A novel selection algorithm for maximizing genetic response while constraining the rate of inbreeding is presented. It is shown that the proposed method controls the rate of inbreeding by maintaining the sum of squared genetic contributions at a constant value and represents an improvement on previous procedures. To maintain a constant rate of inbreeding the contributions from all generations are weighted equally and this is facilitated by modifying the numerator relationship matrix. By considering the optimization of the contributions of many generations the initial mating proportions (the genetic contributions to the next generation) are not equal to their long-term values, but are set equal to the expected long-term contributions given the current information. This is confirmed by the regression of the long-term contributions on the assigned mating proportions being close to one. The gain obtained from the selection algorithm is compared with the maximum theoretical genetic gain under constrained inbreeding. It is concluded that this theoretical upper bound is in general unattainable, but from this a concept of genetic efficiency in terms of resources and constraints is derived.

Three selection indexes for the UK hill sheep sector are derived to suit the extremes of hill production systems. These are: (i) intensive, where all surplus lambs not required for breeding are finished for slaughter, (ii) extensive, where all surplus ‘store’ lambs are sold to other farmers for finishing, and (iii) semi-intensive, which is intermediate between the two extremes, i.e. farms finish some lambs for slaughter and sell others as store lambs. Parameters for 12 breeding goal and index traits were estimated using a total of 3962 lamb records and 5944 ewe lambing records from Scottish Blackface sheep on two Scottish Agricultural College experimental hill farms. The breeding goal comprised carcass, maternal and survival traits. The evaluation of these indexes showed that improvements in maternal traits are possible, along with more modest improvements in carcass quality traits. Responses to selection are expected to be lower for the extensive farm in general, compared with the intensive farm. Evaluations of alternative indexes show that an index using measurements of fat and muscle on ewes rather than on lambs may be more cost-effective to implement in practice, compared with the original index, although this change results in a higher (i.e. undesirable) gain in mature size. Sensitivity analyses showed that in general, the indexes are robust to changes in economic values and to changes in heritability estimates.

A dynamic selection algorithm for maximizing annual genetic response while constraining the rate of inbreeding per generation in populations with overlapping generations is presented. The procedure gives the optimum number of individuals to be selected and the progeny they each produce. The solution to the problem was obtained by using BLUP estimated breeding values, the augmented numerator relationship matrix and lifetime breeding profiles. The procedure was able to constrain the rate of inbreeding per generation to a predefined level across generations of selection by considering all gene flow pathways. The optimization procedure represents an improvement on standard truncation BLUP selection, as it yielded substantially more genetic response (up to 35%) at the same rate of inbreeding.

The value of a parameter such as heritability (h2) or intra-dass correlation in best linear unbiased prediction (BLUP) with the animal model or a family selection index affects both the rate of response achieved and the rate of inbreeding. If in BLUP an estimate of h2 is used which is biased upwards above its actual value, the rate of inbreeding can be substantially reduced with little reduction in the rate of response. Further, by mating individuals from families in which many are selected to others from families with few selected (compensatory mating), rates of inbreeding can be further reduced without substantial effect on response.

This paper reports the selection responses achieved, and related results, following 9 years of index selection for lean growth in Suffolk sheep. The breeding goal of the index used comprised carcass lean weight and carcass fat weight at a constant age, with relative economic values of + 3 and -1 per kg. The selection criteria were live weight (LWT), ultrasonic fat depth (UFD) and ultrasonic muscle depth (UMD) adjusted to a constant age of 150 days. By year 9, responses in LWT, UFD and UMD in both sexes, as judged by the divergence between selection and control line performance, amounted to 4·88 kg, -1·1 mm and 2·8 mm respectively ; these responses are between 7 and 15% of the overall means of the traits concerned. Although selection was originally on index scores based on phenotypic records, the retrospective analyses reported here used the mixed model applications of residual maximum likelihood to estimate parameters and best linear unbiased prediction to predict breeding values. The statistical model comprised fixed effects plus random effects accounting for direct additive, maternal additive and temporary environmental variation. Estimated genetic trends obtained by regressing estimated breeding values on year of birth were similar to annual responses estimated by comparing selection and control line means. Estimates of direct heritabilities were 0·054, 0·177, 0·286, 0·561 and 0·410 for birth weight (BWT), weaning weight (WWT), LWT, UFD and UMD respectively. Corresponding estimates of maternal heritabilities were 0·287, 0·205, 0·160, 0·083 and 0·164. Phenotypic correlations between all pairs of traits were positive and usually moderately high. There were low negative direct additive correlations between BWT and WWT, and between BWT and LWT, but higher positive maternal additive correlations between all other pairs of weight traits.