We used a gene-flow model of the Canadian Holstein population coupled to a non-linear optimisation program to determine the optimum economic weights for milk, fat and protein yield in each of the next 20 years, for alternative scenarios of evolving market demand. The results indicate that the appropriate index for sire selection today is,

where milk, fat and protein are ETA or EBV for yield expressed in kg. The recommended index reflects our assessment of most likely future demand and management (fat demand relative to protein decreasing at 1% per annum; dairy management causing increased milk, fat and protein output per cow of 1% per annum), and is also robust to our predictions being wrong. The index promotes slow decrease in the fat to protein ratio while achieving substantial increases in milk fat and protein yield per cow, and maximises profit to dairy producers. We recommend that this index be incorporated into the LPI and the proposed total economic value index (TEV), after appropriate scaling.

There is considerable uncertainty about future markets and prices for dairy products in Canada. However, market trend analysis at the University of Guelph, and elsewhere, all point to a continuing decrease in the demand for fat relative to protein. The ratio of fat to protein in milk produced currently matches demand almost exactly; but in the future we can expect an increasing excess of fat production. There is already a huge excess production of milk volume (water) and lactose.

Defining breeding objectives in a shifting market is a complex non-linear problem which has to take into account the flow of genes through the population over time in relation to the shifting demand (and hence prices achieved) when genes are expressed in lactating cows. Dekkers, Birke and Gibson (1995) showed how a non-linear optimisation program could be used to find optimum selection index weights over time when profit was a non-linear function of genotype. We have extended that method to deal with the multiple pathways of selection and overlapping generations inherent in dairy cattle breeding, to determine the selection index weights for milk, fat and protein yield that would maximise long-term profit in the Canadian dairy industry.

Briefly, we first constructed an approximate gene-flow model of genetic improvement in Canadian dairy cattle breeding. In this model it is assumed that selection in each of four paths of selection takes place on multiple trait selection indexes, with information on milk, fat and protein yield on the individual (if female) and close relatives. This is a very close approximation to using a selection index of single trait estimated breeding values for milk, fat and protein. Annual cohorts of cows and bulls are born into and move through the population, lactating and/or reproducing at stages and in proportions matching those observed in practice.

The goal of genetic improvement is to maximize the cumulative net present value of profit (CNPV) over a planning horizon of T years, given the initial state of the population at time zero, and selection decisions taking place annually. That is,

where,

f_t(m_t)is the average profit in year t as a nonlinear function of m_t, which is a q * 1 vector of population means of q economic traits in year t, and
w_t is the discount factor applied to future profit expressed as (1+r)^-t, where r is the discount rate.

To simplify the problem, we assumed an approximation of asymptotic response, derived from the gene flow model, such that m_t is derived recursively from m_o, the original population performance, as

where,

K is the number of selection paths k, that contribute to the genetic response,
G_k is a n * q matrix of additive genetic (co-)variances between n information sources and q economic traits,
P_k is a n * n matrix of phenotypic (co-)variances among information sources,
i_k is the selection intensity for selection path k,
L_k is the generation interval for selection path k, and
b_tk is a n * 1 vector of index weights used in selection path k, leading to expression of genes in progeny at time t.

As defined, a solution to 1 would require finding the optimum set of nKT index weights. To reduce the scale of the problem, we further assumed that each year there is an optimum set of n economic weights, v_t, leading to index weights using the standard formula,

The problem was then to find the qT set of economic weights which maximize CNPV.

This problem was solved by the derivative-free nonlinear programming optimizer developed by Birke (1993). The method, which is called Nonlinear Programming via Sequential Smoothed-out Minimax (NLPSSM), transforms the general nonlinear programming problem into an equivalent minimax problem (Bandler and Charalambous, 1974). The resulting minimax problem is sequentially smoothed-out in the spirit of Charalambous and El-Turky (1979) to produce robust and accurate resolution of the objective function. The derivative-free method applied to the decision variable strategy (in this case the decision variables are the set of relative economic weights) uses a direct search method, namely, Vectored-Step-Exploratory Pattern Search (VESPS). The vectored steps within VESPS create high pseudo-gradient information, which is utilized by the pattern moves. Motivation for the philosophy of VESPS is found in Dennis and Torezon (1991). VESPS possesses the distinct advantage of ease of problem modelling, allowing changes in market conditions, quota policy and management decisions during the observed time period, to be taken into account.

