Non-additive Measures Research Articles

Base topologies and convergence in nonadditive measure

The existing literature on convergence in nonadditive measure has exclusively focused on the construction of weak base topologies whereby different authors have used different notions of balls in their respective constructions. Whenever the nonadditive measure fails to satisfy specific structural properties, some notions of balls might fail to be open sets. Building on existing results, we show that these weak base topologies are equivalent for finite nonadditive measures regardless of whether the different notions of balls are open sets or not. As our main contribution we complement the existing literature through the construction of base topologies which are finer than the corresponding weak base topologies. In contrast to weak base topologies, a decision theoretic modeler can directly ensure through a base topology that his/her preferred notions of balls are always open sets for arbitrary nonadditive measures.

A Novel Multi-Criteria Decision-Making Method Based on Rough Sets and Fuzzy Measures

Rough set theory provides a useful tool for data analysis, data mining and decision making. For multi-criteria decision making (MCDM), rough sets are used to obtain decision rules by reducing attributes and objects. However, different reduction methods correspond to different rules, which will influence the decision result. To solve this problem, we propose a novel method for MCDM based on rough sets and a fuzzy measure in this paper. Firstly, a type of non-additive measure of attributes is presented by the importance degree in rough sets, which is a fuzzy measure and called an attribute measure. Secondly, for a decision information system, the notion of the matching degree between two objects is presented under an attribute. Thirdly, based on the notions of the attribute measure and matching degree, a Choquet integral is constructed. Moreover, a novel MCDM method is presented by the Choquet integral. Finally, the presented method is compared with other methods through a numerical example, which is used to illustrate the feasibility and effectiveness of our method.

Recursive Aggregation and Its Fusion Process for Intuitionistic Fuzzy Numbers Based on Non-Additive Measure

In this paper, the recursive aggregation of OWA operators for intuitionistic fuzzy numbers (IFN) based on a non-additive measure (NAM) with σ−λ rules is constructed and investigated in light of the σ−λ rules of a non-additive measure (NAM). Additionally, an integrator is designed by drawing on the genetic algorithm and the process of calculation is elaborated by an example.

A flexible class of dependence-aware multi-label loss functions

The idea to exploit label dependencies for better prediction is at the core of methods for multi-label classification (MLC), and performance improvements are normally explained in this way. Surprisingly, however, there is no established methodology that allows to analyze the dependence-awareness of MLC algorithms. With that goal in mind, we introduce a class of loss functions that are able to capture the important aspect of label dependence. To this end, we leverage the mathematical framework of non-additive measures and integrals. Roughly speaking, a non-additive measure allows for modeling the importance of correct predictions of label subsets (instead of single labels), and thereby their impact on the overall evaluation, in a flexible way. The well-known Hamming and subset 0/1 losses are rather extreme special cases of this function class, which give full importance to single label sets or the entire label set, respectively. We present concrete instantiations of this class, which appear to be especially appealing from a modeling perspective. The assessment of multi-label classifiers in terms of these losses is illustrated in an empirical study, clearly showing their aptness at capturing label dependencies. Finally, while not being the main goal of this study, we also show some preliminary results on the minimization of this parametrized family of losses.

A Generalization of the Concave Integral in Terms of Decomposition of the Integrated Function for Bipolar Scales

In the context of the multiple-criteria decision aid (MCDA), several fuzzy integrals concerning capacities (non-additive measures) have been introduced by various researchers in the last sixty years. Recently, Lehrer has proposed a new integral for capacities known as concave integral. The concave integral is based on the decomposition of random variables into simple ingredients. The concave integral concerning capacity is defined as the maximum value obtained among all its decompositions. The paper aims to model a new integration based on the decomposition of random variables into simple ingredients for multi-criteria decision making support when underlying scales are bipolar. This paper proposes a generalization of the concave integral in terms of decompositions of the integrated function to be suitable for bipolar scales. We show that the random variable is analyzed as a combination of indicators, where each allowed decomposition has a value determined by the bi-capacity. Lastly, we illustrate our framework by a practical example.

