Choquet Integral Research Articles

Fermatean Hesitant Fuzzy Choquet Integral Aggregation Operators

Fermatean fuzzy hesitant sets provide a flexible and powerful tool for decision making, allowing decision makers to incorporate uncertainty and hesitation into their decision making process, and enabling them to make more informed and effective decisions in complex and uncertain environments. Considering that many factors are interdependent in reality while existing methods cannot solve this problem, we propose Fermatean hesitant fuzzy Choquet integral ordered aggregation operators based on the Fermatean hesitant fuzzy set and Choquet integral, and establish their related properties. These operators including averaging and geometric operators not only handle situations where decision criteria or preferences are interdependent, but also provides the decision under ideal and non-ideal situation. Additionally, we present a multi-attribute decision-making method for Fermatean hesitant fuzzy information using these operators. We have validated the proposed method, and its practicality and effectiveness are demonstrated through numerical examples from previous research regarding Fermatean Hesitant Fuzzy set, including sensitivity analysis and comparison.

Quadrature-Inspired Generalized Choquet Integral in an Application to Classification Problems

Quadrature-Inspired Generalized Choquet Integral in an Application to Classification Problems

A Novel Planetary Gearbox Fault Diagnosis Method for Nuclear Circulating Water Pump With Class Imbalance and Data Distribution Shift

Deep learning (DL) has demonstrated splendid performance in fault diagnosis with sufficient samples and ideal operating environments. However, in practice, it is hard to acquire adequate fault samples from high-reliability equipment, thus inducing challenges in applying DL for intelligent diagnosis. Moreover, data distribution shift occurs due to working load variation or environment noise interference, severely degrading DL models’ performance. Aiming at the above problems, this article proposes a method called Q networks calibrated ensemble (QCE). Specifically, two improved deep Q networks are employed to enhance the attention to critical feature samples and fault samples by quantifying feature distribution and utilizing fault prior knowledge. Afterward, the calibrated Choquet integral is adopted to achieve the models’ ensemble, further enhancing the generalization ability and robustness of the diagnosis results. The proposed method has been verified on the nuclear circulating water pump (NCWP) test bench. Results indicate the superiority and reliability of the QCE method. In addition, results when working load changes demonstrate the outstanding generalization ability of the proposed method for cross-domain fault diagnosis with class imbalance. Furthermore, gradient-weighted class activation mapping (Grad-CAM) visualization verifies that the proposed method effectively learns fault features instead of noise.

Interval Valued q- Rung Orthopair Hesitant Fuzzy Choquet Aggregating Operators in Multi-Criteria Decision Making Problems

Abstract: In this paper, we introduce Interval valued q- Rung Orthopair Hesitant fuzzy sets (IVq-ROHFS) with motivation of Interval valued pythagorean Hesitant fuzzy sets [44] as a new concept. Then, we give some basic operations as complement, union, intersection, addition, scalar multiplication, scalar power. Also, we combine to (IVq-ROHFS) and choquet integral , together with aggregating operators, and develop to Interval valued q- rung orthopair hesitant fuzzy Choquet averaging operator (IVq-ROHCA) and Interval valued q- rung orthopair hesitant fuzzy Choquet geometric operator (IVq-ROHCG). Then, we offer to indicate soft approach of proposed IVq-ROHCA and IVq-ROHCG an example adopted from Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS (IVIHCI) [43]. The obtained results are agreement with IVIHCI but presented IVq-ROHCA and IVq-ROHCG have more advantages than existing structures as Interval-valued intuitionistic hesitant fuzzy sets (IVIHFS), interval-valued Pythagorean Hesitant fuzzy sets (IVPHFS) with reasons changing according to need, requirement, prefer of decision makers and moreover, being IVIHFS and IVPHFS are special cases of IVq-ROHFS. It is open from comparative analysis that the while some of offered approaches are giving no solution for some values, our operators present to needed results.

