Optimizing the Herd Calving Pattern with Linear Programming and

Optimizing the Herd Calving Pattern with Linear Programming and Dynamic Probabilistic Simulation A. W. JALVINGH,’ ~2 A. A. DIJKHUIZEN,l and J A M . VAN ARENDONK2 Wageningen Agricultural University 6706 KN Wageningen, The Netherlands ABSTRACT

INTRODUCTION

Until recently, little attention has been paid to the influence of seasonal variation in performance and prices on the optimal calving pattern of a herd. A method was developed to determine the herd calving pattern that is fann-specific and optimal with use of linear programming. The required technical and economic parameters are calculated with a dynamic probabilistic simulation model of the dauy herd. The approach was illustrated with a situation in which the objective was to maximize the gross margin of the herd and the annual milk production of the herd was restricted, resulting in an optimal calving pattern: all heifers calved during August. When, in addition, only home-reared replacement heifers were allowed to enter the herd, heifer calvings took place from July to October. The gross margin was reduced by only Dfl. .13/100 kg of milk ($1 US = 1.80 Dfl.) as a result of the additional constraint. The sensitivity of the optimal calving pattern of herd was determined for lower reproductive performance and when seasonal price variation was ignored. The method described herein is a flexible tool for determining the optimal calving pattern of herd, taking into account farm-specific inputs and constraints. (Key words: calving pattern, economics, linear programming, simulation)

For individual animals, optimal decisions on insemination and replacement can be obtained using dynamic programming (1, 10) and taking into account seasonal variation in performance and prices. DeLorenzo et al. (1) determined the optimal calving pattern based on the results from the dynamic programming model, assuming immediate replacement of the culled animals, i.e., constant herd size throughout the year. The restriction of constant herd size is a consequence of using the dynamic programming technique. Jalvingh et al. (4) developed a dynamic probabilistic simulation model and compared different calving patterns of herds with use of decisions for insemination and replacement of individual animals calculated by the dynamic programming model of Van Arendonk (10). This simulation model allowed for variation in herd size during the year. Jalvingh et al. (4) found that, for Dutch conditions, a herd with all heifers calving during October is the most profitable in the case of maximization of the gross margin per average cow in the herd. A herd with all heifers calving during October, however, is not very realistic. For farms using home-reared young stock, the availability of replacement heifers depends on the calving pattern of the herd during previous years, the selection of calves reared for replacement, and the quality of reproductive management. In addition, d a q producers may not prefer such a concentration of calvings because of restrictions, for example, in the amount of available labor or roughage. Until recently, little attention has been paid to the determination of the optimal calving pattern of the herd, taking into account herd and farm restrictions. A linear programming approach is described herein to determine the optimal calving pattern of herd, taking into account herd restrictions. The objective was to choose a calving pattern to maximize the gross margin of the herd. Maximization is achieved with the annual milk production of the herd restricted

Abbreviation key: SME = single month equilibrium.

Received January 8, 1993. Accepted January 24, 1994. ‘Department of Farm Management. *Department of Animal Breeding. 1994 J Dairy Sci 77:1719-1730

1719

situation with milk quota, the gross margin per kilogram of milk rather than per cow should be maximized (2, 11). Various types of additional constraints can be taken into account, e.g., culling, calvings, and feed intake. The parameters of the objective function and constraints are obtained from the results of the dynamic probabilistic simulation model described by Jalvingh et al. (4). A few constraints are examined herein to illustrate the approach. Other applications of the model are discussed. MATERIALS AND METHODS Simulation Model

Jalvingh et al. (4) developed a dynamic probabilistic simulation model for dairy herds that simulated the technical (e.g., reproductive performance) and economic consequences of various decisions concerning production, reproduction, replacement, and calving patterns. Central to the simulation model is the simulation of herd dynamics, using the Markov chain approach (3). For the herd dynamics module, the herd is described in terms of possible states for cows and transitions among these states. The time between state transitions equals 1 mo. The state variables defined in the model were lactation number, stage of lactation, time of conception, level of milk production for present lactation, and month of calving. Uncertainty of future performances was included in four groups of transition probabilities: production, reproduction, disposal, and replacement. The transition probabilities depended on biological variables (e.g., conception rate and estrus detection rate) on the one hand and the management strategies of dairy producers (e.g., with respect to insemination and replacement) on the other. The model has the ability to derive, for a given set of biological variables and management strategies, the steady state of the herd, representing stable size and age structures of the herd. Such an equilibrium distribution of animals over states was equal to the distribution of replacement heifers calving during different months over all possible states in the herd during their lifetime. A steady state herd Journal of Dairy Science Vol. 77. No. 6. 1994

