An iterative procedure for deriving selection indexes with constant restrictions

The objective of this study was to present an iterative procedure for deriving selection indexes with constant restrictions. Constant restriction means that the genetic responses of the restricted traits are preset to actual amounts for a given selection intensity (ī). Results of this study show that an index with constant restriction alone or in combination with other types of restrictions possesses three distinctive characteristics: 1) the coefficient matrix of the index equations is not symmetric and is nonlinear; 2) the coefficient matrix contains unknown ī, indicating that the index coefficients (b) to be derived depends on the value of ī predetermined before selection; and 3) the coefficient matrix contains unknown b, thus requiring iterative methods to solve the index equations. As a result of these unique characteristics, the index coefficients, genetic responses of the index traits, and overall genetic gain in net merit change nonlinearly with varying levels of ī, which is in sharp contrast to both unrestricted and restricted indexes reported in the literature. The construction of a constant-restricted index requires predetermining the value of ī intended for a selection program to derive the corresponding b. An index with constant restrictions has no meaning unless it is associated with a specific value of ī. Numerical examples are given to illustrate the construction of the index with constant restrictions and to validate the theoretical development proposed. The derived equations have yielded an index that maximized the total merit and fulfilled constant restriction at the same time.