac{1}{\mathrm{m}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_1+}}\right|}^2}}\ast \sqrt{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}{\left|\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_2+}-\frac{1}{\mathrm{m}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{{\mathrm{t}}_2+}}\right|}^2}}}}$$

(45)

$${\delta }_{{\mathrm{t}}_1}={\displaystyle \frac{{\displaystyle \sum _{{\mathrm{t}}_2=1}^{\mathrm{T}}\left(1-\mathrm{R}\left({\mathrm{d}}_{{\mathrm{t}}_1},{\mathrm{d}}_{{\mathrm{t}}_2}\right)\right)}}{{\displaystyle \sum _{{\mathrm{t}}_1=1}^{\mathrm{T}}{\displaystyle \sum _{{\mathrm{t}}_2=1}^{\mathrm{T}}\left(1-\mathrm{R}\left({\mathrm{d}}_{{\mathrm{t}}_1},{\mathrm{d}}_{{\mathrm{t}}_2}\right)\right)}}}}$$

(46)

Step 6. Calculate the individual regret value–rejoice value of alternative ${\mathrm{o}}_{\mathrm{i}}$ according to Equations (47)–(49).

$$\mathrm{R}\mathrm{E}{\mathrm{G}}_{\mathrm{i}}^{\mathrm{t}}={\displaystyle \frac{1-\exp \left\{-\beta \left({\mathrm{u}}_{\mathrm{i}}^{\mathrm{t}}-{\mathrm{u}}^{\mathrm{t}+}\right)\right\}+1-\exp \left\{-\beta \left({\mathrm{v}}^{\mathrm{t}+}-{\mathrm{v}}_{\mathrm{i}}^{\mathrm{t}}\right)\right\}}{2}}$$

(47)

$$\mathrm{R}\mathrm{E}{\mathrm{J}}_{\mathrm{i}}^{\mathrm{t}}={\displaystyle \frac{1-\exp \left\{-\beta \left({\mathrm{u}}_{\mathrm{i}}^{\mathrm{t}}-{\mathrm{u}}^{\mathrm{t}-}\right)\right\}+1-\exp \left\{-\beta \left({\mathrm{v}}^{\mathrm{t}-}-{\mathrm{v}}_{\mathrm{i}}^{\mathrm{t}}\right)\right\}}{2}}$$

(48)

$${\mathrm{U}}_{\mathrm{i}}^{\mathrm{t}}=\mathrm{R}\mathrm{E}{\mathrm{G}}_{\mathrm{i}}^{\mathrm{t}}+\mathrm{R}\mathrm{E}{\mathrm{J}}_{\mathrm{i}}^{\mathrm{t}}$$

(49)

where ${\mathrm{u}}_{\mathrm{i}}^{\mathrm{t}}={\left(\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}-}\right)}^{\alpha },v_{\mathrm{i}}^{\mathrm{t}}={\left(\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}+}\right)}^{\alpha }$, ${\mathrm{u}}^{\mathrm{t}+}={\left(\underset{\mathrm{i}}{\max }\left\{\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}-}\right\}\right)}^{\alpha }$, ${\mathrm{v}}^{\mathrm{t}+}={\left(\underset{\mathrm{i}}{\min }\left\{\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}+}\right\}\right)}^{\alpha }$, ${\mathrm{u}}^{\mathrm{t}-}={\left(\underset{\mathrm{i}}{\min }\left\{\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}-}\right\}\right)}^{\alpha }$, and ${\mathrm{v}}^{\mathrm{t}-}={\left(\underset{\mathrm{i}}{\max }\left\{\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{t}+}\right\}\right)}^{\alpha }$.

Step 7. Compute their comprehensive regret–rejoice values based on Equation (50).

$${\mathrm{U}}_{\mathrm{i}}={\displaystyle \sum _{\mathrm{t}=1}^{\mathrm{T}}{\delta }_{\mathrm{t}}{\mathrm{U}}_{\mathrm{i}}^{\mathrm{t}}}$$

(50)

5.3. The Specific Decision Procedures for the PIT2HF-MAGDM Method

Based on the above analysis, the steps of the proposed method are shown in Figure 3, and the Algorithm 1 is shown as follows:

Algorithm 1: The algorithm of the proposed PIT2HF-MAGDM model based on the GS-PIT2HFPBCI operator and modified regret theory

Input: The decision matrices $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,\dots ,{\mathrm{R}}_{\mathrm{n}}\right\}$ about alternatives $\left\{{\mathrm{o}}_1,{\mathrm{o}}_2,\dots ,{\mathrm{o}}_{\mathrm{m}}\right\}$ given by DMs $\left\{{\mathrm{d}}_1,{\mathrm{d}}_2,\dots ,{\mathrm{d}}_{\mathrm{T}}\right\}$ under attributes $\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,\dots ,{\mathrm{C}}_{\mathrm{n}}\right\}$, the attribute partition structure $\left\{{\mathrm{P}}_1,{\mathrm{P}}_2,\dots ,{\mathrm{P}}_{\mathrm{d}}\right\}$, parameters $\alpha$, $\beta$, and $\sigma$, and the fuzzy measures $\left\{\rho \left({\mathrm{C}}_1\right),\rho \left({\mathrm{C}}_2\right),\dots ,\rho \left({\mathrm{C}}_{\mathrm{n}}\right)\right\}$ of $\mathrm{n}$ attributes.

Output: The ranking of all alternatives.

Begin

(Step1) For $\mathrm{j}=1$ to $\mathrm{n}$, do

Normalize and standardize the decision matrices $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,\dots ,{\mathrm{R}}_{\mathrm{n}}\right\}$ using Equation (41) and Algorithm 1 in [46]

(Step2) For $\mathrm{j}=1$ to $\mathrm{n}$, $\mathrm{r}=1$ to $\mathrm{d}$, do

Determine the parameter $\lambda$ using Equation (13).

Determine the fuzzy measure $\rho$ using Equation (14)

Determine the generalized Shapley value $\phi$ using Equation (15)

(Step3) For $\mathrm{j}=1$ to $\mathrm{n}$, do

Aggregate the decision matrices $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,\dots ,{\mathrm{R}}_{\mathrm{n}}\right\}$ under $\mathrm{n}$ attributes based on the GS-PIT2HFBCI operator

(Step4) For $\mathrm{i}=1$ to $\mathrm{m}$, $\mathrm{t}=1$ to $\mathrm{T}$, do

Calculate the Normalized Bidirectional Projection (NBP) for all alternatives using Equations (43) and (44)

(Step5) For $\mathrm{t}=1$ to $\mathrm{T}$, do

Determine the weights of experts ${\delta }_{\mathrm{t}}$ based on Equations (45) and (46)

(Step6) For $\mathrm{i}=1$ to $\mathrm{m}$, $\mathrm{t}=1$ to $\mathrm{T}$, do

Calculate the individual regret value of alternative ${\mathrm{o}}_{\mathrm{i}}$ using Equation (47)

Calculate the individual rejoice value of alternative ${\mathrm{o}}_{\mathrm{i}}$ using Equation (48)

Calculate the individual regret–rejoice value of alternative ${\mathrm{o}}_{\mathrm{i}}$ using Equation (49)

