The random generation of fuzzy measures under complex linear constraints holds significance in various fields, including optimization solutions, machine learning, decision making, and property investigation. However, most existing random generation methods primarily focus on addressing the monotonicity and normalization conditions inherent in the construction of fuzzy measures, rather than the linear constraints that are crucial for representing special families of fuzzy measures and additional preference information. In this paper, we present two categories of methods to address the generation of linearly constrained fuzzy measures using linear programming models. These methods enable a comprehensive exploration and coverage of the entire feasible convex domain. The first category involves randomly selecting a subset and assigning measure values within the allowable range under given linear constraints. The second category utilizes convex combinations of constrained extreme fuzzy measures and vertex fuzzy measures. Then we employ some indices of fuzzy measures, objective functions, and distances to domain boundaries to evaluate the coverage performance of these methods across the entire feasible domain. We further provide enhancement techniques to improve the coverage ratios. Finally, we discuss and demonstrate potential applications of these generation methods in practical scenarios.

Measures of diversity, spread and inequality can be important indicators in domains as diverse as ecology, economics and health. One of the key characteristics of such indices is the Pigou–Dalton (P-D) principle, also known as the principle of progressive transfers, whereby proportional redistribution from larger to smaller arguments should increase (or decrease, depending on the context) the overall measure of social or economic welfare, diversity and so on. Previous studies have identified the ordered weighted averaging operators as being appropriate for welfare measurement, subject to conditions on the weighting vectors. We propose the Choquet integral, defined with respect to a capacity or fuzzy measure, as a candidate for defining nonsymmetric measures of welfare. This allows for importance and interaction to be modelled between inputs while still satisfying the P-D principle. We extend the buoyancy concept to fuzzy measures and characterise the resulting classes of buoyant and antibuoyant fuzzy measures. We then turn to the problem of optimisation of the Choquet integral subject to linear constraints, which in the case of antibuoyant fuzzy measures permits an efficient linear programming solution.

This paper addresses reliable and efficient calculation of the mode of a multivariate sample, which is a classical fusion function. In particular, we focus on the inputs given on the unit simplex, when aggregating elements of Atanassov intuitionistic fuzzy sets, interval-valued fuzzy sets and their extensions, as well as compositional data. We outline the use of a specially designed 2-additive fuzzy measures and the Choquet integral for the purposes of reducing computational complexity in higher dimensions. We present computational analysis and benchmark four different methods of density-based mode estimation.

Integrals defined with respect to fuzzy measures (capacities) are powerful tools in multicriteria decision making. Monotonicity is a basic property of capacity, which means that the marginal contribution of any single criterion to any subset of criteria is always nonnegative. In this paper, we present the capacity-based decision making theory in terms of marginal contributions, which provides an alternative perspective to this widely used decision making strategy. We construct the marginal contribution representations of the equivalent transformations of capacities, some particular capacities, three types of nonlinear integrals, and discuss the capacity identification methods. We also introduce some new concepts and representations, such as nonadditivity and nonmodularity indices, 0 to 1 variables-based linear constraints of k-maxitive capacity, a special representation of the Choquet integral and pan integral. We discuss constraints on marginal contributions which ensure supermodularity of capacities. Finally, an illustrative example is given to show the use of marginal contribution presentation in capacity-based decision making methods.

The discrete Sugeno integral is an aggregation function particularly suited to aggregation of ordinal inputs. It accounts for inputs interactions, such as redundancy and complementarity, and its learning from empirical data is a challenging optimisation problem. The methods of ordinal regression involve an expensive objective function, whose complexity is quadratic in the number of data. We formulate ordinal regression using a much less expensive objective computed in linear time by the pool-adjacent-violators algorithm. We investigate the learning problem numerically and show the superiority of the new algorithm.

