Economic Optimization of Growth Trajectory for Broilers

Agricultural & Applied Economics Association

Economic Optimization of a Growth Trajectory for BroilersAuthor(s): Hovav Talpaz, Shmuel Hurwitz, Jose R. de la Torre and Peter J. H. SharpeSource: American Journal of Agricultural Economics, Vol. 70, No. 2 (May, 1988), pp. 382-390Published by: Oxford University Press on behalf of the Agricultural & Applied EconomicsAssociationStable URL: http://www.jstor.org/stable/1242079 .Accessed: 18/03/2014 01:53

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Economic Optimization of a Growth Trajectory for Broilers Hovav Talpaz, Shmuel Hurwitz, Jose R. de la Torre, and Peter J. H. Sharpe A dynamic model was constructed to select the economically optimal growth trajectory of broilers and compute the feeding schedule designed to satisfy the nutritional requirements along this trajectory. Alterations in the normal growth rates are possible by utilization of the phenomenon of compensatory growth which follows a period of feed restriction. The optimal trajectory is solved for by a reduced-gradient nonlinear programming algorithm. The model calculates the daily optimal growth rates along with the corresponding requirements of total protein, amino acids, and energy in obtaining the optimal diets. A substantial increase in profits can be achieved by following this methodology.

Key wtords: dynamic optimal trajectory, Gompertz curve, least-cost diets, reduced-gradient algorithm.

The poultry industry has become highly capital intensive. Competition in the industry has improved ways to cut production expenses and to market better quality products. This development has reinforced the need for pre- cise management and short- and long-run planning. For broiler production, planning in- volves such controls as diet composition, environmental temperature, marketing schedule, and housing density, all in the context of economic optimization. Because of the nonlinear nature of broiler growth in response to the controls and interactions among these controls, optimization requires a dynamic simulation of the biological response in terms of growth rates and the corresponding feed in- takes. The simulated biological functions can

then be used as the basis for economic optimization.

A simulation model aimed at improving management, but not including feed formulation, was documented previously (Hurwitz, Talpaz, and Waibel). Recently, Talpaz et al. developed a dynamic broiler-feeding model capable of computing least-cost rations along a predetermined growth path while accounting for dynamic changes in the nutritional requirements. This significant step has not considered a major control decision available to the grower-the definition of the optimal growth path, namely, the profit-maximizing body-weight trajectory.

Biological experiments demonstrated compensatory growth following an early-age feed restriction (Plavnik and Hurwitz 1985). This natural capability suggests potential savings of feed required for maintenance which is non- linearly and positively related to body weight. To include this aspect in the optimization, a dynamic feedback control component must be introduced into the growth model. In this study we describe a computerized algorithm to optimize a broiler operation, including formulation of feed needed to support the optimal growth trajectory. The use of this algorithm may lead to an improvement of 8%-10% in profitability over present practices.

Hovav Talpaz and Shmuel Hurwitz are in the Departments of Statistics and Poultry Science, respectively, Agricultural Re- search Organization, The Volcani Center, Bet Dagan, Israel. Tal- paz was on a visit to the Department of Agricultural Economics, Texas A&M University. Jose de la Torre and Peter Sharpe are at the Biosystems Research Division, Texas A&M University.

Contribution from the Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel, No. 1940-E, 1987 series. Texas Agricultural Experiment Station technical paper TA No. 22411.

Financial support was provided jointly by a grant from the United States-Israel Binational Agricultural Research and Devel- opment Fund-BARD (No. US-777-86), by the Agricultural Re- search Organization and by the Texas Agricultural Experiment Station.

The authors are grateful for the constructive and helpful comments provided by the two anonymous Journal reviewers.

Copyright 1988 American Agricultural Economics Association


Talpaz, Hurwitz, de la Torre, and Sharpe Optimal Broiler Growth 383

The Problem Statement

The problem is developed in terms of the objective function and growth paths, as dis- cussed in the next two subsections.

Objective Function Consider a broiler growing operation with fixed equipment and housing capacity. The grower seeks to maximize the profit from these fixed assets.