(Step7) For $\mathrm{i}=1$ to $\mathrm{m}$, do

Calculate the comprehensive regret–rejoice value of alternative ${\mathrm{o}}_{\mathrm{i}}$ using Equation (50)

End

In Algorithm 1, the time complexity of Step 1 is $\mathrm{O}\left(\mathrm{T}\mathrm{n}\mathrm{m}\right)$. In Step 2, determining the parameter $\lambda$ causes the time complexity $\mathrm{O}\left(\mathrm{d}\right)$, determining the fuzzy measure $\rho$ and the generalized Shapley value $\phi$ cause the same time complexity $\mathrm{O}\left(\mathrm{d}\left(2^{\mathrm{n}}-2\right)\right)$. The time complexity of Step 3 is $\mathrm{O}\left(\mathrm{n}\right)$. The time complexity of Step 4 is $\mathrm{O}\left(\mathrm{m}\mathrm{T}\right)$. The time complexity of Step 5 is $\mathrm{O}\left(\mathrm{T}\right)$. The time complexity of Step 6 is $\mathrm{O}\left(\mathrm{m}\mathrm{T}\right)$. The time complexity of Step 7 is $\mathrm{O}\left(\mathrm{m}\right)$.

6. Case Study

Manufacturing enterprises play an important role in China’s industry development [53,54]. At present, the total number of manufacturing enterprises in China has exceeded 6 million, and their added value accounts for more than 30% of the global proportion, making great contributions to the promotion of national economic development and technological innovation. However, with the increasing global environmental challenges and the gradual clarification of the goal of sustainable development, China is at a critical stage of transition from a “manufacturing power” to a “manufacturing power”. In this process, China’s manufacturing enterprises are faced with the serious challenge of realizing green and intelligent transformation while improving production efficiency. In this context, sustainable supplier selection has become a key link for manufacturing enterprises to optimize supply chain management, enhance competitiveness and achieve sustainable development goals. Therefore, selecting excellent suppliers that meet environmental standards, social responsibility requirements and economic benefits [55] is of great significance for enterprises to reduce the waste of resources, reduce the burden on the environment, and build a green supply chain to enhance the market competitiveness of enterprises.

To ensure the validity and fairness of the case study, a panel of three experts was formed to handle the decision-making process. The selection of experts was based on the following conditions: (1) each expert possesses substantial knowledge and experience in sustainable supply chain management, multi-attribute decision-making, or supplier evaluation; (2) the experts come from diverse backgrounds, including academia, manufacturing enterprises, and consulting practice, to ensure both theoretical and practical perspectives are represented; and (3) all experts participate independently in providing their evaluations to avoid potential bias. Their assessments were subsequently aggregated using the proposed method to guarantee scientific rigor and objectivity in the decision-making process.

Suppose an enterprise is planning to build an intelligent factory. The enterprise intends to purchase a lot of equipment and building materials, and there are five suppliers that can supply them. The five supplier alternatives are denoted by $\mathrm{O}=\left\{{\mathrm{o}}_1,{\mathrm{o}}_2,{\mathrm{o}}_3,{\mathrm{o}}_4,{\mathrm{o}}_5\right\}$. Six attributes of each supplier are considered in the SSS process: ${\mathrm{C}}_1$: technology, ${\mathrm{C}}_2$: management skill, ${\mathrm{C}}_3$: honesty, ${\mathrm{C}}_4$: geographical position, ${\mathrm{C}}_5$: economic factors, and ${\mathrm{C}}_6$: social factors. To select an appropriate supplier, the enterprise invites three experts denoted by $\mathrm{d}=\left\{{\mathrm{d}}_1,{\mathrm{d}}_2,{\mathrm{d}}_3\right\}$ to evaluate the five suppliers. The experts use linguistic terms (shown in Table 3) to evaluate the alternative ${\mathrm{o}}_{\mathrm{i}}\in \mathrm{O}$ with respect attribute ${\mathrm{C}}_{\mathrm{j}}\in \mathrm{C}$. Table 3 defines the linguistic terms and their corresponding IT2FS representations. To further enhance interpretability, the semantic meanings of these linguistic terms under each evaluation criterion are summarized in

Table 4. Semantic anchoring of linguistic terms.

Linguistic Term	C1 Technology	C2 Management Skill	C3 Honesty	C4 Geographical Position	C5 Economic Factors	C6 Social Factors
VG	Industry-leading, highly intelligent	Well-structured, highly digitalized	Consistently reliable	Excellent logistics access	Significant cost advantage	Outstanding social responsibility
G	Mature and stable	Sound and coordinated	Generally trustworthy	Good accessibility	High cost–performance	Stable compliance
LG	Upper-medium, partial dependence	Basically standardized	Minor disputes	Acceptable logistics	Average cost level	Basic compliance
M	Conventional technology	Experience-based	Moderate stability	Distant location	Limited advantage	Minimum compliance
LB	Relatively outdated	Poorly organized	Past dishonesty	High logistics cost	Low cost–performance	Weak responsibility
B	Weak capability	Disordered	Frequent defaults	Poor accessibility	High price level	Existing violations
VB	Severely lagging	No effective system	Serious dishonesty	Severely constrained	High financial risk	Serious violations

. Based on the intrinsic characteristics of the evaluation attributes and the decision context, the attributes are partitioned into two subsets in an expert-driven manner. Specifically, ${\mathrm{P}}_1$ represents micro-level attributes reflecting suppliers’ internal capabilities, including human resource ${\mathrm{C}}_1$, management skill ${\mathrm{C}}_2$ and honesty ${\mathrm{C}}_3$. ${\mathrm{P}}_2$ represents macro-level attributes associated with external conditions and overall impacts, including geographical position ${\mathrm{C}}_4$, economic factors ${\mathrm{C}}_5$ and social factors ${\mathrm{C}}_6$.

6.1. Decision Analysis

To solve the SSS PIT2HF-MAGDM problem, we employ the proposed method in Section 5. In the proposed method, the parameters $\alpha$, $\beta$ and $\sigma$ are very important. For this example, we assume that $\alpha =0.3$, $\beta =0.88$ and $\sigma =0.5$. Then, the main steps are presented as follows:

Step 1. Construct the individual decision matrices. Due to space, the original decision matrices are shown in Appendix F.

