Combinatorial Structure of the Polytope of 2-Additive Measures

Miranda, Pedro; García-Segador, Pedro

doi:10.1109/tfuzz.2019.2945243

Cited by 9 publications

(1 citation statement)

References 38 publications

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“…Furthermore, the additional constraints, such as supermodularity, and in our case antibuoyancy, lead to an even more restricted and complicated polytope. Some recent works on generating special types of capacities include References [40–43].…”

Section: Construction Of Antibuoyant Capacitiesmentioning

confidence: 99%

Choquet integral‐based measures of economic welfare and species diversity

Beliakov

Liu

2021

Int J of Intelligent Sys

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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.

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Section: Construction Of Antibuoyant Capacitiesmentioning

confidence: 99%

Choquet integral‐based measures of economic welfare and species diversity

Beliakov

Liu

2021

Int J of Intelligent Sys

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An Approximation Algorithm for Random Generation of Capacities

Grabisch

Labreuche

Sun

2023

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Efficient random walks for generating random fuzzy measures in Möbius representation in large universe

Beliakov,

Baz,

2024

Comp. Appl. Math.

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Random generation of fuzzy measures plays a pivotal role in large-scale decision-making and optimization. Random walks ensure uniform generation and adequate coverage. The Möbius representation of set functions is a valuable tool for establishing the sparse structure of fuzzy measures, with its non-negativity closely linked to monotonicity and convexity/supermodularity checking. We propose three efficient methods for monotonicity verification and convexity/supermodularity verification applicable to random walks in Möbius representations, specifically tailored for universal sets larger than ten inputs and k-order fuzzy measures. We first present the baseline methods by directly inspecting monotonicity and convexity constraints. Building on the observation that the majority of initially generated values exhibit non-negativity, we intentionally track negative Möbius values to enhance the computational performance of these baseline approaches. Further, we introduce the more agile methods that employ insertion and merge sorting techniques for both monotonicity and convexity checks in random walks that involve small perturbations of fuzzy measures. To address sparsity in large-scale scenarios, we focus on two major types of measures: k-additive and k-interactive measures, demonstrating their effectiveness through theoretical analysis and experimental results.

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Combinatorial Structure of the Polytope of 2-Additive Measures

Cited by 9 publications

References 38 publications

Choquet integral‐based measures of economic welfare and species diversity

Choquet integral‐based measures of economic welfare and species diversity

An Approximation Algorithm for Random Generation of Capacities

Efficient random walks for generating random fuzzy measures in Möbius representation in large universe

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