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258 Graham R. Wood, Duangdaw Sirisatien, and Patrick C.H. Morel2.1 DomainAn animal “diet", specifying nutrition for a certain growth period, and required as input toa simple growth model, can be described using only three parameters, p, r and d, defined asfollows:p = proportion of the ad libitum digestible energy intaker = minimum lysine to digestible energy ratio, in grams per megajouled = digestible energy density, in megajoules per kilogramThe ad libitum digestible energy intake is determined by a standard National Research Council(NRC) [3] curve, relating digestible energy to live weight of the animal. Parameter p determinesthe proportion (typically around 0.8) of that amount to be fed. Lysine is an aminoacid, required for growth, and generally the amino acid found to be limiting in a diet. Forthat reason we specify the level of lysine required using parameter r and specify the otheramino acids needed for growth (in a particular ratio to lysine - providing so-called “balanced"protein) using the lysine level. Finally, the energy density d of the diet must be within limitsin order to ensure palatability.We term a “feeding schedule" to be a finite sequence of diets, indexed by i, with the ith dietfed for a predetermined time of T i days (fixed at the outset). Thus we write a feeding scheduleasF = (p 1 , r 1 , d 1 ; p 2 , r 2 , d 2 ; . . . ; p n , r n , d n )Typically T i = 7 for each i, so diets are fed for a week, and n = 15 diets are offered. This totalperiod of 105 days, for New Zealand conditions, amply covers the usual time from weanerarrival (it is assumed that the producer buys in weaners) to slaughter date. Typical parameterranges used are [0.8, 1] for p, [0.2, 1.2] for r and [12, 17] for d. Our aim will be to find the optimalfeeding schedule and slaughter date, so the domain of the problem isP 1 × R 1 × D 1 × P 2 × R 2 × D 2 × . . . × P n × R n × D n × {1, 2, . . . , T }where P i = [0.8, 1], R i = [0.2, 1.2] and D i = [12, 17] for each i = 1, . . . , n and T = ∑ i T i.2.2 Objective functionThe objective function to be maximised is profit, or gross margin at market, given byg(F, x) = Gross Return(F, x) − Feed cost(F, x) − Weaner Costwhere F is a feeding schedule and x the slaughter date. Throughout, we measure this as“profit per pig" (an alternative is to measure “profit per pig place per year"). The weanercost is fixed, typically in New Zealand at NZ$70. Feed cost FC is a function of F and x, as isthe gross return GR. In Table 1 we give a typical linear programme using New Zealand data,showing ingredients, their cost, their composition and the associated constraints, when usinga diet with r and d. In Table 2 we show a typical New Zealand price schedule; the return on apig depends on backfat thickness and carcass weight.Objective function: Single pig The flow chart in Figure 1 details how, given a feedingschedule F and slaughter date x, together with the p i and genotype parameters ofminLP = minimum allowable lipid to protein ratioP dmax = maximum daily protein depositionThe gross return GR(F, x) and total feed cost F C(F, x) can be computed. In turn, the objectivefunction g(F, x) is then calculated. An iteration of the routine uses r i and d i to complete the