Step 2. Normalize and standardize the individual decision matrices. As the six attributes are all benefit attributes, we only need to standardize the individual decision matrices. The results are shown in

Table 5. The standardized decision metric ${\mathrm{R}}_1$ under attribute ${\mathrm{C}}_1$.

	d1	d2	d3
${\mathrm{o}}_1$	{(B, 1/2), (M, 1/2)}	{(M, 1/2), (M, 1/2)}	{(LB, 1/2), (M, 1/2)}
${\mathrm{o}}_2$	{(LB, 1/2), (M, 1/2)}	{(M, 1/3), (LG, 2/3)}	{(M, 1/2), (M, 1/2)}
${\mathrm{o}}_3$	{(G, 1/3), (VG, 2/3)}	{(VG, 1/2), (VG, 1/2)}	{(G, 1/2), (VG, 1/2)}
${\mathrm{o}}_4$	{(M, 1/2), (M, 1/2)}	{(M, 1/2), (LG, 1/2)}	{(LG, 2/3), (G, 1/3)}
${\mathrm{o}}_5$	{(LG, 1/2), (LG, 1/2)}	{(LG, 1/3), (G, 2/3)}	{(LG, 1/2), (G, 1/2)}

,

Table 6. The standardized decision metric ${\mathrm{R}}_2$ under attribute ${\mathrm{C}}_2$.

	d1	d2	d3
${\mathrm{o}}_1$	{(B, 1/2), (B, 1/2)}	{(VB, 1/3), (B, 2/3)}	{(B, 1/2), (LB, 1/2)}
${\mathrm{o}}_2$	{(B, 1/2), (LB, 1/2)}	{(LB, 1/3), (M, 2/3)}	{(M, 1/2), (M, 1/2)}
${\mathrm{o}}_3$	{(VG, 1/2), (VG, 1/2)}	{(G, 1/3), (VG, 2/3)}	{(G, 1/2), (G, 1/2)}
${\mathrm{o}}_4$	{(M, 2/3), (LG, 1/3)}	{(LG, 1/2), (G, 1/2)}	{(LG, 1/2), (LG, 1/2)}
${\mathrm{o}}_5$	{(LG, 1/2), (G, 1/2)}	{(G, 1/2), (G, 1/2)}	{(LG, 2/3), (G, 1/3)}

,

Table 7. The standardized decision metric ${\mathrm{R}}_3$ under attribute ${\mathrm{C}}_3$.

	d1	d2	d3
${\mathrm{o}}_1$	{(M, 1/2), (M, 1/2)}	{(B, 1/2), (B, 1/2)}	{(B, 1/2), (M, 1/2)}
${\mathrm{o}}_2$	{(LB, 1/2), (M, 1/2)}	{(M, 1/2), (M, 1/2)}	{(M, 1/2), (LG, 1/2)}
${\mathrm{o}}_3$	{(G, 1/2), (VG, 1/2)}	{(VG, 1/2), (VG, 1/2)}	{(LG, 1/3), (G, 2/3)}
${\mathrm{o}}_4$	{(G, 1/2), (G, 1/2)}	{(G, 1/2), (G, 1/2)}	{(M, 1/2), (LG, 1/2)}
${\mathrm{o}}_5$	{(LG, 1/2), (LG, 1/2)}	{(LG, 1/2), (LG, 1/2)}	{(LG, 1/2), (G, 1/2)}

,

Table 8. The standardized decision metric ${\mathrm{R}}_4$ under attribute ${\mathrm{C}}_4$.

	d1	d2	d3
${\mathrm{o}}_1$	{(M, 1/2), (LG, 1/2)}	{(LB, 1/2), (LB, 1/2)}	{(M, 1/2), (M, 1/2)}
${\mathrm{o}}_2$	{(LG, 1/2), (LG, 1/2)}	{(M, 1/3), (LG, 2/3)}	{(LB, 1/2), (M, 1/2)}
${\mathrm{o}}_3$	{(G, 1/3), (VG, 2/3)}	{(G, 1/2), (G, 1/2)}	{(G, 1/2), (VG, 1/2)}
${\mathrm{o}}_4$	{(LG, 1/3), (G, 2/3)}	{(LG, 1/2), (G, 1/2)}	{(G, 1/2), (G, 1/2)}
${\mathrm{o}}_5$	{(G, 1/2), (G, 1/2)}	{(G, 1/2), (G, 1/2)}	{(G, 2/3), (VG, 1/3)}

,

Table 9. The standardized decision metric ${\mathrm{R}}_5$ under attribute ${\mathrm{C}}_5$.

	d1	d2	d3
${\mathrm{o}}_1$	{(LB, 1/2), (M, 1/2)}	{(M, 1/2), (M, 1/2)}	{(M, 1/3), (LG, 2/3)}
${\mathrm{o}}_2$	{(M, 1/2), (M, 1/2)}	{(LG, 1/2), (LG, 1/2)}	{(LB, 1/3), (M, 2/3)}
${\mathrm{O}}_3$	{(LG, 1/3), (G, 2/3)}	{(LG, 1/2), (LG, 1/2)}	{(G, 1/2), (G, 1/2)}
${\mathrm{o}}_4$	{(LG, 1/2), (LG, 1/2)}	{(LG, 2/3), (G, 1/3)}	{(M, 1/2), (M, 1/2)}
${\mathrm{o}}_5$	{(G, 1), (G, 1/2)}	{(G, 2/3), (VG, 1/3)}	{(LG, 1/2), (LG, 1/2)}

and

Table 10. The standardized decision metric ${\mathrm{R}}_6$ under attribute ${\mathrm{C}}_6$.

	d1	d2	d3
${\mathrm{o}}_1$	{(M, 1/2), (LG, 1/2)}	{(M, 1/2), (M, 1/2)}	{(M, 1/3), (LG, 2/3)}
${\mathrm{o}}_2$	{(LB, 1/2), (LB, 1/2)}	{(LB, 1/2), (M, 1/2)}	{(B, 1/2), (LB, 1/2)}
${\mathrm{o}}_3$	{(LG, 1/3), (G, 2/3)}	{(LG, 1/2), (G, 1/2)}	{(LG, 1/2), (LG, 1/2)}
${\mathrm{o}}_4$	{(G, 1/2), (G, 1/2)}	{(M, 1/2), (LG, 1/2)}	{(LG, 1/2), (LG, 1/2)}
${\mathrm{o}}_5$	{(G, 1/2), (VG/2)}	{(G, 2/3), (VG, 1/3)}	{(VG, 1/2), (VG, 1/2)}

.

Step 3. Identify the fuzzy measure $\rho \left({\mathrm{C}}_{\mathrm{j}}\right)$ for each attribute ${\mathrm{C}}_{\mathrm{j}}\left(\mathrm{j}=1,2,\dots ,6\right)$, which represents the importance of each attribute ${\mathrm{C}}_{\mathrm{j}}\left(\mathrm{j}=1,2,\dots ,6\right)$. According to the experts’ assessments, the fuzzy measure of each attribute is given as:

$\rho \left({\mathrm{C}}_1\right)=0.28$, $\rho \left({\mathrm{C}}_2\right)=0.31$, $\rho \left({\mathrm{C}}_3\right)=0.35$, $\rho \left({\mathrm{C}}_4\right)=0.25$, $\rho \left({\mathrm{C}}_5\right)=0.30$, and $\rho \left({\mathrm{C}}_6\right)=0.33$.

Since there are two groups of unrelated attributes, we need to calculate the fuzzy measures and generalized Shapley values of these two groups of attributes, respectively.

Using Equation (13), parameter ${\lambda }_1$ is calculated to be ${\lambda }_1=0.20$. Based on Equation (14), we have

$\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2\right)=0.61$, $\rho \left({\mathrm{C}}_1,{\mathrm{C}}_3\right)=0.65$, $\rho \left({\mathrm{C}}_2,{\mathrm{C}}_3\right)=0.68$, and $\rho \left({\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right)=1$.

