Pub Date : 2025-11-19DOI: 10.1016/j.fss.2025.109664
Serafina Lapenta , Sebastiano Napolitano
We investigate the computational complexity of various satisfiability problems in Łukasiewicz logic, restricting attention to valuations in the standard MV-algebra [0,1]. Specifically, we focus on maximal r-satisfiability – the task of maximizing the number of formulas whose valuation is at least a given rational r ∈ (0, 1]. We also consider the decisional and weighted versions of this problem, as well as the partial (weighted) r-satisfiability problem.
{"title":"Computational complexity of some MaxSAT problems in Łukasiewicz logic","authors":"Serafina Lapenta , Sebastiano Napolitano","doi":"10.1016/j.fss.2025.109664","DOIUrl":"10.1016/j.fss.2025.109664","url":null,"abstract":"<div><div>We investigate the computational complexity of various satisfiability problems in Łukasiewicz logic, restricting attention to valuations in the standard MV-algebra [0,1]. Specifically, we focus on maximal <em>r</em>-satisfiability – the task of maximizing the number of formulas whose valuation is at least a given rational <em>r</em> ∈ (0, 1]. We also consider the decisional and weighted versions of this problem, as well as the partial (weighted) <em>r</em>-satisfiability problem.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109664"},"PeriodicalIF":2.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-19DOI: 10.1016/j.fss.2025.109676
Gleb Beliakov , Peiqi Sun , Jian-Zhang Wu
Random generation of fuzzy measures plays a pivotal role in large-scale decision-making and optimization that involve fuzzy integrals as a model to aggregate dependent inputs. We address the problem of random generation of fuzzy measures with specific additional constraints on their values and their combinations that reflect decision maker preferences. We present a range of approaches to handle sparse linear equality constraints and analyse their computational complexity. Some approaches involve random walks in the affine subspaces while others are based on projecting random points in an order polytope onto those affine spaces. We also examine special cases of linear constraints that arise in generation of k-additive fuzzy measures, and provide recommendations on the applicability of the approaches that we examined.
{"title":"Random projections of constrained fuzzy measures","authors":"Gleb Beliakov , Peiqi Sun , Jian-Zhang Wu","doi":"10.1016/j.fss.2025.109676","DOIUrl":"10.1016/j.fss.2025.109676","url":null,"abstract":"<div><div>Random generation of fuzzy measures plays a pivotal role in large-scale decision-making and optimization that involve fuzzy integrals as a model to aggregate dependent inputs. We address the problem of random generation of fuzzy measures with specific additional constraints on their values and their combinations that reflect decision maker preferences. We present a range of approaches to handle sparse linear equality constraints and analyse their computational complexity. Some approaches involve random walks in the affine subspaces while others are based on projecting random points in an order polytope onto those affine spaces. We also examine special cases of linear constraints that arise in generation of k-additive fuzzy measures, and provide recommendations on the applicability of the approaches that we examined.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109676"},"PeriodicalIF":2.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-19DOI: 10.1016/j.fss.2025.109684
Alessandro Gallo, Francesca Adele Giambona
Economic insecurity has gained increasing attention over the last decade, particularly in terms of its measurement and how it affects everyday life. This paper contributes to the literature on measurement by proposing a new individual-level, multidimensional index based on a fuzzy sets approach. The fuzzy logic moves beyond the classic binary framework of set theory, which classifies elements strictly as 0 or 1. In the fuzzy sets approach, each set is defined by a membership function that indicates the degree to which each element belongs to the set. This flexibility makes it particularly well suited for capturing complex socio-economic conditions such as economic insecurity. The proposed measure incorporates a range of economic insecurity indicators and offers some advantages. First, it produces an individual score that can be easily aggregated for geographical and socio-demographic comparisons. Second, the methodology allows for precise estimation of the variance, which is useful for assessing the reliability of aggregate estimates. The new index is applied to the Italian context using the most recent EU-SILC data. Aggregate estimates by region and socio-demographic group are derived and compared. Results indicate that the well-known North-South gradient persists and that economic insecurity is higher among the most disadvantaged sub-populations. In particular, individuals with low educational attainment and those who are unemployed or inactive experience the highest levels of economic insecurity.