tiom heifers calving only in a specific month. Consequently, 12 different SME herds could be distinguished. The corresponding technical and economic results of an SME herd were determined by combining the equilibrium distribution over states with information regarding milk production, feed intake, slaughter value, and prices. The information on milk production, feed intake, and slaughter value was separately simulated by the performance module of the dynamic probabilistic simulation model. The technical and economic results of a herd with heifer calvings during more than 1 mo could be derived by weighing the results of the SME herds according to the proportion of heifer calvings during each month for that herd (4). The technical and economic results of the SME herds and the weighing of their results to derive the results of a herd with any calving pattern were the major ingredients in determining the optimal calving pattern of the herd. Input variables for reproduction, disposal, milk production, feed intake, slaughter value, and prices were taken from Jalvingh et al. (4) and are summarized in the Appendix. For simulating herd dynamics, the applied management strategies on insemination and replacement were based on the decisions for individual animals calculated by dynamic programming, taking into account the seasonal variation in performance and prices (4, lo). The major technical and economic results of the 12 SME herds are presented in Table 1, which shows that a heifer calving during October realizes, on average, 3.78 calvings during her life. The average herd life of the same heifer is 3.34 yr. The average herd life is shortest when the heifer calves during March. The gross margin for herds was calculated as the difference between the revenues from milk, calves, and culled cows on the one hand and the total costs for feed and replacement heifers on the other. If the gross margin was considered per average cow present in the herd, the SME herd for October has the highest gross margin and for March the lowest. When gross margin was considered per 100 kg of milk, the SME herd with all heifers calving during August has the highest, and February the lowest.

ECONOMIC OPTLMIZATION OF HERD CALVING PAITERN

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Linear Programming Approach

AN

mp. m

The technical and economic results of the 12 SME herds and the weighing of these results to obtain results for a herd with any calving pattern formed the basic ingredients of the linear programming approach. The monthly numbers of heifer calvings were used as decision variables in the linear programming approach. The objective was to choose the calving pattern for heifers that maximized the gross margin of the herd, taking into account herd and farm constraints. The parameters for the objective function and constraints were obtained from the technical and economic results of the SME herds. Therefore, the influence of biological variables and management strategies on herd dynamics were taken into account for optimizing the herd calving pattern, as was the influence of seasonal variation in performance and prices. Different constraints could be used simultaneously, provided that the information for each constraint is calculated in the simulation model that generates the SME herds. The optimal solution of the linear programming approach represents the optimal calving pattern for heifers. The optimal herd calving pattern and the corresponding technical and economic results can be derived by weighing the results of the SME herds according to the optimal calving pattern for heifers. The gross margin per kilogram of milk is maximized, and the annual milk production of the herd is restricted to 500,000 kg. The following linear programming problem was used to determine the optimal calving pattern for heifers:

& & m a

-. m -\o

N w -m

12

Maximize Z =

gmixi i= I

.g

Subject to

.I

Xi

1 0, for all i

PI

in which Journal of Dairy Science Vol. 77, No. 6, 1994

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JALVMGH ET AL.

xi = number of heifers calving during

month i; gmi = gross margin of the SME herd expressed per heifer calving, in case the heifer calves during month i (see Table 1); and mpi = milk production of the SME herd expressed per heifer calving, in case the heifer calves during month i (see Table 1).

for all j

[41

IL

max ahsixi,

5

Sets of Additional Constraints

i=l

Set 1. The linear programming problem presented is referred to as set 1. This set can be extended with additional constraints. For three different sets of additional constraints, the optimal calving patterns for heifers and herds are determined. Set 2. For set 2, an additional constraint was used, which specifies that all replacement heifers entering the herd should come from calves that were born 24 mo earlier in the herd:

for all j

151

where hsi, = herd size for SME herd during month j, in case one heifer calves during month i; ahsi = average annual size of herd for SME herd, in case one heifer calves during month i (see Table 1);