Similarly, parameter ${\lambda }_2$ is calculated to be ${\lambda }_2=0.45$, and the fuzzy measures are presented as follows:

$\rho \left({\mathrm{C}}_4,{\mathrm{C}}_5\right)=0.58$, $\rho \left({\mathrm{C}}_4,{\mathrm{C}}_6\right)=0.62$, $\rho \left({\mathrm{C}}_5,{\mathrm{C}}_6\right)=0.68$, and $\rho \left({\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right)=1$.

Now, according to Equation (15), the generalized Shapley value of each combination can be computed as follows:

$${\phi }_{\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}}\left(\rho ,\mathrm{N}\right)=1, {\phi }_{\left\{{\mathrm{C}}_1,{\mathrm{C}}_3\right\}}\left(\rho ,\mathrm{N}\right)=0.68, {\phi }_{\left\{{\mathrm{C}}_2,{\mathrm{C}}_3\right\}}\left(\rho ,\mathrm{N}\right)=0.70;$$

$${\phi }_{\left\{\varnothing \right\}}\left(\rho ,\mathrm{N}\right)=0, {\phi }_{\left\{{\mathrm{C}}_1\right\}}\left(\rho ,\mathrm{N}\right)=0.3, {\phi }_{\left\{{\mathrm{C}}_2\right\}}\left(\rho ,\mathrm{N}\right)=0.33, {\phi }_{\left\{{\mathrm{C}}_3\right\}}\left(\rho ,\mathrm{N}\right)=0.37;$$

$${\phi }_{\left\{{\mathrm{C}}_4,{\mathrm{C}}_5,{\mathrm{C}}_6\right\}}\left(\rho ,\mathrm{N}\right)=1, {\phi }_{\left\{{\mathrm{C}}_4,{\mathrm{C}}_5\right\}}\left(\rho ,\mathrm{N}\right)=0.63, {\phi }_{\left\{{\mathrm{C}}_4,{\mathrm{C}}_6\right\}}\left(\rho ,\mathrm{N}\right)=0.66, {\phi }_{\left\{{\mathrm{C}}_5,{\mathrm{C}}_6\right\}}\left(\rho ,\mathrm{N}\right)=0.72;$$

${\phi }_{\left\{{\mathrm{C}}_4\right\}}\left(\rho ,\mathrm{N}\right)=0.29$, ${\phi }_{\left\{{\mathrm{C}}_5\right\}}\left(\rho ,\mathrm{N}\right)=0.34$, and ${\phi }_{\left\{{\mathrm{C}}_6\right\}}\left(\rho ,\mathrm{N}\right)=0.33$.

Step 4. Aggregate the decision metrics ${\mathrm{R}}_{\mathrm{j}}\left(\mathrm{j}=1,2,\dots ,6\right)$ under different attribute based on the GS-PIT2HFBCI operator (Algebraic). For convenience, the aggregated result is shown in Appendix G.

Step 5. Calculate the normalized bidirectional projection measure for all $\mathrm{h}{\left(\mathrm{p}\right)}_{\mathrm{i}}^{\mathrm{t}}\left(\mathrm{i}=1,2,\dots ,6;\mathrm{t}=1,2,3\right)$ in $\mathrm{R}$ based on Equations (47) and (48). The results are shown in Figure 4 and Figure 5.

Step 6. Determine the weights of experts based on Equations (49) and (50) described in Section 5.2. The computed expert weights are shown as follows:

${\delta }_1=0.224$, ${\delta }_2=0.431$, and ${\delta }_3=0.345$.

Step 7. Calculate the individual regret–rejoice value of alternative ${\mathrm{o}}_{\mathrm{i}}$ according to Equations (47)–(49). The results are shown in Figure 6.

Step 8. Calculate the comprehensive regret–rejoice value of alternative ${\mathrm{o}}_{\mathrm{i}}$ according to Equation (50). The results are shown as follows:

${\mathrm{U}}_1=-1.175$, ${\mathrm{U}}_2=-0.792$, ${\mathrm{U}}_3=0.585$, ${\mathrm{U}}_4=-0.078$, and ${\mathrm{U}}_5=0.197$.

Step 9. As a result, the ranking order of alternatives is ${\mathrm{o}}_3>{\mathrm{o}}_5>{\mathrm{o}}_4>{\mathrm{o}}_2>{\mathrm{o}}_1$, which indicates that ${\mathrm{o}}_3$ is the best alternative.

6.2. Sensitivity Analysis

To prove the stability and validity of the proposed method, we conducted sensitivity analysis to explore the influence of parameters on the final ranking order. Since three parameters $\left(\sigma ,\alpha ,\beta \right)$ and fuzzy measures are involved in this paper, the following experiment is divided into four parts.

First, the sensitivity analysis of parameter $\sigma$ is shown in Figure 7 and

Table 11. The comprehensive regret–rejoice value with different $\sigma$ values.

	O1	O2	O3	O4	O5
$\sigma =0.1$	${\mathrm{U}}_1=-0.930$	${\mathrm{U}}_2=-0.919$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.004$	${\mathrm{U}}_5=0.366$
$\sigma =0.2$	${\mathrm{U}}_1=-0.994$	${\mathrm{U}}_2=-0.870$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.023$	${\mathrm{U}}_5=0.352$
$\sigma =0.3$	${\mathrm{U}}_1=-1.050$	${\mathrm{U}}_2=-0.837$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.041$	${\mathrm{U}}_5=0.347$
$\sigma =0.4$	${\mathrm{U}}_1=-1.109$	${\mathrm{U}}_2=-0.813$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.060$	${\mathrm{U}}_5=0.306$
$\sigma =0.5$	${\mathrm{U}}_1=-1.175$	${\mathrm{U}}_2=-0.792$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.078$	${\mathrm{U}}_5=0.197$
$\sigma =0.6$	${\mathrm{U}}_1=-1.251$	${\mathrm{U}}_2=-0.794$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.099$	${\mathrm{U}}_5=0.120$
$\sigma =0.7$	${\mathrm{U}}_1=-1.225$	${\mathrm{U}}_2=-0.829$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.121$	${\mathrm{U}}_5=0.074$
$\sigma =0.8$	${\mathrm{U}}_1=-1.147$	${\mathrm{U}}_2=-0.855$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.140$	${\mathrm{U}}_5=0.035$
$\sigma =0.9$	${\mathrm{U}}_1=-1.090$	${\mathrm{U}}_2=-0.873$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.158$	${\mathrm{U}}_5=-0.001$
$\sigma =1.0$	${\mathrm{U}}_1=-1.040$	${\mathrm{U}}_2=-0.889$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.180$	${\mathrm{U}}_5=-0.035$

. Based on the results in Figure 7 and Table 11, we can observe that the total ranking order of the five alternatives has no change when setting different values of $\sigma$, but the scores change significantly. The reason is that the parameter $\sigma$ is used to determine the weights of the CI and RCI. There are differences in the attributes considered for the CI and RCI. For example, ${\mathrm{C}}_{\tau \left(3\right)}=\left\{{\mathrm{C}}_3,{\mathrm{C}}_4,{\mathrm{C}}_5\right\}$ and $\mathrm{R}{\mathrm{C}}_{\tau \left(3\right)}=\left\{{\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3\right\}$. Obviously, ${\mathrm{C}}_{\tau \left(3\right)}$ has attributes ${\mathrm{C}}_3$, ${\mathrm{C}}_4$ and ${\mathrm{C}}_5$, but not attributes ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$. $\mathrm{R}{\mathrm{C}}_{\tau \left(3\right)}$ has attributes ${\mathrm{C}}_1$, ${\mathrm{C}}_2$ and ${\mathrm{C}}_3$, but not attributes ${\mathrm{C}}_4$ and ${\mathrm{C}}_5$.