(1) Maximize Z= Y2{Pv[R2(q) -C,(q, #, 7) ]}; i = 1, .. . ,N,

where Pv is the present value operator; R, is revenue from batch i; q is total live weight produced per square meter of floor area per batch; C, is the total variable cost function for batch i (each batch corresponds to a hatch); Q( is total feed used per square meter of floor area per batch i; 7 is the marketing age in days (including 7 days needed for clearning between batches); and N is the number of batches per year. Since N = 365/7 may not be an integer, an alternative yet equivalent formulation is applied assuming identical batches:

(2) Maximize Z* = OPv[R(q) - C(q, I, r)],

where ?n is the capital recovery factor (i.e., Fleischer, p. 16), given by (3) f" = p(1 + p)'l/[(1 + p)' - 1], and p is the daily interest rate. For a fixed 7 the capital recovery factor fn does not affect the optimal solution; hence, it is removed until the optimal marketing age is considered.

In practice, it is customary to raise broilers at a fixed projected q (the target density at market age) by controlling the initial chick density per floor area.1 Hence equation (2) is simplified to

(4) Minimize Z = C(0), for a given set of (q, 7).

This means that feed rates over time and their combinations (denoted by () comprise the major portion of dynamic control elements.

Because these rates determine growth, it becomes crucial to select the optimal growth pat- tern out of the feasible growth rates. This is the objective of this study. Before doing this, let us consider the issue in detail.

Growth Paths

The potential growth rate is defined as that rate achieved by a specific strain and sex under free feeding (ad libitum) conditions, while satisfying all nutritional requirements. The corresponding live-weight path is designated as W,. Such a path is best described by a Gompertz curve, given by (5) WP = a exp[-f exp(-yt)], where t is age in days, and a, /3, y are parameters as estimated previously (Talpaz et al.), using the experimental data obtained by Plav- nik and Hurwitz (1982).

Dynamic least-cost rations for the above W,, subject to the nutritional requirements, can be calculated using the methodology developed by Talpaz et al. However, alternative growth paths with early restriction followed by compensatory growth, as demonstrated by Plavnik and Hurwitz (1985), could be more profitable. To demonstrate the general idea, refer to figure 1, where W, is the potential growth curve equation (5). The other curves represent growth patterns resulting from vari- ous feed restriction regimes. The most restricted regime allows for a minimum of 15% of the potential growth during restriction. Such feeding severely restricts growth for t, days, but the broilers retain the ability to re- cover with compensatory growth rates once free feeding is resumed at age t, + 1 days.

w S//W

ts tp t AGE AGE

Figure 1. Potential (W,) and compensatory live-weight trajectories (We, s)

The commonly practiced target density of 30 Kg/m' is assumed here. Higher levels have shown increased vulnerability for diseases, while lower levels provide indifferent per-bird perfor- mances.


384 May 1988 Amer. J. Agr. Econ.

Compensatory growth is defined in relation to the potential growth; let a broiler's weight at age t,, be w1 (on the W. curve), while another broiler on the We curve reaches that weight at age t(t > t,) due to early feed restriction. Then, the situation where We > W,, such that W1 = W, defines compensatory growth. This situation is depicted in figure 1, where tan(o) > tan(r).

Because maintenance expenditure of energy is a function of the integral of the live-weight path, then the area between curves W, and W1 represents the reduction in potential energy consumption. However, this is not a net savings because at any given market age, the gap (W, - Wa) calls for more broilers to be raised along with the associated costs. The larger this final gap, the higher the cost. Hence, the problem is to find the optimal body-weight trajectory Wa by minimizing Z in equation (4), since C(Q) depends very closely on Wa, to be dis- cussed below.

Growth and Nutritional Requirements

A detailed description of the growth model, assuming potential path W,, was given in Tal- paz et al. A brief summary is presented here.

Body weight is given by equation (5). The fat fraction in body weight is

(6) F = X exp[-r exp(-pt)], where X, rl, and 9 are parameters.

The feather fraction of body weight is

(7) F, = #[1 - exp(-vt)], where h and v are parameters.

The requirements of essential amino acids are given by the sum of the maintenance, the gains of body protein, and feather requirements (Hurwitz, Sklan, and Bartov; Hurwitz, Plavnik, and Bornstein). The requirements for maintenance (AARm) are given by (8) AARm = AAm * .0233 * Wa,2/3, where AAm is a twelve-element vector con- taining the essential amino acid profile for maintenance, and .0233 * W,2/3 is daily protein required for maintenance. The requirements for gain of carcass protein (AARc) are based on protein content Rct of the tissue gained, and by an absorption rate of 0.85: (9) AARc = [Rc, * AAc * W,]/.85,

where AAc is a twelve-element vector contain- ing the amino acid profile for carcass protein, and Wa is daily gain in body weight.

