Economic replacement of a heterogeneous herd.

The goal of this paper is to determine the importance of considering heterogeneity of animal growth in making herd replacement decisions. As a first step, the net returns of basing the optimal slaughter age on the herd average growth curve are compared with results obtained when basing the optimal slaughter age on the heterogeneous growth curves in the herd. For comparability, the entire herd is marketed on a single day in both the average growth and heterogeneous growth cases. In addition, revenues in both cases are calculated on the basis of the true heterogeneous weights of the animals at slaughter. As a second step, it is demonstrated that marketing animals in truckload batches over time rather than on a single day, corresponding to common producer practice in the industry, can further increase profit. The advantage of this approach is that faster growing animals can be marketed earlier than slower growing animals, potentially avoiding some discounts for overweight and underweight animals that occur when marketing on a single day.

Heterogeneity is particularly important in the swine industry where packer payment programs frequently include significant price discounts for over- and under-weight animals, and where the majority of producers with over 100 animals empty an entire production unit for cleaning before replacing the animals with a new herd under a system called All-In/All-Out production (USDA 2005). The present analysis focuses upon the replacement decision for a hog producer using an All-In/All-Out (AIAO) grow/finish production system in which animals are fed from age 50 days to slaughter in a barn with a capacity of 1,000 head. As per the AIAO system, all pigs must be marketed and the barn cleaned and disinfected before the next group of animals can be brought in. It is further assumed that animals are marketed through a cash based carcass merit payment system. Under this system, packers set prices that are transparent and known prior to delivery. As is common in practice, this study assumes that the packer applies price discounts to animals whose weight and/or percent leanness is outside a desired range with larger discounts for weights/lean percentage further from the desired range. An important aspect of these discounts is that they are (discontinuous) step functions of live weight and leanness.

Traditional analysis of the livestock replacement problem has focused on a representative animal or the mean of a group of animals as the unit of analysis. For example, Chavas, Kliebenstein, and Crenshaw (1985) apply this type of analysis to the evaluation of nutrition and marketing decisions. With a few exceptions, within-herd heterogeneity of animal growth has been ignored in the production literature. Some exceptions include Greer and Trapp (2000) who examine the impact of alternative pricing grids on the optimal feeding period for groups of animals. Lusk et al. (2003) evaluate the use of ultrasound technology to choose the marketing method for cattle, but they do not consider the optimal timing of marketing. Brorsen et al. (2002) evaluate the economic impact of banning the use of antibiotics at sub-therapeutic levels in swine production, accounting for the decrease in ending weight variation with antibiotics.

This paper goes beyond previous research in two important ways that lead to new insights. First, we follow common industry practice wherein swine are marketed in semi-tractor trailer loads. We evaluate the strategy of optimally marketing the herd over time, with the potential to ship some truckload batches of animals to slaughter several days before the rest of the herd is shipped, and taking into account that the entire herd must be shipped before a new herd of feeder pigs can be brought into the facilities.

Models

A simulation model was developed that incorporates an algorithm to determine the optimal slaughter weights of pigs that are raised in a large (1,000 head) barn, and are potentially marketed in truckload groups on different days. The purpose of this analysis is to consider herd replacement in a fixed capacity facility, and consequently the objective is to maximize average daily returns to the facility and operator labor. In order to assess the economic outcomes of the alternative approaches to analyzing marketing decisions, three variants of the model are developed. These models are presented below, followed by a discussion of the model parameters.

Homogeneous Herd

In this model, the producer bases the shipping decision upon a "mean animal" whose growth curve is equal to the average of the growth curves of all animals in the herd. It is assumed that all animals will be shipped on one day, and that the selected day will be the one for which the discounted average daily profit generated by this mean animal is maximized (Dillon and Anderson 1990). As profits received by the producer are based on actual weights rather than on the mean animal's weight, the heterogeneous herd information is used to determine actual profits on the selected shipping date.

This model can be written as

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where t + k denotes the number of days the production facility is in use for the production cycle (animal growth and finishing, t, plus turnover time, k), [beta] denotes the discount factor used to convert future revenue to present value, [[bar.W].sub.t] is the herd average hot carcass weight on day t, [[bar.L].sub.t] denotes the herd average leanness (%), n denotes total number of individual pigs (n = 1,000), [[bar.VC].sub.t] denotes the present value of average cumulative variable production costs per pig on day t, P denotes the base hot carcass weight price ($/kg), and d([[bar.W].sub.t][[bar.L].sub.t]) denotes the price discount for the herd average hot carcass weight and leanness ($/kg).

Heterogeneous Herd with a Single Shipping Decision

Similar to the homogeneous herd model, this model optimizes the day on which all animals are marketed. This analysis differs, however, in that this model explicitly recognizes the effect of herd heterogeneity on revenues; here the shipping decision independently considers each animal's weight and its associated discounted price. This model can thus be specified as

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where i indexes the animals in the herd, [L.sub.it] denotes the leanness of pig i on day t, [W.sub.it] denotes the hot carcass weight of pig i on day t, [VC.sub.it] denotes the present value of cumulative variable production costs for pig i on day t, and d([W.sub.it], [L.sub.it]) denotes the price discount for the hot carcass weight of pig i on day t ($/kg).

Assumptions concerning producer information and prices are the same as those described in the homogeneous herd model above. The net returns for the homogeneous herd model are calculated as in (2) despite (1) being used as the objective function for the optimization. Thus, the homogeneous herd model is optimizing with respect to the wrong objective due to the assumption that it is adequate to analyze the mean animal. By taking the heterogeneity of animal weights into account when evaluating revenue, the heterogeneous herd with a single shipping decision model uses a more accurate measure of the price discounts.

Heterogeneous Herd with Multiple Shipping Decisions

The heterogeneous herd with a multiple shipping decision model also treats the herd as a heterogeneous group. In this model, however, the artificial restriction that the entire herd of animals must be shipped on the same day is relaxed. Thus in the heterogeneous herd with a multiple shipping decision model, truckload batches of animals may be shipped on multiple shipping dates, with the possibility that more than one batch will be shipped on any given day. For each herd of 1,000 animals, it is assumed that truckloads are shipped full (170 head), with the exception of the final, sixth load which contains the remaining 150 head. Changes in animal growth due to reduced crowding following shipments are not considered in this analysis; as such, the benefits of multiple shipment dates are likely understated.

While producers are assumed to have perfect knowledge of animal weights, they are not assumed to have knowledge concerning the leanness of each animal. (Only a small fraction of producers use ultrasound imaging to assess carcass leanness at the farm, and even those producers do not evaluate every market hog.) Because an important reason for early shipments is to reduce discounts due to animal weight and/or leanness, lack of carcass quality information limits producer ability to optimally market their animals. Thus, we model the producer's selection of animals for shipment within a load by the simple rule that they will opt to ship the heaviest animals first.

As argued by Burt (1965, 1993) and Dillon and Anderson (1990), the decision of the length of the complete production cycle should reflect the opportunity cost of the facility. However, when multiple shipping dates are optimal, this opportunity cost only plays a role in the choice of the final shipping date. Shipments on earlier dates should be chosen so as to maximize the net return to the load. A flow chart which describes the algorithm for determining the optimal shipping dates for each load is presented in figure 1.

[FIGURE 1 OMITTED]

The algebraic statement of this model is

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]