A novel framework for resilient overhead power distribution networks

The natural disasters that are classified as high impact low probability (HILP) events have a major effect on the power system infrastructure, in particular, the overhead power distribution networks (OPDNs) causing significant challenges in maintaining continuity of service to critical loads (CLs). A resilient OPDN should maintain operational normalcy by quickly absorbing, adapting and recovering from the HILP events resulting in an improved customer satisfaction through continuity of electricity supply and minimal safety hazards. Modern OPDS predominantly consists of overhead electric lines and poles with diverse characteristics by virtue of their age, maintenance, location etc. Unlike the normal operating conditions, quick and accurate decision making from the network operator becomes very critical in maintaining the resiliency of a OPDN during HILP events. This paper proposes a novel framework for resiliency assessment, quantification, enabling and enhancement that facilitates resilient reconfiguration of an OPDN during HILP events. Two novel physical system metrics with multiple sub- factors specific to the overhead networks are also proposed for resiliency assessment followed by resiliency quantification using diamond pairwise comparison, kth-order additive fuzzy measure and Choquet integral. The resiliency enabling and enhancement algorithms for contingency scenarios as well as cost-benefit analysis for network planning strategies are presented in this work. The novel resiliency framework presented in this work is used to enable the resiliency of a OPDN and provide the relevant information about the feasible network options with their resiliency value. The performance of the proposed framework is illustrated on the 415V OPDN in Nayabad area of the CESC Limited, Kolkata, India through simulation case studies.

On the extended Choquet-Sugeno-like operator

In this short paper, we introduce and discuss the extended Choquet-Sugeno-like operator. It generalizes most of modifications of the Choquet integral on a finite set known in the literature providing a partially positive answer to the question posed by H. Bustince et al. in 2021. It also includes unknown Choquet-like and Sugeno-like operators. The standard optimization problem in cluster analysis is formulated using multivariable extension of the new operator.

Coherent Upper Conditional Previsions Defined through Conditional Aggregation Operators

Conditional aggregation operators are introduced, and coherent upper conditional previsions are constructed by sub-additive, positively homogenous, and shift-invariant conditional aggregation operators. Composed operators defined by positively homogenous conditional aggregation operators are proven to be conditional aggregation operators and they are involved in the construction of coherent upper conditional previsions. The given results show that the composed conditional aggregation operator obtained as the supremum of the class of the Choquet integrals with respect to Hausdorff outer measures defined by bi-Lipschitz equivalent metrics is a coherent upper conditional prevision.

Social network utility consensus model with empathic and fuzzy interactions

In social network group decision-making, interactions between decision makers are the basis for the formation of a sufficient consensus. Based on the traditional minimum cost consensus models, this study adopts Choquet integral and empathy theory to consider interaction constraints that exist in the integration of a consensus opinion and the evolution of opinion adjustments. We employ social network structures, interaction characteristics, and influence indices as weight constraints to study the convergence law of social network cost consensus, and apply piecewise-linear utility functions to fit the heterogeneity and stage characteristics of decision makers’ attitudes. Three cost consensus models of social utility maximization are constructed with consideration of fuzzy interaction, empathic interaction, and a mixture of fuzzy and empathic interactions, respectively. And this paper extracts the correlation between traditional cost consensus models and cost consensus models with interaction. A case analysis of emergency material reserve decision-making explains the good applicabilities the constructed models in this paper.

TOPSIS Method Based on Hamacher Choquet-Integral Aggregation Operators for Atanassov-Intuitionistic Fuzzy Sets and Their Applications in Decision-Making