The requirements for feather protein (AARf) are based on 85% protein content of the feathers; hence, (10) AARf = [.85 * AAf* DFE]/.85

= AAf * DFE, where AAfis a twelve-element vector contain- ing the amino acid profile for feather protein, and DFE is daily gain in feather weight. The requirements for nonessential amino acids are obtained as the difference between the required protein and the sum of the essential amino acids.

The energy intake, E,, is based on the energy conversion formula presented by Hur- witz et al.:

(11) E, = H(T) * Wa2/3 + 8, * W, where the first term is the energy maintenance requirement, affected by temperature, T, through a temperature-dependent coefficient H(T), which is also proportional to the degree of feed restriction. The second term is due to daily growth, Wa, and is dependent through a variable, 6,, on the fat content of the tissue deposited during growth: (12) 8, = 0.6 + 9.35 * (F + W, * F/Wa), where F is the time derivative of F in equation (6).

Growth Under Control

Growth is by definition the time derivative of the live-weight path. The potential growth is derived from equation (5): (13) W, = 0 W,,/8t = yf3W, exp(-yt).

To obtain the growth for any of the W, curves, referral to figure 1 again will facilitate the discussion. Consider the curve designated Wa. At age t the weight is w,; the same weight is achieved at age t, on the W, curve. Then, t,, can be eliminated using equation (5) by sub- stituting t0 for t:

(14) t, = -loge(-log(w,/ac)/fl3)/y,

where loge is the natural logarithm operator. Then, with no compensatory growth, (15) W0 = 0 Wa/t = y/3 W, exp(-yt,).



Within the framework of the model, growth can be controlled either directly or through restricted energy intake. The two approaches are equivalent. Let xt be the fraction of growth controlled by the decision maker, with 0 xt

-

1. When xt = 1, growth rate is at its full potential rate; when xt = 0, growth rate is set to zero; 0 < xt < 1 is the fraction of the full rate. Thus, under controlled (restricted) growth, or before any restriction is initiated, feed is consumed according to equations (6)- (12), where (16) Wa = a W,/at* Xt = yP W, exp(-yt) * xt.

When free feeding is resumed, following a period of restriction, broilers exhibit higher- than-potential growth rates, or compensatory growth. Several sigmoidal functional forms have been evaluated using the data obtained by Plavnik and Hurwitz (1985) in order to for- mulate this phenomenon. The selected functional representation of growth along the en- tire path,2 which includes restriction and compensatory growth, is

(17) Wa = yP Wa exp(-yt,) * Cf, where the compensation factor Cf is given by

(18) Cf =

[(W, - Wa)/W, + 1)]6; for xt

= 1.0

xt ; for xt < 1.0 where 0 is the compensation coefficient, estimated using a revised Marquardt's method for nonlinear regression. Before restriction begins We = W,, hence Cf = 1. Following a restriction period, Cf increases with the difference between the potential and actual body weight due to more severe or longer restrictions. In terms of figure 1, at age t, Cf = [(w2 - 2 + 1]6.

The potential - W - curve is based on the experimental results for White Rock broiler males obtained by Plavnik and Hurwitz (1982). Data on feed ingredients in terms of nutritional composition, prices, and other constraints are similar to those of Talpaz et al. To estimate the compensation coefficient 6, data from Plavnik and Hurwitz (1985) were used in applying nonlinear regression on equations (17) and (18) and numerical integration to obtain the pre-

dicted We curve.3 Estimation yielded the value of 0 = 0.61 with standard deviation of 0.11.

When feed is restricted, fat accumulation is diminished. Detailed information on the rela- tionship between the severity of restriction and fat behavior is not available. In this study it is assumed that fat concentration in carcass remains unchanged (F = 0) whenever xt < 1.0. Then, following the restriction period, fat de- position is equal to the derivative of equation (6) using t. instead of t.

Trajectory Optimization This problem can be classified as an optimal control case. However, derivation of the equations of Pontriyagin's maximum principle is impractical due to the model's nonlinear com- plexities. The problem is approached as a nonlinear mathematical program, with nonlinear constraints. Formally, the problem is stated as follows:

(19) Minimize Z = C[G(xt); for any relevant value of 7 (final age)

subject to(a) 0 s xt 1 forall t = 1,...,r7; (b) equations (6)-(18); and (c) least-costs diets along the optimal trajectory, where P(xt) is the feed and its composition at age t-days, as controlled by the decision variable xt.