{"title":"Measuring economic insecurity using a fuzzy sets approach","authors":"Alessandro Gallo, Francesca Adele Giambona","doi":"10.1016/j.fss.2025.109684","DOIUrl":"10.1016/j.fss.2025.109684","url":null,"abstract":"<div><div>Economic insecurity has gained increasing attention over the last decade, particularly in terms of its measurement and how it affects everyday life. This paper contributes to the literature on measurement by proposing a new individual-level, multidimensional index based on a fuzzy sets approach. The fuzzy logic moves beyond the classic binary framework of set theory, which classifies elements strictly as 0 or 1. In the fuzzy sets approach, each set is defined by a membership function that indicates the degree to which each element belongs to the set. This flexibility makes it particularly well suited for capturing complex socio-economic conditions such as economic insecurity. The proposed measure incorporates a range of economic insecurity indicators and offers some advantages. First, it produces an individual score that can be easily aggregated for geographical and socio-demographic comparisons. Second, the methodology allows for precise estimation of the variance, which is useful for assessing the reliability of aggregate estimates. The new index is applied to the Italian context using the most recent EU-SILC data. Aggregate estimates by region and socio-demographic group are derived and compared. Results indicate that the well-known North-South gradient persists and that economic insecurity is higher among the most disadvantaged sub-populations. In particular, individuals with low educational attainment and those who are unemployed or inactive experience the highest levels of economic insecurity.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109684"},"PeriodicalIF":2.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145584369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-19DOI: 10.1016/j.fss.2025.109673
Bo Xu , Changzhong Wang , Shuang An , Yang Huang
Fuzzy rough set theory offers an effective approach for feature selection; however, traditional methods lack an adaptive learning mechanism to adjust feature weights, making it difficult to accurately measure the contribution of each feature to classification. To address this issue, this paper introduces a novel dynamic optimization feature selection method based on maximum likelihood estimation. The method leverages the fuzzy similarity relation strategy from fuzzy rough sets to handle data uncertainty, while employing maximum likelihood estimation to assess feature importance. Specifically, the proposed model treats class labels as observed data and sample features as hidden variables, evaluating the classification ability of features by constructing a maximum likelihood function. Feature weights and class variances are integrated into the fuzzy similarity relation, and they are dynamically adjusted in accordance with the data characteristics through collaborative optimization. The inclusion degrees of samples are utilized to derive the empirical estimation of the conditional probability of classes relative to features. Finally, maximum likelihood estimation is applied to optimize the weighted features, assess their impact on the target variable, and select those that best explain the variation of the target variable. In this way, the model combines the strengths of fuzzy similarity relations in addressing uncertainty and the power of maximum likelihood estimation in parameter estimation, significantly enhancing the accuracy and robustness of feature selection. The experimental results show that the proposed algorithm has significant advantages over mainstream comparison methods on 18 benchmark data sets and provides a novel solution for feature selection in the field of uncertain data.
{"title":"Feature selection driven by maximum likelihood estimation and fuzzy similarity relation learning","authors":"Bo Xu , Changzhong Wang , Shuang An , Yang Huang","doi":"10.1016/j.fss.2025.109673","DOIUrl":"10.1016/j.fss.2025.109673","url":null,"abstract":"<div><div>Fuzzy rough set theory offers an effective approach for feature selection; however, traditional methods lack an adaptive learning mechanism to adjust feature weights, making it difficult to accurately measure the contribution of each feature to classification. To address this issue, this paper introduces a novel dynamic optimization feature selection method based on maximum likelihood estimation. The method leverages the fuzzy similarity relation strategy from fuzzy rough sets to handle data uncertainty, while employing maximum likelihood estimation to assess feature importance. Specifically, the proposed model treats class labels as observed data and sample features as hidden variables, evaluating the classification ability of features by constructing a maximum likelihood function. Feature weights and class variances are integrated into the fuzzy similarity relation, and they are dynamically adjusted in accordance with the data characteristics through collaborative optimization. The inclusion degrees of samples are utilized to derive the empirical estimation of the conditional probability of classes relative to features. Finally, maximum likelihood estimation is applied to optimize the weighted features, assess their impact on the target variable, and select those that best explain the variation of the target variable. In this way, the model combines the strengths of fuzzy similarity relations in addressing uncertainty and the power of maximum likelihood estimation in parameter estimation, significantly enhancing the accuracy and robustness of feature selection. The experimental results show that the proposed algorithm has significant advantages over mainstream comparison methods on 18 benchmark data sets and provides a novel solution for feature selection in the field of uncertain data.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109673"},"PeriodicalIF":2.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-17DOI: 10.1016/j.fss.2025.109683
Qian Hu , Jiapeng Bai , Jun Zhang , Yafei Song , Jusheng Mi
Outlier detection, as an important direction of data mining, aims to identify data objects that deviate from normal patterns and is widely used in fields such as financial fraud, network security, and medical diagnosis. Functioning as an essential tool in knowledge acquisition and data mining, granular computing provides a novel framework that emulates human cognitive patterns for resolving large-scale complex problems. However, traditional outlier detection methods based on granular computing are difficult to balance data diversity and fuzziness. Therefore, this article constructs an outlier detection model based on fuzzy neighborhood combination entropy using neighborhood fuzzy granules and combination entropy. Firstly, the fuzzy neighborhood combination entropy of the information system is defined, and the relative fuzzy neighborhood combination entropy of the object is defined by the change in neighborhood fuzzy entropy caused by the object. Secondly, the relative fuzzy cardinality of the object is defined by the difference degree between its fuzzy neighborhoods, and the anomaly factor of the object is measured by its relative fuzzy neighborhoods combination entropy and relative fuzzy cardinality. Then, an outlier detection model based on the combination entropy of fuzzy neighborhoods is constructed and the relevant algorithm is designed. Finally, the effectiveness and efficiency of the proposed method were verified through publicly available datasets.