Adaptive Artificial Intelligence for Resource-Constrained Connected Vehicles in Cybertwin-Driven 6G Network

The emerging technology of cybertwin is expected to bring revolutionary benefits to the sixth-generation (6G) network in respect of communication, resources allocation, and digital asset management. Empowered by ubiquitous artificial intelligence (AI), cybertwin is capable of adjusting the requests for computing resources to support network services by analyzing user’s demands for quality of experience and resource scarcity in the market. For resource-constrained applications, such as connected vehicles in the 6G network, cybertwin can intelligently determine the time-varying requests of computing resources for various vehicles at different times. However, the current service architecture executes AI algorithms with universal configurations for all vehicles. This causes the difficulty of customizing the complexity of AI algorithms to maintain adaptive to cybertwin’s decisions on dynamic resources allocation. In this article, we propose an adaptive AI framework based on efficient feature selection to cooperate with cybertwin’s resource allocation. This proposed framework can adaptively customizing AI model complexity with available computing resources. Specifically, we systematically characterize the aggregated impacts of all feature combinations on the modeling outcomes of AI algorithms. By utilizing nonadditive measures, the interactions among features can be quantified to indicate their contributions to the modeling process. Then, we propose an efficient algorithm to obtain accurate interaction measures for adaptive feature selection to balance the tradeoff between modeling accuracy and computational overhead. Finally, extensive simulations are conducted to validate that our proposed framework substantially reduces the overhead of AI algorithms while guaranteeing desired modeling accuracy for cybertwin-driven connected vehicles in 6G.

Knapsack problems with dependencies through non-additive measures and Choquet integral

In portfolio selection problems the items often depend on each other, and their synergies and redundancies need to be taken into account. We consider the knapsack problem in which the objective is modelled as the Choquet integral with respect to a supermodular capacity which quantifies possible synergies. We provide various formulations which lead to the standard linear mixed integer programs, applicable to small and large portfolios. We also study scalability of the solution methods and compare large problems defined with respect to 2-additive capacities which model pairwise interactions, and linear knapsack with respect to the Shapley values of these capacities.

Simplest non-additive measures of quantum resources

Given an arbitrary state $\rho$ and some figure of merit ${\cal E}(\rho)$, it is usually a hard problem to determine the value of ${\cal E}(\rho^{\otimes N})$. One noticeable exception is the case of additive measures, for which we simply have ${\cal E}(\rho^{\otimes N}) = Ne$, with $e\equiv {\cal E}(\rho)$. In this work we study measures that can be described by ${\cal E}(\rho^{\otimes N}) =E(e;N) \ne Ne$, that is, measures for which the amount of resources of $N$ copies is still determined by the single real variable $e$, but in a nonlinear way. If, in addition, the measures are analytic around $e=0$, recurrence relations can be found for the Maclaurin coefficients of $E$ for larger $N$. As an example, we show that the $\ell_1$-norm of coherence is a nontrivial case of such a behavior.

When a combination of convexity and continuity forces monotonicity of preferences

We consider arbitrary subsets L of random variables defined on an arbitrary non-additive probability space (Ω,F,ν). A topology τ on L satisfies Condition BU if every open set in this topology which contains X∈L as a member also contains as a subset some (c,ϵ)–ball around X, defined as Bc,ϵ(X)={Y∈L|ν(|X−Y|≥c)<ϵ}. Condition BU is satisfied by any topology of convergence in non-additive measure ν[21,18] but also by all coarser topologies. Next we consider preference relations that are continuous with respect to any topology τ satisfying Condition BU. For non-atomic ν we prove that any convex and τ-continuous preference relation over the random variables in a given local cone L must satisfy monotonicity in the local cone order. This monotonicity result comes with surprisingly strong decision theoretic implications: (i) The only τ-continuous and convex preference relation defined over all random variables is the indifference relation; (ii) Any τ-continuous and convex preference relation defined over all positive random variables must satisfy payoff-monotonicity; (iii) Any convex and payoff-monotone preference relation defined over all loss random variables must violate τ-continuity.

The topology on the space of measurable functions that is compatible with convergence in nonadditive measure

In this paper, given a nonadditive measure μ, a topology, which is compatible with convergence in μ-measure, is defined on the real linear space of all (equivalence classes of) measurable functions by using the distance introduced by Dunford and Schwartz. Some fundamental properties of the topology, such as the open sphere condition, the Hausdorff separation axiom, separability, approximation by continuous functions, linearlization, and metrizability, are related to the characteristics of a nonadditive measure.