The (c)-type constraints require a set of alternative ingredients (with their nutritional composition and prices) available to the decision maker. The major reason for complexity of the system lies in the fact that at different growth rates, as determined by xt, both feed quantity and composition are affected. For example, lowering growth at an early age saves feed but leads to a relative reduction in protein and amino acids requirements for that period. These are relatively expensive nutritional components of the diet. Later, these requirements are significantly increased to support compensatory growth. Adjustments in diet composition must be computed to support the desired one in order to be economically efficient.

With two qualifications, the problem stated in (19) is a convex program with bounded constraint set. The first qualification is that C[t(xt)] is not continuous at x, = 1 where xt-1

2 Space limitation does not permit a discussion of the other functional forms tried but not chosen.

A detailed account of this estimation procedure is under prep- aration and will be published elsewhere.



< 1. Consider again the definition of Cf in (18). When xt = 1 at the end of the feed restriction period, compensation commences abruptly, leading to a discontinuity in the growth function with a sudden positive jump of the daily growth. To overcome this problem a proper smoothening approximation is introduced (appendix 1). The second qualification is the pos- sibility of a change in the optimal basis of the feed mix ingredients along an optimal trajectory. Such a basis change introduces a possible discontinuity in the derivative of C[F(xt)]. This may become a serious problem when such a change brings a substantial difference in the feed ration's composition and costs. However, if the competing ingredients are not sharply different from their next in-line candi- dates; or if such a change occurs away from the optimal solution, then no special action is necessary.

The overall optimization of system (19) is carried out by using a reduced-gradient methodology with the computer code MINOS (Murtagh and Saunders) as depicted in figure 2. A special simulator (named BRL) was designed as a set of subroutines to be called by MINOS in order to solve the system of equations (6)-(18) over the decision-planing hori- zon t = 0, 1, .. , 7 under the corresponding trajectory vector, xt, which is being selected by MINOS. Another subroutine is called from BRL to solve for the least-cost diet each day (t), using the linear programming technique, once the nutritional requirements correspond with the (6)-(18) system and the associated x,. The algorithm iteratively converges to the optimal set, xt, while simultaneously solving for

SETUP: Initial values; Parameters; Technological data

( MINOS )

BRL

GROWTH ENERGY NUTRITION

LP: DUAL E - LEAST-COST

RATION

Figure 2. The overall optimization system

the growth rates, nutritional requirements and optimal diets.

With the data and parameters determined at the SETUP phase, MINOS calls BRL with the initial guess values XA with its elements x) for each t 0

..... 7, which in turn calculates

growth, energy, and other nutritional requirements. Then the LP subroutine is called to compute the least-cost diet for age t. Control then shifts back to BRL to calculate feedbacks (compensation, etc.), proper accounting and so on, until t = 7. At this point BRL transfers back the objective function value, F(X), to MINOS to change the values of X' for the kth iteration cycle, and so on until convergence to the optimal solution is achieved. The gradients needed for the reduced-gradient technique are computed numerically in MINOS by for- ward and, if necessary, central differencing. By the time convergence is achieved many calls (thousands) to BRL are required. This process is time consuming even for a fast mainframe computer unless efficient design is achieved. Two major remarks on this issue are as follows. First, it is not necessary to decide all of the elements of x, by the optimization process. Toward the end of the growing period x, must be 1.0 in order to allow for compensation to take place. Thus, one should shorten the decision vector x, for the period t, < t< r,, where t, is the earliest age possible for any feeding restriction (this is usually given by the biologists), and t, is an age beyond the economically feasible one for any feeding restriction, allowing for at least one day of free feeding to be decided by the algorithm. It is important that tf is determined with a "margin of safety" to ensure a true optimal trajectory. Obviously, x, - 1.0 for all t > t(. Second, the number of diet calculations per optimizing run equals the number of calls to BRL times marketing-age 7 which is over 50,000: approxi- mately 30 revised simplex iterations are needed to solve for the daily (each t) optimal diet, or a total of about 1.5 million simplex iterations. This would have required a prohibitive and impractical amount of computer time. Thus the LP problem was solved in its dual form rather than the more natural primal form.