{"title":"FNCEOD: Fuzzy neighborhood combination entropy-based outlier detection","authors":"Qian Hu , Jiapeng Bai , Jun Zhang , Yafei Song , Jusheng Mi","doi":"10.1016/j.fss.2025.109683","DOIUrl":"10.1016/j.fss.2025.109683","url":null,"abstract":"<div><div>Outlier detection, as an important direction of data mining, aims to identify data objects that deviate from normal patterns and is widely used in fields such as financial fraud, network security, and medical diagnosis. Functioning as an essential tool in knowledge acquisition and data mining, granular computing provides a novel framework that emulates human cognitive patterns for resolving large-scale complex problems. However, traditional outlier detection methods based on granular computing are difficult to balance data diversity and fuzziness. Therefore, this article constructs an outlier detection model based on fuzzy neighborhood combination entropy using neighborhood fuzzy granules and combination entropy. Firstly, the fuzzy neighborhood combination entropy of the information system is defined, and the relative fuzzy neighborhood combination entropy of the object is defined by the change in neighborhood fuzzy entropy caused by the object. Secondly, the relative fuzzy cardinality of the object is defined by the difference degree between its fuzzy neighborhoods, and the anomaly factor of the object is measured by its relative fuzzy neighborhoods combination entropy and relative fuzzy cardinality. Then, an outlier detection model based on the combination entropy of fuzzy neighborhoods is constructed and the relevant algorithm is designed. Finally, the effectiveness and efficiency of the proposed method were verified through publicly available datasets.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"526 ","pages":"Article 109683"},"PeriodicalIF":2.7,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-16DOI: 10.1016/j.fss.2025.109677
José R. González de Mendívil , Zorana Jančić , Aitor González de Mendívil Grau , Ivana Micić , Stefan Stanimirović
Minimization of fuzzy deterministic finite automata (FDfAs) is a challenging problem due to two main reasons. First, the graded nature of transitions and state memberships makes traditional minimization techniques difficult to apply. Second, a minimal FDfA is not necessarily unique, as multiple equivalent FDfAs of the same size may exist. In this paper, we focus on finding a polynomial-time minimization method that constructs a minimal FDfA for a given FDfA. Our approach is based on establishing isomorphisms between well-known polynomial-time constructions, providing a mathematical foundation for the proposed method. Specifically, we introduce the notion of quasi-deterministic fuzzy finite automata (QDFfAs) and explore their isomorphism properties with the Myhill-Nerode automaton of a fuzzy language. We show that the determinization via factorization of a QDFfA preserves strong isomorphism with the generalized Myhill-Nerode automaton of the recognized fuzzy language. This insight enables the development of an efficient minimization method by leveraging the interpretable backward replica of an FDfA.