After finding the optimum economic weights, the corresponding selection index weights were calculated for sire selection based on a progeny test of 50 effective daughters for milk, fat and protein yield, as described by Gibson et al. (1992). It is these index weights which are presented in the results.

The profit function, expressed for the total industry, based on the Ontario dairy market with a multiple component pricing system and regulation of fat production through a quota policy, is:

where, R_Mt, R_Ft and R_Pt are the average net returns (= returns - costs) from the milk, fat and protein produced per cow within the fat quota; RQ_Mt, RQ_Ft, RQ_Pt are the average net returns per cow from milk, fat and protein produced over the fat quota; C_m is the maintenance cost per average animal and year; and n_ft and n_pt are the number of cows required to meet the domestic demand for fat and protein in year t. We assume that the market will be regulated (politically or by free market forces) such that demand for both fat and protein will be met by dairy producers. Thus, the second term of the equation above describes the profit made when producing over the fat quota to meet the protein demand. In all situations presented here, RQ_Pt = R_Pt, and n_ft < n_pt.

Table 1 gives net returns and average production/cow/yr for milk, fat and protein as well as the demand for fat and protein in year zero. Returns for milk, fat and protein reflect the Ontario milk market. The production cost have been adapted from Gibson et al. (1992), except those for feed costs for protein, which have been increased following the arguments of Dado et al. (1994). Details of cost and return calculations as well as the average production level for the Canadian dairy population will be fully described by Greimel et al. (1995 unpublished).

Fat and protein demand in year zero are met exactly by the same number of cows (n_f0 = n_p0 = 1,017,500 cows), which reflects the current balance between supply and demand in the Canadian dairy market.

We assumed that annual selection starts 5 years (= average generation interval) prior to the observed time period so that t₁ = 1 is the year the first changes in production due to selection (5 years earlier) occur. A time horizon of 20 years from first expression at t=1 was assumed with an inflation free discount rate of r = .05 per annum.

Heritabilities, genetic and phenotypic correlations and coefficients of variance for the Canadian dairy population were the same as used previously in the research leading to the original LPI (Gibson, Graham, and Burnside, 1992). No correction was made for the effects of selection on genetic parameters in the index calculations. Other biological coefficients used to calculate selection intensity and accuracy will be described by Greimel et al. (1995 unpublished); but closely reflect current breeding programs.

Based on the data presented above many alternatives were derived which differed in, a) the absolute and relative demand for fat and protein over the time horizon, b) the profit function and c) the non-genetic management factors which affect production per cow. In the present report, results are given for three alternative scenarios:

The optimum index weights at years 2 and 20 are presented for the three scenarios in Table 2. The weights are scaled relative to a weight on protein of +10. In all three scenarios the index weights change over time, but only by a relatively small amount. Thus, provided market conditions are anticipated correctly, an appropriate index would not have to be updated frequently. (Note that, for some scenarios, the index looks a little different in year 1 compared to year 2 and beyond. This is due to the unique condition of market demand for fat and protein being exactly equal to supply in the first year, and in real life indexes for year 2 and beyond would be appropriate. We have thus shown only the year 2 results as representative of appropriate indexes for selection in the early years.)

If market and management conditions remain constant (scenario S1), the index gives a heavy negative weight to volume, moderate positive weight to fat and heavy positive weight to protein. As shown in Table 3, the index promotes genetic improvement in milk, fat and protein yield per cow, with somewhat more rapid increase in fat than protein. Table 4 shows the number of cows required to meet the national demand for fat and protein at years 2 and 20. By year 20 slightly fewer cows are required to meet fat demand than protein demand, so that a very small excess fat production is predicted to develop.