Let the diet problem, in its primal form, be defined as

(20) Min. C' U, subject to

AU ? B; U > 0,



where C is the vector of unit price of ingredients, Uis the unknown ingredients vector, A the matrix of nutritional fraction compositions and other technical coefficients, and B the vector of constraints' levels. The dual is then given by

(21) Max. B'V, subject to

A'V V C; V 0,

where V-the solution here-is the vector of shadow prices of the primal problem (20). The vector U is solved for here as the shadow prices of the dual problem.

Because the solution of (20) is identical to that of (21), the dual should yield a substantial savings in calculations, since the constraint inequalities of (21) remain unchanged along the growth trajectory, keeping the feasible space constant over time. Thus, starting each day's diet problem with the solution of the previous day's optimal diet (where only values of B in the objective function change from day to day) may keep the same corner point solution as optimal, or move to a nearby corner. This means that less than a full iteration is required when the same corner proves optimal, while only one or two iterations are needed to move to another corner. This was indeed the case in this study (see below). Dur- ing a single run, only twice, and rarely three times for some trajectories, was more than a single iteration required for daily diet compu- tation, except for the first day.

Economic Analysis

Motivation for early feeding restriction, in terms of its severity and duration, exists in view of the resulting saving in feed. Compen- satory growth is a function of the severity and duration of feed restriction. However, the final gap between W, and Wa also increases with such restrictions. This gap forces the purchase and handling of a larger number of broilers. Hence, an optimal path is sought.

As a point of departure a basic run was designed with no restrictions and marketing time, 7, of seventy days. The results are given in appendix 2, with more details and discussion in Talpaz et al.

Optimal Trajectories for Different r's Finding the optimal marketing age cannot be achieved by the reduced-gradient algorithm because 7 is an integer and BRL simulates with time steps of exactly one day. Hence, a proper search must be conducted. Practically, such a search is a simple operation since growers would be satisfied with the optimal marketing age in weeks or half weeks. Hence, a line search to maximize equation (2) was performed by repeated runs with the marketing age lying in the interval 35

_

r -

70. Note that the capital recovery factor permits a proper comparison and selection of the optimal 7. Figure 3 demonstrates the effect of 7 on the conditionally optimal daily profit given by equation (2).

The optimal 7 is at 47 days, or 6.7 weeks. The shape of the curve has an important effect in practice. Shortening the marketing age by 2-3 days leads to relatively low reduction in profit as compared with the extension of it by the same increment. The corresponding optimal trajectories for these i's are given in figure 4.

The curves show that the duration of a severe feed restriction increases with marketing age. However, marketing weight also increases, reflecting the longer realization of compensatory growth rates which becomes possible with higher marketing age. These optimal trajectories lead to about 8% savings in production expenses for low marketing age and about 10% for high marketing age compared with the corresponding best free-feeding scheme.

The levels of the decision values xt's for each of the above 7 sets are shown in figure 5. The upper value for any xt is 1.0, while its floor was set at 0.15 because of biological consid- erations. Lower levels of xt's were optimal

io~o

90 [- 8.5

g 8.0

7.5

7.0

6.5 1 40 45 50 55 60 65 70

MARKETING AGE (DAYS)

Figure 3. Maximum daily profit as a function of marketing age



3000

74 70 63

2500 200056 49

(g) 150 0

1000

0 10 20 30 40 50 60 70 AGE (days)

Figure 4. Potential versus optimal trajectories as a function of marketing age - t

with higher levels of 7. The kinks on the increasing parts of the curves are the result of discontinuities in the optimal diet calculations at a change in the basis.

With the higher 7, feed restriction is both longer and more severe. This is understandable in view of longer time available for compensatory growth. The reason for relatively low optimal 7(=45) lies in the diminishing growth rates for larger broilers.

Price Sensitivity Analysis

The magnitude of any restriction depends on the price ratio between feed ingredients and the cost of baby chicks. The higher are feed prices, the more severe and longer is the feeding restriction. To analyze this issue, a sensitivity analysis with respect to relative feed price levels (keeping all other prices constant) was conducted, as shown in figure 6.

For the r = 45 days batch, two other cases were optimized: with 20% and 50% price-level increases (the 1.2 and 1.5 lines, respectively). With higher feed prices, the optimal W, paths are lower as a result of an increase in the corresponding feed savings.

2.0

1.0 l 15 1.2

0.5

0.0 ,

? , ? , ? , ? '. . .