12

fjyijxi 2 xj, i= 1

for all j

PI

where yy = number of herd calvings during month j in SME herd corresponding to one heifer calving during month i, and fj = factor giving the number of 24-mo-old replacement heifers per calving during month j that become available 2 yr later during month j (fj is set to .4 for all months). All replacement heifers were assumed to calve at 24 mo of age but this age can easily be altered to permit variation. Set 3. A concentration of calvings within a few months results in a large fluctuation in the monthly herd size. For set 3, variation in monthly herd size was restricted by using lower and upper limits, between which monthly herd size was allowed to vary. The limits were formulated in terms of a proportion of the average annual size of herd. In formula Journal of Dairy Science VoI. 77, No. 6, 1994

min = lower limit of the variation in herd size per month, expressed as a proportion of the average annual size of herd; and max = upper limit of the variation in herd size per month. The lower and upper limits for variation in monthly herd sizes are set to 95 and 105% of the annual average size of herd, respectively. The constraints used for set 2 also hold for set 3. Set 4. To study the impact of allowing variation in herd size during the year, as for the previous sets, the optimal calving pattern of herd was determined for set 4 for the situation in which all culled cows are immediately replaced. A constant herd size can be formulated by the constraint:

c 12

cijxi = xj,

i=l

for all j

[GI

where qj = number of culled cows in SME herd d u n g month j, in case one heifer calves during month i. The linear programming model and the simulation model were programmed in Turbo

ECONOMIC OPTIMIZATION OF HERD CALVING PATTERN

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Pascal 6.0@(Borland International, Inc., Scotts Valley, CA) and run on a personal computer. The linear programming problems were solved with the simplex method described by Press et al. (8).

calves born in the herd during each month (set 2), heifer calvings occurred from August to October. The resulting herd calvings were still concentrated from August to October. The gross margin was reduced by only Dfl. .13/100 kg of milk, which was Dfl. 651 at the herd level. The reduction in gross margin was a Sensitivity Analysis result of the reduction in milk and calf For the sensitivity analysis, the o p W revenues. The milk revenues were reduced becalving pattern of herd belonging to the four cause of the reduction in average realized sets of constraints was determined in case of a monthly deviation in base price of milk, lower reproductive performance of the herd. In whereas the revenues from calves were that case, the initial conception rates after in- reduced because of the reduction in the number semination and estrus detection rates (see the of calvings in the herd. For set 2, the monthly Appendix) were reduced by 15%, and the aver- herd size varied from 90% during June to age interval from calving to first insemination 113% during September of the average annual was increased by 11 d. Moreover, tbe optimal size of herd (Table 2). For set 3, the monthly herd size was recalving pattern of herd was determined for the stricted to vary from 95 to 105% of the aversituation in which seasonal variations in prices age annual size of herd, resulting in an optimal for milk, calves, heifers, and slaughter values calving pattern that was spread over a longer were ignored. Seasonal variations in milk period than for set 2. The gross margin was production, feed costs, and reproduction were reduced by Dfl. .36/100 kg of mdk compared still present. For both situations, the decisions with set 1, which equals Dfl. 1784 at the herd on insemination and replacement of individual level. For the case of immediate replacement animals were recalculated using the dynamic (set 4), the heifer calvings occurred during all programming model; consequently, the cor- months because culling took place during all responding SME herds were also recalculated. months. However, peaks in heifer calvings were observed during June, July, October, and November, the months with the highest RESULTS proportion of voluntary culling, which made The optimal calving patterns for heifers and up approximately 50% of all culling. The gross herds for the different sets of constraints are margin was reduced by Dfl. 1.06/100 kg of presented in Table 2 with technical and eco- milk, which was Dfl. 5304 at the herd level. Results are presented for the herd with nomic results corresponding to the herds with the optimal calving pattern. As expected, the lower reproductive performance in Table 3. gross margin per 100 kg of milk was highest Compared with the basic situation, the average when only the milk production of the herd was calving interval was increased by approxirestricted (set 1). In that instance, all heifer mately 2 wk to about 385 d. For set 1, the calvings took place during August, which optimal results were obtained when all heifers would be expected from the information calved during July. The gross margin per 100 presented in Table 1. The resulting herd calv- kg of milk was Dfl. .40 lower than for set 1 in the basic situation, mainly because of the ings, including individual heifer calvings, took reduction in milk revenues and the increase in place mostly from July to October. The replacement costs. With a constraint on availaproportional monthly production of milk var- bility of replacement heifers (set 2), calvings ied from 4.3% during June to 11.3% during took place during 8 different mo, compared September. The variation in monthly milk with only 4 mo for the basic situation. Because production was much smaller than the vafia- of lower reproductive performance, the herd tion in monthly herd calvings. The monthly calvings and available replacement heifers herd size, expressed as a percentage of the resulting from a calving were spread over a average annual size of herd, varied from 87% longer period, which directly affected availaduring July to 117% during August. birity of replacement heifers. The reduction in When the number of heifers calving per gross margin per 100 kg of milk was Dfl. .61, month was restricted by the number of female compared with set 1 for the same low Journal of Dairy Science Vol. 77, No. 6 , 1994