Second, the sensitivity analysis of parameter $\alpha$ is shown in Figure 8 and

Table 12. The comprehensive regret–rejoice value with different $\alpha$ values.

	O1	O2	O3	O4	O5
$\alpha =0.1$	${\mathrm{U}}_1=-1.105$	${\mathrm{U}}_2=-0.769$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.264$	${\mathrm{U}}_5=-0.083$
$\alpha =0.2$	${\mathrm{U}}_1=-1.142$	${\mathrm{U}}_2=-0.781$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.156$	${\mathrm{U}}_5=0.096$
$\alpha =0.3$	${\mathrm{U}}_1=-1.175$	${\mathrm{U}}_2=-0.792$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.078$	${\mathrm{U}}_5=0.197$
$\alpha =0.4$	${\mathrm{U}}_1=-1.205$	${\mathrm{U}}_2=-0.802$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=-0.021$	${\mathrm{U}}_5=0.254$
$\alpha =0.5$	${\mathrm{U}}_1=-1.231$	${\mathrm{U}}_2=-0.809$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=0.020$	${\mathrm{U}}_5=0.285$
$\alpha =0.6$	${\mathrm{U}}_1=-1.253$	${\mathrm{U}}_2=-0.815$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=0.050$	${\mathrm{U}}_5=0.299$
$\alpha =0.7$	${\mathrm{U}}_1=-1.272$	${\mathrm{U}}_2=-0.819$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=0.070$	${\mathrm{U}}_5=0.302$
$\alpha =0.8$	${\mathrm{U}}_1=-1.287$	${\mathrm{U}}_2=-0.821$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=0.084$	${\mathrm{U}}_5=0.299$
$\alpha =0.9$	${\mathrm{U}}_1=-1.300$	${\mathrm{U}}_2=-0.823$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=0.093$	${\mathrm{U}}_5=0.290$
$\alpha =1.0$	${\mathrm{U}}_1=-1.310$	${\mathrm{U}}_2=-0.824$	${\mathrm{U}}_3=0.585$	${\mathrm{U}}_4=0.097$	${\mathrm{U}}_5=0.279$

. From Figure 8 and Table 12, we can observe that the total ranking order of the five alternatives does not change when setting different values of $\alpha$, but the scores changed significantly. The reason is that the normalized bidirectional projection of each alternative is a constant. Moreover, ${\mathrm{x}}^{\alpha }$ and $1-{\mathrm{e}}^{-\beta \left(\mathrm{x}-{\mathrm{x}}^{\ast }\right)}$ monotonically increase with respect to the projection value, and $1-{\mathrm{e}}^{-\beta \left({\mathrm{x}}^{\ast }-\mathrm{x}\right)}$ monotonically decreases with respect to the projection value. Therefore, the ranking order of five comprehensive utility values is constant.

For example, according to Table 10 and Table 11, we can find that $\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_1^{1-}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_2^{1-}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_4^{1-}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_5^{1-}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_3^{1-}$ and $\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_3^{1+}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_5^{1+}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_4^{1+}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_2^{1+}<\mathrm{N}\mathrm{B}\mathrm{p}\mathrm{o}{\mathrm{j}}_1^{1+}$. Then, we can deduce ${\mathrm{u}}_1^1<{\mathrm{u}}_2^1<{\mathrm{u}}_4^1<{\mathrm{u}}_5^1<{\mathrm{u}}_3^1$ and ${\mathrm{v}}_3^1<{\mathrm{v}}_5^1<{\mathrm{v}}_4^1<{\mathrm{v}}_2^1<{\mathrm{v}}_1^1$. Further, we have $\mathrm{R}\mathrm{E}{\mathrm{G}}_1^1<\mathrm{R}\mathrm{E}{\mathrm{G}}_2^1<\mathrm{R}\mathrm{E}{\mathrm{G}}_4^1<\mathrm{R}\mathrm{E}{\mathrm{G}}_5^1<\mathrm{R}\mathrm{E}{\mathrm{G}}_3^1$ and $\mathrm{R}\mathrm{E}{\mathrm{J}}_1^1<\mathrm{R}\mathrm{E}{\mathrm{J}}_2^1<\mathrm{R}\mathrm{E}{\mathrm{J}}_4^1<\mathrm{R}\mathrm{E}{\mathrm{J}}_5^1<\mathrm{R}\mathrm{E}{\mathrm{J}}_3^1$. Then, we can get ${\mathrm{U}}_1^1<{\mathrm{U}}_2^1<{\mathrm{U}}_4^1<{\mathrm{U}}_5^1<{\mathrm{U}}_3^1$.

Third, the sensitivity analysis of parameter $\beta$ is shown in Figure 9 and

Table 13. The comprehensive regret–rejoice value with different $\beta$ values.

	O1	O2	O3	O4	O5
$\beta =0.1$	${\mathrm{U}}_1=-0.085$	${\mathrm{U}}_2=-0.050$	${\mathrm{U}}_3=0.095$	${\mathrm{U}}_4=0.016$	${\mathrm{U}}_5=0.044$
$\beta =0.2$	${\mathrm{U}}_1=-0.180$	${\mathrm{U}}_2=-0.109$	${\mathrm{U}}_3=0.181$	${\mathrm{U}}_4=0.026$	${\mathrm{U}}_5=0.082$
$\beta =0.3$	${\mathrm{U}}_1=-0.286$	${\mathrm{U}}_2=-0.177$	${\mathrm{U}}_3=0.259$	${\mathrm{U}}_4=0.029$	${\mathrm{U}}_5=0.114$
$\beta =0.4$	${\mathrm{U}}_1=-0.404$	${\mathrm{U}}_2=-0.255$	${\mathrm{U}}_3=0.330$	${\mathrm{U}}_4=0.025$	${\mathrm{U}}_5=0.141$
$\beta =0.5$	${\mathrm{U}}_1=-0.535$	${\mathrm{U}}_2=-0.344$	${\mathrm{U}}_3=0.393$	${\mathrm{U}}_4=0.016$	${\mathrm{U}}_5=0.162$
$\beta =0.6$	${\mathrm{U}}_1=-0.680$	${\mathrm{U}}_2=-0.444$	${\mathrm{U}}_3=0.451$	${\mathrm{U}}_4=0.000$	${\mathrm{U}}_5=0.177$
$\beta =0.7$	${\mathrm{U}}_1=-0.841$	${\mathrm{U}}_2=-0.556$	${\mathrm{U}}_3=0.503$	${\mathrm{U}}_4=-0.022$	${\mathrm{U}}_5=0.188$
$\beta =0.8$	${\mathrm{U}}_1=-1.109$	${\mathrm{U}}_2=-0.681$	${\mathrm{U}}_3=0.551$	${\mathrm{U}}_4=-0.051$	${\mathrm{U}}_5=0.195$
$\beta =0.9$	${\mathrm{U}}_1=-1.216$	${\mathrm{U}}_2=-0.822$	${\mathrm{U}}_3=0.593$	${\mathrm{U}}_4=-0.086$	${\mathrm{U}}_5=0.197$
$\beta =1.0$	${\mathrm{U}}_1=-1.434$	${\mathrm{U}}_2=-0.977$	${\mathrm{U}}_3=0.632$	${\mathrm{U}}_4=-0.127$	${\mathrm{U}}_5=0.194$