The collection of Hamacher t-norms was created by Hamacher in 1970, which played a critical and significant role in computing aggregation operators. All aggregation operators that are derived based on Hamacher norms are very powerful and are beneficial because of the parameter 0≤ζ≤+∞. Choquet first posited the theory of the Choquet integral (CI) in 1953, which is used for evaluating awkward and unreliable information to address real-life problems. In this manuscript, we analyze several aggregation operators based on CI, aggregation operators, the Hamacher t-norm and t-conorm, and Atanassov intuitionistic fuzzy (A-IF) information. These are called A-IF Hamacher CI averaging (A-IFHCIA), A-IF Hamacher CI ordered averaging (A-IFHCIOA), A-IF Hamacher CI geometric (A-IFHCIG), and A-IF Hamacher CI ordered geometric (A-IFHCIOG) operators; herein, we identify their most beneficial and valuable results according to their main properties. Working continuously, we developed a multi-attribute decision-making (MADM) procedure for evaluating awkward and unreliable information, with the help of the TOPSIS technique for order performance by similarity to the ideal solution, and derive operators to enhance the worth and value of the present information. Finally, by comparing the pioneering information with some of the existing operators, we illustrate some examples for evaluating the real-life problems related to enterprises, wherein the owner of a company appointed four senior board members of the enterprise to decide what was the best Asian company in which to invest money, to show the supremacy and superiority of the invented approaches.

D-XC Integrals: On the Generalization of the Expanded Form of the Choquet Integral by Restricted Dissimilarity Functions and Their Applications

Restricted dissimilarity functions (RDFs) were introduced to overcome problems resulting from the adoption of the standard difference. Based on those RDFs, Bustince <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> introduced a generalization of the Choquet integral (CI), called d-Choquet integral, where the authors replaced standard differences with RDFs, providing interesting theoretical results. Motivated by such worthy properties, joint with the excellent performance in applications of other generalizations of the CI (using its expanded form, mainly), this article introduces a generalization of the expanded form of the standard Choquet integral (X-CI) based on RDFs, which we named d-XC integrals. We present not only relevant theoretical results but also two examples of applications. We apply d-XC integrals in two problems in decision making, namely a supplier selection problem (which is a multicriteria decision-making problem) and a classification problem in signal processing, based on motor-imagery brain-computer interface (MI-BCI). We found that two d-XC integrals provided better results when compared to the original CI in the supplier selection problem. Besides that, one of the d-XC integrals performed better than any previous MI-BCI results obtained with this framework in the considered signal processing problem.

A New Consensus Model Based on Trust Interactive Weights for Intuitionistic Group Decision Making in Social Networks

A promising feature for group decision making (GDM) lies in the study of the interaction between individuals. In conventional GDM research, experts are independent. This is reflected in the setting of preferences and weights. Nevertheless, each expert's role is played through communication, collaboration, and cooperation with other individuals. The interaction from others may affect the power of an expert as well as his/her opinion. Furthermore, it is noted that a link path with the highest degree of trust is the most efficient information transmission channel. Inspired by these findings, an optimal trust-induced consensus process is designed with the usage of intuitionistic fuzzy preference relation. The comprehensive weight of each expert is decomposed into two portions, namely: 1) the individual weights and 2) interactive weights. Three optimization models are constructed to achieve weight parameters under different decision situations, where the weight parameters are represented through a 2-order additive fuzzy measure and the Shapley value. To reflect the interaction, the Choquet integral is employed for aggregating opinions, and a novel distance measure is adopted for accomplishing a consensus index. An illustrative example and comparison are put in practice to show the effectiveness and improvements of the proposed method.

An MCDM approach based on some new Pythagorean cubic fuzzy Frank Muirhead mean operators

