A simultaneous procedure for deriving selection indexes with multiple restrictions.

The formulas given in literature for the construction of restricted indexes were designed only for the imposition of a single restriction (zero, fixed, or proportional). This study presents both the theory and the methods of a simultaneous procedure for constructing indexes with single or multiple restriction(s). Numerical examples are given to verify the theoretical development and to demonstrate the implementation of the procedure. The simultaneous procedure presented brings the construction of various restricted indexes into a simple computational scheme. In addition to the use of the proposed procedure to handle multiple traits, it can be used to modify the growth curve of meat animals or the lactation curve of dairy animals, which generally requires simultaneous imposition of different restrictions on different stages of the curves. A misconception in the literature is that the variance of an index (b'Pb) is not equal to the covariance between an index and its net merit (b'Ga) when the index is a restricted one. This study showed generally that b'Pb and b'Ga are equal in the restricted or unrestricted case only when elements of b represent the original solutions from the index equations and are not equal when elements of b are expressed in terms of proportions.