. According to Figure 9 and Table 13, we can find that the total ranking order of the five alternatives does not change when setting different values of $\beta$, but the scores changed significantly. The reason is similar to the sensitivity analyses of parameter $\alpha$ because $1-{\mathrm{e}}^{-\beta \left(\mathrm{x}-{\mathrm{x}}^{\ast }\right)}$ monotonically increases with respect to the projection value and $1-{\mathrm{e}}^{-\beta \left({\mathrm{x}}^{\ast }-\mathrm{x}\right)}$ monotonically decreases with respect to the projection value. Thus, the ranking order of the five comprehensive utility values is constant with different values of $\beta$.

Finally, the sensitivity analysis of fuzzy measure is shown in

Table 14. The comprehensive regret–rejoice value with different fuzzy measures.

g(C1)	g(C2)	g(C3)	g(C4)	g(C5)	g(C6)	U1	U2	U3	U4	U5
0.28	0.33	0.26	0.30	0.32	0.29	−1.184	−0.787	0.585	−0.085	0.195
0.27	0.34	0.31	0.25	0.29	0.32	−1.175	−0.794	0.585	−0.079	0.211
0.30	0.26	0.35	0.25	0.31	0.27	−1.179	−0.785	0.585	−0.082	0.17
0.29	0.33	0.25	0.32	0.28	0.34	−1.173	−0.802	0.585	−0.809	0.205
0.31	0.27	0.30	0.33	0.26	0.29	−1.174	−0.797	0.585	−0.089	0.174
0.26	0.34	0.29	0.31	0.27	0.32	−1.179	−0.795	0.585	−0.077	0.202
0.32	0.28	0.30	0.25	0.34	0.27	−1.178	−0.786	0.585	−0.092	0.185
0.29	0.33	0.26	0.31	0.28	0.35	−1.171	−0.804	0.585	−0.088	0.209
0.30	0.27	0.32	0.28	0.34	0.26	−1.184	−0.780	0.585	−0.085	0.172
0.28	0.31	0.29	0.34	0.25	0.33	−1.173	−0.802	0.585	−0.082	−0.197

. According to Figure 9 and Table 13, we can find that the total ranking order of the five alternatives does not change when setting different fuzzy measures, but the scores have some differences. The differences in scores arise from varying fuzzy measures, leading to distinct GS values, while the identical rankings result from the monotonic nature of the utility and regret functions. This further demonstrates that the model proposed in this paper achieves a balance between flexibility and stability.

The sensitivity analysis demonstrates that the proposed method yields validity and reasonable results and presents suitable outcomes to support DM in decision-making. In real decision environments, DMs can choose appropriate parameters according to their actual needs.

6.3. Comparative Analysis

To further validate the effectiveness and rationality of the proposed method, this paper selected several representative decision-making methods for comparative analysis, including the PIT2HFPWA method [46], Wang’s method [42], Liu’s method [15], and three classical multi-attribute decision-making methods: TOPSIS [56], VIKOR [55], and COPRAS [57]. All methods were applied to the same supplier selection problem to ensure fairness and consistency in the comparison results. As Wang’s method and Liu’s method are constructed under an interval type-2 fuzzy environment, we utilized the following definition to convert PIT2HFN into IT2FN.

Definition 21.

Let $\mathrm{h}\left(\mathrm{p}\right)={\displaystyle {\cup }_{\mathrm{i}=1}^{\mathrm{T}}\left\{⌈{\mathrm{A}}_{\mathrm{i}},{\mathrm{p}}_{{\mathrm{A}}_{\mathrm{i}}}⌉\right\}}$ be a PIT2HFN. Then, the trapezoidal IT2FN $\mathrm{A}$ of $\mathrm{h}\left(\mathrm{p}\right)$ can be computed as

$$\mathrm{A}=\underset{\mathrm{i}=1}{\overset{\mathrm{T}}{\oplus }}{\mathrm{A}}_{\mathrm{i}}{\mathrm{p}}_{\mathrm{i}}=\left(\begin{array}{l}\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}1}^{\mathrm{u}}{\mathrm{p}}_{\mathrm{i}}},{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{u}}{\mathrm{p}}_{\mathrm{i}}},{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{u}}{\mathrm{p}}_{\mathrm{i}}},{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{u}}{\mathrm{p}}_{\mathrm{i}}};\underset{\mathrm{i}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{u}}\right)\right\}\right),\\ \left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}1}^{\mathrm{l}}{\mathrm{p}}_{\mathrm{i}}},{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}2}^{\mathrm{l}}{\mathrm{p}}_{\mathrm{i}}},{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}3}^{\mathrm{l}}{\mathrm{p}}_{\mathrm{i}}},{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}{\mathrm{a}}_{\mathrm{i}4}^{\mathrm{l}}{\mathrm{p}}_{\mathrm{i}}};\underset{\mathrm{i}}{\min }\left\{\mathrm{h}\left({\mathrm{A}}_{\mathrm{i}}^{\mathrm{l}}\right)\right\}\right)\end{array}\right)$$

(51)

Table 15. Ranking results of the proposed method and other decision-making methods.

Method	Ranking Order
PIT2HFPWA [45]	${\mathrm{o}}_5\succ {\mathrm{o}}_3\succ {\mathrm{o}}_4\succ {\mathrm{o}}_2\succ {\mathrm{o}}_1$
Wang’ method [38]	${\mathrm{o}}_3\succ {\mathrm{o}}_5\succ {\mathrm{o}}_4\succ {\mathrm{o}}_1\succ {\mathrm{o}}_2$
Liu’s method [15]	${\mathrm{o}}_3\succ {\mathrm{o}}_4\succ {\mathrm{o}}_5\succ {\mathrm{o}}_1\succ {\mathrm{o}}_2$
TOPSIS [56]	${\mathrm{o}}_4\succ {\mathrm{o}}_2\succ {\mathrm{o}}_3\succ {\mathrm{o}}_1\succ {\mathrm{o}}_5$
VIKOR [55]	${\mathrm{o}}_4\succ {\mathrm{o}}_3\succ {\mathrm{o}}_5\succ {\mathrm{o}}_2\succ {\mathrm{o}}_1$
COPRAS [57]	${\mathrm{o}}_3\succ {\mathrm{o}}_4\succ {\mathrm{o}}_5\succ {\mathrm{o}}_2\succ {\mathrm{o}}_1$
GS-PIT2HFABCI	${\mathrm{o}}_3\succ {\mathrm{o}}_5\succ {\mathrm{o}}_4\succ {\mathrm{o}}_2\succ {\mathrm{o}}_1$
GS-PIT2HFEBCI	${\mathrm{o}}_3\succ {\mathrm{o}}_5\succ {\mathrm{o}}_4\succ {\mathrm{o}}_1\succ {\mathrm{o}}_2$
GS-PIT2HFHBCI $\left(\gamma =1\right)$	${\mathrm{o}}_3\succ {\mathrm{o}}_5\succ {\mathrm{o}}_4\succ {\mathrm{o}}_2\succ {\mathrm{o}}_1$
GS-PIT2HFHBCI $\left(\gamma =2\right)$	${\mathrm{o}}_3\succ {\mathrm{o}}_5\succ {\mathrm{o}}_4\succ {\mathrm{o}}_1\succ {\mathrm{o}}_2$
GS-PIT2HFFBCI $\left(\tau =1.01\right)$	${\mathrm{o}}_3\succ {\mathrm{o}}_5\succ {\mathrm{o}}_4\succ {\mathrm{o}}_2\succ {\mathrm{o}}_1$