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JALVINGH ET AL.

parameters for reproduction. Because heifer calvings were distributed over more months for set 2, the monthly fluctuations of herd size were not as large as for the basic situation. Therefore, the difference in results for sets 2

and 3 was rather small (Dfl..04/100 kg of milk) compared with that for the basic situation. With immediate replacement, the gross margin per 100 kg of milk was Dfl.1.00 lower than for set 1.

TABLE 2. Results for the optimal calving pattern of herd for different sets of constraints for the basic situation. Set of constraints1 Herd milk production, kg Calving pattern for heifers, % Jan Feb MW APr May Jun Jul Aug SeP Oct Nov DeC Calving pattern for herd, %

Jan Feb Mar APr May Jun Jul Aug SeP Oct Nov DeC Average cows, no. Range herd size, % of average Calvings, no. A M U culling ~ rate, % Calving interval, d Milk per average cow, kg Average monthly deviation in base price milk, Dfl.1100 kg EEonomic results, M1./100kg of milk Revenues Mlk Calves Culling

1

2

3

4

500,OOO

500,000

500,000

500,000

0 0 0 0 0 0 0

0 0 0 0

0 0 0

100.0 0

0 0 0 1.6 .9 .5 .6 1.8 4.9 13.1 43.2 14.5 10.2 5.6 3.1 72.6 87-1 17 84.3 31.7 373 6891

0 0 23.9 32.4 37.2 6.5 0 0

2.1 1.1

.6 .8 2.1 5.5 16.4 22.2 25.5 13.2 6.6 3.9 72.1 90-113 83.7 31.8 372 6932

0 1.5 13.7 21.8 23.7 10.2 19.6 9.6 0

4.4 3.9 3.9 4.1 4.5 14.2 12.1 5.4 9.7 19.0 13.2 5.6

2.6 1.5 1.o 1.6 4.3 10.6 15.0 16.4 15.5 16.8 10.1 4.6 71.7 95-1 05 83.0 32.0 372 6972

5.8 4.7 4.0 4.4 6.1 9.9 9.3 7.9 12.6 15.7 11.8 7.8 71.0 100-1M) 82.3 33.2 37 I 7039

1.32

1.31

1.06

.64

84.44 6.14 6.93

84.37 6.00 6.94

83.98 5.95 6.99

83.31 5.67 7.28

22.41 12.13 62.97 314,843

22.37 12.10 62.84 314,192

22.25 12.06 62.61 313,059

22.06 12.30 61.91 309,539

costs

Feed Heifers Gross marein Gross mar& herd, M1.

%et 1, only milk production herd constrained; set 2, 1 plus home-reared replacement heifers have to be available; set 3, set 2 plus variation in herd size restricted; set 4, set 1 plus immediate replacement of culled animals.

Journal of Dairy Science Vol. 77, No. 6, 1994

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ECONOMIC OPTIMIZATION OF HERD CALVING PATTERN

Results are presented for the situation in which the seasonal variation in prices of milk, calves, heifers, and slaughter values are ignored in Table 4. Seasonal effects on milk

production, feed costs, and reproduction were still present. For set 1, all heifer calvings took place at the optimal moment, i.e., March. Because milk production per average cow in the

TABLE 3. Results for the optimal calving pattern of herd for different sets of constraints for the situation with a lower reproductive performance.' Set of constraints2

Herd milk production, kg Calving pattern for heifers, 8 Jan Feb Mar Apr

May Jun Jul Aug SeP Oct Nov Dec Calving pattern for herd, % Jan Feb Mar APr May Jun Jul

