The use of mathematical programming to control inbreeding in selection schemes
Authors: Fernández By J. , Toro M. A. 
Published: March 2003

IntroductionSelection on the best estimate of the breeding value of individuals should, in large populations, provide the maximal response in breeding value. However, many breeders deal with the selection of small numbers of animals from relatively small populations and therefore there is a trend for inbreeding to rise because of genetic drift. Moreover, as the evaluation of candidates is traditionally based on methodologies including information from relatives [selection indices, best linear unbiased predictor (BLUP)] more individuals are selected from the best families and so closely related individuals will generate most of the offspring. This effect is more important for traits with low heritability as phenotype gives little information on the breeding value of the individuals and more weight is given to relatives’ data.The need for controlling inbreeding refers not only to a better use of the genetic variability available and to a reduced inbreeding depression in the selected trait, but also to a reduced depression of fitness‐related traits, which may be the most serious drawback at present due to the increase in inbreeding in domestic populations (M euwissen and W
oolliams 1994).In recent years considerable work has been carried out on the design of strategies to maintain genetic diversity in selection programmes. These strategies are aimed at simultaneously optimizing genetic gain and inbreeding, either by reducing the rate of inbreeding (or variance of response) while keeping genetic gains at a predetermined level, or by increasing selection response under a restriction on inbreeding (or on variance of response). Following T oro and P&
eacute;
rez‐E
nciso (1990) the different strategies can be classified according to the factor on which they act: (i) the selection criterion used; (ii) the mating system imposed; (iii) the number of selected individuals and their contribution to the next generation.The first group of strategies proposes the use of a suboptimal selection criterion that reduces the weight given to family information or the use of an upward‐biased heritability in BLUP evaluation (T oro and P&
eacute;
rez‐E
nciso 1990; see G
rundy et al. 1998a for the latest development of this idea). The second group of strategies proposes action on the mating system including factorial mating designs, minimum co‐ancestry mating (using linear programming) or compensatory mating (see review by C
aballero et al. 1996).The third group of strategies includes the ones considered in the present work. The first possibility is to modify the contribution of the selected individuals of generation t to the selected individuals of generation t + 1, by practising some form of within‐family selection with respect to BLUP values. Two strategies of this type were considered: modified within‐family selection (MWFS) and restricted co‐ancestry selection (RCS). The second possibility is to modify the contribution of the selected individuals of generation t to the evaluated individuals of generation t + 1 (instead of to the selected individuals) by a strategy called weighted selection (T
oro and N
ieto 1984). Three strategies were considered in this case: weighted selection (WS), restricted co‐ancestry weighted selection (RCWS) and pair weighted selection (PWS). More specifically, the aim of the present paper is to show how these five strategies can be implemented using mathematical programming techniques. A small example comparing all of these strategies with standard truncation selection (TS) is also given for illustration.