Pythagorean cubic fuzzy sets (PCFSs) are the most convenient aid to depict ambiguous data in practical decision-making situations. The Muirhead mean (MM) possesses the ability to capture the interrelationship between the criteria. Further, the MM generalizes several significant operators, for instance, the Bonferroni mean (BM) and the Maclaurin symmetric mean (MSM), etc. The Frank triangular norms can offer significant adaptability and robustness due to the presence of an additional parameter as compared to other families of triangular norms. Therefore, Muirhead mean and Frank operations can be combined for obtaining suitable results during decision-making under Pythagorean cubic fuzzy information with interrelated criteria. The primary goal of this study is to introduce some new MM operators based on Frank operations, namely, PCFFMM and PCFFGMM. Then some of the properties of PCFFMM and PCFFGMM are described. Moreover, an MCDM approach is formulated using PCFFMM and PCFFGMM operators. Eventually, the projected MCDM approach is demonstrated with a numerical example. The phenomenon of interrelationships among criteria of real-life problems can be suitably handled by the PCFFMM and PCFFGMM. The overall ranking result obtained by utilizing the proposed operators PCFFMM or PCFFGMM is more appropriate as compared to the ones obtained by using the BM, MSM, Heronian mean (HM), Choquet integral (CI), etc. Moreover, PCFFMM or PCFFGMM can be preferred ahead of BM, MSM, HM, and CI due to its computation simplicity, accuracy, flexibility, and robustness during the aggregation process of multiple correlated input data.

Spherical Fuzzy Choquet Integral-Based VIKOR Method for Multi-Criteria Group Decision-Making Problems

Inter-dependency among the decision criteria and difficulty of handling the information in form of “yes,” “abstain,” “no,” and “refusal” are two important issues to be addressed in multicriteria group decision-making (MCGDM) problems in the environment of uncertainty. Recently, spherical fuzzy set (SFS) has gained the attention of the researchers for MCGDM problems due to their capability of handling decision makers’ preferences in the form “yes,” “abstain,” “no,” and “refusal” more efficiently than picture fuzzy set. Choquet integral operator has an edge over traditional aggregation operator in the modeling of interaction among the preferences of decision makers and decision criteria and therefore, we define Choquet integral operator for SFS. The main objective of this study is to define the spherical fuzzy Choquet integral (SFCI) operator and to extend the VIsekriterijumska optimizacija I Kmpromisno Resenje (VIKOR) for spherical fuzzy environment. In this study we define spherical fuzzy Choquet average, spherical fuzzy Choquet geometric and spherical fuzzy Choquet integral distance (SFCID) operators and propose SFCI-based VIKOR method for MCGDM problems. Proposed SFCI-based VIKOR method is applied on warehouse selection problems and results are compared with existing MCGDM methods.

CUSTOMER PREFERENCE ANALYSIS FROM ONLINE REVIEWS BY A 2-ADDITIVE CHOQUET INTEGRAL-BASED PREFERENCE DISAGGREGATION MODEL

Online reviews have become an important data source for analyzing consumers’ preferences. Consumer preference analysis assists product managers to understand consumers’ propensity for different product attributes and make consumer-oriented market strategies. Existing studies on consumer preference analysis used simple additive algorithms to represent the relationship between overall ratings and attribute ratings, but ignored the interactions between attributes. In addition, not all attribute ratings were given by consumers when calculating the overall ratings of a product. To fill these gaps, a preference model based on the extended 2-additive Choquet integral is constructed. The 2-additive Choquet integral can reflect the importance of attributes and the interactions between pairs of attributes when integrating attribute ratings. In cases where consumers choose only a subset of product attributes to rate a product, we introduce the scale parameter into the 2-additive Choquet integral to characterize the relationship between different attribute subsets. Afterwards, a preference disaggregation paradigm based on nonlinear programming is provided to solve the preference model. Finally, the proposed method is validated by experimental analysis using the dataset collected from TripAdvisor.com. Experimental outcomes indicate that our approach can deduce consumers’ preferences and approximate the evaluation behavior of consumers efficiently.

Cross-platform hotel evaluation by aggregating multi-website consumer reviews with probabilistic linguistic term set and Choquet integral.