Aug SeP

OCt

1

2

3

4

500,000

500,000

500,000

500,000

0 0 0 0 0 0

100.0 0 0 0 0 0

2.1 1.4 .8 .8 1.8 8.0 44.3 12.7 10.6 8.4 5.5 3.6 73.4 86-1 18 83.8 35.6 386 6807

Nov Dec Average cows, no. Range herd size, % ' of average Calvings, no. Annual culling rate, % Calving interval, d Milk per average cow, kg Average monthly deviation in base .95 price milk, Dfl.1100 kg Economic results, Dfl./100 kg of milk Revenues 84.12 Mlk 6.30 Calves 8.15 Culling costs 22.30 Feed 13.71 Heifers 62.57 Gross margin Gross margin herd, Dfl. 3 12,850

0 0 2.2 5.3 6.9 9.4 12.9 17.8 22.6 23.0 0 0

0 0 3.5 5.8 7.6 10.1 13.5 18.1 21.6 15.8 4.0 0

3.8 3.4 3.5 5.1 5.3 15.9 14.4 6.9 12.2 14.5 9.8 5.I

3.6 2.7 2.9 4.2 5.5 7.4 10.2 14.1 17.9 18.2 7.8 5.5 72.3 96107 82.1 35.9 385 691 1

3.5 2.7 3.5 4.6 6.0 8.0 10.7 14.3 17.1 15.3 8.8 5.3 72.4 95-105 82.2 36.0 385 6907

5.7 5.0 4.4 4.9 6.0 10.7 10.9 9.3 12.0 13.3 10.3 7.5 72.1 100-100 82.0 36.6 385 6933

.84

.57

83.95 5.68 8.23

83.85 5.71 8.26

83.42 5.64 8.38

22.24 13.65 61.96 3W,818

22.21 13.69 61.92 309,623

22.10 13.78 61.57 307,827

.91

IConception rate and estrus detection rate are reduced by 15%, and average interval calving first insemination was increased by 1 1 d. *Set 1, only milk production herd constrained; set 2,set 1 plus home-reared replacement heifers have to be available; set 3, set 2 plus variation in herd size restricted; set 4, set 1 plus immediate replacement of culled animals. Journal of Dairy Science Vol. 77, No. 6, 1994

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JALVINGH ET AL.

herd was approximately 300 kg higher than for set 1, fewer heifers were needed to achieve the same production for the herd. Because of the absence of the seasonal variation in the milk

price, the revenues per 100 kg of milk were Dfl. 2.46 lower than for the basic situation, and the gross margin per 100 kg of milk was Dfl. 1.43 lower. For set 2, heifer calvings took

TABLE 4. Results for the optimal calving pattern of herd for different sets of constraints for the situation in which seasonal variation in prices of milk, calves, heifers, and slaughter values was ignored. Set of constraints'

Herd milk production, kg Calving pattern for heifers, % Jan Feb Ma APr May JUn JUl A%! SeP Oct Nov Dec Calving pattern for herd (%) Jan Feb Ma APr May Jun Jul Aug SeP

1

2

3

4

500,000

500,000

500,000

500,000

0 0 100.0 0 0 0 0 0 0 0 0 0

21.7 429 35.4 0

6. I 17.3 47.6 12.8 7.1 3.3 1.5 .7 .5

18.7 29.5 24.3 9.2 4.9 2.2 1.0 .4 .4 .9 2.2 6.4 69.6 86-1 11 79.7 31.5 370 7182 0

OCt Nov

.4

.8 DK 2. I Average cows, no. 69.9 Range herd size, 5% of average 83-1 15 Calvings, no. 80.5 32.9 Annual culling rate, 8 369 Calving interval, d 7149 Milk per average cow, kg Average monthly deviation in base 0 price milk, Dfl.1100 kg Eoonomic results, Dfl.1100 kg of milk Revenues 81.98 Milk 5.49 calves 7.2 Culling costs 21.16 Feed 1l.% Heifers 61.54 Gross margin Gross margin 307,708 - herd, Dfl.