0 10 20 30 40 50

Age (days)

Figure 6. Optimal trajectories under different feed price levels

1.00

0.90

0.80 40

0.70

0.60 X(t) 42 0.50

0.40 ts

0.30 t 0.20

0.10 5 10 15 20 25 30 35

Age (days)

Figure 5. The decision variables xt as a function of marketing age - t

Effect of Consumers' Preference on Carcass Fatness

Modern consumers express their dislike for high fat content in meat as requested by a downward-sloping demand curve with respect to the fraction of fat in the carcass. In order to measure this effect on the optimal trajectories, the usual stairway shape of live-weight prices P, was smoothened out and approximated by a linear curve, given by (22) P, = Po - s(F - 0.10), where P0 is the live-weight price at 0.10 of the fat's carcass fraction - F; s is the slope of the curve. The above model was accordingly amended and equation (2) was maximixed with R(q) = qP, subject to equation (22). Op- timal solutions were found for values of s = 0.1, 0.25, 0.5, 1.0. Figure 7 shows the effect of increasing slope values of s on the simulated optimal final live weight and fraction of carcass fat at age nine weeks. As expected, both variables decrease at a diminishing rate with respect to s. However, at the above as-

20001 12.0

. 1990-1 \11.9

Weight 1980 11 8

% Fat

1970 ... . . ' ? ...... . w r 11.7 0.0 0.2 0.4 0.6 0.8 1.0

SLOPE

Figure 7. Final live weight and fraction of carcass fat at age sixty-three days as a function of the slope of the demand for fat carcass fraction



sumed quality preferences, little effect is expected on the optimal trajectory. This is primarily because of the low optimal marketing age at the given input-output price ratio, which naturally leads to low fat birds. Other- wise, the quality preferences may have a stronger effect on the optimal trajectory.

Effect of Interest Rate

The optimal marketing age is relatively insen- sitive to changes in interest rates. Increasing the interest rate, used in equations (2) and (3), from 5% to 20% annually, led to a one-day reduction in marketing age. This lack of sensitivity is explained as follows. There is a slight increase in the growth trajectory after three weeks of age with the above increases in interest rates. This is understandable since higher feeding rates toward the end of the production cycle are more severely discounted. Furthermore, the free feeding following the restriction period is close in time to the final sale, hence partially offsetting each other. The short production period does not allow for a more dramatic effect. This is a part of the adjustment process caused by the optimization process.

Concluding Comments

In the present study, management ideas gen- erated by animal physiologists are considered in management practices leading to 8%-10% savings in production expenses above the best free-feeding dynamic plan while keeping total production unchanged. The level of these savings depends on the market's input/output price ratios. The higher the feed ingredient price, the lower is the corresponding optimal growth trajectory, which means more severe feed restrictions. The consumers' quality preference in terms of fat content may affect it too. In particular, fat dislike, leading to lower product prices for fatty broilers, should lower the optimal trajectory, resulting in smaller, low fat birds.

The optimizing technique selected, namely, the reduced-gradient method, which drives a complete and nearly continuous growth simulator, is an efficient and convenient tool that may be exploited to fine tune other live- stock management practices.

[Received February 1987; final revision received October 1987. ]

References

Fleischer, G. A. Capital Allocation Theory: The Study of Investment Decisions. New York, Meredith Corp., 1969.

Glover, F. "Integer Programming and Combinatorics." Handbook of Operations Research, ed. J.J. Moder and S. E. Elmaghraby, chap. II-2. New York: Van Nostrand Reinhold Co. 1978.

Hurwitz, S., I. Plavnik, and S. Bornstein. "The Amino Acid Requirements for Chicks: Experimental Valida- tion of Model-Calculated Requirements." Poultry Sci. 59(1980):2470-79.

Hurwitz, S., N. Shavir, and A. Bar. "Protein Digestion and Absorption in the Chick: Effect of Ascardia Galli." Amer. J. Clin. Nutrition 25(1972):311-16.

Hurwitz, S., D. Sklan, and I. Bartov. "New Formal Approaches to the Determination of Energy and Amino Acid Requirements." Poultry Sci. 57(1978): 197-205.

Hurwitz, S., H. Talpaz, and I. E. Waibel. "The Use of Simulation in the Evaluation of Economic and Man- agement of Turkey Production: Dietary Nutrient Density, Marketing Age on Environmental Tempera- ture." Poultry Sci. 64(1985):1415-23.

Hurwitz, S., M. Weisselberg, U. Eisner, I. Bartov, G. Riesenfeld, M. Sharvit, A. Niv, and S. Bornstein. "The Energy Requirements and Performance of Growing Chickens and Turkeys as Affected by Envi- ronmental Temperature." Poultry Sci. 59(1980): 2290-99.