These results for S1 differ from the calculations leading to the original LPI in putting slightly more negative emphasis on milk and less much less positive emphasis on fat. This reflects, a) the relative increase in protein prices since 1990, b) the heavier penalties on over quota production of fat, and c) the non-linear optimisation technique used here appropriately anticipating the increasing divergence between supply and demand that could develop over time due to greater genetic change in fat than in protein.

When the more likely scenario of a continuing 1% decline in fat demand relative to protein demand is anticipated (scenarios S2 and S3), the index gives heavy negative weight to milk, a small negative weight to fat and high positive weight to protein. Compared to S1 genetic responses for milk are a little lower, for fat are 20 to 30% lower and are about the same for protein. After 20 years, the number of cows required to meet fat demand is considerably lower than to meet protein demand, indicating a substantial excess production of fat. Thus, although these indexes promote a decreased fat to protein ratio in milk, they do not maintain an exact balance between supply and demand for fat and protein. Rather, the indexes recognise that more profit can be achieved by having a balance between a high efficiency cow (due to high output per cow) and an optimum fat to protein ration cow.

Allowing for the effects that non-genetic management will likely continue to improve output of milk, fat and protein per cow at about 1% per annum, has rather little effect on the indexes and responses (Tables 2 and 3, S2 vs. S3).

When market demand for fat remained constant while demand for protein increased at 1% per annum, indexes were essentially identical to those for S2 and S3. Thus it is the changing relative demand for fat versus protein that determines the index rather than the absolute demands for fat and protein.

Based on assessments of market trends by ourselves and others, and on a comparison between Canada and most other western countries, the most likely scenario is S3, with decreasing relative demand for fat vs. protein and continuing improvement in non-genetic management leading to higher yields per cow. This alone would point to S3 as an appropriate index. In addition, we tested the efficiency of indexes for all scenarios against the possibility that each of the alternative scenarios was realised in practice. The index for S3 had the highest average efficiency and the highest minimum efficiency, indicating that it was the most robust index to our predictions failing to come true.

A number of approximations had to be made in these calculations. The gene flow model is only an approximation of genetic improvement in Canada. In particular, the derivation of a single set of economic weights across all paths of genetic improvement is not optimum, since different paths have different time horizons for gene expression in descendants. In particular, selection of sires for use to produce dairy replacements should have a considerably shorter term perspective (around 5 to 7 years) compared to sires of sons (around 11 to 15 years). The fact that indexes derived here do not change markedly over time suggests, however, that both sire selection paths will have similar indexes (we plan to check this, but results will not be available for some time).

Our pricing systems assume values similar to the present for fat and protein that can be sold in the Canadian market, with penalties only for production in excess of demand (this is effectively how the current quota and pricing systems determine net costs, even though this fact is not transparent to the farmer). There is however, some pressure to substantially lower fat prices and raise protein prices in the hope that this will stimulate extra demand for fat. This would lead to rather similar indexes to those here if demand for fat were increased, or to much lower weighting (i.e. more negative) if demand for fat were not affected.

There is also uncertainty about the appropriate way to deal with milk volume. We have used the current Ontario blend price and various deductions for volume with penalties as currently applied to production over the fat quota. In reality there is currently a massive over production of fluid (most water is extracted and discarded), and so the marginal price of milk should probably be zero, with appropriate deductions and costs leading to a lower negative net value than used here. On the other hand, it is not clear that several of the current deductions should be applied specifically to milk volume, and there is considerable uncertainty about the correct marginal costs of producing milk volume. We have probably erred on the high side for costs, so that the net result is that our net values for volume are probably not far from the true values.

These results point to immediate implementation of index S3, or something very similar, for maximum economic progress of milk production traits in Canada. In relative terms, the index would be,

where milk fat and protein are sire EBV expressed in kg. For implementation, this index would need to scaled so that the index was expressed in dollars, and could be incorporated into total economic value index (TEV; see this year's report by Dekkers et al.).

* Values within brackets are net returns for production over the fat quota.

* s _g = 672.1 kg, 26.2 kg and 19.7 kg for milk, fat and protein

Table 4. Number of cows required to meet market demand for fat and protein in years 2 and 20.