{"title":"Quasi-deterministic fuzzy automata: Isomorphisms and fuzzy deterministic automata minimization","authors":"José R. González de Mendívil , Zorana Jančić , Aitor González de Mendívil Grau , Ivana Micić , Stefan Stanimirović","doi":"10.1016/j.fss.2025.109677","DOIUrl":"10.1016/j.fss.2025.109677","url":null,"abstract":"<div><div>Minimization of fuzzy deterministic finite automata (FDfAs) is a challenging problem due to two main reasons. First, the graded nature of transitions and state memberships makes traditional minimization techniques difficult to apply. Second, a minimal FDfA is not necessarily unique, as multiple equivalent FDfAs of the same size may exist. In this paper, we focus on finding a polynomial-time minimization method that constructs a minimal FDfA for a given FDfA. Our approach is based on establishing isomorphisms between well-known polynomial-time constructions, providing a mathematical foundation for the proposed method. Specifically, we introduce the notion of quasi-deterministic fuzzy finite automata (QDFfAs) and explore their isomorphism properties with the Myhill-Nerode automaton of a fuzzy language. We show that the determinization via factorization of a QDFfA preserves strong isomorphism with the generalized Myhill-Nerode automaton of the recognized fuzzy language. This insight enables the development of an efficient minimization method by leveraging the interpretable backward replica of an FDfA.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109677"},"PeriodicalIF":2.7,"publicationDate":"2025-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145584368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the impact of nonlinear components is studied for cooperative load transportation systems with any number of quadrotors and a single slung load suspended by ropes. The main goal is to control and estimate constraints caused by the nonlinear term of the load transportation system. A novel distributed control strategy is proposed for cooperative systems based on adaptive fuzzy wavelet networks (AFWNs). Distributed AFWNs are employed to compensate for nonlinear effects. Another result is the expansion of the system’s attraction region for the initial state values. Also, by employing an integral term in the control law, the formation error of the agents converges to zero. These expansions allow the system to significantly improve its robustness to disturbances. The simulation results illustrate that the proposed method can keep the agents in desired formation and guide the load in right direction.
{"title":"Adaptive fuzzy wavelet network control for nonlinear cooperative load transportation systems","authors":"Matin Fadavi , Majdeddin Najafi , Farid Sheikholeslam","doi":"10.1016/j.fss.2025.109681","DOIUrl":"10.1016/j.fss.2025.109681","url":null,"abstract":"<div><div>In this paper, the impact of nonlinear components is studied for cooperative load transportation systems with any number of quadrotors and a single slung load suspended by ropes. The main goal is to control and estimate constraints caused by the nonlinear term of the load transportation system. A novel distributed control strategy is proposed for cooperative systems based on adaptive fuzzy wavelet networks (AFWNs). Distributed AFWNs are employed to compensate for nonlinear effects. Another result is the expansion of the system’s attraction region for the initial state values. Also, by employing an integral term in the control law, the formation error of the agents converges to zero. These expansions allow the system to significantly improve its robustness to disturbances. The simulation results illustrate that the proposed method can keep the agents in desired formation and guide the load in right direction.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109681"},"PeriodicalIF":2.7,"publicationDate":"2025-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-15DOI: 10.1016/j.fss.2025.109682
Hua-Peng Zhang , Yao Ouyang
After mining some inherent properties of the class of uninorms on a bounded lattice, we present structural representation theorems for uninorms in by distinguishing two cases. As an application of the representation theorems, we propose several construction methods for uninorms in . These results can be easily translated into their counterparts for uninorms in via the Duality Principle in poset theory.
{"title":"Representation and construction of the classes Umin1 and Umax0 of uninorms on a bounded lattice","authors":"Hua-Peng Zhang , Yao Ouyang","doi":"10.1016/j.fss.2025.109682","DOIUrl":"10.1016/j.fss.2025.109682","url":null,"abstract":"<div><div>After mining some inherent properties of the class <span><math><msubsup><mi>U</mi><mrow><mi>min</mi></mrow><mrow><mspace></mspace><mn>1</mn></mrow></msubsup></math></span> of uninorms on a bounded lattice, we present structural representation theorems for uninorms in <span><math><msubsup><mi>U</mi><mrow><mi>min</mi></mrow><mrow><mspace></mspace><mn>1</mn></mrow></msubsup></math></span> by distinguishing two cases. As an application of the representation theorems, we propose several construction methods for uninorms in <span><math><msubsup><mi>U</mi><mrow><mi>min</mi></mrow><mrow><mspace></mspace><mn>1</mn></mrow></msubsup></math></span>. These results can be easily translated into their counterparts for uninorms in <span><math><msubsup><mi>U</mi><mrow><mi>max</mi></mrow><mrow><mspace></mspace><mn>0</mn></mrow></msubsup></math></span> via the Duality Principle in poset theory.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"529 ","pages":"Article 109682"},"PeriodicalIF":2.7,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-14DOI: 10.1016/j.fss.2025.109679
M.D.M. Bibiloni-Femenias , O. Valero
In the literature there are two different approaches that extend the classical crisp notion of equivalence relation to the fuzzy framework. On the one hand, one can find the notion of indistinguishability operator and a few of its generalizations. These can be understood as a kind of measurement of the degree of similarity or indistinguishability between objects. On the other hand, fuzzy (quasi-)metrics measure such a degree with respect to a parameter. The study of both types of the aforesaid notions has been carried out independently without any connection between them. As a consequence, the notion of modular indistinguishability operator has been introduced recently. Such a notion unifies under the same framework both aforesaid similarity concepts. In this paper, we explore the aggregation problem for modular indistinguishability operators and for several generalizations. Hence we introduce the notions of modular fuzzy pre-order, modular fuzzy partial order and modular equality and we characterize the functions that are able to fuse all these different types of modular similarities. The aforementioned characterizations are stated in terms of triangular triplets or related notions, monotony and dominance. In contrast to the non-modular case, the class of those functions that merge modular fuzzy pre-orders (modular fuzzy partial orders) is shown to match the class of modular indistinguishability operators (modular equalities). Furthermore, the relationships between the non-modular aggregation problem, the modular one and the fuzzy metric aggregation problem are explored and the differences between them are clarified by means of appropriate examples.