Fuzzy Measures and Choquet Integrals Based on Fuzzy Covering Rough Sets

Fuzzy sets and fuzzy rough sets are widely applied in data analysis, data mining, and decision-making. So far, the common method is to use rough approximate operators to induce aggregation functions when fuzzy rough sets are used for multicriteria decision-making (MCDM). However, they are parametric linear and the corresponding weights are additive measures. In this article, we give a novel method for MCDM based on fuzzy covering rough sets by using the nonadditive measure [i.e., fuzzy measure (FM)] and the nonlinear integral [i.e., Choquet integral (ChI)]. First, two nonadditive measures are presented by fuzzy covering lower and upper approximation operators, respectively. Moreover, both of them are FMs which are called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -neighborhood approximation measures. Second, two types of ChIs with respect to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -neighborhood approximation measures are constructed. A novel method, which considers the association, is presented to solve the problem of MCDM under the fuzzy covering rough set model. Third, a new approach based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -neighborhood approximation measures is proposed for attribute reductions in a fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -covering information table. This approach of attribute reductions is used in MCDM. Finally, both new methods above are compared with other methods through some numerical examples and UCI datasets, respectively.

Nonadditive measures and nonlinear integrals —focusing on a theoretical aspect—

Nonadditive measures are monotonically increasing set functions that are not necessarily additive and nonlinear integrals are the integrals with respect to nonadditive measures. The main theme of the theoretical study of nonadditive measures and nonlinear integrals is the refinement of measure theory and is accomplished by finding necessary and sufficient conditions (or sufficient conditions weak enough) under which various important theorems in measure theory remain valid for nonadditive measures. In this expository article, some of the basic results are described in exact detail in terms of refining measure theory, each of which is the cornerstone of developing the study of nonadditive measures and nonlinear integrals.

Measuring SMEs’ Propensity for Open Innovation Using Cognitive Mapping and MCDA

Open innovation (OI) has captured increasing interest over recent decades as this approach is linked to higher level organizational ambidextrous strategy, and has become a prerequisite for achieving competitive advantages. Assessing firms' propensity for OI has, however, turned out to be an increasingly challenging endeavor. This is particularly true for small and medium-sized enterprises (SMEs) due to the myriad of factors that influence these companies' capability for innovation. To overcome this challenge, this paper sought to integrate two operational research/management science techniques-cognitive mapping and the Choquet integral (CI) (a nonadditive measure and information aggregator)-to identify and prioritize relevant criteria for evaluating SMEs' propensity for OI, and improving their organizational ambidexterity. To facilitate a real-world application, information was first collected from SME managers and entrepreneurs who agreed to participate in face-to-face group meetings, allowing realism to be incorporated in the decision-making process. The results were validated by both the panel members and project director of COTEC Portugal-a leading think-and-action network for advancing technology diffusion and business innovation cooperation. The findings include that cognitive mapping facilitates the identification and understanding of cause-and-effect relationships between the determinants of OI in SMEs. The CI, in turn, introduces realism into the construction of value functions and the respective assessments of SMEs. The limitations and implications of the proposed system are also discussed.

Toward Fast and Accurate SOH Prediction for Lithium-Ion Batteries

For timely maintenance and replacement in lithium-ion battery system, it is crucial to achieve fast and accurate State of Health (SOH) prediction. SOH is a dynamic status parameter of a battery indicated by its available capacity compared to the initial condition. SOH tends to decrease in the long-term because a battery is aged by its charging and discharging cycles due to the degradation of chemical composition. To ensure a battery has sufficiently good conditions, it is significant to predict the battery SOH precisely so that the depleted or weak battery can be replaced in time. However, the prediction of battery SOH is a difficult and complex task due to the following reasons (1) SOH degradation is a dynamic process because of the time-varying nature of both battery electrochemistry and working condition; (2) SOH prediction model usually involves numerous input attributes that lead to sheer complexity and computational cost for real-time battery management; (3) the balance of overhead-accuracy tradeoff is challenging to adapt to various application scenarios. In this paper, we propose a theoretical framework for fast and accurate SOH prediction based on the non-additive measure and overhead-accuracy balancing. Specifically, we characterize the interactions among battery attributes and their aggregated impacts on SOH prediction. By choosing the most significant subset of battery attributes, the computational complexity of SOH prediction can be substantially reduced while maintaining high accuracy. The tradeoff between accuracy and computational overhead can be balanced by adjusting the number of attributes and volume of training data. The experimental results validate the effectiveness and efficiency of the proposed framework.