-

0 0 0 0 0 0 0 0

21.2 3.1 26.5 6.4 2.0 0 0 0 0 6.4 17.9 10.5

10.7 8.1 6.1 6.5 5.6 4.3 4.1 4.0 4.4 10.2 21.3 14.7

18.6 13.6 18.1 9.0 4.6 1.9 .8 .7 7.3 12.2 12.9 69.7 95-105 79.7 31.2 372 7170

12.8 11.7 9.1 7.4 5.5 3.8 2.8 2.1 3.5 11.5 15.5 14.2 70.2 100-100 80.8 32.6 374 7125

0

0

.4

81.97 5.44 6.79

82.13 5.45 6.68

82.29 5.52 6.97

21.33 11.4 61.47 307,328

21.59 11.32 61.35 306,727

21.76 11.89 61.13 305,634

'Set 1, only milk production herd constrained; set 2, set 1 plus home-reared replacement heifers have to be available; set 3, set 2 plus variation in herd size restricted; set 4, set 1 plus immediate replacement of culled animals. Journal of Dairy Science Vol. 77, No. 6, 1994

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place from January to March. The gross margin per 100 kg of milk was reduced by only Dfl. .07.In case of immediate replacement, the gross margin was reduced by Dfl. .41. The reduction in the gross margin between set 1 and the other sets was much smaller than for the basic situation because of the smaller difference in gross margin per 100 kg of milk for the SME herds. This difference was a direct effect of the lack of seasonal variation in prices. For the basic situation, the difference in gross margin per 100 kg of milk between the highest (August) and the lowest (February) SME herd was Dfl. 3.92. With no seasonal variation in prices, the difference between the highest SME herd (March) and the lowest (August) was Dfl. 1.50.

In The Netherlands, dairy producers are advised to produce the milk quota with as few cows as possible (9). Based on the results in Table 2, when seasonal variation in performance and prices is taken into account, keeping a few more cows for producing the milk quota will be more profitable. In those cases, additional costs for maintenance are outweighed by the increased returns because of seasonality. Additional calculations in which the gross margin per average cow is maximized, while the annual milk production of the herd is still restricted, showed similar results. All heifer calvings then take place in October, which could be expected from the information in Table 1. The average number of cows needed to produce the quota in case of optimization per average cow is decreased by DISCUSSION 2.3, compared with set 1 in Table 2 (maximization per kilogram of milk). Although fewer When only annual milk production was cows are required, the gross margin of the herd constrained, all heifers calved during August is decreased by Dfl. 2104 (compared with set 1 (set 1). For set 2 (home-reared replacements in Table 2). only), heifer calvings took place from July to In simulation of herd dynamics in the SME October, and gross margin at herd level was herds, the optimal insemination and replacereduced by Dfl. 651 for a herd with an annual milk production of 500,000 kg (Table 2). This ment decisions calculated by the dynamic difference could be considered to be the maxi- programming model derived from Van Arenmum amount to spend on changing the rearing donk (10) were used. The optimal decisions period of heifers to have enough replacement were based on immediate replacement assumheifers available during August. In case of a ing an unlimited supply of heifers. In the SME reduction in reproductive performance, the herds, however, culled animals were not difference between set 1 and set 2 was much replaced immediately, because all heifers calve larger (Dfl. 3032 at the herd level; Table 3). In during only one specific month. In real life, case of reduced reproductive performance, immediate replacement also will not take place fewer calvings occurred during the month in in all cases because the supply of replacement which the heifer entered the herd. Modifica- heifers is limited. Ideally, the optimal decitions in the length of the rearing period of sions for individual cows would be determined heifers might yield a larger profit in the case taking into account the restrictions on number reproductive performance was lower. How- of available replacement heifers. This determiever, this measure has to be repeated annually. nation can be formulated as a multicomponent An improvement in reproductive performance model, in which decisions concerning a cow of the herd would be more effective and would depend not only on the state of that particular yield a higher income because more cows cow, but also on the states of the other cows in could have a 1-yr calving interval, and, conse- the herd. Such a model is far too large to be quently, more opportunities would exist to solved by a known method (7). Kristensen (7) concentrate calvings within a few months. In used an approximate method, which combined case modifications of the length of the rearing dynamic programming and stochastic simulaperiod are preferred, the linear programming tion to determine the optimal strategy when the approach will prove flexible enough to include supply of heifers was limited. He found little these modifications. The calving pattern for differences in results with and without restricheifers will be optimized, taking into account tion on heifer supply when replacement costs, differences in rearing costs of heifers that calve i.e., cost of heifer minus average carcass value, at different ages. were >2000 Dkr., which was equivalent to the Journal of Dairy Science Vol. 77, No. 6, 1994