{"title":"Modular indistinguishability: The aggregation problem","authors":"M.D.M. Bibiloni-Femenias , O. Valero","doi":"10.1016/j.fss.2025.109679","DOIUrl":"10.1016/j.fss.2025.109679","url":null,"abstract":"<div><div>In the literature there are two different approaches that extend the classical crisp notion of equivalence relation to the fuzzy framework. On the one hand, one can find the notion of indistinguishability operator and a few of its generalizations. These can be understood as a kind of measurement of the degree of similarity or indistinguishability between objects. On the other hand, fuzzy (quasi-)metrics measure such a degree with respect to a parameter. The study of both types of the aforesaid notions has been carried out independently without any connection between them. As a consequence, the notion of modular indistinguishability operator has been introduced recently. Such a notion unifies under the same framework both aforesaid similarity concepts. In this paper, we explore the aggregation problem for modular indistinguishability operators and for several generalizations. Hence we introduce the notions of modular fuzzy pre-order, modular fuzzy partial order and modular equality and we characterize the functions that are able to fuse all these different types of modular similarities. The aforementioned characterizations are stated in terms of triangular triplets or related notions, monotony and dominance. In contrast to the non-modular case, the class of those functions that merge modular fuzzy pre-orders (modular fuzzy partial orders) is shown to match the class of modular indistinguishability operators (modular equalities). Furthermore, the relationships between the non-modular aggregation problem, the modular one and the fuzzy metric aggregation problem are explored and the differences between them are clarified by means of appropriate examples.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"526 ","pages":"Article 109679"},"PeriodicalIF":2.7,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-14DOI: 10.1016/j.fss.2025.109678
Jih-Jeng Huang , Chin-Yi Chen
We introduce a unified framework extending the classical Choquet integral by incorporating Stieltjes-type accumulation functions and dual set-functions. This construction, termed the dual Choquet–Stieltjes (DCS) integral, broadens non-additive integral theory, allowing simultaneous treatment of threshold-dependent behaviors and asymmetric interactions. We prove fundamental properties including well-definedness, monotonicity, and comonotonic additivity under precisely specified conditions. We establish convergence theorems (monotone convergence, Fatou’s lemma, dominated convergence) with complete proofs, and demonstrate applications in decision-making. Our framework generalizes existing extensions under a single, coherent approach that maintains theoretical properties while enhancing modeling flexibility. Through parameter recovery studies, we demonstrate the theoretical soundness of our approach and identify scenarios where the full DCS framework is necessary to capture complex interdependencies and threshold effects.
{"title":"The choquet–Stieltjes integral with dual set-Functions: a unified theory and applications","authors":"Jih-Jeng Huang , Chin-Yi Chen","doi":"10.1016/j.fss.2025.109678","DOIUrl":"10.1016/j.fss.2025.109678","url":null,"abstract":"<div><div>We introduce a unified framework extending the classical Choquet integral by incorporating Stieltjes-type accumulation functions and dual set-functions. This construction, termed the dual Choquet–Stieltjes (DCS) integral, broadens non-additive integral theory, allowing simultaneous treatment of threshold-dependent behaviors and asymmetric interactions. We prove fundamental properties including well-definedness, monotonicity, and comonotonic additivity under precisely specified conditions. We establish convergence theorems (monotone convergence, Fatou’s lemma, dominated convergence) with complete proofs, and demonstrate applications in decision-making. Our framework generalizes existing extensions under a single, coherent approach that maintains theoretical properties while enhancing modeling flexibility. Through parameter recovery studies, we demonstrate the theoretical soundness of our approach and identify scenarios where the full DCS framework is necessary to capture complex interdependencies and threshold effects.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"526 ","pages":"Article 109678"},"PeriodicalIF":2.7,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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