Use of a stochastic algorithm to make selection and reproduction
decisions under operational and cost constraints.

In preparation: Kinghorn, Grundy, Shepherd, and Woolliams

The problem addressed here is, given candidates of each sex:

• Which animals to select as parents
• Which animals to boost reproductively (AI, MOET, Ooctyte harvesting, etc)
• What pattern of mate allocation to use.

The objective aims for high merit of progeny, low costs of reproductive technology and low levels of inbreeding both in the short term and in the long term. [Not addressed here is generating matings most conducive to high accuracy in subsequent genetic evaluations].

Basic Approach

The basic approach follows Naomi Wray, Mike Goddard, Theo Meuwissen and others …

First consider maximising x'G + l x'Ax with respect to x - where x is a vector of contributions from candidates plus predicted contributions from existing juveniles. G is a vector of EBV's and A is the NRM for candidates and juveniles. x'G reflects predicted genetic merit of progeny and x'Ax reflects coancestry among parents (which leads to inbreeding in later generations). Vector x should sum to the number of matings (or, more precisely, number of pregnancies) for each sex of candidates, and (number of matings)/(generation interval) for each sex of existing juveniles. The latter is an approximation to the ideal handling of juveniles and overlapping generations.

A stochastic algorithm was developed to give integer solutions for x. This has features of both differential evolution and genetic algorithms. Integer solutions are important here, as selections and matings are discrete. Looping through values of l gives the top curve in Figure 1, in which x'G (~ genetic merit) and x'Ax (~ long-term inbreeding) are scaled as if x elements sum to 0.5 for each sex of candidates. It is also possible to set a constraint on x'Ax, by setting x'G as the criterion to maximise, and allocating a low fitness value to candidate solutions which result in x'Ax being above the constraint.

Using the stochastic algorithm, it is also possible to set maximum contributions from each candidate or juvenile. Brian Grundy has also developed Theo Meuwissen's 'real' method to set maximum contributions. There is a NAG subroutine which can also be used.

Accommodating costs

Costs associated with using novel reproductive techniques can be simply accommodated in the stochastic method. The middle and lowest curves in Figure 1 assume three levels of reproduction for each sex (Males: natural mating, fresh AI, frozen AI; Females: natural mating, MOET, Oocyte harvesting) with a cost per parent and a cost per pregnancy within each level. Other cost patterns can be adopted with relative ease. The fitness criterion becomes x'G + l x'Ax - Cost, or x'G - Cost with a constraint on x'Ax.

The lowest curve in figure 1 involves such high costs that only natural matings were recommended. The middle curve involves lower costs, such that some AI comes in at low values of x'Ax (lower animal relationships means more parents therefore less reproductive boosting), then more AI plus MOET come in at higher values of x'Ax. Wherever AI or MOET are invoked, there is usually close to maximal use of the ensuing gametes for each parent, to capitalise on the cost per parent.

Mate allocation

Following animal selection, mate allocation can be made in order to minimise inbreeding in the progeny, and/or maximise merit for non-linear traits. Figure 2 shows predicted inbreeding coefficient values for ensuing progeny, from the run with no reproductive costs considered. The upper curve is from random mate allocation and the lower curve is for mate allocation to minimise inbreeding, using the full NRM and following Jansen and Wilton's LP approach. The curves join at the point where only one male is selected. In practice, this mate allocation would be run after harvesting of gametes, as allocation depends on the genetic resources available at mating itself.

The rationale for this 2-step approach (selection then mate allocation), rather than simultaneous mate selection, comes from other work using mate selection on the index x'G + l ₁x'Ax + l ₂(F in progeny). Very low values of l ₂ gave favourable effects on inbreeding and genetic gain, in both short and long terms, and this is equivalent to the 2-step procedure used here. The optimal value of l ₂ is higher than 'very low' but unfortunately mate selection on such an index is impractically slow for data sets of any practical size.

Use of method

The method is driven by a pedigree file (ID, sire, dam) which includes fields for EBV index, sex, and candidate status (0=dead, -1=juvenile, other integers = maximum number of matings possible). The output is a list of recommended selections, number of uses (implying reproductive manipulation) and mate allocations. As such, the method can readily be used on real data as a service to breeders.

Brian Kinghorn, September 1997.