Marquardt, D. W. "An Algorithm for Least-Squares Es- timation of Nonlinear Parameters." J. Soc. and In- dust. Appl. Math. 11(1963):431-41.

Murtagh, B. A., and M. A. Saunders. MINOS 5.0 User's Guide. TR SOL 83-20, Stanford University, 1983.

National Research Council. Nutrient Requirements of Poultry, 7th ed. Washington DC: National Academy of Science, 1977.

Plavnik, I., and S. Hurwitz. "Organ Weights and Body Composition in Chickens as Related to the Energy and Amino Acid Requirements: Effects of Strain, Sex, and Age." Poultry Sci. 62(1982):152-63.

. "The Performance of Broiler-Chicks During and Following a Severe Feed Restriction at an Early Age." Poultry, Sci. 64(1985):348-55.

Talpaz, H. "Handling Setup Costs in Nonlinear Pro- gramming Problems." Manage. Sci. in press.

Talpaz, H., J. R. de la Torre, P. J. H. Sharpe, and S. Hur- witz. "Dynamic Optimization Model for Feeding of Broilers." Agr. Systems 20(1986):121-32.

Appendix 1

A Smoothed Approximation to the No-Control Transition

A key to computing the gradient of the objective function, which is needed for the reduced-gradient optimization, is



the derivative of equation (17) with respect to xt, or Wa, which exists for xt < 1.0 and for xt = 1.0 but not at that instance when xt becomes 1.0 from below. At that point the function is discontinuous with respect to xt, disallow- ing the use of the convenient and otherwise efficient algorithm. This problem is quite similar to the so-called "setup cost" problem (see for example, Glover). Re- cently, some proper approximations to a true function have been suggested by Talpaz to avoid the problem with- out any measurable loss in accuracy. For our problem, consider the following approximation to equation (18):

(18a) Cf = x,[x,120(W - Wa)/Wp + 1]; for x, ? 1.0. Notice that for xt < 1, xt1'20 0 and Cf O xt; likewise for xt = 1, C, = [(Wa - Wa)/Wp + 1]o, preserving the original Cf. Only when x, S 1 does this approximation become crude (i.e., 0.95120 = .0021); however, this is even a desir- able property because it means that partial compensation begins already at near free-feeding situations. Equation (18a) is applied in the model. Values greater than 120 may lead to a computer's underflow problem.

Appendix 2. Results of the "Basic Model" Case

Table B 1. Ingredient Contents in Optimal Diets for the "Basic" Case (grams/bird/day) Age in Weeks

Description 1 2 3 4 5 6

Corn 0.00 0.00 0.00 0.00 0.53 3.64 Sorghum 18.22 29.89 45.58 62.82 79.50 92.25 Soybean 5.00 9.55 14.45 18.36 20.52 20.80 Sopstk 0.31 0.52 0.79 1.07 1.33 1.54 DCP 0.47 0.77 1.17 1.61 2.04 2.46 Salt 0.08 0.14 0.21 0.28 0.35 0.41 DL-Methionine 0.05 0.11 0.17 0.22 0.25 0.25 Diet 24.13 40.97 62.37 84.37 104.52 121.34 Price($/ton) 187.30 189.40 189.00 187.40 185.30 183.20 Cost(c/bird/day) 4.52 7.76 11.79 15.81 19.37 22.23