In order to adequately utilize and integrate both ratings and comments from multiple websites, this paper proposes a new hotel evaluation model with probabilistic linguistic information processing. Taking consumers' possible psychological activities when leaving their reviews into consideration, this paper adapts the Weber-Fechner Law with the linguistic scale function and develop a novel unbalanced linguistic scale function. This paper also attempts to develop a method that enables adjusting linguistic-term formations among different websites to make full use of information. Then, we learn the decision criteria and the corresponding weight of hotel evaluation based on analyzing rating rules and consumer comments. Moreover, considering the interrelationships among criteria, this paper extends the Choquet integral to the probabilistic linguistic term set (PLTS) environment and designs some novel fusion operators. Furthermore, considering the fact that different websites mostly focus on heterogeneous hotel criteria, this paper puts forward a weighted averaging linear assignment based ranking method with the aid of PLTS Choquet integral. Finally, a case study of hotel evaluation is given to illustrate the validity and applicability of our proposed approach.

A complex spherical fuzzy CRADIS method based Fine-Kinney framework for occupational risk evaluation in natural gas pipeline construction

Occupational risk evaluation is one of the most indispensable issues in the risk prevention and control process for the natural gas pipeline construction (NGPC) project. The Fine-Kinney model, recognized as an effective occupational risk evaluation technique, has limited capability to handle the occupational risk analysis problem under the complex spherical fuzzy (CSF) environment. Accordingly, a synthetical Fine-Kinney framework based on the compromise ranking of alternatives from distance to ideal solution (CRADIS) method is developed to overcome these downsides of occupation risk analysis in the NGPC project within the CSF context. A prioritized weighted average (PWA) operator for complex spherical fuzzy numbers (CSFNs) is incorporated into the group risk evaluation matrix generation process, which can take the priority degrees of experts into account. Then, the extended CRADIS method-based Fine-Kinney framework is generated, in which the Choquet integral for CSFNs is incorporated to reflect the impact of interactive risk factors. Next, the detailed solution procedures of the framework for handling the occupational risk evaluation problem are presented. Finally, the described framework is employed as an empirical example of occupational risk analysis for the NGPC project to demonstrate its feasibility in practice. After that, a sensitivity analysis of the parameter is investigated to testify to the stability and rationality of the reported synthetical Fine-Kinney framework. Subsequently, to further display the advantages of the developed Fine-Kinney framework, a comparative study is implemented to discuss the evaluation result of occupational risk derived from the proposed framework and those of the existing similar Fine-Kinney frameworks. The analysis results indicate that the occupational risk (attack by human or animal) with the maximum risk priority value (1.000) using the framework is identified as the most serious risk for the NGPC project.

Secure Intelligent Fuzzy Blockchain Framework: Effective Threat Detection in IoT Networks

Integrating blockchain into the Internet of Things (IoT) for security is a new development in computational communication systems. While security threats are changing their strategies and constructing new threats on blockchain-based IoT systems. Also, in combining blockchain with IoT networks, malicious transactions and active attacks deliver more vulnerabilities, privacy issues, and security threats. The concept of blockchain-based IoT attacks is a hot topic in both IoT and blockchain disciplines. Network attacks are a type of security and privacy threat and cover the exact scope of threats related to the combination of IoT and blockchain. Even though blockchain has potential security benefits, new cyberattacks have emerged that make blockchain alone insufficient to deal with threats and attacks in IoT networks since vagueness and ambiguity issues are unavoidable in IoT data. The heterogeneous nature of IoT sources has made uncertainty a critical issue in IoT networks. Deep Learning (DL) models have difficulty dealing with uncertainty issues and cannot manage them efficiently as an essential tool in security techniques. Thus, we need better security, privacy, and practical approaches, such as efficient threat detection against network attacks in blockchain-based IoT environments. Also helpful to consider fuzzy logic to tackle deterministic issues when DL models face uncertainty. This paper designs and implements a secure, intelligent fuzzy blockchain framework. This framework utilizes a novel fuzzy DL model, optimized adaptive neuro-fuzzy inference system (ANFIS)-based attack detection, fuzzy matching (FM), and fuzzy control system (FCS) for detection of network attacks. The proposed fuzzy DL applies the fuzzy Choquet integral to have a powerful nonlinear aggregation function in the detection. We use metaheuristic algorithms to optimize the attack detection error function in ANFIS. We also validate transactions via FM to tackle fraud detection and efficiency in the blockchain layer. This framework is the first secure, intelligent fuzzy blockchain framework that identifies and detects security threats while considering uncertainty issues in IoT networks and having more flexibility in decision-making and accepting transactions in the blockchain layer. Evaluation results verify the efficiency of the blockchain layer in throughput and latency metrics and the intelligent fuzzy layer in performance metrics (Accuracy, Precision, Recall, and F1-Score) in threat detection on both blockchain and IoT network sides. Additionally, FCS demonstrates that we obtain an effective system (stable model) for threat detection in blockchain-based IoT networks.