presents the decision results of the proposed method, several representative fuzzy methods, the recently proposed MARCOS method, and three classical multi-criteria decision-making methods, with a visual representation shown in Figure 10.

As shown in Table 13, although the ranking results obtained by different methods exhibit certain discrepancies in individual alternatives, the overall ranking trends demonstrate good consistency. Notably, both the proposed method and most fuzzy-information-based comparison methods identify supplier ${\mathrm{o}}_3$ as the optimal option, indicating that the proposed method effectively captures the comprehensive performance of suppliers in complex uncertain environments, demonstrating high stability and reliability.

From a management perspective, further analysis of the final ranking reveals that supplier ${\mathrm{o}}_3$ secured the top position primarily due to its balanced strengths across multiple key sustainability dimensions. Specifically, ${\mathrm{o}}_3$ demonstrates mature technical capabilities and a well-structured and standardized management system, along with consistent performance in integrity records and social responsibility. This indicates lower operational and reputational risks in long-term collaborations, reflecting strong overall sustainability. In contrast, the second-ranked supplier ${\mathrm{o}}_5$ excels in social responsibility and compliance, demonstrating positive external impact and controllable risks. However, its technical capabilities and management maturity slightly lag behind ${\mathrm{o}}_3$, resulting in a marginally lower overall evaluation. The mid-tier supplier ${\mathrm{o}}_4$ meets basic requirements in technical and managerial capabilities. However, it lacks distinct advantages in economic factors and risk prevention, limiting its long-term partnership stability. Regarding the lower-ranked suppliers ${\mathrm{o}}_2$ and ${\mathrm{o}}_1$, while ${\mathrm{o}}_2$ possesses a certain technical foundation, their performance in critical dimensions such as integrity, social factors, and economic risk is relatively weak. Meanwhile, ${\mathrm{o}}_1$ exhibits significant shortcomings in multiple areas including technology, management, and social responsibility, presenting the highest overall risk level. This makes them unsuitable for supporting the enterprise’s long-term sustainable development goals within the context of smart factory construction.

To quantitatively analyze the consistency among the ranking results of different methods, this paper further employs Spearman’s rank correlation coefficient to compare the ranking outcomes of each method. The formula for calculating Spearman’s rank correlation coefficient is as follows:

$${\mathrm{S}}_{\mathrm{c}\mathrm{c}}=1-{\displaystyle \frac{6{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{l}}_{\mathrm{i}}^2}}{\mathrm{n}({\mathrm{n}}^2-1)}}$$

(52)

where $\mathrm{n}$ denotes the number of alternatives and ${\mathrm{l}}_{\mathrm{i}}={\mathrm{p}}_{\mathrm{i}}-{\mathrm{q}}_{\mathrm{i}}$ with ${\mathrm{p}}_{\mathrm{i}}$ and ${\mathrm{q}}_{\mathrm{i}}$ representing the ranking positions of alternative ${\mathrm{o}}_{\mathrm{i}}$ obtained by two different decision-making methods. The results listed in

Table 16. Spearman’s rank correlation coefficient for sorting results obtained by different methods.

Methods	GS-PIT2HFABCI	PIT2HFPWA	Wang’s Method	Liu’s Method	TOPSIS	VIKOR	COPRAS
GS-PIT2HFABCI	1.00	0.60	0.90	0.80	−0.40	0.70	0.90
PIT2HFPWA	-	1.00	0.50	0.60	0.00	0.90	0.70
Wang’s method	-	-	1.00	0.90	0.00	0.60	0.80
Liu’s method	-	-	-	1.00	0.10	0.80	0.90
TOPSIS	-	-	-	-	1.00	−0.10	−0.30
VIKOR	-	-	-	-	-	1.00	0.90
COPRAS	-	-	-	-	-	-	1.00

.

Table 16 demonstrates that the proposed method exhibits overall high rank correlation coefficients compared to the benchmark methods, with most coefficients exceeding 0.7, indicating strong consistency in ranking trends. Concurrently, the incorporation of decision-maker psychological factors and inter-attribute correlations during modeling introduces unique characteristics to the proposed method, resulting in ranking discrepancies for certain suppliers relative to other approaches.

In what follows, we further investigate the merits the proposed method by comparing and analyzing the characteristics and differences between the proposed method and some existing methods. Table 17 summarizes the comparison between the proposed PIT2HFS-based method and existing methods in terms of linguistic term processing, psychological state consideration, attribute interrelation, multi-expert handling, and similarity measures.

(1)

Regarding preference information representation, this paper employs PIT2HFSs to characterize decision information, whereas some existing methods [10,11,15,38,56,57,58,59,60,61] primarily rely on T1FSs, IT2FSs or IT2HFSs. Compared to T1FSs, IT2FSs, and IT2HFSs, PIT2HFSs can more precisely capture the uncertainty and hesitation inherent in linguistic evaluations. This enables a more comprehensive reflection of the complex subjective judgment involved in supplier selection, thereby enhancing the reliability of decision outcomes.

(2)

In modeling decision-makers’ psychological states and risk attitudes, this paper introduces an improved regret–rejoice function that explicitly incorporates psychological behaviors when facing different options in the decision process. Existing methods [10,15,45,56,57,58,59,60,61] failing to account for decision-makers’ psychological preferences and risk perceptions. Given that supplier selection typically involves uncertainty and risk, and decision-makers often exhibit bounded rationality, the proposed method provides a more reasonable representation of real-world decision-making behavior. Furthermore, compared to Wang’s method [38], the regret–relief model in this paper simultaneously considers both optimal and worst-case reference scenarios, resulting in a more comprehensive characterization of psychological behavior.

(3)

Regarding attribute correlation modeling, this paper introduces the PIT2HFPBCI operator and the GS-PIT2HFPBCI operator to characterize interdependencies among attributes. In contrast, most comparative methods [11,38,55,56,57,59,60,61], assume attributes are mutually independent, failing to explicitly account for attribute correlations and grouping structures. In reality, within the vendor selection problem, internal capability attributes, such as technology, management, and integrity, often exhibit intrinsic connections, while external environmental attributes also possess certain interdependencies. The structural modeling approach in this paper better aligns with actual decision-making scenarios.

