A method for finding optimum breeding schemes which maximize genetic gain under index selection with constraints on the rate of inbreeding is derived. The selection index includes information on the candidate and its sibs. Optimization is for the numbers of males and females to be selected and for the index weights when fixed numbers of offspring per generation, heritabilities and time horizons are considered. The expected rate of gain after a number of generations of selection is combined with the expected asymptotic rate of inbreeding (ΔF) in a single objective function which is maximized for finding the optimum solutions. Under restricted inbreeding, optimum designs are very similar for maximizing gains at different time horizons. The optimum number of selected males (for giving maximum gains) increases with the size of the scheme and with the severity in restricting ΔF and decreases with the heritability. Low heritability, less severe restrictions on ΔF and large schemes lead to increases in the relative weights given to performance of relatives in the index. The presence of common environmental effects leads to increases in optimum mating ratio when the heritability is low, to increases in the number of selected males and to more intense selection within families. Gains from index selection are compared with gains from mass selection. Under restricted inbreeding the advantage of optimized index selection over mass selection is only notable when the heritability is low and the scheme is large (in which case indices put more emphasis on family information than mass selection) and when the heritability is high and the scheme is small (in which case indices put less emphasis on family information).