The probabilistic simultaneous interaction index has been widely adopted to measure the interaction among the decision criteria. However, this type of indices sometimes fails to reflect the kind of interaction associated with the nonadditivity of a fuzzy measure (capacity). For example, any simultaneous interaction index of the universal set of criteria w.r.t. a strictly superadditive capacity is not always positive. The main reason is that the simultaneous interaction index generalizes the notion of value by replacing the marginal contribution of a single criterion with the marginal simultaneous interaction of criteria subset. In this paper, we reform the generalization process and replace the marginal contribution with the marginal bipartition interaction, which can better reflect the kind of interaction associated with the nonadditivity, for example, superadditivity, subadditivity, strict-superadditivity, or strict-subadditivity. We construct a family of probabilistic bipartition interaction indices of subsets of criteria and study its properties. We discuss the issue of capacity identification based on the bipartition interaction index and demonstrate that the new type of interaction index can be adopted as a feasible alternative to describing the interaction phenomenon among the decision criteria.

Nonadditive robust ordinal regression (NAROR) is a widely adopted approach to analyze and reveal the dominance relationships among all decision alternatives based on nonadditive measures, called capacities. In this paper, we first investigate some advantages of the nonadditivity index as an explicit interaction index, as compared with the traditional probabilistic simultaneous interaction indices, and show that nonadditivity index can serve as an equivalent representation of a capacity. Then we enhance the NAROR method by using nonadditivity index as well as multiple-goal linear programming, where the former is used to replace the traditional interaction index to more naturally represent the decision maker's preferences, and the latter aims to replace the 0 to 1 mixed integer programming to enhance the ability to detect and adjust contradictory and redundant preference information. The updated NAROR's steps are constructed and discussed in detail and illustrated with a practical example.

This paper advocates the use of weighted ordered weighted averaging (WOWA) functions in decision-making processes, where the alternatives are not directly comparable. In particular, WOWA allows one to compare the strongest points of each alternative, also weighted by the importance of each criterion. Four different approaches to applying weights in OWA functions are reviewed. Torra's method based on interpolating regular increasing monotone quantifier and the pruned n-ary tree are compared to the WOWA obtained from recently proposed implicit averaging. Computationally efficient algorithms are outlined. The use of WOWA is illustrated in several examples.

We propose a new generalization of the discrete Choquet integral based on an arbitrary bivariate symmetric averaging function (mean). So far only the means with a natural multivariate extension were used for this purpose. In this paper, we use a general method based on a pruned binary tree to extend symmetric means with no obvious multivariate form, such as the logarithmic, identric, Heronian, Lagrangean, and Cauchy means. The generalized Choquet integral is built by computing the extensions of the bivariate means of the ordered inputs, and includes some existing extensions as special cases. Our construction is illustrated with multiple examples.

We propose weighted aggregation algorithms for creating general idempotent weighted aggregators of n variables derived from related symmetric idempotent aggregators of two variables. This computational method, together with interpolative aggregation, can be used for the development of general idempotent logic aggregators that satisfy a variety of conditions necessary for building decision models in the area of weighted compensative logic.

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. It is also a limiting property that results in many important nonmonotonic averaging functions being excluded from the theoretical framework. This work proposes a definition for weakly monotonic averaging functions, studies some properties of this class of functions, and proves that several families of important nonmonotonic means are actually weakly monotonic averaging functions. Specifically, we provide sufficient conditions for weak monotonicity of the Lehmer mean and generalized mixture operators. We establish weak monotonicity of several robust estimators of location and conditions for weak monotonicity of a large class of penalty-based aggregation functions. These results permit a proof of the weak monotonicity of the class of spatial-tonal filters that include important members such as the bilateral filter and anisotropic diffusion. Our concept of weak monotonicity provides a sound theoretical and practical basis by which (monotonic) aggregation functions and nonmonotonic averaging functions can be related within the same framework, allowing us to bridge the gap between these previously disparate areas of research.

This article discusses a range of regression techniques specifically tailored to building aggregation operators from empirical data. These techniques identify optimal parameters of aggregation operators from various classes (triangular norms, uninorms, copulas, ordered weighted aggregation (OWA), generalized means, and compensatory and general aggregation operators), while allowing one to preserve specific properties such as commutativity or associativity. © 2003 Wiley Periodicals, Inc.