(4)

Regarding similarity measures, this paper proposes a normalized bidirectional projection measure based on PIT2HFSs, simultaneously considering both distance and directional information between schemes. Compared to distance-based measures in [10,15,45,55,56,57,58,59], this metric more comprehensively reflects the relative closeness between schemes. Furthermore, unlike the single reference point or utility ratio-based evaluation mechanisms in Wang’s method [38], our approach enhances the robustness and interpretability of scheme comparisons through dual reference points and normalization.

(5)

Table 15 demonstrates that when $\gamma$ = 1, the results of the HAMAC method are fully consistent with those of the algebraic process. Similarly, when $\gamma$ = 2, the results of the HAMAC process are fully consistent with those generated by the Einstein process. This equivalence stems from the parametric nature of the HAMAC process. When $\gamma$ = 1, it coincides with the quantum process; when $\gamma$ = 2, it perfectly aligns with the results of the Einstein process. This equivalence stems from the parametric principle of the HAMAC process. The HAMAC process can be defined within a more general framework, where it constitutes a special case, with the algebraic process and the identity process appearing as its particular instances. Furthermore, the HAMAC process can be integrated into an even broader framework, where both the algebraic process and the Einstein process emerge as its special cases.

(6)

According to the comparative analysis shown in Table 15, as the parameter $\tau$ approaches zero, the results of the Frank function converge toward the algebraic fan. They converge gradually. This behavior is as follows: as the parameter $\tau$ tends toward infinity, the Frank function and the higher-order polynomial functions become identical.

Table 17. A comparison of the characteristics and differences for different methods.

Methods	Ways of Processing Linguistic Term	Psychological State and Risk Attitude	Interrelation Between Attributes	Multi Expert and Multi Attribute	Similarity Measure
PIT2HFPWA [45]	PIT2HFSs	No	Yes	Yes	Distance
PIT2HFPWG [45]	PIT2HFSs	No	Yes	Yes	Distance
Hu’s method [61]	IT2HFSs	No	No	No	No
Wang’ method [38]	IT2FSs	Yes	No	No	Projection
Xing’s method [10]	IT2FSs	No	Yes	Yes	Distance
Liu’ method [15]	IT2FSs	No	Yes	No	Bidirectional projection
Qin’s method [11]	IT2FSs	Yes	No	No	Distance
TOPSIS [56]	IT2FSs	No	No	Yes	Euclidean distance to ideal
VIKOR [55]	Interval valued number	Yes	No	Yes	L1 or L∞ distance with compromise ranking
COPRAS [57]	Fuzzy number	No	No	Yes	Normalized weighted sums
Q-ROFWBMO [58]	Q-ROFS	No	Yes	No	Distance
SFWAO [59]	SFS	No	No	No	Distance
NMWAAO [60]	NM	No	No	No	Possibility degree
The proposed method	PIT2HFs	Yes	Yes	Yes	Normalized bidirectional projection

Table 17. A comparison of the characteristics and differences for different methods.

Methods	Ways of Processing Linguistic Term	Psychological State and Risk Attitude	Interrelation Between Attributes	Multi Expert and Multi Attribute	Similarity Measure
PIT2HFPWA [45]	PIT2HFSs	No	Yes	Yes	Distance
PIT2HFPWG [45]	PIT2HFSs	No	Yes	Yes	Distance
Hu’s method [61]	IT2HFSs	No	No	No	No
Wang’ method [38]	IT2FSs	Yes	No	No	Projection
Xing’s method [10]	IT2FSs	No	Yes	Yes	Distance
Liu’ method [15]	IT2FSs	No	Yes	No	Bidirectional projection
Qin’s method [11]	IT2FSs	Yes	No	No	Distance
TOPSIS [56]	IT2FSs	No	No	Yes	Euclidean distance to ideal
VIKOR [55]	Interval valued number	Yes	No	Yes	L1 or L∞ distance with compromise ranking
COPRAS [57]	Fuzzy number	No	No	Yes	Normalized weighted sums
Q-ROFWBMO [58]	Q-ROFS	No	Yes	No	Distance
SFWAO [59]	SFS	No	No	No	Distance
NMWAAO [60]	NM	No	No	No	Possibility degree
The proposed method	PIT2HFs	Yes	Yes	Yes	Normalized bidirectional projection

7. Conclusions and Outlook

Section 7 comprises two sections. The first part presents our conclusions, and the second part outlines future research directions, which are further divided into theoretical prospects and applied prospects.

7.1. Conclusion

This paper puts forward the novel SSS method based on the Choquet integral, the generalized Shapley value, regret theory and PIT2HFSs, which can be used to express the DM’s uncertainty evaluation opinion more accurately and effectively. The main contributions of this paper are organized as follows:

(1)

As the attributes can be divided into some partitions and the attributes of each partition are related, while no relationships exist between the partitions, we define the PIT2HFPBCI operator and the GS-PIT2HFPBCI operator and study the properties and special cases of these novel operators. Compared with the methods [10,11,52,61], our method not only considers attribute relationships within partitions but also accounts for relationships between different partitions, making it more suitable for complex multi-attribute decision-making problems.

(2)

As existing methods [48,49,50] based on regret theory only compare with the best alternative but ignore the worst alternative when calculating the regret–rejoice value, we utilized the improved regret–rejoice function in the literature [51,56], which simultaneously considers optimal and suboptimal values, thereby making the ranking results more comprehensive and reasonable.

(3)

Because the projection measure is a powerful tool to measure the closeness degree, we define the projection measure of PIT2HFSs. At the same time, we further define the normalized bidirectional projection measure of PIT2HFSs. Compared with the distance measure [10,11,61] and projection measure [38], the normalized bidirectional projection measure not only takes both distance and angle into account but also two reference points, so as to reflect the actual situation more realistically.

(4)

We have implemented the bidirectional idea throughout our overall research framework, resulting in a more scientific and robust ranking of final alternative options. This provides an effective support tool for multi-attribute group decision-making.

7.2. Outlook

Future research about theoretical prospects may incorporate machine learning [62] and intelligent optimization methods [63] to achieve adaptive determination of attribute partitioning [64], parameter learning [65], and fuzzy measures [66], thereby reducing reliance on expert experience. Simultaneously, the PIT2HFS–Choquet model can be extended to dynamic evolutionary decision environments by constructing time-varying attribute weights [67], interaction relationships [68], and preference functions [69] to support long-term supplier evaluation and real-time decision-making. Building upon this foundation, incorporating tripartite decision-making, interval information systems [70], and social network [71] structures holds promise for enhancing the model’s robustness and interpretability under conditions of group conflict and incomplete information.

Regarding the applied prospects, this method can be extended beyond manufacturing to complex multi-attribute decision scenarios. For instance, in logistics and supply chain management, it can be applied to supplier performance evaluation, transportation plan optimization, and multi-stage risk control [72]. In energy and environmental fields, it can assist energy procurement, carbon emission management, and renewable energy allocation decisions [73]. In healthcare and public services, it can support pharmaceutical or equipment procurement, medical resource allocation, and emergency response plan evaluation [74]. In drone and intelligent system evaluations, it enables mission plan selection and comprehensive assessments of performance and safety [75].