Article Contentsp. [382]p. 383p. 384p. 385p. 386p. 387p. 388p. 389p. 390

Issue Table of ContentsAmerican Journal of Agricultural Economics, Vol. 70, No. 2 (May, 1988), pp. 219-512Front MatterThe Demand for Dairy Products: Structure, Prediction, and Decomposition [pp. 219-228]Generic Fluid Milk Advertising, Demand Expansion, and Supply Response: The Case of New York City [pp. 229-236]Chickens, Eggs, and Causality, or Which Came First? [pp. 237-238]Food Processor Price Behavior: Firm-Level Evidence of Sticky Prices [pp. 239-244]Input Substitution and the Distribution of Surplus Gains from Lower U.S. Beef-Processing Costs [pp. 245-254]The Value of Ideal Contingency Markets in Agriculture [pp. 255-267]Hedging under Output Price Randomness [pp. 268-272]Intermediation Costs in an Agricultural Development Bank: A Cost-Function Approach to Measuring Scale Economies [pp. 273-280]Market Distortions and Benefits from Research [pp. 281-288]Tax Policy and U.S. Agriculture: A General Equilibrium Analysis [pp. 289-302]A Nonparametric Analysis of Agricultural Technology [pp. 303-310]A Nonparametric Investigation of Agricultural Production Behavior for U.S. Subregions [pp. 311-317]A Model of Production with Supply Management for the Canadian Agricultural Sector [pp. 318-329]Multiproduct Supply and Input Demand in U.S. Agriculture [pp. 330-337]Education, Experience, and Allocative Efficiency: A Dual Approach [pp. 338-345]The Effects of Suburbanization on Agriculture [pp. 346-358]Arbitrage Pricing, Capital Asset Pricing, and Agricultural Assets [pp. 359-365]The Biological Control of Cassava Mealybug in Africa [pp. 366-371]Use of Dichotomous Choice Nonmarket Methods to Value the Whooping Crane Resource [pp. 372-381]Economic Optimization of a Growth Trajectory for Broilers [pp. 382-390]Objective Function Approximation: An Application to Spatial Price Equilibrium Models [pp. 391-396]Approximating Linear Programs with Summary Functions: Pseudodata with an Infinite Sample [pp. 397-402]Ranking of Agricultural Economics Departments: Influence of Regional Journals, Joint Authorship, and Self-Citations [pp. 403-409]Comments and ReplyReducing Moral Hazard Associated with Implied Warranties of Animal Health: Comment [pp. 410-412]Reducing Moral Hazard Associated with Implied Warranties of Animal Health: Reply [pp. 413-414]A Note on Qualitative Forecast Evaluation: Comment [pp. 415-416]

ProceedingsPrivate Agricultural Markets in a Socialist EconomySoviet Union: The Anomaly of Private-cum-Socialist Agriculture [pp. 417-422]People's Republic of China: Systemic and Structural Change in a North China Township [pp. 423-430]Indirect and Direct Taxation of Agriculture in Sudan: The Role of the Government in Agriculture Surplus Extraction [pp. 431-436]Soviet Union: The Anomaly of Private-cum-Socialist Agriculture: Discussion [pp. 437-438]People's Republic of China: Systematic and Structural Change in a North China Township [p. 439]Indirect and Direct Taxation of Agriculture in Sudan: The Role of Government in Agriculture Surplus Extraction: Discussion [pp. 440-441]

New Approaches in Agricultural Policy ResearchMeasuring Economic Welfare: Is Theory a Cookbook for Empirical Analysis? [pp. 442-447]Making Economic Welfare Analysis Useful in the Policy Process: Implications of the Public Choice Literature [pp. 448-453]Social Welfare and Interpersonal Utility Comparisons in Applied Policy Research [pp. 454-458]New Approaches in Agricultural Policy Research: Discussion [p. 459]New Approaches in Agricultural Policy Research: Discussion [pp. 460-461]

Conceptualization of Research Problems in AgribusinessAssessing Opportunities in Food and Fiber Processing and Distribution [pp. 462-468]Industrial Organization: Some Applications for Managerial Decisions [pp. 469-474]The Design and Impact of Strategic Management Information Systems [pp. 475-479]Assessing Opportunities in Food and Fiber Processing and Distribution: Discussion [pp. 480-481]Industrial Organization: Some Applications for Managerial Decisions: Discussion [pp. 482-483]The Design and Impact of Strategic Management Information Systems: Discussion [pp. 484-485]

Books ReviewedRetrospective ReviewReview: untitled [pp. 486-488]Review: untitled [pp. 488-489]Review: untitled [p. 489]Review: untitled [pp. 489-490]Review: untitled [pp. 490-491]Review: untitled [pp. 491-492]Review: untitled [pp. 492-494]Review: untitled [pp. 494-495]Review: untitled [pp. 495-496]Review: untitled [pp. 496-497]Review: untitled [pp. 497-498]Review: untitled [pp. 498-499]Review: untitled [pp. 499-500]Review: untitled [pp. 500-501]Review: untitled [pp. 501-502]Review: untitled [pp. 502-503]Review: untitled [pp. 503-504]Review: untitled [pp. 504-505]

Back Matter [pp. 506-512]

Economic Optimization of Growth Trajectory for Broilers

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dynamic optimal trajectory

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daily optimal growth

normal growth rates

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dynamic model

dynamic broiler

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