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value of 800 kg milk. Consequences for rank- mediate replacement). However, this practice ings of cows were not reported. In the Dutch results in variation in herd size, implying that situation, replacement costs are rather high. more cow places will be needed than are used Using optimal decisions based on an unlimited on average. The linear programming approach supply of replacements is currently the best is able to handle restrictions on cow places available option, and it is expected to have (e.g., set 3). In the comparison of sets 1 to 4, a large variation in herd size was observed r a limited consequences on the results found. The linear programming approach was ap- ble 2). The effect of this within-year variation plied for a situation with output restrictions, in on the economic results, through its effect on which returns minus costs per kilogram of fixed costs, is not accounted for, because, under a quota system, many farms have excess milk were maximized. In the underlying SME facilities. In other situations, however, the opherds, the insemination and replacement poli- portunity costs for fixed assets should be incies used for individual cows, however, were cluded. based on a maximization of returns minus costs per cow place and were, therefore, not CONCLUSIONS necessarily optimal for a situation with milk The approach described herein combines quota. Optimal insemination and replacement decisions for a situation with milk quota might simulation and optimization and can be considbe obtained when the objective function in the ered to be a promising tool for on-farm decidynamic programming would be modified, for sion support. It provides the possibility of dewhich Kristensen (6) made a start. In the cur- termining the optimal calving pattern of the rent paper, the dynamic programming model herd, taking into account herd and farm restrictions. The impact of possible constraints was for a situation without quota was considered to illustrated in this study. Additional constraints be the best available option for individual cow can be defined, such as restrictions on roughadvice, but the approach was flexible enough age supply or amount of available labor, to include other insemination and replacement provided that the monthly numbers of heifer strategies. If, in the linear programming ap- calvings are used as decision variables and that proach, the basic constraint on annual milk the information about the parameters needed is production was replaced by a constraint on calculated by the simulation model. If these maximum herd size, the objective functions of precise coefficients are not available, estimates the linear programming and dynamic program- provided by the dairy producer or advisor can ming were equal. The maximization of gross also be used. The approach offers the possibilmargin per cow place resulted in immediate ity of quantifying the economic consequences replacement of culled cows, which equaled the of various constraints and, moreover, of situation assumed for dynamic programming. managing different objective functions. Based on the results of the calculations with different The resulting optimal calving patterns of heif- sets of constraints, the desired calving pattern ers and herd were equal to the patterns derived of the herd can be defined. Subsequently, the for set 4 (Table 2) in the case of maximization model of Jalvingh et al. (5) can be used to of gross margin per kilogram of milk and a compare strategies that actually change the constraint on maximum herd size. The max- current calving pattern of the herd to the imization of gross margin per cow place equals desired one. The comparison of the current the situation studied by DeLorenzo et al. (1). calving pattern and the optimal calving pattern If gross margin is maximized per cow place results in the maximum possible benefits of a with a restriction on maximum herd size, in- change. The strategy that is used to actually stead of a restriction on annual milk produc- change the calving pattern, determines to a tion, immediate replacement of culled animals large extent the final profit that can be realized based on the optimal decisions calculated by (5). the dynamic programming model results in the ACKNOWLEDGMENTS optimal calving pattern of heifers and herds. In a situation with milk quota, however, it is most The costs of this study were partially coprofitable to leave cow spaces temporarily vered by financial support from the Developopen for reasons of seasonality (i.e., nonim- ment and Reconstruction Fund for Agriculture. Journal of Dairy Science Vol. 77, No. 6, 1994

TABLE A l . Base price and monthly deviations in price of milk, calves, replacement heifers, and carcass weight. Base Price