Modeling Interactive Multiattribute Decision-Making via Probabilistic Linguistic Term Set Extended by Dempster–Shafer Theory

In multiattribute decision-making (MADM), more and more attention is paid to the interaction between attributes when considering the actual decision environment. As a result, interactive MADM has become an emerging and challenging area of research whose success will greatly facilitate the development of decision-making. This paper models the interactive MADM problem and its contribution is multifaceted. First, the concept of probabilistic linguistic term set (PLTS) is extended by Dempster–Shafer theory (DST), which helps to express more uncertain information, followed by some basic operations, such as score function and aggregation operator. In virtue of evidential best-worst method and the principle of maximum entropy, then a novel nonadditive measure determination method is developed based on the 2-order additive measure to better model the interaction between attributes. Further, the generalized PLTS-based Choquet integral is defined by which generalized PLTSs on a nonadditive measure can be reasonably aggregated. Finally, an interactive MADM model is constructed and the technical details are described. The proposed approach is implemented to select the supplier for medical devices, and its effectiveness is emphasized by comparison with other methods.

A New Multicriteria Decision-Making Method for the Selection of Sponge City Schemes with Shapley–Choquet Aggregation Operators

The construction of sponge cities is of great strategic significance to solving the urban water resource problem in the future. According to the policy guidance of sponge city construction, the evaluation index system of sponge city construction projects is constructed. In order to overcome the interference caused by the interaction between indexes, a nonadditive measure and Shapley function are combined to determine the weights of attribute indexes, and the generalized Shapley interval-valued intuitionistic uncertain linguistic Choquet averaging (GS-IVIULCA) operator is used to calculate the comprehensive evaluation value of the schemes. On this basis, a new evaluation method of sponge city construction project selection under an uncertain information environment is presented and empirically evaluated. The results show that the index weight of rainwater collection and utilization is the largest, indicating that decision makers pay more attention to the ecological and environmental benefits of this item in the sponge city construction process.

Egoroff's theorems for random sets on non-additive measure spaces

<abstract> We present four versions of Egoroff's theorems for measurable closed-valued multifunctions on non-additive measure spaces. The conditions provided for each of these four versions are not only sufficient, but also necessary. In our discussions the continuity of non-additive measures is not required. The previous related results are improved and generalized. </abstract>

Combining scenario approach and Choquet integral in decision making

Scenario-based approach is applied to evaluate alternatives. Using the ideas of Dempster-Schaeffer a capacity distribution is defined over scenarios. This allows describing possible future developments in terms of individual and integrated expert judgments and transforming expert judgments into scenario assessments. To get a final numerical estimate of alternatives, we use integration over non-additive measures applying the Choquet integral.

Aggregation of sequence of fuzzy measures

In this paper we present construction of new fuzzy measures by applying extended aggregation function on a sequence of fuzzy measures. According to properties of applied aggregation function and properties of initial fuzzy measures, some properties of constructed fuzzy measure are proved. Additionally, one new extended aggregation function named extended weighted arithmetic mean of distorted arguments is introduced, and its relevant properties are proved. It is shown that this extended aggregation function, with appropriatedparameters, can be suitable for construction of new fuzzy measures. Other types of non-additive measures can be constructed in the same way, by applying aggregation function on the initial sequence of non-additive measures.

Multi-criteria and medical diagnosis for application to health insurance systems: a general approach through non-additive measures

In this contribution, we propose a healthcare decision support system. Nowadays, it is commonly recognized that quantitative tools and decision support can increase the benefits and the performances of healthcare systems, and for this, different multiple criteria methods were proposed in many branches of medicine. The approach we propose is based on non-additive measures and the Choquet integral. This methodology has been intensively applied in many real-world applications, given its capability to represent interactions among criteria, and thus to model a wide range of preference structures. Considering that the diagnosis procedure needs also to take the clinical expertise into account, this method appears particularly tailored for a diagnosis support, mainly when statistical models cannot be applied and/or available data are scarce and knowledge can be inferred by physicians’ opinions. In particular, we propose a disease risk evaluation and compute some associated indicators. Furthermore, an error estimation is performed. As an application, a cardiovascular risk diagnosis model is presented. The proposed methodology, that allows to quantify the disease risk taking into account individual’s medical conditions, can be used for improving healthcare service quality or for pricing and reserving health insurance policies. An application to health insurance pricing is provided.