This paper investigates the problem of optimal partitioning of a measurable space among a finite number of individuals. We demonstrate the sufficient conditions for the existence of weakly Pareto optimal partitions and for the equivalence between weak Pareto optimality and Pareto optimality. We demonstrate that every weakly Pareto optimal partition is a solution to the problem of maximizing a weighted sum of individual utilities. We also provide sufficient conditions for the existence of core partitions with non-transferable and transferable utility.
The image de-noising is a practical application of image processing.
Both linear and nonlinear filters are ušed for the noise reduction. The filters which are realizable in Lukasiewicz algebra with square root were analyzed first and then they were used for the 2D image de-noising. There is a set of quality measures recommended for the evaluation of de-noising quality. In čase of various quality measures we can find the best filter. The Pareto optimality principle and the AIA technique were used for this purpose. The procedures were demonstrated on a set of MRI biomedical images.
To overcome the shortage of cadaveric kidneys available for transplantation, several countries organize systematic kidney exchange programs. The kidney exchange problem can be modelled as a cooperative game between incompatible patient-donor pairs whose solutions are permutations of players representing cyclic donations. We show that the problems to decide whether a given permutation is not (weakly) Pareto optimal are NP-complete.