Logistic autonomous vehicles assessment using decision support model under spherical fuzzy set integrated Choquet Integral approach

Autonomous vehicles (AVs) are the newest products in the intelligent transportation system that can move around with minimal human intervention. These products continue their path with all kinds of sensors with different parts. Effective use of these technologies in the logistics industry can create a competitive advantage. Nowadays, there are many AVs, some of which are superior to others in terms of build quality, variety of features, and design. Choosing an efficient, optimal, and reliable vehicle is one of the most important challenges in logistics planning. Therefore, choosing an AV based on a series of criteria can be considered a multi-criteria decision-making (MCDM) problem. Due to the complication of decision-making issues, criteria are usually not independent of each other and there are relationships between them. Therefore, this study develop an extended MCDM framework based on Choquet integral (CI) under group decision-making with a Spherical fuzzy set (SFS) for assessing logistics AVs. The CI technique is expanded with SFS to increase the power of CI. Furthermore, the combination of CI with SFS leads to greater freedom for decision makers to express opinions and use three independent membership functions. Accordingly, the interactions between the criteria are considered and the skepticism and uncertainty present during the decision are controlled. The proposed approach is implemented in selecting the best AVs in the logistics industry, and the results are compared with Pythagorean fuzzy CI and Intuitionistic fuzzy CI. Moreover, sensitivity analysis is done by changing the weights and creating different scenarios to confirm and check the robustness of the proposed approach results. The results indicate the suggested approach's efficiency and the ranking's stability in different scenarios.

Macsum aggregation learning

Identification of a multi-input/single-output system enables the creation of a mathematical model that as accurately as possible represents the input-output relation induced by a physical system. When there is little information about the physical laws related to the system or when the system is too complex, methods such as parametric identification are used to define the system model. In that case, preliminary assumptions can be made about the system leading to a parametric aggregation function based model. This parametric model may be learned by estimating the parameters of this aggregation function from a representative set of inputs/outputs. A major difficulty is to design a model that is relatively simple yet precise enough to meet the user's needs. Linear models are commonly used because they meet these two contradictory constraints. However, use of a linear model is often at the expense of the accuracy of input-output relationship description. In a recent paper, a new modeling approach was proposed, under the name of macsum modeling [31], which aims at replacing the linear model concept by a set of linear models. The obtained aggregation function leads to an interval-valued output. This represents the lack of accuracy of the model to predict the system output when the inputs are known. An interesting feature of this model is that it is ruled by a single precise parametric vector whereas the output is imprecise. Moreover, the vector dimension is equal to that of the input space. In this paper, we propose a method to learn such a model so that it best reflects the input-output relationship of the considered system. This method is based, as in the case of linear modeling, on a regression method. This is particularly new approach because the macsum aggregation is based on a Choquet integral and very few authors have proposed learning of an aggregation function based on the Choquet integral.

Coincidences of nonlinear integrals related to ordered pair of non-additive measures

This note studies the coincidences among the Choquet, concave and pan-integral in the setting of an ordered pair of monotone measures. We further investigate some properties of the minimal atoms and (M)-property of monotone measures, and optimal measures. By using these properties we present several sufficient and/or necessary conditions of coincidences among these three integrals, in the context relating to two monotone measures. These generalized versions, as special cases, cover respectively the coincidences in the usual sense among the pan-integral, the Choquet integral, and the concave integral on finite spaces.