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A new criterion of fitness evaluation for Genetic Algorithms is introduced where the fitness value of an individual is determined by considering its own fitness as well as those of its ancestors. Some guidelines for selecting the weighting coefficients for quantifying the importance to be given to the fitness of the individual and its ancestors are provided. This is done both heuristically and automatically under fixed and adaptive frameworks. The Schema Theorem corresponding to the proposed concept is derived. The effectiveness of this new methodology is demonstrated extensively on the problems of optimizing complex functions including a noisy one and selecting optimal neural network parameters.