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.

The useful balance of flexibility and robustness provided by fuzzy integrals for classification and regression tasks has been well established in the fuzzy research community. With the rise of artificial intelligence and neural networks, we have witnessed great advances in data analysis and decision-making, however the issue of explainability has become more prominent in recent times. It is in this regard that the theory of fuzzy integrals and other sophisticated aggregation frameworks have a lot to offer. In this contribution, a neural network architecture is proposed based on the inclusion-exclusion integral, which is defined with respect to a fuzzy measure and a triangular norm. After presenting the model and learning methodology, we provide some applications of the network to real datasets. The key benefit of this approach is in the ability to derive interpretations using the Shapley value and interaction indices, read directly from the model. However, we also highlight the predictive performance, which is comparable with state-of-the-art machine learning techniques. Our study hence contributes to the aims of explainable artificial intelligence, whereby model flexibility and accuracy is achieved without sacrificing interpretability.

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.