Monthly deviations in price Jan

Feb

Mar ~

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

~~~

rn.) Milk.1 100 kg -2.90 Calves: kg 10.55 Replacement heifer 2600 Carcass weight.3 kg 6.40

+3.7 +1.8 -.7 -1.5 4 . 5 4 . 5 4 . 5 -3.7 -.7 +5.3 +7.3 +7.0 -.74 -1.44 -1.68 -.36 +1.76 +2.02 +2.36 +.31 -.38 +.08 -.63 -1.30 -37 -57 -66 4 5 +13 +13 +27 +46 +44 +65 +20 +24 -.38 -.23 +.05 +.07 +.32 +.33 +.23 +.15 +.lo -.03 -.21 -.30

'Prices of milk fat and protein were Dfl. 8.00 and 14.50kg. respectively. Price of 100 kg milk was based on the negative base price, the monthly deviation in base price, and the price of fat and protein contents in 1 0 0 kg of milk. Mature equivalent milk production was set at 7750 kg of milk, 4.35% of fat, and 3.39% of protein. *Refers to male calves Base price for female calves was Dfl. 6.60. 3Refers to price of a kilogram of carcass weight of a fmt parity heifer 210 d in lactation.

REFERENCES 1 DeLorenzo, M. A,, T. H. Spreen, G. R. Bryan, D. K.

Beede, and J.A.M. Van Arendonk. 1992. Optimizing model: insemination, replacement, seasonal production, and cash flow. J. Dairy Sci. 75:885. 2 Gibson, J. P. 1989. Altering milk composition through genetic selection. J. Dairy Sci. 72:2815. 3 Hillier, F. S., and G. J. Liebennan. 1990. Introduction to Operations Research. Holden-Day, San Francisco, CA . 4 Jalvingh, A. W.. J.A.M. Van Arendonk, and A. A. Dijkhuizen. 1994. Dynamic probabilistic simulation of dairy herd management practices. I. Model description and outcome of different seasonal calving patterns. Livest. Prod. Sci. 37:107. 5 Jalvingh, A. W., J.A.M. Van Arendonk, A. A. Dijkhuizen, and J. A. Renkema. 1994. Dynamic probabilistic simulation of dairy herd management practices. II. Comparison of strategies in order to change a herd's calving pattern. Livest. Prod. Sci. 37: 133. 6Kristensen, A. R. 1989. Optimal replacement and ranking of dairy heifers under milk quotas. Acta Agric. Scand. 39:311. 7 Kristensen, A. R. 1992. Optimal replacement in the dairy herd: a multicomponent system. Agric. Syst. 39: 1. 8 Press, W. H., B. P.Flannery, S. A. Teukolsky, and W. T. Vetterling. 1989. Numerical Recipes in Pascal.The Art of Scientific Computing. Cambridge Univ. Press, Cambridge, MA. 9 Snoek, H., ed. 1988. Boeren met quotum. Publikatie nr. 55. Proefstation voor de Rundveehouderij, Schapenhouderij en Paardenhouderij. Lelystad, Neth. 10 Van Arendonk, J.A.M. 1986. Studies on the replacement policies in dairy cattle. IV.Influence of seasonal variation in performance and prices. Livest. Prod. Sci. 14:15. 1 1 Van Arendonk, J.A.M.. and E. W. Brascamp. 1990. Economic considerations in dairy cattle breeding. Proc. 4th World Congr. Genet. Appl. Livest. Prod.. Edinburgh, Scotland, XIV:78.

APPENDIX

See Jalvingh et al. (4) for more details on input variables and for a complete overview. The given input values were assumed to represent typically Dutch herds, but they could be easily modified to suit other farm and price conditions. Herd Dynamics Model

Proportions of first insemination for mo 2 to 5 after calving were 44, 41, 11, and 4, respectively. After second calving and later, these proportions were 49, 38, 10, and 3%. Conception rate after insemination depended on lactation number. Conception rate per lactation number was weighed according to an average herd composition and was 62%. Estrus detection rate was 70%. Probability of involuntary disposal was 12% for lactation 1 and increased to 23% for lactation 10.

TABLE A2. Energy content and prices of different kinds of feed. ~

Energy content

Feed

0' Grass Silage Concentrates

95 1 850 1045

Price (Dfl./lOOO VEM) .22 .30 .35

1VEM = Dutch Feed Unit; lo00 VEM = 6.9 MJ of NEL.

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Performance Model

In Table AI, the base prices of i l k , calves, replacement heifers, and carcass weight are presented with the monthly deviation in prices.

Journal of Dairy Science Vol. 77. No. 6, 1994

In Table A2, energy content and price of grass, silage, and concentrate are presented. During summer (May to October), heifers were fed grass and concentrates. During winter, the ration consisted of silage and concentrates.